From cfd453e15caa313d1c8f09de39b8c4468e4c67c4 Mon Sep 17 00:00:00 2001
From: Samraat Pawar <mhasoba@gmail.com>
Date: Tue, 14 Jan 2025 08:47:23 +0000
Subject: [PATCH] Update documentation

---
 .buildinfo                                    |    2 +-
 .../2024_11_16_HPC_worksheet.html             |   33 +-
 .../HPC_practical_mark_scheme_2024.html       |   33 +-
 Stats-Intro.html                              |   22 +-
 ...257f1e3dd61651f3a9d9a9350f32e0b5ebc84a.png |  Bin 0 -> 34597 bytes
 _images/func_diff1.png                        |  Bin 19480 -> 0 bytes
 _images/log_base_02.png                       |  Bin 18117 -> 26908 bytes
 _sources/intro-EnE_Modelling.md               |   27 +
 _sources/intro-Stats.md                       |   28 +
 .../01_Making_mathematical_statements.ipynb   | 1029 ---
 .../02_Describing_shapes_and_patterns.ipynb   |  969 ---
 ...Analysing_change,_one_step_at_a_time.ipynb | 1004 ---
 ...table_arrangements_to_powerful_tools.ipynb |  384 --
 ...arrangements_to_powerful_tools_part2.ipynb |  396 --
 .../06_Analysing_change_continuously.ipynb    |  507 --
 ..._Analysing_change_continuously_part2.ipynb |  860 ---
 .../08_Calculating_accumulated_change.ipynb   |  336 -
 .../tutorials/lecture01/Tutorial_01.ipynb     |  203 -
 .../tutorials/lecture05/Tutorial_05.ipynb     |  275 -
 .../tutorials/lecture06/Tutorial_06.ipynb     |  224 -
 _sources/notebooks/Appendix-Maths.ipynb       |    6 +-
 .../MathsForBiologists.ipynb                  | 5883 ++++++++++++++++
 _sources/notebooks/Ecosystems.ipynb           |  178 +
 _sources/notebooks/Interactions.ipynb         |  201 +
 .../MathsForBiologists.ipynb                  | 5931 +++++++++++++++++
 _sources/notebooks/MetabolicBasis.ipynb       |  160 +
 _sources/notebooks/ModelFitting-NLLS.ipynb    | 4021 -----------
 _sources/notebooks/Populations.ipynb          |   56 +
 _sources/notebooks/Space.ipynb                |   44 +
 _sources/notebooks/TurbulentMarriage.ipynb    |  631 ++
 genindex.html                                 |   29 +-
 intro-EnE_Modelling.html                      |  575 ++
 intro-Stats.html                              |  570 ++
 intro.html                                    |   31 +-
 .../01_Making_mathematical_statements.html    |  796 ---
 .../02_Describing_shapes_and_patterns.html    |  895 ---
 ..._Analysing_change,_one_step_at_a_time.html |  851 ---
 ..._table_arrangements_to_powerful_tools.html |  823 ---
 ..._arrangements_to_powerful_tools_part2.html |  727 --
 .../06_Analysing_change_continuously.html     |  812 ---
 ...7_Analysing_change_continuously_part2.html |  808 ---
 .../08_Calculating_accumulated_change.html    |  775 ---
 .../tutorials/lecture01/Tutorial_01.html      |  651 --
 .../tutorials/lecture05/Tutorial_05.html      |  719 --
 .../tutorials/lecture06/Tutorial_06.html      |  653 --
 notebooks/Appendix-Assessment.html            |   29 +-
 notebooks/Appendix-Data-Python.html           |   33 +-
 notebooks/Appendix-Databases.html             |   39 +-
 .../Appendix-HighPerformanceComputing.html    |   33 +-
 notebooks/Appendix-JupyIntro.html             |   37 +-
 notebooks/Appendix-Maths.html                 |   55 +-
 .../MathsForBiologists.html                   | 3047 +++++++++
 notebooks/Appendix-MiniProj.html              |   29 +-
 notebooks/Appendix-NLLS-Python.html           |   29 +-
 notebooks/Bayesian.html                       |   29 +-
 notebooks/Data_R.html                         |   37 +-
 notebooks/Ecosystems.html                     |  688 ++
 notebooks/ExpDesign.html                      |   29 +-
 notebooks/Git.html                            |   29 +-
 notebooks/Interactions.html                   |  678 ++
 notebooks/LaTeX.html                          |   33 +-
 notebooks/MLE.html                            |   29 +-
 .../MathsForBiologists.html                   | 3002 +++++++++
 notebooks/MetabolicBasis.html                 |  636 ++
 notebooks/ModelFitting-NLLS.html              | 2461 -------
 notebooks/ModelSimp.html                      |   33 +-
 notebooks/MulExpl.html                        |   29 +-
 notebooks/MulExplInter.html                   |   29 +-
 notebooks/NLLS.html                           |   29 +-
 notebooks/Populations.html                    |  542 ++
 notebooks/Python.html                         |   49 +-
 notebooks/R.html                              |   35 +-
 notebooks/ShellScripting.html                 |   29 +-
 notebooks/Space.html                          |  518 ++
 notebooks/TurbulentMarriage.html              | 1394 ++++
 notebooks/Unix.html                           |   29 +-
 notebooks/anova.html                          |   29 +-
 notebooks/glm.html                            |   29 +-
 notebooks/regress.html                        |   29 +-
 notebooks/t_F_tests.html                      |   29 +-
 objects.inv                                   |  Bin 5341 -> 5111 bytes
 search.html                                   |   29 +-
 searchindex.js                                |    2 +-
 83 files changed, 25097 insertions(+), 21906 deletions(-)
 create mode 100644 _images/0e3205167ee6fe026989b4ed40257f1e3dd61651f3a9d9a9350f32e0b5ebc84a.png
 delete mode 100644 _images/func_diff1.png
 create mode 100644 _sources/intro-EnE_Modelling.md
 create mode 100644 _sources/intro-Stats.md
 delete mode 100644 _sources/mathscourse/lecture01/01_Making_mathematical_statements.ipynb
 delete mode 100644 _sources/mathscourse/lecture02/02_Describing_shapes_and_patterns.ipynb
 delete mode 100644 _sources/mathscourse/lecture03/03_Analysing_change,_one_step_at_a_time.ipynb
 delete mode 100644 _sources/mathscourse/lecture04/04_From_table_arrangements_to_powerful_tools.ipynb
 delete mode 100644 _sources/mathscourse/lecture05/05_From_table_arrangements_to_powerful_tools_part2.ipynb
 delete mode 100644 _sources/mathscourse/lecture06/06_Analysing_change_continuously.ipynb
 delete mode 100644 _sources/mathscourse/lecture07/07_Analysing_change_continuously_part2.ipynb
 delete mode 100644 _sources/mathscourse/lecture08/08_Calculating_accumulated_change.ipynb
 delete mode 100644 _sources/mathscourse/tutorials/lecture01/Tutorial_01.ipynb
 delete mode 100644 _sources/mathscourse/tutorials/lecture05/Tutorial_05.ipynb
 delete mode 100644 _sources/mathscourse/tutorials/lecture06/Tutorial_06.ipynb
 create mode 100644 _sources/notebooks/Appendix-MathsForBiologists/MathsForBiologists.ipynb
 create mode 100644 _sources/notebooks/Ecosystems.ipynb
 create mode 100644 _sources/notebooks/Interactions.ipynb
 create mode 100644 _sources/notebooks/MathsForBiologists/MathsForBiologists.ipynb
 create mode 100644 _sources/notebooks/MetabolicBasis.ipynb
 delete mode 100644 _sources/notebooks/ModelFitting-NLLS.ipynb
 create mode 100644 _sources/notebooks/Populations.ipynb
 create mode 100644 _sources/notebooks/Space.ipynb
 create mode 100644 _sources/notebooks/TurbulentMarriage.ipynb
 create mode 100644 intro-EnE_Modelling.html
 create mode 100644 intro-Stats.html
 delete mode 100644 mathscourse/lecture01/01_Making_mathematical_statements.html
 delete mode 100644 mathscourse/lecture02/02_Describing_shapes_and_patterns.html
 delete mode 100644 mathscourse/lecture03/03_Analysing_change,_one_step_at_a_time.html
 delete mode 100644 mathscourse/lecture04/04_From_table_arrangements_to_powerful_tools.html
 delete mode 100644 mathscourse/lecture05/05_From_table_arrangements_to_powerful_tools_part2.html
 delete mode 100644 mathscourse/lecture06/06_Analysing_change_continuously.html
 delete mode 100644 mathscourse/lecture07/07_Analysing_change_continuously_part2.html
 delete mode 100644 mathscourse/lecture08/08_Calculating_accumulated_change.html
 delete mode 100644 mathscourse/tutorials/lecture01/Tutorial_01.html
 delete mode 100644 mathscourse/tutorials/lecture05/Tutorial_05.html
 delete mode 100644 mathscourse/tutorials/lecture06/Tutorial_06.html
 create mode 100644 notebooks/Appendix-MathsForBiologists/MathsForBiologists.html
 create mode 100644 notebooks/Ecosystems.html
 create mode 100644 notebooks/Interactions.html
 create mode 100644 notebooks/MathsForBiologists/MathsForBiologists.html
 create mode 100644 notebooks/MetabolicBasis.html
 delete mode 100644 notebooks/ModelFitting-NLLS.html
 create mode 100644 notebooks/Populations.html
 create mode 100644 notebooks/Space.html
 create mode 100644 notebooks/TurbulentMarriage.html

diff --git a/.buildinfo b/.buildinfo
index 658ed5f4..37eae655 100644
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index 1a3ad6c3..fd8cfdbd 100644
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+    <title>&lt;no title&gt; &#8212; MulQuaBio</title>
   
   
   
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+    <img src="../../_static/logo.png" class="logo__image only-light" alt="MulQuaBio - Home"/>
+    <script>document.write(`<img src="../../_static/logo.png" class="logo__image only-dark" alt="MulQuaBio - Home"/>`);</script>
   
   
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 <li class="toctree-l1"><a class="reference internal" href="../../notebooks/Python.html">Biological Computing in Python</a></li>
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-<ul class="nav bd-sidenav">
-<li class="toctree-l1"><a class="reference internal" href="../../mathscourse/lecture01/01_Making_mathematical_statements.html">01 - Making mathematical statements</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../../mathscourse/lecture02/02_Describing_shapes_and_patterns.html">02   - Describing shapes and patterns</a></li>
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-<li class="toctree-l1"><a class="reference internal" href="../../mathscourse/tutorials/lecture06/Tutorial_06.html">Tutorial 06</a></li>
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 <p aria-level="2" class="caption" role="heading"><span class="caption-text">Basic Data Analyses and Statistics</span></p>
 <ul class="nav bd-sidenav">
-<li class="toctree-l1"><a class="reference internal" href="../../Stats-Intro.html">Basic Data Science and Statistics: Introduction</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../intro-Stats.html">Introduction</a></li>
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 <li class="toctree-l1"><a class="reference internal" href="../../notebooks/ExpDesign.html">Experimental design and Data exploration</a></li>
 <li class="toctree-l1"><a class="reference internal" href="../../notebooks/t_F_tests.html">Basic hypothesis testing: <span class="math notranslate nohighlight">\(t\)</span> and <span class="math notranslate nohighlight">\(F\)</span> tests</a></li>
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 <ul class="nav bd-sidenav">
 <li class="toctree-l1"><a class="reference internal" href="../../notebooks/glm.html">Generalised Linear Models</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../../notebooks/ModelFitting-NLLS.html">Model Fitting using Non-linear Least-squares</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../notebooks/NLLS.html">Model Fitting using Non-linear Least-squares</a></li>
 </ul>
 <p aria-level="2" class="caption" role="heading"><span class="caption-text">Appendices</span></p>
 <ul class="nav bd-sidenav">
 <li class="toctree-l1"><a class="reference internal" href="../../notebooks/Appendix-JupyIntro.html">Introduction to Jupyter</a></li>
 <li class="toctree-l1"><a class="reference internal" href="../../notebooks/Appendix-Data-Python.html">Data analyses  with Python &amp; Jupyter</a></li>
 <li class="toctree-l1"><a class="reference internal" href="../../notebooks/Appendix-Maths.html">Mathematical models in Jupyter</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../notebooks/Appendix-MathsForBiologists/MathsForBiologists.html">Maths for Biologists</a></li>
 <li class="toctree-l1"><a class="reference internal" href="../../notebooks/Appendix-Databases.html">Databases <span class="tocSkip"></span></a></li>
 <li class="toctree-l1"><a class="reference internal" href="../../notebooks/Appendix-NLLS-Python.html">Model fitting in Python</a></li>
 <li class="toctree-l1"><a class="reference internal" href="../../notebooks/Appendix-MiniProj.html">The Computing Miniproject</a></li>
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+      <li><a href="https://github.com/MulQuaBio/MQB" target="_blank"
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    title="Source repository"
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-      <li><a href="https://github.com/mhasoba/TheMulQuaBio/issues/new?title=Issue%20on%20page%20%2FHPC/Instructions/2024_11_16_HPC_worksheet.html&body=Your%20issue%20content%20here." target="_blank"
+      <li><a href="https://github.com/MulQuaBio/MQB/issues/new?title=Issue%20on%20page%20%2FHPC/Instructions/2024_11_16_HPC_worksheet.html&body=Your%20issue%20content%20here." target="_blank"
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@@ -1336,7 +1323,7 @@ <h2> Contents </h2>
   <div class="footer-item">
     
 <p class="component-author">
-By TheMulQuaBio collective!
+By MulQuaBio collective!
 </p>
 
   </div>
diff --git a/HPC/Instructions/HPC_practical_mark_scheme_2024.html b/HPC/Instructions/HPC_practical_mark_scheme_2024.html
index 687b8a15..91c207e5 100644
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+    <title>&lt;no title&gt; &#8212; MulQuaBio</title>
   
   
   
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+    <img src="../../_static/logo.png" class="logo__image only-light" alt="MulQuaBio - Home"/>
+    <script>document.write(`<img src="../../_static/logo.png" class="logo__image only-dark" alt="MulQuaBio - Home"/>`);</script>
   
   
 </a></div>
@@ -186,23 +186,9 @@
 <li class="toctree-l1"><a class="reference internal" href="../../notebooks/Python.html">Biological Computing in Python</a></li>
 <li class="toctree-l1"><a class="reference internal" href="../../notebooks/R.html">Biological Computing in R</a></li>
 </ul>
-<p aria-level="2" class="caption" role="heading"><span class="caption-text">Maths for Biologists</span></p>
-<ul class="nav bd-sidenav">
-<li class="toctree-l1"><a class="reference internal" href="../../mathscourse/lecture01/01_Making_mathematical_statements.html">01 - Making mathematical statements</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../../mathscourse/lecture02/02_Describing_shapes_and_patterns.html">02   - Describing shapes and patterns</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../../mathscourse/lecture03/03_Analysing_change%2C_one_step_at_a_time.html">03   - Analysing change, one step at a time</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../../mathscourse/lecture04/04_From_table_arrangements_to_powerful_tools.html">04 - From table arrangements to powerful tools</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../../mathscourse/lecture05/05_From_table_arrangements_to_powerful_tools_part2.html">05 - From table arrangements to powerful tools - part 2</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../../mathscourse/lecture06/06_Analysing_change_continuously.html">06 - Analysing change, continuously</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../../mathscourse/lecture07/07_Analysing_change_continuously_part2.html">07 - Analysing change, continuously (Part 2)</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../../mathscourse/lecture08/08_Calculating_accumulated_change.html">08 - Calculating accumulated change</a></li>
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-<li class="toctree-l1"><a class="reference internal" href="../../mathscourse/tutorials/lecture05/Tutorial_05.html">Tutorial 05</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../../mathscourse/tutorials/lecture06/Tutorial_06.html">Tutorial 06</a></li>
-</ul>
 <p aria-level="2" class="caption" role="heading"><span class="caption-text">Basic Data Analyses and Statistics</span></p>
 <ul class="nav bd-sidenav">
-<li class="toctree-l1"><a class="reference internal" href="../../Stats-Intro.html">Basic Data Science and Statistics: Introduction</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../intro-Stats.html">Introduction</a></li>
 <li class="toctree-l1"><a class="reference internal" href="../../notebooks/Data_R.html">Data Management and Visualization</a></li>
 <li class="toctree-l1"><a class="reference internal" href="../../notebooks/ExpDesign.html">Experimental design and Data exploration</a></li>
 <li class="toctree-l1"><a class="reference internal" href="../../notebooks/t_F_tests.html">Basic hypothesis testing: <span class="math notranslate nohighlight">\(t\)</span> and <span class="math notranslate nohighlight">\(F\)</span> tests</a></li>
@@ -215,13 +201,14 @@
 <p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced Data Analyses and Statistics</span></p>
 <ul class="nav bd-sidenav">
 <li class="toctree-l1"><a class="reference internal" href="../../notebooks/glm.html">Generalised Linear Models</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../../notebooks/ModelFitting-NLLS.html">Model Fitting using Non-linear Least-squares</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../notebooks/NLLS.html">Model Fitting using Non-linear Least-squares</a></li>
 </ul>
 <p aria-level="2" class="caption" role="heading"><span class="caption-text">Appendices</span></p>
 <ul class="nav bd-sidenav">
 <li class="toctree-l1"><a class="reference internal" href="../../notebooks/Appendix-JupyIntro.html">Introduction to Jupyter</a></li>
 <li class="toctree-l1"><a class="reference internal" href="../../notebooks/Appendix-Data-Python.html">Data analyses  with Python &amp; Jupyter</a></li>
 <li class="toctree-l1"><a class="reference internal" href="../../notebooks/Appendix-Maths.html">Mathematical models in Jupyter</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../../notebooks/Appendix-MathsForBiologists/MathsForBiologists.html">Maths for Biologists</a></li>
 <li class="toctree-l1"><a class="reference internal" href="../../notebooks/Appendix-Databases.html">Databases <span class="tocSkip"></span></a></li>
 <li class="toctree-l1"><a class="reference internal" href="../../notebooks/Appendix-NLLS-Python.html">Model fitting in Python</a></li>
 <li class="toctree-l1"><a class="reference internal" href="../../notebooks/Appendix-MiniProj.html">The Computing Miniproject</a></li>
@@ -281,7 +268,7 @@
       
       
       
-      <li><a href="https://github.com/mhasoba/TheMulQuaBio" target="_blank"
+      <li><a href="https://github.com/MulQuaBio/MQB" target="_blank"
    class="btn btn-sm btn-source-repository-button dropdown-item"
    title="Source repository"
    data-bs-placement="left" data-bs-toggle="tooltip"
@@ -298,7 +285,7 @@
       
       
       
-      <li><a href="https://github.com/mhasoba/TheMulQuaBio/issues/new?title=Issue%20on%20page%20%2FHPC/Instructions/HPC_practical_mark_scheme_2024.html&body=Your%20issue%20content%20here." target="_blank"
+      <li><a href="https://github.com/MulQuaBio/MQB/issues/new?title=Issue%20on%20page%20%2FHPC/Instructions/HPC_practical_mark_scheme_2024.html&body=Your%20issue%20content%20here." target="_blank"
    class="btn btn-sm btn-source-issues-button dropdown-item"
    title="Open an issue"
    data-bs-placement="left" data-bs-toggle="tooltip"
@@ -550,7 +537,7 @@ <h2> Contents </h2>
   <div class="footer-item">
     
 <p class="component-author">
-By TheMulQuaBio collective!
+By MulQuaBio collective!
 </p>
 
   </div>
diff --git a/Stats-Intro.html b/Stats-Intro.html
index 61db371a..9a757012 100644
--- a/Stats-Intro.html
+++ b/Stats-Intro.html
@@ -59,7 +59,7 @@
     <link rel="index" title="Index" href="genindex.html" />
     <link rel="search" title="Search" href="search.html" />
     <link rel="next" title="Data Management and Visualization" href="notebooks/Data_R.html" />
-    <link rel="prev" title="Tutorial 06" href="mathscourse/tutorials/lecture06/Tutorial_06.html" />
+    <link rel="prev" title="Biological Computing in R" href="notebooks/R.html" />
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@@ -186,20 +186,6 @@
 <li class="toctree-l1"><a class="reference internal" href="notebooks/Python.html">Biological Computing in Python</a></li>
 <li class="toctree-l1"><a class="reference internal" href="notebooks/R.html">Biological Computing in R</a></li>
 </ul>
-<p aria-level="2" class="caption" role="heading"><span class="caption-text">Maths for Biologists</span></p>
-<ul class="nav bd-sidenav">
-<li class="toctree-l1"><a class="reference internal" href="mathscourse/lecture01/01_Making_mathematical_statements.html">01 - Making mathematical statements</a></li>
-<li class="toctree-l1"><a class="reference internal" href="mathscourse/lecture02/02_Describing_shapes_and_patterns.html">02   - Describing shapes and patterns</a></li>
-<li class="toctree-l1"><a class="reference internal" href="mathscourse/lecture03/03_Analysing_change%2C_one_step_at_a_time.html">03   - Analysing change, one step at a time</a></li>
-<li class="toctree-l1"><a class="reference internal" href="mathscourse/lecture04/04_From_table_arrangements_to_powerful_tools.html">04 - From table arrangements to powerful tools</a></li>
-<li class="toctree-l1"><a class="reference internal" href="mathscourse/lecture05/05_From_table_arrangements_to_powerful_tools_part2.html">05 - From table arrangements to powerful tools - part 2</a></li>
-<li class="toctree-l1"><a class="reference internal" href="mathscourse/lecture06/06_Analysing_change_continuously.html">06 - Analysing change, continuously</a></li>
-<li class="toctree-l1"><a class="reference internal" href="mathscourse/lecture07/07_Analysing_change_continuously_part2.html">07 - Analysing change, continuously (Part 2)</a></li>
-<li class="toctree-l1"><a class="reference internal" href="mathscourse/lecture08/08_Calculating_accumulated_change.html">08 - Calculating accumulated change</a></li>
-<li class="toctree-l1"><a class="reference internal" href="mathscourse/tutorials/lecture01/Tutorial_01.html">Tutorial 01</a></li>
-<li class="toctree-l1"><a class="reference internal" href="mathscourse/tutorials/lecture05/Tutorial_05.html">Tutorial 05</a></li>
-<li class="toctree-l1"><a class="reference internal" href="mathscourse/tutorials/lecture06/Tutorial_06.html">Tutorial 06</a></li>
-</ul>
 <p aria-level="2" class="caption" role="heading"><span class="caption-text">Basic Data Analyses and Statistics</span></p>
 <ul class="current nav bd-sidenav">
 <li class="toctree-l1 current active"><a class="current reference internal" href="#">Basic Data Science and Statistics: Introduction</a></li>
@@ -215,7 +201,7 @@
 <p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced Data Analyses and Statistics</span></p>
 <ul class="nav bd-sidenav">
 <li class="toctree-l1"><a class="reference internal" href="notebooks/glm.html">Generalised Linear Models</a></li>
-<li class="toctree-l1"><a class="reference internal" href="notebooks/ModelFitting-NLLS.html">Model Fitting using Non-linear Least-squares</a></li>
+<li class="toctree-l1"><a class="reference internal" href="notebooks/NLLS.html">Model Fitting using Non-linear Least-squares</a></li>
 </ul>
 <p aria-level="2" class="caption" role="heading"><span class="caption-text">Appendices</span></p>
 <ul class="nav bd-sidenav">
@@ -485,12 +471,12 @@ <h2>Readings &amp; Resources<a class="headerlink" href="#readings-resources" tit
                   
 <div class="prev-next-area">
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diff --git a/_sources/intro-EnE_Modelling.md b/_sources/intro-EnE_Modelling.md
new file mode 100644
index 00000000..d0b3b746
--- /dev/null
+++ b/_sources/intro-EnE_Modelling.md
@@ -0,0 +1,27 @@
+# Introduction
+
+This MQB section is aimed at teaching you core concepts, theories, mathematical models, and modelling methods in Ecology and Evolution. 
+
+## Overview
+
+We will cover key ecological and evolutionary concepts, models, understand how they behave, and learn to interpret and classify their behaviour using dynamical systems theory and bifurcation analysis. Dynamical systems theory plays a major role in modern theoretical approaches to ecological concepts and phenomena such as competition, predation, metapopulation dynamics and disease spread. This section will introduce some of the key basics of dynamical systems theory in application to these topics. We will look at ordinary differential equations and difference equation models and will use stability analysis and bifurcation analysis as tools to understand the qualitative behaviour of ecological models.
+
+### Structure of this section
+
+We will go from individual (single cells, organisms) to ecosystem levels, integratng ecology and evolution (as seamlessly as possible) at each level. To lay the foundation for this, we will start with a philosphically- and historically-flavored chapter on the (brief) history of Ecology and Evolutionary biology and theory, the interlinkages between the two fields, and the status of their unification into a single discipline. There you will learn that its basically a matter of time scales, and whether and when ecology and evolutionary dynamics play out at the same timescales, and when they can be separated.  
+
+* Real world applications in
+  * Microbial ecology and engineering
+  * Fisheries
+  * Global change
+  * Agriculture
+  * Disease
+  * Etc
+
+## Pre-requisites
+  * Maths
+    * Each chapter linked to specific maths appendix sections for pre-requisites (Move current maths materials to a single Appendix)
+    * Bifurcation analyses / theory
+  * Computing
+    * Python
+  * Jupyter
diff --git a/_sources/intro-Stats.md b/_sources/intro-Stats.md
new file mode 100644
index 00000000..3aedd11d
--- /dev/null
+++ b/_sources/intro-Stats.md
@@ -0,0 +1,28 @@
+# Introduction
+
+This MQB section builds on the computing component, introducing you to basic
+Data Science including visualization and statistical analyses of data. 
+
+You will learn basic to somewhat advanced
+[*frequentist*](https://en.wikipedia.org/wiki/Frequentist_inference) statistical
+methods ad inference in a hands-on on way, interspersed with lectures on
+underlying concepts.
+
+```{note}
+The chapters in this Section assume that you have already worked through at least the basic sections of the [R Chapter](./notebooks/R.ipynb) of this book.
+```
+
+It is important that you work through the problems in each chapter, particularly
+as some of the questions ask you to find out about commands and functions not
+introduced in the chapter's text itself, but which will be relied on in later
+chapters.
+
+## Readings & Resources
+
+Look up the Readings directory on [MulQuaBio](https://github.com/MulQuaBio/MQB).
+
+* Bolker, B. M.: Ecological Models and Data in R (eBook and Hardcover available).
+
+* Beckerman, A. P. & Petchey, O. L. (2012) Getting started with R: an introduction for biologists. Oxford, Oxford University Press. (*Good, short, general introduction*)
+
+* Crawley, R. (2013) The R book. 2nd edition. Chichester, Wiley.Excellent but enormous reference book, with code and data [available online](http://www.bio.ic.ac.uk/research/mjcraw/therbook/index.htm)
\ No newline at end of file
diff --git a/_sources/mathscourse/lecture01/01_Making_mathematical_statements.ipynb b/_sources/mathscourse/lecture01/01_Making_mathematical_statements.ipynb
deleted file mode 100644
index 8b583ffd..00000000
--- a/_sources/mathscourse/lecture01/01_Making_mathematical_statements.ipynb
+++ /dev/null
@@ -1,1029 +0,0 @@
-{
- "cells": [
-  {
-   "cell_type": "code",
-   "execution_count": 3,
-   "metadata": {
-    "tags": [
-     "remove-cell"
-    ]
-   },
-   "outputs": [],
-   "source": [
-    "from ipywidgets import interact\n",
-    "import matplotlib.pyplot as plt\n",
-    "import numpy as np\n",
-    "\n",
-    "def fun(a,b):\n",
-    "    x = np.linspace(-2,2)\n",
-    "    plt.plot(x,a*x+b)\n",
-    "    plt.ylim(-8, 8)\n",
-    "    plt.show()\n",
-    "    \n",
-    "# create a slider\n",
-    "\n",
-    "interact(fun, a=(-6.0,4.0),b=(-1.0,1.0))"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 62,
-   "metadata": {
-    "tags": [
-     "remove-cell"
-    ]
-   },
-   "outputs": [
-    {
-     "data": {
-      "image/png": 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\n",
-      "text/plain": [
-       "<Figure size 432x432 with 1 Axes>"
-      ]
-     },
-     "metadata": {
-      "needs_background": "light"
-     },
-     "output_type": "display_data"
-    }
-   ],
-   "source": [
-    "import matplotlib as mpl\n",
-    "import matplotlib.pyplot as plt\n",
-    "import numpy as np\n",
-    "\n",
-    "\n",
-    "# Create figure and add axes\n",
-    "fig = plt.figure(figsize=(6, 6))\n",
-    "ax = fig.add_subplot(111)\n",
-    "\n",
-    "\n",
-    "m = np.linspace(-2, 2, 11)\n",
-    "\n",
-    "# Get colors from coolwarm colormap\n",
-    "colors = plt.get_cmap('coolwarm', 11)\n",
-    "# Create variable reference to plot\n",
-    "f_d, = ax.plot([], [], linewidth=2.5)\n",
-    "# Add text annotation and create variable reference\n",
-    "temp = ax.text(1, 1, '', ha='right', va='top', fontsize=24)\n",
-    "\n",
-    "# Move left y-axis and bottim x-axis to centre, passing through (0,0)\n",
-    "ax.spines['left'].set_position('center')\n",
-    "ax.spines['bottom'].set_position('center')\n",
-    "\n",
-    "# Eliminate upper and right axes\n",
-    "ax.spines['right'].set_color('none')\n",
-    "ax.spines['top'].set_color('none')\n",
-    "\n",
-    "# Show ticks in the left and lower axes only\n",
-    "ax.xaxis.set_ticks_position('bottom')\n",
-    "ax.yaxis.set_ticks_position('left')\n",
-    "\n",
-    "\n",
-    "\n",
-    "labels = ['-2','-1.6','-1.2','-0.8','-0.4','0.0','0.4','0.8','1.2','1.6','2']\n",
-    "\n",
-    "for i in range(len(m)):\n",
-    "    x = np.linspace(-2, 2, 100)\n",
-    "    y = m[i]*x\n",
-    "    ax.plot(x, y, color=colors(i), linewidth=2.5, label=labels[i])\n",
-    "    \n",
-    "ax.tick_params(axis='both', which='major', labelsize=13)\n",
-    "\n",
-    "    \n",
-    "# Add legend\n",
-    "\n",
-    "leg = ax.legend(bbox_to_anchor=(1.05, 0.3),loc='lower left', \n",
-    "          frameon=True, labelspacing=0.2, fontsize=13)\n",
-    "leg.set_title(r\"$m$\", prop = {'size':'x-large'})\n",
-    "\n",
-    "plt.tight_layout()\n",
-    "plt.savefig('slope_01.png')"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 59,
-   "metadata": {
-    "tags": [
-     "remove-cell"
-    ]
-   },
-   "outputs": [
-    {
-     "data": {
-      "image/png": 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\n",
-      "text/plain": [
-       "<Figure size 432x360 with 1 Axes>"
-      ]
-     },
-     "metadata": {
-      "needs_background": "light"
-     },
-     "output_type": "display_data"
-    }
-   ],
-   "source": [
-    "import matplotlib.ticker as plticker\n",
-    "\n",
-    "# Create figure and add axes\n",
-    "fig = plt.figure(figsize=(6, 5))\n",
-    "ax = fig.add_subplot(111)\n",
-    "\n",
-    "\n",
-    "# Move left y-axis and bottim x-axis to centre, passing through (0,0)\n",
-    "ax.spines['left'].set_position('zero')\n",
-    "ax.spines['bottom'].set_position('zero')\n",
-    "\n",
-    "# Eliminate upper and right axes\n",
-    "ax.spines['right'].set_color('none')\n",
-    "ax.spines['top'].set_color('none')\n",
-    "\n",
-    "# Show ticks in the left and lower axes only\n",
-    "ax.xaxis.set_ticks_position('bottom')\n",
-    "ax.yaxis.set_ticks_position('left')\n",
-    "\n",
-    "\n",
-    "loc = plticker.MultipleLocator(base=3.0) # this locator puts ticks at regular intervals\n",
-    "ax.xaxis.set_major_locator(loc)\n",
-    "loc = plticker.MultipleLocator(base=2.0) # this locator puts ticks at regular intervals\n",
-    "ax.yaxis.set_major_locator(loc)\n",
-    "\n",
-    "x = np.linspace(3.1, 11, 100)\n",
-    "y = 2 + 1/(x-3)\n",
-    "ax.plot(x, y, color='b', linewidth=2.5)\n",
-    "\n",
-    "x = np.linspace(-6, 2.9, 100)\n",
-    "y = 2 + 1/(x-3)\n",
-    "ax.plot(x, y, color='b', linewidth=2.5)\n",
-    "\n",
-    "ylim = ax.get_ylim()\n",
-    "plt.vlines(3, ylim[0], ylim[1], ls='dashed',alpha=0.3)\n",
-    "\n",
-    "xlim = ax.get_xlim()\n",
-    "plt.hlines(2, xlim[0], xlim[1], ls='dashed',alpha=0.3)\n",
-    "\n",
-    "ax.tick_params(axis='both', which='major', labelsize=13)\n",
-    "    \n",
-    "# Add legend\n",
-    "\n",
-    "#ax.legend(bbox_to_anchor=(1.05, 0.3),loc='lower left', \n",
-    "#          frameon=False, labelspacing=0.2, fontsize=13)\n",
-    "\n",
-    "plt.tight_layout()\n",
-    "plt.savefig('rational_01.png')"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 66,
-   "metadata": {
-    "tags": [
-     "remove-cell"
-    ]
-   },
-   "outputs": [
-    {
-     "data": {
-      "image/png": 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\n",
-      "text/plain": [
-       "<Figure size 432x360 with 1 Axes>"
-      ]
-     },
-     "metadata": {
-      "needs_background": "light"
-     },
-     "output_type": "display_data"
-    }
-   ],
-   "source": [
-    "import matplotlib as mpl\n",
-    "import matplotlib.pyplot as plt\n",
-    "import numpy as np\n",
-    "\n",
-    "\n",
-    "# Create figure and add axes\n",
-    "fig = plt.figure(figsize=(6, 5))\n",
-    "ax = fig.add_subplot(111)\n",
-    "\n",
-    "alphaset = [-2.5,-2,-1,-0.5, 0.0, 0.5, 1.0, 2.0, 2.5]\n",
-    "colors = plt.get_cmap('coolwarm', 9)\n",
-    "\n",
-    "labels = ['-5/2','-2','-1','-1/2','0','1/2','1','2','5/2']\n",
-    "\n",
-    "plt.ylim(top=10)\n",
-    "\n",
-    "ax.spines['right'].set_color('none')\n",
-    "ax.spines['top'].set_color('none')\n",
-    "\n",
-    "for i in range(len(alphaset)):\n",
-    "    \n",
-    "    x = np.linspace(0.1, 2, 100)\n",
-    "    y = x**alphaset[i]\n",
-    "    ax.plot(x, y, color=colors(i), linewidth=2.5, label=labels[i])\n",
-    "    \n",
-    "ax.tick_params(axis='both', which='major', labelsize=13)\n",
-    "\n",
-    "    \n",
-    "# Add legend\n",
-    "\n",
-    "leg = ax.legend(bbox_to_anchor=(1.05, 0.3),loc='lower left', \n",
-    "          frameon=True, labelspacing=0.2, fontsize=13,)\n",
-    "leg.set_title(r\"${\\alpha}$\", prop = {'size':'x-large'})\n",
-    "\n",
-    "plt.tight_layout()\n",
-    "plt.savefig('powerlaw_01.png')"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 65,
-   "metadata": {
-    "tags": [
-     "remove-cell"
-    ]
-   },
-   "outputs": [
-    {
-     "data": {
-      "image/png": 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Znm7eDmm+rkT1sdtpPqsO5acIym2ThBVBsXAUUsT9xYjNopsCJcdTZC8nfAUJlAPIEkOnOd18KWv4frp5V91eKHevZTXrtrpRl3MObdXbGpTUBg4+o5SvmV7u/2iP2F2QQDmEZTdXNY5cyd8CFVQYEmbv6KLLKT6rBpVvcerTKJXBNavO4nIEtTQHoD0MEvXpPf/fL7G7IIFyiO6Omunm2danszajJymiqMVVb2pQ+WJrI6i6QbEd7qXduFqqWsy7B1y7jlNoaTMdGQxDiri/goQgtgIJlEOEgxKiIdO+3QZ1qN4agXIzJRmLWSaJ1v5MvBoUqy/VrHnv7nftOk7AdQ16bhmAiJ7IXk74DRIoB7GiqGzRgKr5O83XY1rNVQ3IFt27Tsyymbe4UVfzaNWGsTxnP5Z8LlBaZklYTwEoSf/Xy4jdBwmUg7TTEkMrxQe4m+azTBJl1YCmtW4UlJbN24/dFCir/gT4f8yRvmpGe5IEOdHV2pshiDUggXKQZFSCYmrUUqZ9BGrBRYHyy06o+mWFbqb4ZgEAUrILLOBfyzbn3K4/yR3dYJK8yTMIwntIoBykdonhcl6H4eMlhskYEDD/JrkaQflkq25dDcrNFJ8ZQfk9vWfk0+CaCgBQOv0d6RG7FxIoh+kx61CGAaz4eImhxJg98shNq3nMJxFU3bJCl0YdcW7Ye6DkLn8LlJaumjlIoAi/QgLlMKl4dUDqks/t5l5YzaO+iaDcN0nwbBqoiKjE7xGUJVBSLAkp4O7yRoLYLiRQDqPItVMldF9PlegxRx6lc0DFJddh3GcRlBQJQwoEXLlGnUHCxwJVOz2CoifCz5BAuYCV5hNTJfyb5rMiKA73ZvLFfBJBVcwxR27Wn/TFWoHyb5MupfeIdoEEygVqp0os+tjN11tnNXfnGlajLtDiCCrr/qoNuwcqEALr6HTtOs1iCRQLRSGFaXoE4V9IoFwgFJAQD1tLAf0rUJZJAnCvDmU16gKtbda1GnXdXLVhz+Dr6vPtVAauqWL/EwClk6ZHEP6GBMolujvMGXRljpLqzzRfOMjQERGP3XLyhUISZFn8ESy0cB6fF9t028FiXj8cltJ7hL8hgXKJnto0n4+jKKsO5VazLmOsuhOqhRPNKy4LFFfLMDJirp2fDRJ2/UkOQO5ItfZmCGITSKBcIhZmCAWsPiP/CpQXQ2OtcUfZXDWC+r3f+z3s3bsXiUQCfX19eNvb3oYbN264cn0AqKyIIlugM7HJkdtDX/b/kFhu6NBWFwFY6T369Sf8Db1CXYIxZkdRqwX/Do+1hsaWK0DOpaGxibiwdWdzFftzP/uzP4uzZ88ik8ng+vXr2LNnD975zne6cwMAKitiM2Ogyx3zQv2QWH86+LTVRcAQb5aUTn+KKEHUQgLlIj2JaprPr2aJuqGxLtWhEmY9LlMTQR05cgTJZBKAmAsnSRIuXLjgyvX1YglGsQQACHYl3blGGwyJ1dLmPUoylERPa2+GIBqABMpFklGpOu8u09p1E+vhxVTzjriZ4svW/wz+9m//FslkEvF4HI888gg+/OEP3/LcRx99FKdOncKpU6ewsLBwy9cbwUrvAe6l+CyDBOtIgQX9N5mBG4Zdf1ISPWAyDYcl/A8JlIswxuwoaiVvQNP9l+brjAGK+SpwyyjRsUaKDwAefPBBrK6uYmZmBh/+8Idxxx133PLchx9+GKdPn8bp06fR27u9leRWeg9wL8VnTTGXu/0ZPenZZUA3V96nKL1HtAckUC7TY6a3OPfnjihJYlUnX9rdFF82r6054X1gYAAPPfQQfvzHfxzLy8uOX78ugnIhxce5AX3RFKieQcfP7wR2eo8xKMntCT1BeA0JlMt0xiTI5k/Zr26+vk4hUHMuC5RhrD9NQtM05PN5TE9PO359dTltPw6mnBcoI70EaCI6lHuHHD9/s4jdT+aU9Y5uMMWdWYQE4TQkUC4jScwefbSU8+eOqF5ToDIFoKQ6f39Wig8QaT7DMPDJT34S8/Pij+bk5CR++Zd/GePj4zhy5Ijj169L8bkgUFb0BABSj/8cfEY+DV4pA6D0HtFekEB5gFWHMgyxyNBv9HfWbNd1IYpKxKvz+Cwn39e+9jUcO3YMsVgML3/5yxGNRvHNb34TiqKsd5ptU1l2N8VnLFSjPj+m+CorVYchDYcl2gnn/xoQt9AVlyExwODA4qpu16X8Qm+NQM2vcow6/DcsUfP9ZrMVSJKEr33ta85eZAMqK2aKT5JcmSShL84AAFgsASnqr+GrnHNoK2Z9LN5Ju5+ItoIiKA+Qpeoq+MWs/9J8XR2w62TzLkRQtSm+2l4or7CbdDsTYJLzL3l9QQiU3Ou/6MkoZMBV0QOmpPyXfiSIjSCB8ojepBAo3Yer4OUaJ58bAlWb4su2RKDMMUduOfiW/OvgqyxX62MkUES7QQLlEd1mmg8AFnzYtGs5+dyJoGpqUNnKBke6g2rWoFxx8K0u22veZZ8ZJGrTe1KsE1Iw3OI7IoitQQLlEbJcTfMtZfyX5rMEajXvvJMvFJIRDIqXWksjqJTzUyQMM70HAJLPLOYivScGLAa6/CWeBNEIJFAeYqX5NB+m+fpqnXwuTJSw0nytqUFZKT7np0hYBgnAfyk+K3oCKL1HtCckUB7SFZfBfJrmqxUoV+pQHcIo0YoUn2Uzd6cHynLwdfjKwcc5t+tPlN4j2hUSKA9RfJzmc9/Jd+tOKC/ghoFK2nTxuZDisx18Poue6tJ7FD0RbQoJlMf0JvyZ5pNdnsmXaJFAaatZMQgRQCDlbIrPzzP46tJ7XTQ9gmhP/NUxugvo7qg27S6savYYJD/Ql2SYW+GuzOTraFGKT62ZIuH0LihjdQUwRwhJPhKo+vReElIw0uI7IrbKmTNn7lcU5UOc8wHs3EBCB/C4pmkPnTx5Ul3rABIoj1Fkhq4OGYsZ3W7alSS2+RM9oC/FgOvCyVeucHtlvRO0KsVXN8nc4RSfUWuQ8FGTrpFfrXHv+ee+iMY4c+bM/aFQ6JPj4+NqJBJZkSTJP7UABzEMg01MTLwqnU6/B8Ajax2zU5XZ1/Qlqk27Sz5awdGXdG8mn5XiK5UNlFXvUpv1AuVsBKX7dAZfZbkqnOTeaz8URfnQ+Pi4GovFijtVnABAkiQ+NDSUk2X5F9Y9xrvbISy6OmTbkLCw6iOBctHJZ7n4gFsXF7pJ/aBYZ2tQVv2JReOQYh2Onnu71M3e6+gi914bwjkfiEQipVbfhxcEg8EK53zdd44kUC1Arl3BkdV9s2m3K1F18jldh+po0bgjN1N8VgTlp+hJzy5XV2tQeq9dkXZy5FQLE3036+oQCVSL6EuaS/y4ECk/IEsMvWaab27FnRQfAGSy3gmU6lKKjxtGVaD6hh07b7NoVnqPMQRo9xPR5pBAtYhUTIJiGvjmV/3TtNufEgI167RAtTjFJ0cjkMPOrZowVhaqW3R9IlDcMOzdT3KiB0wJtviOCKI5SKBahCQx9CZEVLGSM1DR/BHRD3QJgcoVgVzRuXtqdYrP8fTe/JT92C8CpWeWAF2IJrn3iEb5yEc+0qcoysmTJ08eXl1d9ZUm+OpmdhvWbD4OYDHjjzTfQKpqlHAyzdfR0ZoUX1WgHHbw2QLFfGMxt917TKLNuUTDvO9971v87Gc/e/nMmTPxL33pS87PA2sCEqgW0hmVEFTMmo9P0ny1AuVkmi8eVew5hJkWpPgcd/CZAiWlesB84JTjugYtLdJ7SmcfmEwtjkRjJJNJ48EHH1wdHBxUz54966uubhKoFsIYQ58ZRa0WDJQ87A9aj3gEiJl/b50UKEliiMe8b9a1t+k6nuKbBADI/SOOnne7aOl5wBBReKDbX2s/CP+j6zrC4bBx7tw5EiiiSn9n9Z3uvA96ohhjVaPEstNGCXPlhqcuvjQAhx185SKM9BIA/9SfKkvCUciUAORET4vvhmg3PvrRj/Zdu3YtfP78+Wir76UWEqgWEwsxxELVNB/nrTdLWGm++TSH7uDE9UTcnMfXghSfk3P46iZI+ECgjEoZemYRAKCkBsEk+rUmGufChQvBj33sY8NveMMb0tPT08GVlRXfvIB8cyO7FZHmE5FFocyRK/lHoHQDWMo4d17LKOFVik8vlmCURNOqkzUofc5fDj6tZrRRoNsfhg2iffjFX/zFsVe/+tWZj370o1Occ5w+fdo3aT4SKB/Q11mdaO6HnijLag4As8vO1cXsgbEepfjqpkh0OleDsh18wRCkzm7Hzrtd7PReKAop5vzWYGLn8sgjj3Q///zzsU9/+tMTd955ZykYDHI/GSVIoHxAOCAhGRX/FfM+SPP1JpntuHPSKOF1iq9+Dp9zKT7NFCi5dxiMtfZXSC/mYBRMI0jXoDU6hiA25ebNm8oHP/jB0Y9+9KM3R0dHtUAggAMHDhSfffZZ39ShSKB8gmWWULXWLzIMKAw9ZsDhZC+UZZLI5TVPtglbm3QBIOhQio9zbkdQvkjvLVXrYeTeI7bCQw89tOeuu+7K/cqv/MqS9bmjR48WX3jhBYqgiHp6E7IdtcylfZDmS4mXhpMRlJXiMwwgX3Dfsagup+3HTqX4jNVlQBWDplstUGIxoRAoKZaEFI619H6I9uGzn/1s5+OPP578i7/4i4nazx8/frxw8eJFEiiiHkVm6DEnnC9mWj/hvN+sQ63mgWLZmXuxUnyAN2k+N1J8Vv8TACj9rRUoPbsMboploLv10RzRPvzcz/1cOpfLPX3gwIG6X8Q/+IM/mM9ms2dbdV8vhQTKRwx0ViecL7R49JEbEyUSHd7O47OadAHnXHx1M/h6WysKlUXzXhij2XvEjoQEykek4tXRR7MtTvO5MZOvbmBs1oMIymzShSRB6XAm/WWPOEp2gYVblwkRo43EYkKlsx9MCWzyDIJoP0igfARjDP2m5TxTMFAot84skYwBYXNbw7RDEyU66lJ87guwulydZO5U86o+Z444anH9qbI8K4p5AAI9lN4jdiYkUD6jdvRRK80SjDEMmnWomaX2TPFppovPMQdfuQhjeR4AIA/sceSc20VbEpEcC4QgJ1rfi0UQbkAC5TNiIQkdEfHfMpfWW9oTNdRdHXnkhGmjLoLyIMVnR1AOOfi0uapBQh4YdeSc28Eo5aHnVgAIa3mre7EIwi3ole1DBsw0X1njLe2JsgRKN4RINUsoKCEUFC85LwbGVkybuWMOvtkb9mOlhRFUpab3SSH3HrGDIYHyIb1JxRc9UUNd1ZfHtMNpPm8iKGcnmeuzNwEALBoH62jNSCHOue3ek6JJyJF4S+6DINaDMXYyEoncHY1G7X9LS0vy5s+8FRIoHxKQGXoT4v9zIaO3bB18dwIImGUjpwQqlRTOi/SqBwK1sAwACPU5U6PRzAhKHtjTspFCemYRvGL2PvX6YxcVQbyUL3/5y5cKhcLT1r/u7u5t9c2QQPmUgZRQBs5bt21XkqpGCacEqjMp6lArLguUXixBzxUAAMHerqbPxysqjEXT1t3C+lNlwayDSTL1PhE7HtoL7VM6oxLCQYaSyjG7omG4S2nJu/ahLoYb8xyzK2I3lCw1dw8pU6DcjqCs6AkAgr3NR1D6/KR4t4DWOfiMShnaqnARKqkBWuu+i/jDRy6MXpvIezbEde9YrPDffu3wze0+/13vetc+TdPY6Oho+QMf+MDsz//8z6c3f9at0CvcpzDGMNip4Np8BfkyR7ZoIBHdVhq3KQZNo4SmA4ur3N62u12qEZTa9L1tRHnenn+JYG+q6fNpM9Xf1VY5+LSlaVskg5Te21Vcm8hHX7yUbYuC45e//OWLP/qjP5oDgL/5m7/p/KVf+qV9sVjs8tve9rYtb5cjgfIx/Z0yrs2LSGM2rbVEoCwnHyDSfP1N/q3vNGtQpbKBYklHJOzO91QbQYWciKDM+hMLRSB1er9SnXMOdUGIpBSO0d6nXcbesVjBj9f78z//864PfOADY9bHhULh6Z/8yZ/MWh8/9NBDK9/61rcSn//857tJoHYYoYCE7g4ZS1kd86s69vdzyLK3ab6+TgZZElbz6SWOuw80d75UZ7UXKr1a8USggg6YJCwHnzww2pJUq55bAS+LvxmBnhHa+7TLaCbd5ibvec97lkx3pyEAACAASURBVN/znvcsb3SMJEnb7uckk4TPGUiJP+C6Acy3YICsLDF7Lp8TI4+sFB/gbpqvvFCb4mvOJMG1CvQF0XvUqvpTZdE0RzAGhfY+ET7lqaeeCn/nO9+JlkolVi6X2ec+97nOf/qnf+p6+9vfvqGIrQdFUD6nOy4jqDCoGsfMiobBlPf/ZYPdDFNLHDNLHAbnkJp4927ZzAF3jRJWBCWFgk0PitUXZgBDvDlohYOPayq0ZWswbB+kQMjzeyCIRpidnQ28//3v3zM1NRUMBAJ8z5495UceeeT6z/zMz6xu/uxbIYHyOYwxDKZkTCxoyBYNZIuGPQrJK6w6lKoByxmgp4m+11RdBOWiQM0LgQr2dTedDqudINGKCKqyOA1wczBsb2tnABLERrzlLW/JvuUtbznn1PkoxdcGDNRETTMr7je4vpR6o0Rzo5c6ayKolbT7Kb5gT/M9UFb9CYEQpK6+ps+3FYQ5wjJoRCF3NP/9EES7QALVBoRNswQAzK16v223v5PBan9qtmE3Eq7O4/MixRfqa/4Puj1Bon/EsbUdjaJnl21zRLC3NQYNgmgVJFBtglV7MgzvJ0sEFGb3P00uNidQjDHbyedqis8UqGabdLmu2TuglMEWpPdMazmYRINhiV0HCVSb0BWXEA6Y+5mWNc/XcAz3VEceGUZz1+5MiDSfWy4+zrndqNusg0+fmwJ08YZAGRpv9ta2hFEpQ0vPiWunBiAFgps8gyB2FiRQbYIwS4goKl/myBS9XcMx0lM1SiysNidQVgTlVopPzxdglMoAmk/xadPX7ceyxwJVWayOVwr0tm7+H0G0ChKoNmIgVV3DMb3sbZpvpLda+2g2zdeZcFegLAcf0LxJQp+5DgBg4SikVG9T59oKnHM7vSdF4pDjNDmC8D/PPvts6I1vfOO+vr6+47FY7O4DBw7c/id/8ifbHr1CAtVGBJX6NRxqxbs0X1+SIWiaCScXnImgVtKqK6nKujl8TU6RsCIoeWjcU4OClp4HV621Gq1b70EQW2FxcVH5oR/6oeyTTz75YjabffrP/uzPJj70oQ+NfOYzn9nWOywSqDZjqKu6hmNmxbsoSpKYbTdvOoIyreYVjaNQdH46hrpYO4dv+xEULxVhLJo1IK/Te/Nm75WsIECTI4g24bWvfW3+d3/3dxfGx8crkiTh/vvvz73qVa/KPPbYYx3bOR816rYZiYiEeJghV+KYXtEw2qs0NdlhK4z0Mlyf45hb4VA1jqCyvevWNeumK4hFnX0Z1k8y375AaTMTAMwVGx4KlF7MQc+K7yHQPUxrNQh86XFtdC7NPVu30d/JCj/1KqXp+X/ZbFY6c+ZM/Hd+53emt/N8euW3GYwxDHcFcGFahapxLGZ09CW9+W8c6ZEAGDA4MLPEMda/PYGqnceXzqgYGYo4dIcCdXHFftyUQNUYJJShsfUPdBg7egIQ7KPJEQQwl+bRqUXeFus2LDRNw9ve9ra9IyMj5fe+971Lmz/jVkig2pDepIwrc2JH0/Sy5qFA1Rslxvq3d55UZ+00CeeNEqoZQcmxKJTY9t906qZAScluSLGEE7e2KVzXUFmaAgDIiR5I4ebmCBI7g/5O5um6jUavt9a6DQAol8vsrW996975+fnAN7/5zUuhUGhbdQESqDZElsQyw5tLGlYLBnIlA/Gw++XEZAyIh4Fcqbk6lOXiA9xp1q32QDW3vKrWIOEVlcUpezBtsM+7qI3wN06k29xgrXUbhUKBvfnNb96fz+flxx577GIymdx2TwyZJNoUyywBAFNL3sznY4zZdvPJxe33YdXWoNIuNOtaKb5mFhUamTR4Tgxg9sogwTmHOj8BwJy7l/R+MSJBNMPq6qr0Iz/yIwdVVWXf/va3mxIngASqbQkHJfTUzOdTNW8s56IOBaxkgXxpe9cMhWREIuLe3YigVAemSNQ36HoTyeirC9W5e31kLSfaj89+9rOpJ598suPMmTPxvr6+u6LR6N3RaPTuBx98cFvFVErxtTHD3QoWszo4F7Wo8b7A5k9qktqG3alFjkMj23fyFYu6KzUoe5J5EwJlNeiCMSgerdhQ58xrygoCPSOeXJMgnOR973vf0vve975tGSLWgiKoNiYZleza0/RypekZeY0wXLN64+ZC82m+dMbZFJ9YT2Gm+Jpo0tWmrgEA5N4hsKD7CwL1QgZ6VqTyAz1kLScIgASqrWGMYaRb/CGr6MD8qvsr4SMhhh7T0HaziYkSVrOu0xGUtpoFr4hzbjeC4oYObVrUguShvY7d20aocxP2YzJHEISABKrN6U3IdsPs5FLFkynne/rFy+bmwvYnm1cjKGcFylqzAWxfoPS5KaAihs0qo/scua+NMCplaMuij1FJ9UMKedaPSRC+hgSqzZEkZjv68mWOdN79KedjfUIQyxVgLr1NgaqZaO5karJ2isR2U3za5BX7sTKyv+l72ozK/I3q1PK+cdevRxDtAgnUDmAopdgbbyc9sJzv6a2+bG7Mb09crJ1Qus6Ryzs3U7AugtrmJHNLoFg07voEc67rqJgr3aVogqaWE0QNJFA7gIDCMNApoqjlnGjcdZOeJBA1fQPbFSgrggKcXbtRJ1DbjaBuXgUgoie3rd6VpSlwzayZDewlazlB1EACtUMY6am6viYX3Y2iGGPYY6b5Jua3J4Z1A2MdbNa1LObA9mpQRmYZPCtcgMqou+k9zjnUOeEWZMEIlNQ2Z0cRhI94xzveMXbgwIHbFUU5+Y53vKMpxw8J1A4hEpTsXVHzqzrKFXejqD194qWTzgGZwtajKMvFBzjr5LOWFSqJOOTQ1lekaze9qz9pK3Pg5SIAIDgwDsbo15Fof+64447iH/3RH9187Wtfm272XPQbsYOwoigOYHLJ3V1RVgQFbC/NVzfuyEEnn7ULatvpvUmR3oOsQB5wb8065xzqrBk9KQEEuodduxZBeMnv//7vzz/wwAOZjo6OpvteqBtwB5GIyEhGJawWDMysaBjrDUCR3alpDHczyBKgG8CNeQPHxrf2XidZOzA27WCKzxpztF2DhBlByYNjYIp7kzn03AqMgpj1F+jdQ425xIZcmCqP5sve7YOKhVjh8HCo5QNq6bdihzHaE8DqjTJ0Q4w/2tPrzh/ZgCI27N5c4JjYYgT127/92/jqV7+KjtE/hhKI40tf/gZ+8v63oatr+6OJLKwUX6hv6+fi5RL0+UkA7tef1BkzUmMSAtSYS2xCvsyj2aLh4T4ofyTXSKB2GF1xCbEQQ77MMblUwXC3AllyJ4ra0ycEamZpaxt2ZVnG5z//efy//18FN6aKyOaAd7/73fjKV77S9D2VZucBAKG+rU8C16av2/1Iyoh7Dbp6IQM9swhAjDWSAluvlRG7i1iIFbwUDXG9zVlvH5RTkEDtMBhjGO0J4PyUiooOzK5oGO52J4ra0yfhP8+JDbtTixx7BxoTqD/8wz8EAPR2P4MbU0X0DR7A1//+sabvRy+rqCyJumx4uG/Lz69r0B12T6Ds6AkMwQFvRikR7Y0f0m1rsdY+KCfxRxxHOEpfUkY4IMTi5pLm2hDZsRqjxFbTfADQ0y2aqRaXyjh+/Piaxzz66KM4deoUTp06hYWFhQ3PV56esx+HhrZu2bYESurqgxTr2PLzG8Eo5aCtzAIAlO5BGmtE7DhKpRIrFApM13Wm6zorFAqsVCptK41DArUDYYxhT4+ImsoV7toQ2XiEodv8Oz4xu3Vbe0+XSG2V1SD+5//8xJrHPPzwwzh9+jROnz6N3t6NpzqUpuftx+GhrUVQXNdtB5+b9vLyzDX7cXDA/Tl/BOE1r3nNaw7FYrETX/nKV7r+8R//sTsWi514zWtec2g756IU3w6lv1PG9QUGVeO4sVhBf6fsypSC8QEJS1kDE/McusG3VO+avPECgA4wScG+/Xc0fS+lmggqvMUISp+9AajmgNixbf0ubYpRLlaHwnb2Q454WPMmCI948sknLzh1LoqgdiiSxDBqruIoqhyLGXeiKKvupGrA9FLjab6/+qu/wuc/9yn748XlctP3Um4igtJuXLIfK2MHm76XtVBnr9kmjOAgRU8EsRkkUDuYwS4FATFcAhML7qziGB+ovoSuzTZ2/j/90z/Fb/7mb+J//PEH7c8tLDXfC2VFUFIwgEB3akvPrUxcFM9NdkNObn/R4XoYagmVRWFhlxPdkGNJx69BEDsNEqgdjCwxjJgOvnyZYzHrfBTVGWNImXWoaw3WoX7t134NmUwGD//Xn7Y/9+DP/FLT92LVoEJDfVtKZ3JDh3bzMgD30nvq7FWAi59PcPCAK9cgiJ0GCdQOZ6hLgWJFUfPuRFF7zShqYk7UoTaDc45KpYKFuaqt+4Mf/h9N34clUFuvP92sqT85n94z1BIqC2b01NENpWNr0R1B7FZIoHY4iux+FLW3v1qHmtlCHUpRJHvthiM1qBlLoLZYfzLTewCg7HE+ghK1JzN6GqLoiSAahQRqFzBcG0W5UIuqq0PNbe3cvV1mL9RyczUozvm2IyjLICEluyB3Olt/EtGT6LGk6IkgtgYJ1C6gLooqOR9FpeIMKdMx3WgdysLqhVpYai6C0jI56HkxnSW0hQiKGzoqLtaf6qMn99fHE4SXPProo6mTJ08ejsfjdyuKctLp85NA7RKGXa5FjW+xDmVhT5NoMoKqb9JtPILSZyeBcgkAoOxxtv5UHz11QelofhguQfiJ7u5u/eGHH174yEc+csON81Oj7i5BkRlGuwO4Nl9BvsyxkNHRl3Tuv3/vAMPTl4FyBZhd5hjuacxF19MtIqj0agVqxUAwsL33TOW6Jt3GI6j6/idnIyh15kpN9ORObxWxOyhee27UKOY8m4slReKFyN47Np3/98ADD2QA4Ktf/aors8FIoHYRw90KJpcqqOjA9fkKehPOTZfY2y8BEKnDq7Mcww0OE7dqUACwvKJioC+8retvN4KyDBJSoguSg/1PRrlQ0/fUQ7UnoimMYi5qFFZ33egREqhdhCwx7OkN4MpsBUWVYzatYzDlzEsg1cHQGRcr4K/OGHj1Mbmh51kRFCDqUNsXqJpBsYMbz+yz4LqOihlBKWMHHR0FVZ6+bE+NCA1T9EQ0hxSJN7T+ol2vtx4kULuMoZSCyUUNZY1jYqGC/qQMyaF9UfsHJfzgkoHrsxwVjSPQwH6onpoIqpk6lBVBKYk4lHisoefo09cA1aw/7b1t29e+5bzFHLSlmpl7NDWCaJJG0m07ETJJ7DIkM4oCxKTzmRXNsXMfGBKCVNGBGwuNGSVeGkFtFyuCCg83nt6rXH3RfhwYP7Lta78Udfqy/Tg4TH1PBLFdSKB2IQMpGeGgEJOJhQo03RlH3/5BCVbMdGW6Mbt5siMAxYy0momgrEGxocHGDRKVa+cBAFLvEKQOZ6IcPb9a3ffUNQQ54s5eKYLwA5qmoVAoMFVVGQAUCgVWKBSYYWx9/c5akEDtQiTGsLdPRFEVHZhcqjhy3miYYbBbiM3l6cZET5KY3Qu16EQE1aBBgpeK0KevAwAC+5xJ73HOUZ40Nw0whhBFT8QO51Of+lR3LBY78cADDxzUdR2xWOxELBY7cenSpeDmz94cEqhdSm9CRjxc3bqras5EUVaab2aJo1BqMM3X5DQJbhgoz4htu41azCsTF2wLeMCh+pOeWYSeFduvA717aFsuseP51V/91SXO+Q9e+u/w4cPNrycACdSuhTGGff3iTY5hiFSfE+wfEi8pDuDKTGNhvlWH2m4Nqjy/BK4Li3ujEZR2zaw/yYojDboiejJn+kkygoM0NYIgmoUEaheTistIxcRLYGZZQ7HcfN54rI/ZO6gaTfM1G0HVLipsdMyRVX9SRvaBBZrPRmjLMzCKWQBilbvkwDkJYrdDArXL2WtGURzA1fnmoyhFZhg3p5tfmTYaGqnUa0ZQxaKOfGHrrsKtrnrXV5dgLJu2dAfSe9zQUZ4S0RNTggj2jzV9ToIgSKB2PR0RCX1JEfIsZnSs5psfJGul+dJ5YCm7+fF1vVDb2Kxb2uKqd63WXu6AQKlz18HNfqrg8AEwmdoLCcIJSKAI7O0LwOrVvTLX/CBZyygBNGY3763thdrGXig7gmIMoYHNZyxZ6T0WiUEeGN3y9WoxKmWoM1cBAFIkjkDPSFPnIwiiCgkUgXBQwki3eNefLRqYX20uiupPMcQj4vGlqc0FqtkIyu6B6uuGFAhseCw3dNsgoYwfBpOa+xVQpy8Dhvh5hUaOgDH6lSJ2D+95z3uGDxw4cHs8Hr+7r6/v+Dvf+c6xubm5xuacNQD9NhEAgNGegG1uuDZf2dLKjJfCGMPBYfHSujIjxh5thNUHBWzPyWdFUKFG6k9T18BLYsxYYP+xLV+r7lzFbHWdRqIHSrLBCbkEsUOQZRmf+cxnri4tLZ09e/bsC9PT08EHH3xw3Knzk0ARAIS5YW+fEIpyhWNyqbkRSIdHxEurogHXZjcWqGhUQTRi1sG2leJrfNV75fLz9uPA/tu3fC0LzjnKN8+bHzGERp0blUQQ7cInP/nJqfvuu68YCoX40NCQ9t73vnf+ySefdGx8ClVzCZuBlIypZYZ8mePGQgUDnTJC29zPdGCIQWKAwYGLUwYOjWx8nr6eEK7fLGBuYesCVbwhBrM2MofPEih5cAxSPLHla1noqwvQM0sAgEDvCOTIrtuEQHhI/qufHdXnpz3r/Jb7hgqxH/+5LQ+o/eY3v9lx6NCholP3QQJF2DDGsH8wiGevl2Fw4OpcBbeNhDZ/4hqEgwxj/QzXZjkuTBp488v4hussBvvDuH6zgJm50pauU0lnoK0Kq2B078YGBSOzDH1+CgAQOLD99B43dJRuVht9aRkh4Tb6/HRUn5nw9bugv/7rv+78whe+0Pv1r3/9glPnJIEi6kjFZPQmZCxkdMyv6hhK6UjGtlfzPDwi4dqsjpUssLgK9Hauf+xgv9gDNTtfAucbi1kthetT9uPo+MYCVbl8zn7cjECpc9fBy+JNYmj4IDXlEq4j9w15up9pq9f7y7/8y9T73//+sS9+8YuXX/WqVzl2ryRQxC3s6w9gKavD4MDlWRUn9oW3tczv8IiEfzstHG4XJg30dq4vdNaiwnxBRzanIdGxsRvPonh90n4cGRve8FgrvcdiCciDexo6/0sx1FK9rby3OZs6QTTCdtJtXvHII490f/CDHxz9+7//+0tveMMb8k6em0wSxC2EgxJGe4RA5Erb3xnVkwRSZlLiwuTGdvOh/uom3a2k+WojqMgGERTXKqhcF6aGwP7bt20HL09eqNrKR28jWzmxq/nIRz7S96EPfWj0n//5ny86LU4ACRSxDqM9CkIBETVdm69sa9o5Y8x2803McZTU9c8xsE2BKk6ICEqORRHsSa17nDZxCaiIHqvtpve0zBK05RkAgJLqh5Lo3tZ5CGKn8Ad/8AejuVxOetOb3nQ4Go3ebf1z6vyU4iPWRJYYDgwEcO6mCk0Hrs2pODy8dcPEoREJ3z9viHThNMex8bVThYN9NQI1v5UISghUZHx4wzRk5fJz4oEkbWu8ETcMlG+8YJ5DJls5QQDgnP/AzfNTBEWsS3eHjK64eInMprc3p2/vAEPAfBt0/ub6ab5ER7UXaksRlJni28ggwTm3BUoZPQAWjjR8fgt17hqMkshghIYOQApu/RwEQWwNEihiXRhjODAYtOf0XZpRYWxxTl9AYfZsvguTxroTKhhjVSdfgwLFOa9GUBsYJPT5KRhps2fp4PGG793CKBegzlwBAEjhOAJ9NK2cILyABIrYkEhQwp5eYZjIlzmmtjFh4uge8TIrloHrc+sLnCVQjab4Kstp6DnhaI2Ory9QlfNP248Dh+9q6NwWnHOUbrwotjoCCI0dbXp+H0EQjUG/acSmjHYriARFFHR9voKiurXFhodHJTsKe3Fi/edaVvOZuVJDE9WLDTr4KhefAQDI/aOQO7dmbNBW5qCvinXySvcQlI6uLT2fIIjtQwJFbIokMRwaMtfDc5Hq28pKjmiIYXxAKNQLN4x104SW1bxUNpDObL48sVDTAxVdJ8WnryxUp0dsNXrSKiibEyOYEkBohIwRBOElJFBEQ3TGZAyYjbYrua2v5LDSfJkCML24tkAN9G3Nal6YqI2g1haoyoWz9uPg4TsbuleL8tRF8IqYDRgaOUITIwjCY0igiIbZ1x+0V3JcmVU3XaNRiyVQgIii1mJwi71QxWsiglIScQRSyTWPUU2BklK9kHqHGr5fLbtcXaXR0Q2lu/HnEsRu4YEHHhhXFOVEbQ/Uxz72sV6nzk99UETDBBTh6ntxUkVFFyJ1pMFhsokYw0gPw+QixwsTBt5w8tZjthtBRcbW7oEysqvQJ6+Jez98V8Pjmrihozxhzu1jEsJjR7c16okgdgMPPPDA0t/93d9NuHFuiqCILdGbqPZGza3qWMo2nuq7zYyiFjPAfPrW6KsjriAeE++ZZhtw8hVNgVpvirkwR4jrBLdQf1KnL9s9T8GhA5DCsYafSxCEc1AERWwJxoRh4qnLJegGcHFaxT0HwlDkzSOM28ck/PsZIWgv3DDQt8bw2KH+MC5ezW0aQXHObRffej1Q6kWR3mPxJOTh8U3vDwD0XBrqrIi6pGgCwYHGnkcQbvLMQ/9tNPv8Rc/2QXUcO1S483/9YUMDav/1X/81lUwmU6lUSrv//vvTf/zHfzydTCa3ZvVdBxIoYsuEAhL2DwRxcVqFqnFcmW1sDFJPkqGvk2E+zfH8NQM/fPxWgRpoUKDUhWXoBbHyYi0Hn1HIQbsm1tIED93Z0FBXbhgoXTc37jKG8N47aBgs4Quyz1+Mrp5+znf7oH791399/hOf+MTk0NCQ9vTTT4ff/e53733Xu9419i//8i/XnDg/CRSxLQY6ZSysSljJG5hN6+hN6uiKb7436o69Er71tI7ZFY75NEdfZ33kNdgnhG52obzhXqi6Hqg1UnyV808DXLyJC9x+qqHvSZ25DKOUAwAEB/dDjji2uZogmqLj2CFP90E1er1Xv/rV9nGnTp0qffzjH7/xYz/2Y4eLxeL1SCSy9QnTL4EEitgWVqrv9BWR6rswpeLUgTACm6T67hgXAgUAz13T8bq761+Cg/1ixp2qGlhaUdHTtXZkVt8DdatAqS+cFvcZT0IZ3b/p96Pn0lBnzNRepAPBgX2bPocgvKLRdFurkcwpK1vpk9zwfI6chdiVhIMS9veL3iBV47g8o276nJ4kw1C3ELHnrhm3vJAH+quCtJFRwlqzAdzaA2Vk02K9BoDg0VObpum4rqN47VkA3EztHadxRgTRAI8++mhqcXFRBoDnnnsu9Fu/9Vujr3vd61aj0agjCkW/hURTDKSqrr75VR0Lq5vP6rtjb9XNN7tc/zoe6q9OCZ+ZK697joLZAxVIJRFI1qfi1BfPwHbvHd08vVeeugBeFpmK4NBByFFK7RFEI3z605/u279//x2RSOTuN77xjYdOnjyZ+8IXvuBI/QmgFB/RJCLVF8LpK0VoOnBxRkUiKtvLDtfi2LiEr5ur4J+9ZmCwu/o+qb4XqrjuOYoT6zv4rPSe1NkNeWjjyeNaZgmV+Rvi+FgnggN7NzyeIIgqTz755AU3z08RFNE0oQDDoUGR6tN04MJ0ecMcdCrOsKdv7TRfNCKjMyGmp0/Prp/iq11UWIueXoQ+Jd7AifTe+kLJNRWla8+KDyQZkb3HqSGXIHwECRThCL1JBX3J6qy+qeWNU31Wmi+dByYX6sVseFBEUTen146gjEoFhatCoOIHx+u+VnmhuuAzsEF6j3OO0vVz1Vl7o4chhT1rMyEIogFIoAjHODgYRNhM7V2dqyBXWr9X79iYBCtYeeZa/XFjI0Iobkyt7XQtXL0JrgkBjB2qT8mp58z0XvcA5L4NdkQtTkJLzwEAlM4+BHpG1z2WIIjWQAJFOIYiMxwZEak+zoEXJ8vrbtDtiDLsM1dwPHvVgKZXj9tjCtRKuoJM9ta1G7kLV+3H8cNVO7g2Nwl9XkRWwdvvWTddZ5RyKN88DwBggRBCY8cotUcQPoQEinCUZFTGWK/w3hTKHFdm19/rdPcByTwOuDRVI1DD1VTbWlFUvkagYoerEZT67Pftx8E7Xr7mNbmho3j1WcAQJo3w3jtojQbhN3TDMHbFOybz+1w31UICRTjOWG8Aiah4ac2saJhfx3p+dI+EoOkjPXO5OnTWSvEBwI3JW+tQuQumCaK3C8GuTgCil0l9/gkAgDJ2aN3NueWbF2AUMgCAQP84lETPVr41gvCCxycmJjrL5XLAqYZXP2IYBltYWEgCeH69Y8hmTjgOYwy3DQfxg6slYT2fVtERkRAJ1r8fCgYYju2VcOaSgQs3OfIljliYYXgwDFlm0HWOiclbIyhLoGrTe5Ur58AL5pii469Y874qK7OoLFiW8iRCw4cc+X4Jwkk0TXsonU6/J5vN/gLnvAs7N5AwADyvadp/Xe8AEijCFcJBCYeHgzh3Q4VuAC/cVHH33hAkqT5zcfd+IVAGF7Woe4/KUBQJQwNh3Jwq4sZLBIpzbqf4YjUCpT5npvcCIQSP3H3L/RjlQnUQrKwgsu9OmhZB+JKTJ0+qAB4x/+1q6DeUcI2eDgXD3eI9UK5k4MrcrfWosX6GlDmj+ekr1VR01clXn+JTF1dQWVkFAMTN+pNRyKFy6TkAQPDIXWDBcN1zuKGjeOUsoItUY3j8DkghspQThN8hgSJcZV9fAB0R8TKbXtYwl66vR0mM4e4Don9qeoljdkWIlOXkm5wpQtOqwlVvkBARlPrCaVt8gsfvveUeyjderNad+sYQSPU78r0RBOEuJFCEq0gSw9GRIBRzE8fFaRX5l/RH3bW/+jI8c8kUqGExk0/Xed1Eidz5Wy3mlntPSnZBGTtYd+7K4iQqi8J6LsU6ERo57MS3RRCEB5BAEa4TDkq4bURMI0k5XAAAIABJREFUKTc4cO5mua7vqauDYa/ZE/X0FQMVjdc7+Wqs5rmL5kqMYADR8WFoMzegz0wAAIJ3vKJucrleyKA08QIAgClBRPbfRXUngmgj6LeV8ISuuIzxXjFjr6hynJ9S62bw3XNIvBSLZeDchFHXC1Xr5LNSfNGD42CyjPLT3xVfYAyhu+6zj+OaiuLl6tLC8L47Ib2kNkUQhL8hgSI8Y0+vYq/mWMrqmFiomiaOjkmImqugnrpoIJkI2ENja40SVgQVP7wPvFyCeu4pAEBg/zFIyS4AAOcGileeAVfF84LDh6Ak1u6LIgjCv5BAEZ7BGMNtIyFEgiKdN7GgYSEjzA2KzHDioHg5TsxxzK0Y2DMi6lC21ZxzFK6KxaLxw3uhPv8koIphr8ETr7avU568AD27JM6bGqAVGgTRppBAEZ6iyAzH9oQgm6+881NV08Q9h2T7uNMXDbsOZaX4jJIKGOLY2MG9dnqPJVII7L8dAFBZmkJlTtSkpEgHwuM0Z48g2hUSKMJzoqEa04QBPH+jDFXj6E4w7B+smiWGh4RAZbIa0qsV6OXqht1IdwT6nHDnhe56FZgkQc+t2M24TAkgcuBuMJl60QmiXSGBIlpCd4eM8T5RYypVOM7dLMMwOF52WERRJRUIdCTs4ycmCzBKVYFSsmJkEZiE0F2vhFEumKYIDjCG8L67qBmXINocEiiiZezpUdBvLjnMFAxcnFZxeBSIi9ITZrJV192NyQJ0U6BCAz0wrovJEYFDx8GicRQvnQHXVPH1PUfJFEEQOwASKKJlMMZwaChoTz6fW9UxtazbUdRilqGrT0RRIoISAhQZTAGacAAGT/4QilfOwiiJQbGB/nEEe2n5IEHsBEigiJYiSQy3j4bsTbzX5yvYP2TYJooDx0YAAFevZ2CUxESJcEQ4/6S+EeioQM8sAgDkZC9NiiCIHQQJFNFyggrDsbEQFPPVOLFYwUlzE0a0M4FwNIT5Z6+C66aDr1sYLJTDx6AtTQEApGhCTCgnxx5B7BhIoAhfEAtJuH1PCIwJn0MqoSEeEYaH4f0DiEzVDIkdiINFYtBDwqHHgmFEDpwgxx5B7DBIoAjf0BmTcWRYrF83OHDPER3hIMfg3j4M5ISlHIwh2hcDhkbAZFnsdjp4isYYEcQOhASK8BV9SQX7+oX9XFGAE4d0RCIy9qhzAIBobxRSMABpdBxgEiIHTkCOxFt4xwRBuAUJFOE5X/ziF/HqV78aiUQCinJrWm6kW8GIuegwFgHuPqihb8XcojvQATY4DBYKI7zvTigdXZ7eO0EQ3kECRXhOKpXCe9/7XnziE59Y8+uMMezrD6DP7JHqKCxAyYkturGBOKS9BxAau50WDxLEDoeqyoTn3H///QCAxx57bN1jGGM4PByEqpWReeYJ+/Ox28YQOnSSep0IYhdAERThWyTGcPY7X0Die/9if67yqvsRoOnkBLErIIEifAvXVDxw1xgKF8R0cqZImD7welycrtQtOyQIYmdCAkX4Eq5VULh4Gtr5Z1CYyYpPhiIAY5hb1XFxWiWRIogdDgkU4TuEOD0FfXURlYsXUFoRm3HLgShy5nLd2TSJFEHsdEigCM/RdR2lUgmqKoa/lkollEolcM7tyMkoZGBcv4LC1Kr9PJUF8IPzMokUQewSSKAIz/nc5z6HSCSC+++/H7quIxKJIBKJ4Ma1KyhceBJGYRVcLYPfuIq8ld4DUNQlFIoGfnBeRlloG2bTOi5MkUgRxE6EBIrwnF/4hV8Q0VLNP10toac4C6NoCtL0NKBpyM2Kj6VkAhqTMXl5BqrG8P1zMmRJDIadW9Xx4qQKg0SKIHYUJFBEyzHKRRTOPwGjKHY6yeEk9CsvAgAKS2LvU+fdRyExhslL02DcgKoxPHdZRiwkRGoho+OFmyoMg0SKIHYKJFBESzFKORQuPAFeLgAAAj3D4JMTgFaBXtaQn1oBAHSevB2hkAStoiMztwAAmFoCAlIAHWHxMl7K6njuRhmaTiJFEDsBEiiiZeiFDArnnwRXxSLCQN8YlI4+qE9/FwBQ0DsBQ+yASr3yBCJhMfro2acmEBUrofDtpw0cHQ0iaW7lTecNPDtRRkUjkSKIdocEimgJWmYJhfNPgGvC7RAcOoDQ6BGUvvNlU5QYClqnfXzq3hOIRIRAVVQdB/vE85aywJlLHHeMhdAVFy/nbNHA09dKKKmGt98UQRCOQgJFeE5laRrFS6cBQwcAhEZvQ2joALSbl1G5cBYAEDz+cqw+ewUAEDu8F6HeLjuCAoDcwhK6E+Lxd57RUa4At+8J2QNmiyrH09fKyJVIpAiiXSGBIjyDcw519hpK154Va3MZQ3jfnQj2j4FzA8Vv/qM4UAkgdN+PYeX7TwMAul55UnxaZhgZigAAnn9xFfefFLOOiyrw2DM6JMZwZDhor+pQNY6z10pYyekef6cEQTgBCRThCZwbKN94AeXJC+IT5ibcQNcgAEA9dxr6jJi5F37565GfWICeE8aJ1CtP2Oc5fpsIm85dyODQMDDeL1x8T5w3sJThYIxh/0DQXnqoG8BzE2XMrmiefJ8EQTgHCRThOlzXULx0BpWFmwAAFgghevhlUBLd4utqCcVvf1l8LZZA+N43YOV7Z+zn1wrUHbclAQC5vI7rNwt40z0ipacbwL89VRWh0Z4AbhsJgjGAA7gwreLaHDX0EkQ7QQJFuIpRLqBw/vvQM4sAACnSgeht90KOJuxjio//K3hW2MkjP/wTYKEwlr/3AwBAsLcLsYPj9rHHjybtx8+9mMFwj4S79ouX8Ys3OS7crNac+pIK7hwLQTFLVzcWNbw4qUKnXimCaAtIoAjX0LLLKLz4f6sNuMleRI+8HFIwbB+jL86g/MQ3xdeH9yJ4573gnGPlP4VApV55Aowx+/g9IxEkO0SN6dkXxJy++0/KCImMHr76pFZnMU/GZNy9N4xIsNrQe/ZaCeUKmScIwu+QQBGuoC7cRPHiU+CamAQR6BtD5MDdYHJ1iTPnHIV/+6KwlTOG6P3vBGMSijemUZqaAwB03Xey7ryMMdxhRlHPvSgEqiPK8PoTIkxayQL/8Vy9KSIaknD33jA6Y+LlnitxnLlawmqBzBME4WdIoAhH4YaB0sQ5lCfOVZ1648cQ3nMbGKt/uVVeOA1t4iIAIHTiNVAG9wBAff3p3hN4KXeYRonZ+TLmF8sAgJcdljDYJaKk7z4nDBO1BBSGO8ZCGEpZDj/gmetlTC+TeYIg/AoJFOEYhlpC4cITVTOEEkTk0D0I9Izcemwxj8K//4M4LtaB8A//hP01S6CkcAjJE0dvea5llACqUZQsMbzlFSKK0gzgn7+v3WKIkBjDwaEgDg4GwCD089KMigtTZZrhRxA+hASKcAQts4TCC9+DkReCIcWSiB59JZSOrjWPL/7734PnMwCAyOsegBSO2l9bfvw0AKDznuOQgsFbnnvkYAeCAREtPXOuui9qT5+EU4fES/rKNMeZy2vXmYa6ArhzPISgmW2cTes4e62MIk2eIAhfQQJFNAXnHOXpK2a9SYwfCvSMIHq43gxRS+Xy81CfewIAoOw/huCxl9lfK83MI/u8SPt1veaeNZ8fDEg4elik+Z58eqXua288JSNhat2/Pqkjk187MkrGZJzYF0YiYo5HKhk4c6WExSyl/AjCL5BAEdvGqJRRvPQDqNOXxCeYJOpN48fApLVfWrxURP5rfys+CIUR+7EH61x6C9943H7c+4ZXr3vtl58QkdnkdBFTM0X78+Egw0/cK0KjUgX4yv+9NdVnEQpIuHM8hOEucbxmAOduqLgyS2s7CMIPkEAR20LLLKLwwvfs/iYWiiJ6271r1ptqKXzrS3bPU/R1D0BKpOq+vvD1/wAABFJJpF5+57rnecWpaurw+2eW6752ZFTCnfvES/vCJMfZq+un7iSJ4cBgEEdHgpDN34bJJQ1nr1PKjyBaDQkUsSW4YaA8eQHFi6fBK8JBp3QNInb0lZCjHRs+V734DNSzIkJSxg8jeNd9dV83NA0L//6fAICeH70PTJZvOYfFgfEYurtEfeqJ08u3fP3NL5MRMzOMX/2+jpXsxhFRb1LBiX1hxMMimssWDfzgSglz6fUjMIIg3IUEimgYvZhD4fz3oc5eE5+QZJHS23u8rr9pLYzsKgr/+/Pig1AY0R//2brUHgCkn3gGWloYJzZK7wGiH8pK8515Lo3yS6KdaJjhv7xS3FO5AvzDd7VN03ZWv5SV8tMN4PyUihcnVVRoCSJBeA4JFLEpnHOocxPCpVcQAmKNLAr0jNwiNLc+30D+q58FL4iJErE3PQg52X3LcQvf+K79uPf+jQUKAF5xUghUqWzg2XPpW75+2x4JLzssXuIT8xz/8dzmKTsr5XdsTwgBM4BbyOj4wWWaik4QXkMCRWyIUS6gePEplG++CHDxBz44sFfM04vEGzpH+anHoF19QTz32MsQvH1td97Cv4n6U+KuowgP9G563nvuStl1o++fWVnzmDfeI6PHHPv37bM6Jhcaqyt1d8g4dSBiL0EsaxzPTpRxaVqFTtEUQXgCCRSxJpxzqAs3kT/3n9CzosbDghFEDr8MoZHD67r0Xoo2dR3Fb30JACAluxG9/51rHleeW8TqmXMAGoueAKAjrth28yd+cGsdCgCCCsPbf0iBLAEGB774mIZCuTGBCSoMx/aEcHAwAOvbnV7RcPpKCek8RVME4TYkUMQtGKW8iJomztlbbwM9I4jdft+6jbdrnqeQQ/5L/0ucQ5IQ+y+/CBaOrHlsrb28742vafgaVprv+s0CZudLax4z1C3hDSdFvi6dN+tRDRofGGMY6grg1P4wklHx61KqcDxzvYyL0yo0iqYIwjVIoAgbbhgoz1x9SdQURuTgSdHbtIkR4qXnyn/lr2BkxHkir3sAysi+dY+36k9KsgOdr7ir4etYAgUA318nigKAVx6VcPuYqJVdnOT4P89uzUIeCYqeqf0DAUhmyW1mRcNTl0tYzFBzL0G4AQkUAQDQcysovPg9qFMX7VpToHcPYre/Ckpy83rQSyk9/jW77hS47SRC9/zI+tcuq5g36089r3slJKVxITy4L46uTrFr4/EnltY9jjGGt96noNuqRz2t49LU1kSKMYaR7gBOHahORlc1jnM3VTx/o4wSrfAgCEchgdrlcE1FaeIcCuefsPc2SeEYIodfhvDY0S1FTRbqi2dQ+u7/FufqHkDsze/a0Om3+I3v2vbygbe+YUvXkiSG19zbAwB46uwK0quVdY8NBxl++kcUBGSxZffv/o+GxdWtp+giQQnHx0I4NBSEYv4G/f/tnXeUHMW97z/V3ZNnNmftSquIhEAIEBKYICyCbQxO4PCwbGPZxtfPEfu9B+f4Ol77XIf7fM954ADXvibY19gYG4t4wZggooRAESEUdldarbSrTbOTQ3e9P3qjtHF2Znd2tz7n1Omenpmu6pqZ/s6v6le/X0fI5LWDcY62p8Y9fKhQKEZHCdQcpd8JYs+W/ujjCA1nzRK8Z05srmkw6eNNRDbfbT9wufFffzPCNXxMvj5a/miLme71UHndhgnXeeVlFQCYpuTZl06O+tqqYo0PXGzPR8WTcN/TKWLjdJoYjBCC6mKDC5Z6qCgcSDt/uDXF9kNxupQThUIxaZRAzUHSoS6i+14m0bS3P6GgXlCKb+XFuGqWjNtD71SsUDfhP/0S0ikQAv8HP4teXj16WyJRWh/+BwCV123A8HlHff1wrDqzkIoyFwB/f65tzNefs0jn8lX2NXb0wB+eTWecBt5pCFbUujh7gas/a280IdnVmODNowniKlySQpExSqDmEFYiRuzQDmL7X+1fcCucbtyLV+NZugbN7cv43DIRJ/ynXyDDdvoLz1U34Fi8csz3tT78D8yoHey15qPvzahuTRNsuMSeJ9v5ZrA/ieFobDhX73eaOHxcsvllc1IhjUr8OmsWu1lYMeBEcbLHZOvBOA2tau2UQpEJSqDmADKdItG8n8ieLaS7TtgHhYazejG+lZfgKK4aMxrEWOcP//lXmCfsoULnuZfgWjOyU8RgWv74CABGUQFlY4Q3Go0r19vDfFLC01vGtqI0Ibj+EoOaUvu6tx+weHrH5IblNE0wv9zBBUvd/cN+UsKR9jRbD8Zo6VRx/RSKiaAEahYjLYtkayORPc/b8fN6vfOM4ip8Z12Ka97SjJwghtQhLSIP30O6cT8AjiVn4333x8YleMnObk7+t73+qfqDV6O7Tk9OOF7OWOyntsZeY/X358cWKACnQ/CJKwyKewNiPLvT4pV9k587cjs0VtS6WL3QRcDd5+1nZ+997ZDtlq6ESqEYGyVQsxApJan2Y0T2PE/i6Fv980yarwjv8nV4Fq9Gcw2/YHai9cSefIDUm9sB0OctwvehzyK0kaOQD+bEX59Cpuy21Xzs2km1RQjBVb3OEvsPhjnaEh3X+wJewaeudvRHPn/0VZPdDdlxcCj06py7yMXyeU5cjoH5qb1Hk+xoSKhoFArFGCiBmkVIKUl1niC690XijbuRSTuygnB5cS9ejXf5OnR/8RhnGX9dsacfJPHaswBoZVX4P/o/EY7xW0Et9z8MgKuqnNL1a8d49dj0efMBPDUOZ4k+ygoEn7zSwGnY7ucPPG+ytzE7zg1CCCqLDNYucbOw0tHvlt4Ts9jZmGBXY5yeqBIqhWI4lEDNAqSUpLpOEH3zReKHd2DF7fVMwuHCtWBlVuaZTq0v9sxDJF59GgCtqJTA//gKmmf8Thbh/YfpeNZO+1794feMmvtpvCyo87JskT1e98iTJyYUhmhemcbGKwwM3Y7Z98fn0uw7kj0PPE0TzC9zsHaZh7oyo9+Roiti8UZDgt1NcXpiSqgUisEogZrB2BbTcdtiOrSjf6GtMBy4as/Ad9ZlOMvrMnYbH6nO+LObSbz8JABaYQn+jbeclhl3LJp++V/9+wtuHj6AbCZ84JoaANraE7zwSvuE3ruoulekBgWWzaZIATh0waJKJ2uXuqkpMej7z9AZtnjjcIJdTXGCyqJSKAAlUDMSaVn9i2zjh3f2W0zoDpzzluI7ez3OqoVZsUqG1CstYk/+ifhLTwAgCorxb7xl2NxOo5EOhWm+145wXnblxfiXL85aG69eX0HAbzt+/PmRYxN+/5IajRs32NHPTQv+8EyanYezLxguh8bSaidrl7ipLh4Qqq6wxY6GBDsa4nSGJ+f6rlDMdJRAzSBkOkXyRAOR3c/Zi2wTtiOAMJw45y3Dv2o9rurFk/bMG7ZuyyT6yH0Dc06FJQQ23oJeVDbhczXf+xDpUASA+i9uzGYzcbt1rr2qCoAde4IcbAhP+BzLajVufOeAJfXn5022vpUbq8bt1FhWYwtVzSChCkYtdjcleP1wnLbg+KOvKxSzCSVQMwArESN+dB/hXc+SaN6PTNkLUYXTjatuOb6z1+OqXpQTYQKQqSSRB/+D5K5XANBKKwl88hvoxRMPIisti8Zf2qnfPQtrqXjP+qy2FeCD19T03+gffLQlo3OcUafxyasGHCc2v2LyzM7cWTRup8bSGifrlrqpLTX680+F45J9zUm2HrDj/Kn0Hoq5hBKoPEVKSTrUSezgG0R2P0eqtak/N5Pm9uOuP9ueY6qsz/pQ3mCscA+h3/07qbd3AqBX1hH4xDfQCjKL1df+j5eJ7G8AoP4LH89J22uqPFy81h52fPKZVnpCIweQHY1F1Rqb3mXgsaMo8fQbJn99ycw4LNJ4cDk0Flc5uXCph/pyR3/a+URKcrg1xStvxzh4PEk0oUIoKWY/SqDyDGmmSZ48SvTNl4jt30q6u7X/OT1Qimfp+XhXXoyjbF5WnR+Gw2w/Tujun2C2NAJg1J9BYOMtaL5AxudsvOM+wA4MW3fT9dlo5rBcf+08ABJJi4efPJ7xeWrLNT73HgdFvQ6Krx+wuPepNPFkbi0ZhyFYUOFg3TIPy6qd/XH+TAuOddp5qHY1xWkPqUW/itlLbsaEFBPGjIVInTxKqqMFzEEJ8ISGUVKNs3IBurdgytqTfHsXkc2/hYS9lsq56iK819w4qWHE7td20/boMwDMu/F9OIoLs9LW4VhzThH1dV4aj0b5w1+b+eA18/B6MrPWKooEn3+vg989neZYh+TQccmvHknx8Q0Oyouy47o/EromqC4xqCrW6QxbHOtI0RWxraeusEVXOInLYUdWryoy+hcEKxSzATHGvy/11yyHSDNNuusEyZPNWJHuIc8JpxtHWR2O8jq0CSx+nXSbpEV8y2P9+ZwA3Jddh/uS90x6HdXW936Gk0++gHA4uHzv43gX1k3o/WvWrOG1114b9+ufeq6N7/3bPgBu/kQ9n/zIggnVdyrJlOSB59PsO2r/LFwOuOFSgxXzp3YgIpKwaOlI0xpMYw4a6RNAaUCnqtigxK9lbd2bYkRUB+cYJVBTjJQSM9xNuuMYqc7j/fNKfegFpTjK52MUlSPE1N74rGiY6MP3kjq42z7gdOG77iacy8efgn0kOrZs45UNtsfegn+6kbNu/86EzzFRgbIsyae/up1DjRH8PoMHfr2u3wU9Uywp+ccOk2d3DijDZWdrXHGujq5N7f0qbUragmlaOtNETslp5TQEVUU6lUUGXpcayc8RSqByjBKoKcJKREl1tJDqaOl3D+9DGE4cZbU4ymvRXBPPh5QNUkcOEHnoP5Eh25LTSirw3/BPY+ZzGg9SSl5+58fpenE7mtvFO/c/hbumcsLnmahAAbzwaju3/WAvAJ/8yHxu/sTCCdc7HG82WTz4QppEr//F/ArBRy4zKPJP/T1LSkkoZnG8K01b0ORUH44Cj0ZlkU55gYHDUPfULKI6M8eoOagcYqUSpLtOkOpowYoEhz4pBEZhOY6yWvSCspw7PIyENE3iLz5O/IXH7NwQgGP5uXjfuxHNnR2xPPnkFrpetAPK1n9xY0bilCkXry1lxbIA+94O8cDmZj583TyKiyY/ZHrmAo3yQgf3P5emtUtypE3y880p3v8Og7Pqp/azFEJQ4NUp8OosrpKcDJoc704TitlWXk/MoidmcfBEihK/TkWhTmlg6i0+hWKiKAsqy8h0klRXK+nOE5ihjtOe17wFOEprMEqq0RyuaWjhAObJ40Q234154oh9QDfwXvVhnOddmrX5Cyud5sULb6Bn5z6MgI93HngaZ2lmAWszsaAAtu3o4pZv7QLsNVLf+MLSjOofjlRa8vg2k637B4b8Vi3UuPZCHa9regUgkrBo7U7T1m2SSA/9KWsalPaKVYlfR1NilQmq03KMsqCygG0ptZLuasUMdXKqrgunG0dJDUZpNboncxftbCFNk8TWp4k993C/x6BWXo3v/ZswKmuzWlfD/7uHnp22o8Kir2/KWJwmw5pzijj/nCK27+zmocdbeNc7KzlreXY8Ih2G4H0XGSyutnjo5TSxBOxqsGg4YfG+i6begWIwPpfGokonCysk3RGLtmCakz0mpgWWZWf8Pdljomt2RuDyAp0SZVkp8ghlQWWIFY+Q7m4j1dV6mgce2JHEjeJKjOJqdH9R3nhUpY81EH3s95htvXHqhMB14VV4LrsWYTiyWlf08FGeW30tViyOb1k9l27fjO7O3GrM1IICaDoa5aavvEYqLVm0wMdv/v08HI7sikcoKnnopTT7mwd+NmfOF1y7zqDAlx+fv2VJOsMmbUGTjrCJdcp6X01AsV+nLGCLlVPNWY2G6pwcowRqnEjLwgx3kQ6eJN3ddpqjAwwSpaJK9EBJ3ogS2B568eceJvH6Fvo+Vq20Et+1n8CozV6w1j6klGx9zyban34JgAufvo/SyyaX82kyAgVw9/1N/Pr3jQB8dmM9N310cm7nwyGl5I2DFo9tM4kn7WMuB2xYrbNuuYah5893wrQknSHbiuoMm0Nc1vso8GiUBOw5K59L5NV3Og9QnZFjlECNgpWMkw62Y/acJN3TMXQBbS/C5cUoqsRRXIHmyx9LqQ9pmiRef574848g472iqhu43/Eu3O94V9atpj6a73uInZtuBaBu04dZdecPJn3OyQpUKmWx6WvbaTgSxWEI7rl9DfNrc+M1GY5JHttmsuvwwF2/rACuWWuwrDb/3L4tS9IZMenoMekImaSGiY3rNASlAY0Sv06RT88rsZ0m5nwH5BolUIOQZhoz1EU61I4Z7BhIY3EKur8IvbACo6gcze3PO1GC3lxR+98g9uxmrI6BcEnGwuV43/VR9NKqnNUdOdjECxdeTzoYwlVVzvpdj2YlasRkBQpgz1tBvvB/diAlLFvs55c/OReXM3eCceCYxaOvpmnvGTi2uEZw9Xk688ryT6jA/u4EoxYdIZPOkEl0hLBOBV6NEp9OkV+jwDMnFwbPuQueaua0QEnLxAx3Y4Y67RLp7ne1HoLuwCgswygsRy8om9LIDhNFSkm64S1iz/4N83hT/3GtuBzPldfjWLoqpzcSMxrjxUs+Smj3fgDO//PPqXr/lVk5dzYECuD23xzijw81A3Dd1VXc+uUzJn3O0UibklffsvjHDrN/3RTAWfUaV6zWcx4uabLEkhadIXvOqjtiDf8T0aDQq1Hss60rn3tODAfO+gucbuaUQEkzjRkJDhKkIMjhokILdH8hekEZRkEZmq8w739sUkrSh98ktuUxzGOH+48Ljw/3xe/Gdf76nA3nDW7Dzk/fyrHf/w2wvfZW/PjWrJ0/WwKVSll85Zs72b3PNmtu+/Iyrr168guSxyIckzy702Tb21b/fI/AFqrLz9GoLM5Pi2owpiUJRiw6wyZdEZNoYvhbhKFDoVen0KtR5NPwu2elhTXrLijfmNUCZSXjtoUU6cIMdWNFexjpkjSPHz1QilFQajs45Ci3UraRlklq3+vEX/n7wHomAIcL99oNuC+8CuH2TElbGn/xe/Z+9fsAlKxfy7onfotmZK8fsyVQAO0dCTZ9bTtQIkreAAARnklEQVSd3SmcDsHPf7SaFcumJhhvZ0jy9zdMdh+2hnwbl9cJLjlLZ0HFzLE+EimL7ohFV691dep6qz40zXa4KPRq9qJiT345jGTIjL+AfGfWCJQ0TcxoD1YkiBnpxox0I5PxEV+vuf3ogRL0QDF6oGTaF81OFCsWIbnrZRLbnsEKdg484XTjXnM5rnVXoHn9U9aelgce442N3wDLwlVTwaVb/4qrcuLZdkcjmwIFsGNPN1/95k5MC4oKHNzxo3Oor/Nl7fxj0dYteW6Xya6GocNmdeWCi1ZonLlgZt3EpZTEk/aaq+6oSXAUwQLwuQQBjy1YAY82E70EZ1RjZyIzUqCkZWHFQpjRIFakBzMSxIqFGbG5QqB5C9D9xf0ln+eRRkJKiXn8CInXnye5dxukByY0hNeP6/z1uC54J5pn6m6yAK2PPsP2G76ETKfRvR7WPXkPxevOyXo92RYogL88eoyf/eogAKUlTn7xo9XMq54ai7OP9qBkyx6THYesIa7efg+sWaaxZqk+LTH+JouUknjKdrjoiZoEo9aIQ4Jgr8HyezQCHo2A2956nHktWnnbsNlC3guUTKcwYyGsaMi2kKI9tnfdKO0WDhe6rxDNV2QLkq8AoeUu62yusaJhknu3ktzx0sAC21600krca6/AefY6xDSIbvvTL7Ht/Z/HSiTRXE4u2HwXZRsuyklduRAogHv/1MRd9zUCUFXh4o5/XU1VhTvr9YxFKCp59S07bFI0MXBcYHv+nb9UY3mdNqMDvqbS0o4NGDXpiVmEYtaw66/60DXwuW3B8rntuSyfS+RLaKa8aMRsJm8ESloWViKCFQvbYhQLYcVCow7TAaAb6N4CW5C8hei+QoTTnc//usaFTCVJHdhFcs9WUof2MmTJvxA4lq7CtWY9Rv3yabvW5t//jd03fxMrmUIYBuc/cDuV127IWX25EiiAO+9t4L4H7Dm8shInP/7WWZyxZHrCUqXSkj2NFq++ZdHcPvQn6HbAynqNcxZp1FfmzY06Y6SURBN2NPZQb1DbSGJ4T8HBeF0Cn8sWLZ/b3nc7ptzamtmdPwOYcoGSlokVj9glFsaKh7FiEaxEZFSrCADdge4tQPMG+kVJuLwzXoz6kMk4qYN7Sb71OqmDeyCVHPK8VlSG85x34Fp1IVrB1Me060NKyYF/uYMD/3IHAMLh4Nx7f0r1De/Jab25FCgpJXf85+F+93OPW+M7/3sFl6zN7jzaRGnpsHj9gMWOw1Z/ZIo+/B5YuUBj5QKNBZVi1sTQsyxJpFe0wnG7ROLWaWlETkXTwOfU8LoEXtfANofDhLOjw/OYnAiUlBKZjNkilIgOCFI8ikzGxnUOze1D8wTQPAF0bwDNG0A4Zr5ldCpWsJPUwd0kD+wm3bj/tGgVwu3Bsfw8nGetxZi/ZMqTGJ5KqivI7i9+h+MPPA6AUVTAmj/fQen6dTmvO5cCBfb39oHNx7j9N4eQ0r7hfebGejbeMB99mp0VUmnJW0ctdh62OHBMnjYs5nHCslqN5XWCJTUanmmOpJ5tpJREk5LIIMGKxOWoThh9CAEep8Dj1PA6BZ5e0fI6NRwGk7mnzK5OzkOyLlDpUCext7eNbQ31NcDlQXP7ba86jx+tt8zkOaPRkMk46SMHSTW8RerwXqz2E6e/yOnGuWwVjuXn4li8Mufrl8ZL+zMvs3PTbcSb7TZ7F9Vxwea78J+xaErqz7VA9fHCq+1896f7iCdsFTh7RQH//PXlzKuaWueJkYjGJXuPWLzZaHHouDzNstAE1JbbQrWkRjCvbPZYV6eSMiXRuD0sGElIogmLaMIieXpUsmHRNbjoDE+m/TM7OzWPyLpAWfEIkT1bhh7UdDSX17aK3D5bhNw+NJcPoc9OIepDxmOkmw+RPnqQ1JEDmC2NnBZCGhD+QhxLz8axdBWOhcvzRpQAkp3dvP2922n6xe/6j1Vcczmrfv2vuMpLpqwdUyVQAAcOh/nuT/fR1GzHL/R4dG7eWM8Hr6nBMPJnQW00YVtW+4/altVwN2anAfVVgvpKe96qplTMKPf1TEilJdGk7TUYTVjEkvY2npRDbmpOQ3DRGRn/8ZjdnZgHZF2gpLRINL/dK0C2KAmHa9YNzQ2HtCysjhOkWxpJH2vAPNaA2dbCsN0oNIzahRiLVuJYdCZ6dd20D9+dipVK0XTn/Rz4/u2kuuyMwLrXw4qf3sb8z310yj/TqRQogHjc5Jf3HObBR1r6jy2o9fKlzyziojWlU9aO8ZI2JY2tkgPHbLFq6x45ykNNqWB+uaCuXGNemaDQN6mhrhmD1btWK5aUxJIWSKgty/jP4OzvsGkmb7z4ZhoyncI8eRyztRmz9SjpE0cxW5shlRjhHQK9Yh5G/RkY9WfgqFsyZREeJooZi9N8z1849LPfEGto7j9eevk6zvr59/AvWzgt7Zpqgepj6+ud/OxXB2k+PjB/evaKAjbeMJ+L1pTkrSddT0Ry+ITF4eP2tnv42McA+Ny2aFWX9BWNkgB5e215guqcHKMEagxkKonZ0YrVcQKzvbecPIbV2Tb6PJvDhVGzAGPeQoy6Jei1i9DcuUntkC2iDUdpvvevNN11P8m2gXT13sXzWfHjW6l83xXT+i97ugQKIJmyePCRY9x9fxOR6EAuioXzvXzovfO48rIKAv78Do8VjEiaWi2a2iRHT0pOdJ4+fzUYQ4eKItFfygvtUhxg1s5pTRDVCTlmzguUlBIZi2AFO7C627G62jG72rG62jA725Ch07PlnobhQC+vQa+ej1G9AL16AXp59Yxw9Ei0ddD6yD9ouf8ROp55ZchznoW1LLplE3WbPozumv7IG9MpUH10BZP86W/N/OXRliFC5XQILruonCsuLeeC1cW43fn/2SfTkpYOuxxrl7R0WLT3jO3fZGfdhdICQUmBoCRgl2I/FPkFLsecuW/PmQudLma1QElpIaMRrHAQGQ5ihbrt0tPVWzqxgl2jDMudikArKrHFqK9U1qGVVswIMQJ7Xin42h46nnuVtieep+ul10+7IxWuOZuFX72J6hvendVgr5MlHwSqj3Akzd+eaOEvj7bQenLo98fl1Ljg3GIuOLeY81cVsaB25qzVS6YlbV2S4132trXb3obHWC8/GI8Linz2vFahT1DgFRT4oNArCHgEAa+dZXim9MkozPgLyHdmlEBJy0TGosh4xLZ6ovZWRkNY0TAyGsaK9CAjPViREDISAmuY1KBjIDw+tKIy9NJKtJIK9JIKtNIq9NLKaQknlCnSsog1HiO4cx/B13bTvW0X3dt2YYZPT1fvLC9h3o3vo/ZTH6Lg7NzmR8qUfBKoPixLsn1XN48+dYItr7STSJ7uoVlS5GDlGQWsWBZgxdIAi+v9FBc5ZtQNOpqQtAclJ4OSjh5Je4+ko8eOzJ4ap0v3YBwG+N3g9wj8bvB5BD63HUDW67YjRfRtPU5wOUHLv/7KuwbNNnIuUFJKMNPIVBKSCWQqgUz2lTgymYBE3N5PxJCJQdt4FBmPIRNRrHgUEhP4GzcKwutHKyhGKyixt0WlaIW9pbgs7+eKBmMlk8Rb2og3nyDa2Ez08FEiB5uIvN1AeN8hzOjIC6N9y+qpvHYDldddQfFF5+a9y38+CtRg4nGTbTu62PJKOy9v76SrOzXiawsDBgvn+6ib56GmysO8ag9VFS4qylyUFDlnjHOClJJwDLrCkq6wpDts7wcjkmDEnvdKjNwN40YIcDvtBclup8DtpLcI3A7bInM7BU4HuAxwOXr3HbYrudOwRdHpyOr82cz4kGYwWRcos+0Y4Qd+hUwlbVFKJUdICphldAPhDaD5/AhfAZqvAOELoPkL7RIoRASK0PyFebXGCOxhNzMSw4zESEeimKEI6XCEdE+YVHeIVDBEujtIsr2bZEcXyfYuEidOkmhtJ9neNb5F0ZpG4MwlFF2wipL1aym7/ELc8ypzf3EjYJomt912G3fffTfxeJyrr76aO++8k7KykUML5btADUZKSePRKK/v6mbHniBvHQhxvG18f7B0XVBS5KCk2ElJkZPCAoddAgYBvwOfVyfgN/B6dLwee+t263hcGi6XPu1RL04lkZL0RKAnKgnFJKGYHRg3HLeTOEZiEIlLIolxr++fFLpmi5XDAK9T8OUPKDfzfCXrAhV7ax+Hvv6/Jn4aif03yTAQugMMB8IwQHcgHPZj+5jTHmbr3cfptAVHNxBC2BZb3zUNvjYp7Ye9z/e/7tTjljVoX4Jl2Y4UpmXvW9bAvmkhTXOgpE1kOo2VNpGpNFYqhUyl7f1kCiuVwkok7RJPYMYTWLEE0pz4MORouOuq8S2ej//MpQTOXEJg5VIKzj0Tw5c/luEPf/hD7rnnHp544glKS0vZtGkT0WiUxx9/fMT3zCSBGo6uYJK3D4VpOBKh4UiUxiMRjh2P092TBRNjEIYhcDk1nE4Nh2FvnQ4NwxAYhsBhaOi6vVjX0AX6oKJpA1tN2G7mmmbPF+maQGi2k4QQAiHoLQP7MOg4vfNMpz7uQwxshLB/dhYCCx1TagMFgSk1+7neY5YUmL2PLSmwyGwNoS4svvepjCPXK4HKMVmfAU/2xGh66mC2T6sAjAI/zrJinKXFOCtLcVWW46oqw1NbjbuuCk9dDd6FteieqU8VMVHuuusuvv3tb7NokR0m6Sc/+QlLliyhsbGR+vr66W1cjigudLLuvBLWnTc0+kY4kubY8Rht7Yn+0tWdpKMrSWd3imBPip5Qqj/00lik05J02hziZTgXMBw6ujG4aOiGjmHoaL37uq6h6fZzmq5h6AB10910xQiMakEJIZ4AMg3nXAa0Z/jeucRc7CcdWA28CQyeJFsNNADBQcfKgPLefQewayoaOMOZi9+pTJlMX7VLKd+dzcYohjLWEF/mJxbiNSnlmpycfBYxF/tJCFEHHAEWSSkbBh1vAr4ppfzdCO+bc32VCaqfxo/qq/wmv4K/KeYKod5t4SnHi4CeKW6LQqHIU5RAKaYcKWU3tgV1Xt8xIcQioAA1hKdQKHrJpUDdlcNzzybmaj/dBdwqhFgohCgAfgz8t5SycYz3KMZG9dP4UX2Vx+RsDkqhGA0hhI4tSjcBLuAp4GYppZrcVygUgBIohUKhUOQpag5KoVAoFHmJEiiFQqFQ5CU5FSghhEsIcacQ4oAQIiSEOCKE+KkQIv9DHUwxQoivCCFeFUJEhRAqFMcghBB67/fmZO/36EEhRKYLyGctQoiPCSG2CCF6hBAZxBifGwghfiyE2NvbTy1CiP8QQpSM/U7FVJNrC8rAXqV9HfYal0uBDdiT44qhtAA/AX443Q3JQ24D3g+sA2p7j903fc3JW7qAXwBfm+6G5DkmsBEoBc7B/k79dlpbpBiWKXeSEEJ8Edtb65wprXiGIIS4CfhnKeWS6W5LvtAbYeL7Usrf9D5eDBwEFo7hlj4nEUJcDvxdSpk/2SbzGCHEe4H/klKeunBcMc1MxxzUFajFmIpxIoQoBOYD2/uOSSkPYUecWDVd7VLMKtQ9KU/JWKCEEHcLIeQo5QfDvOdrwCXANyfT6JlEJv2kGEJB7zZ4yvHuQc8pFBkhhLge+Bzw1elui+J0JjME8CXg1MRPgxmSV1wIcQtwK7BBSnlkEvXONCbUT4rTUHH7FDlBCPFh4E7gfVLK16e7PYrTyVigpJRhIDye1wohvgV8HlgvpdyfaZ0zkYn0k+J0pJTdQoi+uH07QMXtU0weIcSngf8LXCelfHG626MYnpzPQQkhfgp8ljkoThNBCGH0ut877IfCrdzx+8kkbt+co9cd3w04ex+7e4vK/DoIIcRXgH8D3qXEKb/JqRefEGIB0AgkgcF5rZuklCtzVvEMRAjxXeA7px6XUs75m4uK2zc+ej1Ah3OXVt6OgxBCSCANJAYfl1L6p6dFipFQsfgUCoVCkZeoUEcKhUKhyEuUQCkUCoUiL1ECpVAoFIq8RAmUQqFQKPISJVAKhUKhyEuUQCkUCoUiL1ECpVAoFIq8RAmUQqFQKPKS/w8WoJ0g067mpwAAAABJRU5ErkJggg==\n",
-      "text/plain": [
-       "<Figure size 432x360 with 1 Axes>"
-      ]
-     },
-     "metadata": {
-      "needs_background": "light"
-     },
-     "output_type": "display_data"
-    }
-   ],
-   "source": [
-    "import matplotlib as mpl\n",
-    "import matplotlib.pyplot as plt\n",
-    "import numpy as np\n",
-    "import matplotlib.ticker as plticker\n",
-    "\n",
-    "\n",
-    "# Create figure and add axes\n",
-    "fig = plt.figure(figsize=(6, 5))\n",
-    "ax = fig.add_subplot(111)\n",
-    "\n",
-    "alphaset = [-5,-2,-1,1, 2, 5]\n",
-    "colors = plt.get_cmap('coolwarm', 6)\n",
-    "\n",
-    "temp1 = colors(1)\n",
-    "temp0 = colors(0)\n",
-    "\n",
-    "labels = ['-5','-2','-1','1','2','5']\n",
-    "\n",
-    "plt.ylim(top=4)\n",
-    "\n",
-    "# Move left y-axis and bottim x-axis to centre, passing through (0,0)\n",
-    "ax.spines['left'].set_position('zero')\n",
-    "ax.spines['bottom'].set_position('zero')\n",
-    "\n",
-    "# Eliminate upper and right axes\n",
-    "ax.spines['right'].set_color('none')\n",
-    "ax.spines['top'].set_color('none')\n",
-    "\n",
-    "# Show ticks in the left and lower axes only\n",
-    "ax.xaxis.set_ticks_position('bottom')\n",
-    "ax.yaxis.set_ticks_position('left')\n",
-    "\n",
-    "loc = plticker.MultipleLocator(base=1.0) # this locator puts ticks at regular intervals\n",
-    "ax.yaxis.set_major_locator(loc)\n",
-    "\n",
-    "for i in range(len(alphaset)):\n",
-    "    \n",
-    "    x = np.linspace(-2, 2, 100)\n",
-    "    y = np.exp(alphaset[i]*x)\n",
-    "    #if i==0: ax.plot(x, y, color=temp1, linewidth=2.5, label=labels[i])\n",
-    "    #if i==1: ax.plot(x, y, color=temp0, linewidth=2.5, label=labels[i])\n",
-    "    #if i==2 or i==3: ax.plot(x, y, color=colors(i), linewidth=2.5, label=labels[i])\n",
-    "    ax.plot(x, y, color=colors(i), linewidth=2.5, label=labels[i])\n",
-    "    \n",
-    "ax.tick_params(axis='both', which='major', labelsize=13)\n",
-    "\n",
-    "    \n",
-    "# Add legend\n",
-    "\n",
-    "leg = ax.legend(bbox_to_anchor=(1.05, 0.3),loc='lower left', \n",
-    "          frameon=True, labelspacing=0.2, fontsize=13,)\n",
-    "leg.set_title(r\"${\\lambda}$\", prop = {'size':'x-large'})\n",
-    "\n",
-    "plt.tight_layout()\n",
-    "plt.savefig('exp_lamb_01.png')"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 55,
-   "metadata": {
-    "tags": [
-     "remove-cell"
-    ]
-   },
-   "outputs": [
-    {
-     "data": {
-      "image/png": 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\n",
-      "text/plain": [
-       "<Figure size 432x324 with 1 Axes>"
-      ]
-     },
-     "metadata": {
-      "needs_background": "light"
-     },
-     "output_type": "display_data"
-    }
-   ],
-   "source": [
-    "import matplotlib as mpl\n",
-    "import matplotlib.pyplot as plt\n",
-    "import numpy as np\n",
-    "from math import log\n",
-    "import matplotlib.ticker as plticker\n",
-    "\n",
-    "\n",
-    "# Create figure and add axes\n",
-    "fig = plt.figure(figsize=(6, 4.5))\n",
-    "ax = fig.add_subplot(111)\n",
-    "\n",
-    "alphaset = [0.1,0.2,0.5,2,5,10]\n",
-    "colors = plt.get_cmap('coolwarm', 6)\n",
-    "\n",
-    "\n",
-    "labels = ['1/10','1/5','1/2','2','5','10']\n",
-    "\n",
-    "#plt.ylim(top=4)\n",
-    "\n",
-    "# Move left y-axis and bottim x-axis to centre, passing through (0,0)\n",
-    "ax.spines['left'].set_position('zero')\n",
-    "ax.spines['bottom'].set_position('zero')\n",
-    "\n",
-    "# Eliminate upper and right axes\n",
-    "ax.spines['right'].set_color('none')\n",
-    "ax.spines['top'].set_color('none')\n",
-    "\n",
-    "# Show ticks in the left and lower axes only\n",
-    "ax.xaxis.set_ticks_position('bottom')\n",
-    "ax.yaxis.set_ticks_position('left')\n",
-    "\n",
-    "#loc = plticker.MultipleLocator(base=1.0) # this locator puts ticks at regular intervals\n",
-    "#ax.yaxis.set_major_locator(loc)\n",
-    "\n",
-    "\n",
-    "for i in range(len(alphaset)):\n",
-    "    \n",
-    "    x = np.linspace(0.1, 10, 100)\n",
-    "    y = np.log(x) / np.log(alphaset[i])\n",
-    "    #if i==0: ax.plot(x, y, color=temp1, linewidth=2.5, label=labels[i])\n",
-    "    #if i==1: ax.plot(x, y, color=temp0, linewidth=2.5, label=labels[i])\n",
-    "    #if i==2 or i==3: ax.plot(x, y, color=colors(i), linewidth=2.5, label=labels[i])\n",
-    "    ax.plot(x, y, color=colors(i), linewidth=2.5, label=labels[i])\n",
-    "    \n",
-    "ax.tick_params(axis='both', which='major', labelsize=13)\n",
-    "\n",
-    "    \n",
-    "# Add legend\n",
-    "\n",
-    "leg = ax.legend(bbox_to_anchor=(1.05, 0.3),loc='lower left', \n",
-    "          frameon=True, labelspacing=0.2, fontsize=13,)\n",
-    "leg.set_title(r\"$a$\", prop = {'size':'x-large'})\n",
-    "\n",
-    "plt.tight_layout()\n",
-    "plt.savefig('log_base_01.png')"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 53,
-   "metadata": {
-    "tags": [
-     "remove-cell"
-    ]
-   },
-   "outputs": [
-    {
-     "data": {
-      "image/png": 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\n",
-      "text/plain": [
-       "<Figure size 432x324 with 1 Axes>"
-      ]
-     },
-     "metadata": {
-      "needs_background": "light"
-     },
-     "output_type": "display_data"
-    }
-   ],
-   "source": [
-    "import matplotlib as mpl\n",
-    "import matplotlib.pyplot as plt\n",
-    "import numpy as np\n",
-    "from math import log\n",
-    "import matplotlib.ticker as plticker\n",
-    "\n",
-    "\n",
-    "# Create figure and add axes\n",
-    "fig = plt.figure(figsize=(6, 4.5))\n",
-    "ax = fig.add_subplot(111)\n",
-    "\n",
-    "\n",
-    "colors = plt.get_cmap('coolwarm', 2)\n",
-    "\n",
-    "\n",
-    "labels = [r\"$e^x$\",r\"$ln\\ x$\"]\n",
-    "\n",
-    "plt.ylim(-3,3)\n",
-    "plt.xlim(-3,3)\n",
-    "\n",
-    "ax.spines['left'].set_position('zero')\n",
-    "ax.spines['bottom'].set_position('zero')\n",
-    "\n",
-    "# Eliminate upper and right axes\n",
-    "ax.spines['right'].set_color('none')\n",
-    "ax.spines['top'].set_color('none')\n",
-    "\n",
-    "# Show ticks in the left and lower axes only\n",
-    "ax.xaxis.set_ticks_position('bottom')\n",
-    "ax.yaxis.set_ticks_position('left')\n",
-    "\n",
-    "#loc = plticker.MultipleLocator(base=1.0) # this locator puts ticks at regular intervals\n",
-    "#ax.yaxis.set_major_locator(loc)\n",
-    "\n",
-    "xe = np.linspace(-3,3, 100) \n",
-    "y = np.exp(xe)\n",
-    "ax.plot(xe, y, color=colors(0), linewidth=2.5, label=labels[0])\n",
-    "\n",
-    "xl = np.linspace(0.01, 10, 100)\n",
-    "y = np.log(xl)\n",
-    "ax.plot(xl, y, color=colors(1), linewidth=2.5, label=labels[1])\n",
-    "    \n",
-    "ax.plot(xe,xe,ls='dashed',lw=1.5)\n",
-    "\n",
-    "ax.tick_params(axis='both', which='major', labelsize=13)\n",
-    "\n",
-    "    \n",
-    "# Add legend\n",
-    "\n",
-    "leg = ax.legend(bbox_to_anchor=(1.05, 0.3),loc='lower left', \n",
-    "          frameon=True, labelspacing=0.2, fontsize=13)\n",
-    "\n",
-    "plt.tight_layout()\n",
-    "plt.savefig('log_base_02.png')"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": null,
-   "metadata": {},
-   "outputs": [],
-   "source": []
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "# 01 - Making mathematical statements\n",
-    "  "
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "## Sets and relations  \n",
-    "  \n",
-    "Start by watching the [video](https://web.microsoftstream.com/video/a9ad1a8c-3d80-4910-b9f2-fab98c01b40b)."
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "## Polynomial functions  \n",
-    "  \n",
-    "One of the simplest elemntary functions is given by:  \n",
-    "  \n",
-    "  $$ f(x) = a_n x^n + a_{n-1} x^{n-1}+ \\cdots + a_1 x^1 + a_0, $$  \n",
-    "  \n",
-    "where $n$ is a *non-negative integer*, and the coefficients $a_n$, $a_{n-1}$, ..., $a_0$. The highest value of $n$ (for which the coefficient $a_n$ is different from $0$) defines the *degree* of the polynomial.  \n",
-    "  \n",
-    "```{admonition} Examples\n",
-    "$f(x) = 3x^5 + 12x^4 - 6x^3 - x^2 + x - 16$ (Polynomial function of degree 5)  \n",
-    "$g(x) = \\frac{3}{4}x^2 - \\frac{5}{2} $ (Polynomial function of degree 2)  \n",
-    "$h(x) = 1.6x^8 - 0.576x^4 + 1.89x^2 - 0.7$ (Polynomial function of degree 8)  \n",
-    "```  \n",
-    "  \n",
-    "\n",
-    "```{tip}\n",
-    "When the domain of the polynomial function is not explicitly given, it is frequently assumed to be all real numbers, ${\\rm I\\!R}$.\n",
-    "```\n",
-    "```{tip}\n",
-    "When no application is specified, the function name \\'$f$\\' and variable name \\'$x$\\' are usually chosen. For application purposes, the function and variable are often named for what they represent; for example, $A(r)=\\pi r^2$ for the area of a circle dependent on its radius.\n",
-    "```\n",
-    "\n",
-    "Let us discuss three of the most common forms of polynomial functions with degrees $0$, $1$, and $2$.\n",
-    "\n",
-    "### Degree 0  \n",
-    "  \n",
-    "Polynomial functions of degree 0 assume the form:  \n",
-    "  \n",
-    "  $$ f(x) = c, $$  \n",
-    "  \n",
-    "where $c$ is a real number. This function is graphically represented by:  \n",
-    "  \n",
-    "```{image} ./poly_deg0.png\n",
-    "\n",
-    "``` \n",
-    "\n",
-    "```{admonition} Example  \n",
-    "For most mobile phone contracts nowadays a fixed fee is paid for the month with unlimited usage. So the cost as a function of used minutes would be a polynomial of degree zero, since the cost does not depend on the usage.\n",
-    "```\n",
-    "  \n",
-    "  \n",
-    "### Degree 1\n",
-    "\n",
-    "Polynomial functions of degree 1 are also called *linear functions* and are given by:  \n",
-    "  \n",
-    "$$ f(x) = mx + b,$$  \n",
-    "  \n",
-    "where again $m$ and $b$ are real numbers (with $m \\ne 0$).  \n",
-    "  \n",
-    "The constants $m$ and $b$ are frequently referred to as *slope* and *intercept*. The intercept $b$ shows where the function crosses the $y$-axis, while the slope $m$ is related to the inclination of the function.\n",
-    "  \n",
-    "````{panels}  \n",
-    "  \n",
-    "```{image} ./poly_deg1.png\n",
-    "\n",
-    "```  \n",
-    "\n",
-    "---  \n",
-    "  \n",
-    "If we consider two arbitrary points, $(x_1,y_1)$ and $(x_2,y_2)$ on the line, their projections define two intervals on the $y$ and $x$ axes. These intervals have sizes $\\Delta y = y_2 - y_1$ and $\\Delta x = x_2 - x_1$ respectively.  \n",
-    "  \n",
-    "The slope $m$ will be given by the ratio between the changes in $y$ and $x$, for any chosen two points on the line. Thus:  \n",
-    "\n",
-    "  $$m = \\frac{y_2 - y_1}{x_2 - x_1} = \\frac{\\Delta y}{\\Delta x}. $$  \n",
-    "  \n",
-    "Also note that the triangle defined by the sides $\\Delta y$, $\\Delta x$ and this fragment of the function line forms a right triangle. Thus, there is a relationship between the slope $m$ and the angle $\\theta$ between the function line and the $x$-axis (or any horizontal line parallel to it) which is given by:  \n",
-    "  \n",
-    "  $$m =  \\frac{\\Delta y}{\\Delta x} = tan(\\theta).$$\n",
-    "  \n",
-    "````  \n",
-    "  \n",
-    "The sign of the slope $m$ also determines if the function increases or decreases as a function of $x$. We have:  \n",
-    "  \n",
-    "  $$ \\begin{align}\n",
-    "  m>0 &\\rightarrow \\mbox{Increasing function of } x \\\\\n",
-    "  m<0 &\\rightarrow \\mbox{Decreasing function of } x\n",
-    "  \\end{align} $$  \n",
-    "    \n",
-    "    \n",
-    " \n",
-    "````{panels}  \n",
-    "  \n",
-    "```{image} ./slope_01.png  \n",
-    "\n",
-    "```  \n",
-    "\n",
-    "---  \n",
-    "  \n",
-    "The graph on the left show the functions $y = mx$ (intercept $b=0$), for different values of $m$ (see legend colours).  \n",
-    "  \n",
-    "Note that positive values of $m$ give **increasing** functions, while negative values of $m$ return **decreasing** functions.\n",
-    "      \n",
-    "```` \n",
-    "  \n",
-    "One can also ask what is the value of $x$ where the function $f(x)$ crosses de $x$-axis, or, in mathematical notation, what is $x\\ \\mbox{so that}\\ f(x) = 0$. This value of x is called a **root of $f$** and it can be obtained by:  \n",
-    "\n",
-    "$$\\begin{align}\n",
-    "    f(x) &= 0 \\\\\n",
-    "    mx + b &= 0 \\\\\n",
-    "    mx &= -b \\\\\n",
-    "    x &= -\\frac{b}{m}\n",
-    "    \\end{align} $$\n",
-    "    \n",
-    "`````{admonition} Example  \n",
-    "\n",
-    "````{panels}  \n",
-    "  \n",
-    "```{image} ./poly_deg1_02.png\n",
-    "\n",
-    "```  \n",
-    "\n",
-    "---  \n",
-    "  \n",
-    "  Consider the function $y = -2x + 1$.  \n",
-    "  We have:  $m=-2$ and $b=1$. The root of this function will be $x = -\\displaystyle\\frac{b}{m} = -\\displaystyle\\frac{1}{(-2)} = \\displaystyle\\frac{1}{2}$.  \n",
-    "  Additionally, the function crosses the $y$-axis at the intercept $b=1$.\n",
-    "  \n",
-    "````\n",
-    "`````  \n",
-    "\n",
-    "```{admonition} Examples  \n",
-    "The metabolic rate of some organisms depends (approximately) linearly on the body's temperature, with higher rates at higher temperatures.\n",
-    "```\n",
-    "  \n",
-    "### Degree 2\n",
-    "\n",
-    "Now we look at polynomial functions of degree 2, given by the general form:  \n",
-    "\n",
-    "$$f(x) = ax^2 + bx +c,$$  \n",
-    "  \n",
-    "the coefficients $a$, $b$, and $c$ are real numbers (with $a \\ne 0$). The graph of a polynomial function of degree 2 is a *parabola* and many of its properties can be readily known by analysing the coefficients.\n",
-    "\n",
-    "#### Concavity\n",
-    "\n",
-    "````{panels}  \n",
-    "Concave up (positive concavity)\n",
-    "^^^\n",
-    "  \n",
-    "```{image} ./poly_deg2_01.png  \n",
-    "\n",
-    "```  \n",
-    "\n",
-    "---  \n",
-    "  \n",
-    "Concave down (negative concavity)\n",
-    "^^^\n",
-    "  \n",
-    "```{image} ./poly_deg2_02.png  \n",
-    "\n",
-    "```  \n",
-    "  \n",
-    "````\n",
-    "\n",
-    "#### Intercept & Roots  \n",
-    "\n",
-    "````{panels}  \n",
-    "  \n",
-    "```{image} ./poly_deg2_roots.png  \n",
-    "\n",
-    "```  \n",
-    "\n",
-    "---  \n",
-    "  \n",
-    "For the function $f(x) = ax^2 +bx + c$ the coefficient $c$ represents the **intercept**, the point where the function curve crosses the $y$-axis.  \n",
-    "  \n",
-    "The **roots** of the function $f$ are the values of $x$ for which $ax^2 +bx + c = 0$, where the curve crosses the $x$-axis. A polynomial function of degree 2 has a maximum number of two real roots*, given by:  \n",
-    "\n",
-    "$$\\begin{align}\n",
-    "x_1 &= \\frac{-b-\\sqrt{b^2-4ac}}{2a} \\\\\n",
-    "x_2 &= \\frac{-b+\\sqrt{b^2-4ac}}{2a}\n",
-    "\\end{align}$$\n",
-    "\n",
-    "+++\n",
-    "\\* A polynomial function of degree $n$ will always have $n$ roots, but these can be imaginary or multiple roots. It therefore can (but does not have to) have a maximum of $n$ real roots.\n",
-    "  \n",
-    "````  \n",
-    "  \n",
-    "The expression inside the square root distinguish between different properties of this function and therefore receives the name of *discriminant*:  \n",
-    "\n",
-    "$$ \\Delta = b^2 -4ac.$$  \n",
-    "  \n",
-    "Depending on the value of the discriminant a quick conclusion can be reached regarding the roots of the function.\n",
-    "\n",
-    "`````{tabbed} Case 1  -  $\\Delta > 0$  \n",
-    "\n",
-    "````{panels}  \n",
-    "  \n",
-    "```{image} ./poly_deg2_delta01.png  \n",
-    "\n",
-    "```  \n",
-    "\n",
-    "---  \n",
-    "  \n",
-    "When $\\Delta > 0$ there are two real **distinct** roots for the polynomial function.  \n",
-    "  \n",
-    "```{admonition} Examples\n",
-    "Consider the function  $y = 0.5 x^2 - x -1.5$. We have:  \n",
-    "  \n",
-    "  $$\\Delta = (-1)^2 - 4(0.5)(-1.5) = 1 + 3 = 4.$$  \n",
-    "  \n",
-    "  The roots will then be given by:  \n",
-    "    \n",
-    "  $$\\begin{align}\n",
-    "  x_1 &= \\frac{-(-1)-\\sqrt{4}}{2(0.5)} = \\frac{1-2}{1} = -1 \\\\\n",
-    "  x_2 &= \\frac{-(-1)+\\sqrt{4}}{2(0.5)} = \\frac{1+2}{1} = 3 \\end{align}$$  \n",
-    "  \n",
-    "``` \n",
-    "\n",
-    "  \n",
-    "````  \n",
-    "`````  \n",
-    "  \n",
-    "`````{tabbed} Case 2  -  $\\Delta = 0$  \n",
-    "\n",
-    "````{panels}  \n",
-    "  \n",
-    "```{image} ./poly_deg2_delta02.png  \n",
-    "\n",
-    "```  \n",
-    "\n",
-    "---  \n",
-    "  \n",
-    "When $\\Delta = 0$ there are two real **identical** roots for the polynomial function.  \n",
-    "  \n",
-    "```{admonition} Examples\n",
-    "Consider the function  $y = 0.5 x^2 - x +0.5$. We have:  \n",
-    "  \n",
-    "  $$\\Delta = (-1)^2 - 4(0.5)(0.5) = 1 -1 = 0.$$  \n",
-    "  \n",
-    "  The roots will then be given by:  \n",
-    "    \n",
-    "  $$\\begin{align}\n",
-    "  x_1 = x_2 = \\frac{-(-1)\\pm\\sqrt{0}}{2(0.5)} = 1\\end{align}$$  \n",
-    "  \n",
-    "``` \n",
-    "  \n",
-    "````  \n",
-    "`````  \n",
-    "  \n",
-    "`````{tabbed} Case 3  -  $\\Delta < 0$\n",
-    "\n",
-    "````{panels}  \n",
-    "  \n",
-    "```{image} ./poly_deg2_delta03.png  \n",
-    "\n",
-    "```  \n",
-    "\n",
-    "---  \n",
-    "When $\\Delta < 0$ there are no real roots for the polynomial function, since the square root of a negative number is not defined for real numbers.  \n",
-    "  \n",
-    "  ```{admonition} Examples\n",
-    "The graph on the left corresponds to the function  $y = 0.5 x^2 - x + 1$. We see that the curve does not cross the $x$-axis at any point, showing there is no real root for this function.  \n",
-    "\n",
-    "``` \n",
-    "  \n",
-    "````  \n",
-    "`````  \n",
-    "```{admonition} Example\n",
-    "If we throw a ball that is heavy enough so that air resistance is negligible (alternatively we can travel to the moon) the height of the ball as a function of time follows a quadratic equation. Knowing the roots we can calculate when and where the ball will land.\n",
-    "```\n",
-    "\n",
-    "#### Minimum/ maximum  \n",
-    "  \n",
-    "  Depending on the concavity of the parabola (up or down), there will be a point where the function assumes a minimum or a maximum value. This point is called the *vertex* of the parabola. This extreme point can be easily determined by the symmetry properties of the function curve.  \n",
-    "   \n",
-    "   ````{panels}  \n",
-    "  \n",
-    "```{image} ./poly_deg2_vertex.png  \n",
-    "\n",
-    "```  \n",
-    "\n",
-    "---  \n",
-    "  \n",
-    "The coordinate $(x_v,y_v)$ of the vertex is given by:  \n",
-    "  \n",
-    "  $$\\begin{align}\n",
-    "  x_v &= -\\frac{b}{2a} \\\\\n",
-    "  y_v &= -\\frac{(b^2-4ac)}{4a} = -\\frac{\\Delta}{4a}\n",
-    "  \\end{align}$$  \n",
-    "    \n",
-    "  ```{tip}  \n",
-    "  Since the parabola is symmetric in relation to a vertical line passing through the vertex, the $x$ coordinate of the vertex is given by the middle point between the two roots $x_v = (x_1 + x_2)/2$. The $y$ coordinate is then given by substituting $x_v$ in the general form of the polynomial of degree 2, as $y_v = f(x_v) = ax_v^2 + bx_v + c$. Try to obtain the expressions above as an exercise.\n",
-    "  ```\n",
-    " ```{tip}  \n",
-    "The extrema is the point of the parabola where the derivative becomes zero $f'(x)=0$.\n",
-    "  ```\n",
-    "  \n",
-    "````\n",
-    "```{admonition} Example\n",
-    "Imagine the population of a specific region having a negative quadratic relationship - it undergoes initial growth, reaches its peak, and then declines. To determine both the maximum population size and the time when it peaked, we need to identify the extrema of this parabolic curve.\n",
-    "```\n",
-    "\n",
-    "  \n",
-    "Now watch this [video](https://web.microsoftstream.com/video/f64abd8a-9997-4b5e-b11f-0ca09b7603fe) for a short introduction to complex numbers."
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "## Rational functions  \n",
-    "  \n",
-    "  \n",
-    "Rational functions are defined as quotients of polynomial functions:  \n",
-    "  \n",
-    "  \n",
-    "$$f(x) = \\frac{p(x)}{q(x)},\\ \\ \\ q(x) \\ne 0$$  \n",
-    "    \n",
-    "      \n",
-    "````{panels}  \n",
-    "  \n",
-    "```{image} ./rational_01.png  \n",
-    "\n",
-    "```  \n",
-    "\n",
-    "---  \n",
-    "  \n",
-    "The plot on the left shows the graph for the function $y = \\displaystyle\\frac{2x-5}{x-3}$.  \n",
-    "  \n",
-    "Note that the graph approaches a vertical line at $x=3$ when $x$ is close to $3$, and it approaches a horizontal line at $y=2$ when $x$ approaches large values (positive or negative). These lines are called *vertical asymptote* and *horizontal asymptote*, respectively.  \n",
-    "    \n",
-    "Vertical asymptotes appear for the values of $x$ corresponding to the roots of the denominator function ($q(x)$) of the rational function $f(x)$, while horizontal asymptotes to the limits of the function $f(x)$ (if it exists) for large $x$*.  \n",
-    "  \n",
-    "Asymptotes give us a lot of information about the limiting behaviours of the function, and they are especially relevant when modelling biological phenomena.\n",
-    "\n",
-    "+++\n",
-    "\\* We will be talking more about limits on the following lectures.\n",
-    "      \n",
-    "````  \n",
-    "\n",
-    "```{admonition} Example\n",
-    "Rational functions are often used to express ratios, for example of chemicals. Lets assume chemical A increases quadratically $A(t) = 2 t^2$ and chemical B increases linearly $B(t)=t+5$, the ratio will be:\n",
-    "\n",
-    "\n",
-    "$$r(t) = \\frac{A(t)}{B(t)} = 2\\frac{t^2}{t+5}$$\n",
-    "```\n",
-    "        "
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "## Power functions  \n",
-    "  \n",
-    "Another type of function used in many applications are power functions (sometimes also called power laws). These functions have the functional form:  \n",
-    "  \n",
-    "  $$f(x) = Cx^{\\alpha}$$  \n",
-    "    \n",
-    "where $C$ and $\\alpha$ are real constants (with $C \\ne 0$).\n",
-    "  \n",
-    "  \n",
-    "````{panels}  \n",
-    "  \n",
-    "```{image} ./powerlaw_01.png  \n",
-    "\n",
-    "```  \n",
-    "\n",
-    "---  \n",
-    "  \n",
-    "The plot on the left shows the graphs for the functions $y = x^{\\alpha}$ (with $C=1$) for different values of the exponent $\\alpha$ (see the legend colours).  \n",
-    "  \n",
-    "There are a few interesting aspects to note in this plot. First, the graphs were presented only for positive values of $x$. Some of these functions are also defined (on the set of real numbers) for negative values of $x$ which could be presented on the plot. This is the case for $x^{2}$ or $x^{-2}$, but not the case for $x^{1/2}$ or $x^{-1/2}$.  \n",
-    "  \n",
-    "When analysing positive values of $x$ (as it is frequently the case for models in Biology), for $C > 0$, the exponent $\\alpha$ determines the behaviour of the functions, with **increasing** functions of x for $\\alpha > 0$ and **decreasing** functions of x for $\\alpha < 0$.  \n",
-    "      \n",
-    "````  \n",
-    "Many commonly used functions are examples of power functions, e.g. the square root $\\sqrt{x}=x^{\\frac 12}$ or the inverse function $x^{-1}$.\n",
-    "\n",
-    "Watch this [video](https://web.microsoftstream.com/video/bf7415e1-823f-449e-ace1-e379fac8fe0a) about additional properties of power law functions."
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "## Exponential functions  \n",
-    "  \n",
-    "Similar to power functions, but with very different properties are the exponential functions:  \n",
-    "  \n",
-    "  $$f(x) = a^x, $$  \n",
-    "    \n",
-    "where $a$ is a positive constant. The parameter $a$ is called *base* and the variable $x$ is the exponent (remember that for power functions, the variable $x$ is the base, raised to the power of the parameter $\\alpha$).  \n",
-    "  \n",
-    "  Exponential functions are used to model many different real-world applications, from bacterial growth to radioactive decay.  \n",
-    "    \n",
-    "```{admonition} Example  \n",
-    "\n",
-    "Suppose we start a bacterial colony with a single individual on a Petri dish. At each time step, counted as an average reproduction interval, each individual bacterium duplicates, originating two other individuals. The number of bacteria in the colony as a function of time, $N(t)$, can be counted as:  \n",
-    "    \n",
-    "    \n",
-    "| $t$      |      $N(t)$   |\n",
-    "|----------|:-------------:|\n",
-    "| $0$      |  $1\\ (= 2^0)$ |\n",
-    "| $1$      |  $2\\ (= 2^1)$ |\n",
-    "| $2$      |  $4\\ (= 2^2)$ |\n",
-    "| $3$      |  $8\\ (= 2^3)$ |\n",
-    "| $4$      |  $16\\ (= 2^4)$ |\n",
-    "| $5$      |  $32\\ (= 2^5)$ |  \n",
-    "  \n",
-    "If no limitation (on space or resources) is imposed to the colony, the number of bacteria at each time step $t$ can be well modelled by an exponential function of the form:  \n",
-    "  \n",
-    "$$ N(t) = 2^t. $$\n",
-    "  \n",
-    "```  \n",
-    "  \n",
-    "Exponential functions of the form $f(x) = a^x$ (with $a>0$) can never assume negative values and their behaviour depends on the value of $a$.  \n",
-    "  \n",
-    "```{image} ./exponential_01.png  \n",
-    ":align: center\n",
-    "```  \n",
-    "  \n",
-    "```{tip}  \n",
-    "Note that the function *increases* with $x$ if $a>1$ (growth) and *decreases* with $x$ if $0<a<1$ (decay). To remember this, one can directly apply the properties of exponentiation. If $f(x) = (1/3)^x$ (thus $a = (1/3) < 1$), we have:  \n",
-    "  \n",
-    "$$f(x) = \\left(\\frac{1}{3}\\right)^x = \\frac{1}{3^x},$$  \n",
-    "  \n",
-    "and since $3^x$ is an increasing function of $x$, then $1/3^x$ should be a decreasing function of $x$ (as the denominator increases, the fraction decreases).\n",
-    "```\n",
-    "```{admonition} Example  \n",
-    "In the early stages of a new desease, we can approximate the spread as exponential. In this case $a$ is the number of people every infected person infects themselves. For $a<1$ every person infects less than one other person on average and the desease will slow down, for $a>1$ the desease will spread. At the beginning of the COVID-19 pandemic, an estimate for $a$ (or in this case $R$ for reproductive value) was $a=4$. If we start with 1 infected person, how many do we have after 10 infection cycles?\n",
-    "```\n",
-    "  \n",
-    "Two of the most used bases are $a=10$ and $a=e$, where $e=2.71828$ (called *Euler's number* or *Napier's constant*). Exponential functions with $a=e$ are commonly written as:  \n",
-    "  \n",
-    "$$f(x) = e^x = \\mbox{exp}(x) $$  \n",
-    "\n",
-    "The unique charachteristic of the exponential of base $e$ is, that it is its own derivative\n",
-    "\n",
-    "$$\\frac{\\text{d}}{\\text{d}x} e^x = e^x $$\n",
-    "  \n",
-    "````{panels}  \n",
-    "  \n",
-    "```{image} ./exp_lamb_01.png  \n",
-    "\n",
-    "```  \n",
-    "\n",
-    "---  \n",
-    "  \n",
-    "Another common representation of the exponential function is:  \n",
-    "  \n",
-    "$$ f(x) = e^{\\lambda x}, $$  \n",
-    "  \n",
-    "where $\\lambda$ is a real constant, usually called *rate constant*. The absolute value of the parameter $\\lambda$ controls how fast is the growth (in case $\\lambda > 0$) or the decay (for $\\lambda < 0$) of the function $f(x)$ for increasing values of $x$.  \n",
-    "  \n",
-    "For a fixed value of $x$, which $\\lambda$ corresponds to the highest and the lowest values among these exponential functions? Does it change if $x$ is positive or negative? How would it change if, instead of $a=e$, it was a different base $a<1$?\n",
-    "      \n",
-    "````  \n",
-    "The both formulations can be changed into one another via\n",
-    "\n",
-    "\n",
-    "$$f(x) = e^{\\lambda x} = ({\\underbrace{e^\\lambda}_a})^x.$$\n",
-    "\n",
-    "\n",
-    "We have exponential growth when $a>1$ and therefore when $\\lambda>0$ and decay when $0<a<1$ and therefore $\\lambda<0$."
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "## Logarithmic function  \n",
-    "  \n",
-    "Let us start this section by understanding what the expression  \n",
-    "  \n",
-    "$$ \\mbox{log}_a b = x $$  \n",
-    "  \n",
-    "actually means. One possible way to read it is: \"$\\mbox{log}_a b$ is the number with which I have to exponentiate the base $a$ so that I find the number $b$. Since \"$\\mbox{log}_a b = x$, the above expression is equivalent to:  \n",
-    "  \n",
-    "$$ a^x = b.$$  \n",
-    "  \n",
-    "The expression $\\mbox{log}_a b$ is called the *logarithm of b in the base a* and it is also possible to define a logarithmic function given by:  \n",
-    "  \n",
-    "$$f(x) = \\mbox{log}_a x,$$  \n",
-    "  \n",
-    "where $a$ is the base of the logarithm and it is assumed $a>0$ and $a \\ne 1$.  \n",
-    "  \n",
-    "By defining a function like this, for each $x$ we will have a number $y = f(x)$ for which $a^y = x$. Therefore, the logarithmic function is only defined for $x>0$. Like the exponential function, the behaviour of the logarithmic function for increasing $x$ will depend on the value of the base $a$.  \n",
-    "  \n",
-    "````{panels}  \n",
-    "  \n",
-    "```{image} ./log_base_01.png  \n",
-    "\n",
-    "```  \n",
-    "\n",
-    "---  \n",
-    "Note that the logarithmic functions are *increasing* if $a>1$ and *decreasing* if $0<a<1$. For a fixed value of $x$ the functions showing the highest and lowest values $f(x)$ will depend both on the value of the base and on which interval $x$ belongs to ($0<x<1$ or $x>1$). Can you determine these values for each case?  \n",
-    "  \n",
-    "        \n",
-    "````\n",
-    "\n",
-    "````{tip}  \n",
-    "The similarities between the graphs for exponential and logarithmic functions are not coincidental. In fact, the graph for a logarithmic function with base $a$ can be obtained by drawing the graph for an exponential function with base $a$ and reflecting it in relation to the line $y = x$.  [MAKE COMMENT ON INVERSE FUNCTIONS]\n",
-    "  \n",
-    "```{image} ./log_base_02.png\n",
-    "```  \n",
-    "    \n",
-    "(here we use the common nomenclature $\\mbox{log}_e x = \\mbox{ln}\\ x$).  \n",
-    "````"
-   ]
-  }
- ],
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diff --git a/_sources/mathscourse/lecture02/02_Describing_shapes_and_patterns.ipynb b/_sources/mathscourse/lecture02/02_Describing_shapes_and_patterns.ipynb
deleted file mode 100644
index 7c12db74..00000000
--- a/_sources/mathscourse/lecture02/02_Describing_shapes_and_patterns.ipynb
+++ /dev/null
@@ -1,969 +0,0 @@
-{
- "cells": [
-  {
-   "cell_type": "code",
-   "execution_count": 13,
-   "metadata": {
-    "tags": [
-     "remove-cell"
-    ]
-   },
-   "outputs": [
-    {
-     "data": {
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\n",
-      "text/plain": [
-       "<Figure size 864x216 with 1 Axes>"
-      ]
-     },
-     "metadata": {
-      "needs_background": "light"
-     },
-     "output_type": "display_data"
-    },
-    {
-     "data": {
-      "image/png": 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OrPD66xYW3W476NYtaWvSz/LlzSHR3XdP1pasEAmUl9cn5E6gRGSYiPxTRD4QkYUiMk5ETohxnVY4nukImxPB+5/CeO45S13Q4/H667BsmU1h8J2aa6MK//2vfXYP6hO6JG1AO/BD4B3gB8Bs4FDgNhHpp6q/q3HtFcBdZX8vaB8TE0YVnnrKPn/608nakhWefNLSIUOStSMreHgvjKlT4aOPrE/YFyL+hDwK1OGqOrvs78dEZCNMuGoJ1BRVza/XFPHGGzBzJmywgQ9njYNqs0Dts0+ytmSFp5+21F+A4jFmjKVDhnifcBm5C/G1EKeIF4H1O9qW1BJVhqFDvTLEYfJk20l3vfV8gEQcVGH0aPt8wAHJ2pIVHnvM0v33T9aOlJE7garCnsD4GOddJCKNIjJbRP4sIuu0t2GJMHaspUOHJmtHVnjiCUv97TYeb75pGzv26+eTmuOgCo8/bp/32y9ZW1JGHkN8KyEiBwBHAKfUOPVm4D5gFjAYuADYRUQ+o6orqtx7ODAcYEBWNvxralrZg3Jq88gjlnp5xaPce+pUlHfg1eCtt2xe4nrr+aTmFuRaoERkIHAb8A9Vvam1c1X15LI/nxCRCcADwOHAvVWuGQGMABg8eLCutsEdwQsvwPTpNlt9++2Ttib9LF8ODzxgnw8/PFlbssK//mWph/fi8eCDlu63nwt6C3JbGqXw3IPAu8CJbbjFQ8BCIF+TEu65x9Ijj/RwVRzGjrVFT3fc0beMiMPChfDww/b54IOTtSUr3FUaOHzkkcnakUJyKVAi0hO4H+gGHKaqi0LvoaqRR5QNzygOqjBypH0+6qhkbckK//iHpUcckawdWWHUKNsEc889fbh0HGbMsCkf3bvDYYclbU3qyF2IT0S6AH8Htgb2UtUP23ifg4E1gXF1NC9ZXn3Vhpivs44Pl45DQwPcead99rfbeETldeyxydqRFe65x14cDzoIevdO2prUkTuBAv6ATc49E1hHRD5X9t2LqrpMREYDqOoB8Mlgh8HAo9jk3t2B84HngFEdaHv7cv31lh53nK0p57TO/ffDrFnWce0TTmvz0UfmQYnA0UcnbU02+POfLXVBr0geW6mDSulvK3y3OTAFaLk5zWTga8DRQG9gBnALcEG1EXyZY8kSuPVW+/yNbyRrS1aIGo9TTvH+ujjceKMtb3TIITYIx2mdceNs0FLfvi7oVcidQKnqwBjn7Nvi79HA6HYyKR3ccQfMnWuegK/1VZvJk230XpcucGJbxtgUjKYm+OMf7fN3vpOsLVnhT3+y9Gtfgx49krUlpeRykITTgqYmuOwy+3z66cnakhWuuMLK7cQTYX1fhKQm994Lb78Nm21mHpTTOtOmWURDBL75zaStSS0uUEXg3nthwgQYMAC+8pWkrUk/M2ZYuArgnHOStSULNDXBxRfb57PP9u3d43D55RYOPfpoXw+zFYJCfKUBBwcDnwM2AnpggwomAWOBe1V1Tr2NdFaDxkY4/3z7fPbZ0LVrsvZkgQsvhKVLbSj+dtslbU36GTnS9hfbeGM49dSkrUk/777bHN776U+TtSXlxPKgRORrIvIq8G/g+0BP4E3gWWAO8FngeuADEblJRDZvJ3udUEaMMO9pyy19cEQcXn/dRjt27gyXXpq0NelnyRI46yz7fP75sMYaydqTBc49116Ajj3W9xerQU0PSkRexlYCvwU4CXipbBJr+Xl9gC8AXwFeF5Gvq+oddbbXCeGDD5rf0C67zCYDOtVZscI8gKYm6+j3lctrc+mltpfRLrv4C1AcHnsMbr/dhDzqF3aqEifEdyPwJ1Vd2tpJqjoP+CvwVxHZBdiwDvY5bUUVTjvNRu4ddpivHBGHK66AZ5+1UNXPf560Neln3LhmL/N3v/O+p1osWGBTFsBeHDfbLFl7MkDNEJ+qXhWJk4jEWpdOVV9W1YdX1zhnNbj2WnjoIZtjcd11Po+nFi+/DP/7v/b5uutg7bWTtSftzJ9vA24aG+HMM2HvvZO2KN2owre+Zd7m7rtbmM+pSegovsdFxDcsSTtPPw1nnGGfr7kG+vdP1p60M3u2rbW3bJl5nT5MunWamuCrX4VJk2wR3V/8ImmL0s/VV8Ntt0GvXja83AcrxSJUoG4DHhCRVaY9i8gQEXmqPmY5bWbqVPjSl2wduTPOgBNOSNqidLNsGQwbZuX26U9bqMppnQsugH/+07zMe++Fnj2TtijdPPQQ/OhH9vmmm3ybmwCCBEpVvw1cCtwuIt8CEJGdROQ+4Amgb/1NdGLzwQe2B8+HH8L//I/1qTjVaWiwkVRjxsCGG9rCnT4KrXV+/WvzmDp1gr/9zUaHOtV58kl7YVyxAn78YzjmmKQtyhTBE3VV9WLgW8DVIjIWeBHYEdux1vd3TopInCZPtuWM7rzTF4RtjWXL4MtfNk+gb1/bw2jjjZO2Kt1cdZXNpQObyOz7PbXOmDHwhS/YUPxTT/VQaBsIbsFKGwF+ClgB7I3NjdpXVRvrbJsTl/HjrbF47z3YeWfb0dQ7+aszf7691Y4eDWutZeK0885JW5VeVG3UWTRi7w9/gJNOStamtHPXXTaIZPlyOP54G7TkA5WCCfKgRORC4G3gu8AVmNc0GLiy/qY5sRg1Cvbay8Rpjz1snsU66yRtVXp54w3bTG/0aNhgA9sx99OfTtqq9LJggfXRXXqpDSO/+Wb49reTtiq9qNr8pmOPNXE6/XT4y198CH5bUdXYB7Ac229pg7K8/YF5wB1A15D75ekYNGiQdiiNjaoXXKBqVUL1qKNUFy/uWBuyxj33qPbubeW17baqkycnbVGHYVU9kPHjrZzAyu2BB+pvWJ6YM0f1iCOa6+TPf67a1JS0VZkAeEEraU6lzGoHsGWV/N2xPZRGh9wvT0eHCtSbb6oOGWL/fZ06qf7iF6orVnTc72eNefNUv/GN5obj6KNV589P2qoOJUigVqxQveoq1TXWsPLaYQfVSZPaz7g8MHq06mabWXn16aP6j38kbVGmqItAtXYAWwGT63W/rB0dIlBRw9Gjh/3Xbbih6qOPtv/vZpnRo1UHDLDy6tZN9de/LuRbbWyBevtt1X33bRbzk05SXbCgfY3LMvPnq37rW83lNWhQoTzzelFNoOq23YaqvgXsWa/7OS14/nnrO/n+921U0Fe+YgubHnBA0palkw8+sL2cDjjAVo8eNAj++1+bj+Kd1auyZAn87Gc2R2fMGNsD6957rc9pzTWTti59qNpI2R12sJXJu3a1LUf+8x/YYoukrcsPlVSr/AD+AexW67yy89cAfgh8K+41eTjazYOaMUP1619vfkPr31/13nvb57fywNKlqr/8pWqvXlZe3burXnKJ6vLlSVuWKFTzoJqarG9u4MDmZ+yEE1RnzepYA7PESy+pDh3aXF6776768stJW5VpaGuID/gdsAzbWuMMrL+pS4tzNgKOBG7Att94DfhMrXu31wFsj23hvhiYBlwMdI5xXR9scdw52MCPvwLrxvnNugvU/PmqF1/c3KnftavquecWru8kNo2NqrfcorrFFs0Nx5FHWsjKqSxQTzyhuvfezeW1006qY8Z0vHFZYepU1dNOs35fUF13XdVrr7Vnz1kt2ixQdi1bloTqY2z+U2Pp83RgaSmvCXgG+BrQKc592+PAVrOYBjwK/A82qXgR8H8xrn0IeAc4GjgKeAN4Ms7v1k2gFi+2fpJ+/ZobjkMPVX3jjfrcP280NanedZfq9ts3l9d226k+/HDSlqWKlQTq+edVP//55vLq21f16qtVGxqSMzDNTJ+u+r3vWR8mqHburHrGGaoff5y0ZblhtQTqk5OhGzAUOAe4CvgT8HPgq8BmIfdqrwP4SckD6l2Wd07Jm+rdynV7AArsU5b3mVLegbV+d7UFatky1T/+UXWjjZobjj33VH388dW7b15parJhz7vv3lxeAweq3nijN7QVAFRfe82mI0TltdZaqhdeqDp3btLmpZPZs1XPOad5UJKIhT99RGPdqYtAZeHA1gS8vUXegJLQHN7KdRcDMyrkvw1cUet32yxQjY2qN9+suvnmzQ3HbrupjhpVyNFmsXj8cdW99mour/79Va+5xkTeWZU33zSBErHy6tFD9eyzvZ+pGnPnmnCvtVbzM3bkkaqvvJK0ZemksXG126u6ChTQvS3XdcQBfAhcVCF/EXB2K9fdCYypkD8KGFXrd9skUE1NKze0226r+ve/+5ymajz7rOqBBzaX17rrql5+ueqiRUlblk7efdfmf3XubALVtavqd7+rOm1a0palk4ULbYDNOus0P2Of/7zqc88lbVk6WbFC9c47m8Pro0a1+VbVBErsu3iIyL7AzcAmwHzgFeC/2IKx/wXGq2pT7Bu2AyLSgAnRVS3y3wduUdXzqlz3CLBIVY9skf8XYAtVXWUIvYgMB4aX/hxUD/sdx3EKyDhVHdwyM3Qe1DVYX87p2Pp7H2Gj927CxGrh6tlYNyqprlTJb/N1qjpCVQer6uBBgwa1zetbsgRdtixxzzOVx6RJ6PHHoyIooD16oOeei370UfK2pfGYMwc9/3y0Vy8rL0CPOw6dODF6Xv0oPxoa0BtuQAcMaC6vwYPRhx9Gm5qSty+Nx+OPo3vu2VxeG22E/vGPq92GVSN0NfPNgWGqOmqlFlxkbWz4+a6B92sP5gCVlvLuA8ytcd16FfLXrnHd6uH7D63KjBm2Kd6NN9o+Ot262XbZP/mJ7dvkrMzSpbZj6y9/CXPmWN4XvgCXXAK7pqFKpgxVGDkSzjvPFg8Gm3D7f/9nOyv7RO5Vefll22rlkUfs7379rD5++9vQo0f7/W6gyv0XODJxFW/dxieAv7XI25R4gySmV8ifTHsOknCaWbLE+gCizunOnW3eydSpSVuWTpqaVEeOXHmAzdChqk8/vcqptGWx2Dzy0ksrL+W05Zaqf/mLz2WqxsyZ1o8ZDbDp3dvmZ9Z5PiZ1GmZ+InB/yDUdfWDDzD8G1irLO4v4w8yHlOUNpqOGmRed++5buaE9/HDViROTtiq9vPaa6v77N5fXjjva3K8qI6kKL1AffWRr5pVPsr3mmsKvMFKVhgbVK65oXiigSxfVM8+0cmwH6iVQI0oexR3ANiHXdtSBTdSdDjwCHIgNYlhIi4m6wFvADS3yHsKGlX8J61ubREdP1C0aM2eqHn98c0O7ww4+ybY1li+3N9iuXfWTSba//33NuV+FFqi771bdYANdaZJtOzW0ueDll1UHD26uk4ccojphQrv+ZL0E6nWgAVs1YgUwFRgJnA8cQtk+UUke2FJHjwFLSmJ1CS2WOgKmADe1yFsbW+poLjZK8TagX5zfdIFqA3feaW+yoNqzp+qVV/ok29Z48UXVXXZpbji+8Q2bTBqDQgrUrFmqw4Y1l9eQIaqvv560VemloUH1oovMWwLVTTe1yEYHUBeBsvvQvRT6+gY2qu/fwIJItELvl5fDBSqAxYtVhw9vbjgOPNDXzGuNpibVP/yheamdgQNtG5EACidQTz6puvHGVl69epmX6fMLq/Peeyuvy/id79g+ah1ENYEKHcWHqi4DXigdAIiIANsAu4TezykYU6bAF78Ir74K3bvDlVfaSCAfOVWZRYtg+HC47Tb7+5vfhF//2rfAqIYqXHEF/PjHNgJ0zz3hr3+FgQOTtiy9PPYYHHcczJ4N/ftbee23X9JWAeHDzCtSUsCJpcNxKvPcc3D44fDhh/CpT9l+Orv4O01VZs2y8nr2WejVC667Dk44IWmr0ktjI5x+Olx7rf199tnw85/bXk1OZW6+GU47zcruoIPg1lttL7CUECRQItIZOBvYCxt48BqlVSRUdUb9zXNyw6hRMGyYbYx34IFw113Qp0/SVqWXt9+Gz38e3nrL3v4feAC22y5pq9LL4sXmBdx/v80t/Mtf4Oijk7YqvaiaeF9wgf191lnwq19Bp7rtYVsXQj2oX2LbV4wGjgOOjb4QkQ8xoTqsfuY5uWDUKDjqKGhogFNOad6B1KnMlCmwzz62K/Cuu5o49e+ftFXpZelSOPJIm0S6zjpw330W2nOqc8klcOGFJkhXXw3f/W7SFlUkVC6Pw0bsRa8m+5by3sTmGXWum2VOPnj8cfjSlz+ZHHsAABy8SURBVEyczjwTrr/exak1pk2zbeo/+ACGDIGxY12cWqOhAY491sRp/fXhqadcnGpx+eXN4nTbbakVJwgXqPWw1SSixZOWqupd2EKpi4Df1NE2J+u88YaFWZYvh+98B37zGx8M0RpLltgAkrffhsGDzfPs3Ttpq9LNWWeZx7TOOiZSHgZtnXvugXPOsXp4880WFk0xoQL1EdBLbcXyj4B+AKq6CLgcuLC+5jmZZd486+CfM8ca3auvdnFqDVVbb3DcONh8c3jwQRenWtx4oz1XXbvCP/8JO++ctEXp5pVX4Ktftc+//CWceGKy9sQgVKBexIaTg4X19ir7bjo+zNyJOOMM86B22sk6rDt79LdVrr8ebrkFevaEe++1xTid6rz+uk1PAPjjH2GvvVo/v+gsWQLHH2/TFr76VRvhmAFCB0n8lmYRGgH8VkReBd7A1sCbVkfbnKwycqQ1tj16wN//DmutlbRF6ebdd+FHP7LPI0a4J1CLxkY4+WRYtswG3Zx6atIWpZ8LLoAJE2DbbW0YfkaiGUECpaqPAo+W/rwVOBT4G9Yn1QB8ua7WOdnjo49sMinAZZfBNtu0fn7RUbXyWrDARjp+2atQTX79a3jhBdh0U+vXdFrn6adtQnznztbv1J7bY9SZoBCfiNwpIpsCqOoKVT0W2A1bWHUrVR3ZDjY6WeKii2xG+n772cAIp3Xuugseegj69oVrrsnMm21ivP8+/Oxn9vm667yfrhYrVsD3vmcvQueeC5/5TNIWBRHaB3UMsNKYV1V9WVXvAxaJiI/vLDLjx1t/QDS3ImWT/lLH8uW26RvAL37hw8njcP75Nu9p2DCbyOy0zq23wosvwiabwE9/mrQ1wdRsQURkGxHZQURqnfsp4Mn6mOVkkksusTe24cNhxx2Ttib93HADTJ5sYdDTTkvamvQzYYL1bXbtCpdemrQ16aehodnb/MUvbABOxojzins88Cq2tJEC54nIGSIyRETKV6zsAyxtBxudLDBpEtxxhzUe552XtDXpZ8UK60sBuPhi6FKXZTHzzeWXW6jq1FNhyy2Ttib93H67rUqyzTaZ7duMUyuuxLZR3x2b67QjcDDQDWgSkcnAeGx03yvtZKeTdq680hqPk0+2zmundUaOtAm5W27pa8bF4f33bbpCp042OddpHVVbWw9sZfeMTvOo6UGp6gJVfVxVr8BWKz8OWAsTrOHAv4De2Bwpj1MUkfnzbYl+gB/8IFlbssJvf2vpD3+Y2cajQ7n+egtZHX20e09xeOopmyvWvz985StJW9NmQoeZb1/250ulwyk6t91mEwD32ceXmonDm2/a0N9eveCkk5K2Jv00NdmqEdA8hcFpneuus/TrX8/02pe5GmYlIr1F5Gci8pyIzBORGSJyj4h8Ksa1J4uIVji+1RG2Z5o//9lSbzzicfPNlg4b5hsPxmH0aJvMPHBgajbSSzXz5tkEecj8JOa89cwOwLaivwH4KdATW+HiWRHZWVXfi3GP/YElZX+/XXcr88TUqfD88+YNHHVU0takH1Ub+gvWX+fUJgofn3yyT12Iw3332VD8oUNhiy2Stma1yJtAvQNsqaqfCIyIPAm8C5wC/CzGPZ5X1YXtZF/+uPtuSw87LFMz1BPjxRfNG9h4Y9h776StST+NjbYJIdi2Gk5t7rnH0hwMvsmVQJVWVW+Z97GITAXSs49xnrjrLkuPOSZZO7LCffdZ+oUvuDcQh3//25bP2morW0fOaZ3Fi21lErBNHDNO7muIiKwHbIUNhY/DZBFpFJFJIuKdKq0xezb85z/QvTscckjS1mSDyBs4/PBk7cgK//iHpUcc4ctAxeHRR02kBg/OxXSPXHlQVbgCm2R8e43zpgMXAM9hOwOfAPxJRHqqasUVKUVkODbUngEDBtTN4Mwwdqyle+7pnf1xmD7dFjnt0QP23z9pa7JB5A188YvJ2pEVRo+29LDDkrWjTqReoESkDy3W/6uEqk6scO23gROBo1X1oxrXPww8XJb1oIh0B84Xkd+WNmlsec0IbNsRBg8erC2/zz2PPWapN7bxiAR9n328vy4Os2bZ+o49esDnPpe0Ndkgesb23TdRM+pF6gUKGAZcF+O8lfx/Efki8DvgXFW9p42/fRdwLDAQH823Ko8/bqkP/Y3H009bOmRIsnZkhaeesvSzn4Vu3ZK1JQt8/LHtmtutm5VZDkh9H5SqXq+qUusov6a0qvrtwJ9U9fJ6mFGHe+SLGTNs8c6ePeHTn07ammwQCZTv/hqPJ0trT/tox3g8+aRNY/jc53LjoadeoEIRkR2A+4GHgDNW83ZHA7OBqatrV+545hlL99jD327jsGABvPyyLQqbsT15EiMSqH32SdaOrPDEE5YOHZqsHXUkCyG+2IjI+pgwLQSuBj4jzSN/5qvq+NJ5mwGTgVNU9ZZS3t3YAIlXsEESx5WOMyr1PxWeF1+0dNCgZO3ICs88Y0v2DBpkk5qd1lm82J6xzp29/yku48ZZmqPyypVAAdsDm5Q+P97iu7HAvqXPgolQuQc5CZvMu2np+/HASap6a3sZm2kigdptt2TtyAovvGDpHnska0dWGD/etiTZfnsfIRoHVfPQAXbZJVlb6kiuBEpVx9BisESV86a0PE9VzwN8I6O4uECF8dprlu68c7J2ZIVXX7V0p52StSMrvPcezJ0L/frBRhslbU3dyF0flNMBzJpl+/P06gVbb520NdkgEqgddkjWjqwQCZQLejxeKm0sscsuuZrQ7ALlhBN5T7vu6sv1xKGxESaWpultv33r5zrGK6W9T92DikcOw3vgAuW0hejtdtddk7UjK7z1FixfDpttBr17J21NNnAPKoxIoHJWJ12gnHDeesvST9XcZsuB5vDejjsma0dWmDkTPvzQxLyIS4i1hegZy5nH6QLlhBMJlG+9HQ8XqDDGl9Z13mGHXPWntBtNTfDOO/Z5q62StaXOuEA54UyebKkLVDy8/ymMKVMs9ecrHtOmWQh5/fVzNyTfBcoJo6HBNtwTgc03T9qabDC1tBCJl1c8IoEaODBJK7JD5D3l8PlygXLCmDrVJlBusontA+XU5t13LfX+lHi4QIXxdmkd64xv714JFygnjKj/KWex7nZj+XLbB6pTJ9vm3amNC1QY7kE5Tgnvfwrj/fdtGZqNN7aFYp3auECF4R6U45RwgQoj6n/abLNk7cgKDQ0m6iK52LK8Q3CBcpwSUTghh5WhXfD+pzDef9+GTW+8sW/jEhcP8TlOiWnTLN1kk9bPcwz3oMLw8F4YS5danezSJZd10gXKCSMSqBytmNyuuAcVRiToLlDxmD7d0v79c9nH6QLlxKepaeUK4dQmEij3oOLx3nuWev9TPHJeH12gnPh8+KHNgVp3XZ8DFZdIoLzBjceMGZZuuGGydmSFSKByWl4uUE58ovCez+eJT9Tgekg0HjNnWprTBrfuuAflOCVy/rZWd5Ytg3nzrG9g7bWTtiYbRIK+wQbJ2pEVovJygcoGIjJGRLTCsUaMa/cSkWdFZImIvCMiZ3SEzZlh1ixL118/WTuyQnl5+caO8fAQXxg596DyN+zDeBw4r0XestYuEJGtgIeB+4GfAJ8BrhSRxap6fbtYmTWiBne99ZK1Iyt8+KGlLujxiUJ87kHFI3rGclpeeRWoj1X1mcBrzgamASeqaiPwmIgMAC4UkRtUVetuZdZwgQojamxdoOKxaBEsXGgTdPv0SdqabDB7tqX9+iVrRzvhcYdmDgFGlsQp4nZgE8B3mgMXqFDcgwojamzXW883KozLRx9Z6gKVKQ4SkcWl42ER2bm1k0WkF7ApMLHFVxNK6bbtYWTmcIEKIxIoL694fPyxpeuum6wdWSIS9ZyWWR4FaixwJvB5YDgwAHhSRAa2ck00xGpui/w5pbRvpYtEZLiIvCAiL8yKGu88U/6G69QmanBz+nZbd6LyWmedZO3ICo2NMGeOeZt9KzZRmSf1fVAi0geoOURFVSeW0gvLsp8UkUcxz+j7paPV24Tkq+oIYATA4MGD899HFYmwN7jxiBrcnDYedScKV+XUG6g7c0rvz337QufOydrSTqReoIBhwHUxzqsYtFbVGSLyNLB7K9dGnlPLySp9W3xfbKIK4Q1IPMobEKc27kGFkfMBEpCBEJ+qXq+qUuuIc6tWfmMR8B6r9jVFf7fsmyoeTU026RR8hFVcIoHyBjceLlBhuEBlHxHZANgLGFfj1AeBo0Sk3Fc+DhOu19rJvOywYIGJ1Jpr5nLV5HbBPagwPMQXRgGer1wJlIjsLCKjRORkEdlPRL6GDZpoAq4qO2+oiDSKyNCyyy/HhpTfWrr2HOCbwMU+BwqYW4py+pI98fE+qDDcgwqjABGNvL0Kf4T1RV0KrAssAMYAR6rqu2XnCdCZsn4rVX1LRA4GrsS8qRnAj3wViRIuUOEU4A23rnh5heEClS1U9QPg0BjnjaHCoApVfQpb4shpiQtUGOV9dl5m8Zg/39IcN7h1pQAClasQn9OOuECFMW8eqELv3rkdAlx3IoHq3TtZO7KCC5TjlHCBCqMAjUfdcYEKowDPmAuUEw8XqDAWLLB0rbWStSNLuECF4QLlOCUigcpxZagrLlDhuECF4QLlOCUWLrTUG494ROXlAhWPhgZYssQ2duzZM2lrsoELlOOUiBrcNddM1o6s4B5UGFF59e7tW23ExQXKcUq4QIXhAhWGh/fCKUCZuUA58XCBCiMSKC+veBSgsa07BaiTLlBOPApQGeqKe1BhuECFs2iRpb16JWtHO+IC5cQjqgwuUPFwgQojEigvr3g0NNjRqRN07560Ne2GC5QTD/egwnCBCmPxYkv9+YpHVF49e+Z6UIkLlBMPF6gwXKDCKG9wndoUILwHLlBOXFygwnCBCsMFKgwXKMcpIxKonFeIulGQBqRuuECFUZDnywXKqc2KFTbLXwR69EjammywZImlXl7xcIEKwwXKcUqUV4ZO/sjEIhIob3Dj4QIVhguU45QoSGWoK+5BheECFUZB6mSuBEpEBoqIVjkm1bj2oirXHdxR9qcW9wbCcYEKwwUqjIIIVK62fAemA3u0yOsB/At4MMb184CWgjShDnZlG29sw/EyC8MFKgwXqOyhqsuAZ8rzRGQY9u/8W4xbNKrqM7VPKxhLl1q6xhrJ2pElXKDCcIEKIyqvnAtUrkJ8VTgBeEdVn03akMwSNbYuUPHxBjcML68wIg8q5+WVa4ESkd7AIcTzngDWFpHZItIgIi+KyJfa0bzsEHlQ7g3EY8UKWL7chuXneJ20ulKQBrduFKRfONcCBRwJrAHcHuPct4BzgGOBo4FpwN0uUniIL5Ty8srxOml1xT2oMJYtszTnL0Cp74MSkT5A/1rnqerECtknAK+r6qsxrv9Li9+9D/g38L/AyCq2DQeGAwwYMKDWT2QXD/GF4f1P4bhAhVGQl8bUCxQwDLguxnkrvaqKyLrAgcBFbflRVVURGQn8SkQ6q+qKCueMAEYADB48WNvyO5nAQ3xhuECF4wIVRkE8qNSH+FT1elWVWkeFS4/BBDhOeK9VE1bz+uxTkLe1ulGQ/oG64qIeRkHqZOoFajU4AXhOVSe35WIREeAo4OVK3lOh8BBfGN7YhlMQj6BuFKS8shDiC0ZENgL2Bn5U5fuhwGjgAFUdW8obC9wNTAR6Ad8APocNtCg2HuILwwUqnKjB7dYtWTuyQkE8qFwKFDYSD+DOKt8L0JmV+63eAr6PDchoAv4LHKaqcVagyDcFqQx1I+pPcYGKh6oNy4fcewR1wz2o7KKqVwFXtfL9GFoMqlDVU9vZrOziIb4w3IMKo7HRRKpzZzuc2hTkpTHPfVBOvShIZagb0dutl1c8CuIN1JWCPGMuUE5tvA8qDA9XheH9T+FEdTLnz5gLlFMbD/GFEQmUN7jxcEEPpyBRDRcopzYFqQx1wwUqDA/xhVOQMnOBcmrjIb4wXKDCKEhjW1cK8tLoAuXUpiDx7rrhAhWG90GFUxBRd4FyauMNbhje4IbhfVDhuAflOCVcoMLw8gqjIN5A3VAtTJm5QDm1aWiwtGvXZO3ICi5QYRSksa0bjY3Q1GSTmrvkcq2FT3CBcmrjAhWGC1QYHhINoyCTdMEFyomDN7hheHmF4X1QYRRo0JILlFMb96DCcIEKw0N8YRSovFygnNpEAuUNbjxcoMIoUINbFwpUH12gnNpEDa57UPFwgQrD+6DCiAQq5wMkwAXKiYOH+MJwgQrD+6DCKFB9dIFyauMNbhheXmF4iC8MFyjHKaNAFaIuuECF4QIVRmOjpR7icxwK1SlbF1ygwvA+qDAK9MKYKYESkeNEZKSITBcRFZGTq5y3sYjcIyILRWS2iPxeRHrGuH93EblCRD4UkUUiMkpEBtb5n5EtVH2QRCguUGF4eYXhApVajgEGAvdXO0FEugAPA5sBxwFnAsOAETHufzVwMnBW6bf6AY+ISP6nbFdjxQpLO3Wyw6mNN7hhRCGrAjS4daFA5ZW1IOZxqtokImsCp1U5ZxiwHbCVqr4DICINwO0i8jNVfbPSRSKyCXAqcIqq3lLKewV4BzgRuL6+/5SM4OG9cFygwihQg1sXfJh5OlHVphinHQI8H4lTiXuB5cDBrVx3UCkdWfZ7HwBPle5ZTDy8F44LVBgF6vSvCx7iyzTbAhPLM1R1OTC59F1r172vqgtb5E+ocV2+KVBlqBsuUGG4QIVRII8zj09EX2Buhfw5pe/qdp2IDAeGl/5cKCKTSp/7APMqXFItvx8wu4pdofeqlr96dn38MYh09O+3dq9qZZYeu7bfPlV2iUhIedX792s/Y6efbkfa7Fr9e7VPnfznP1e3TraPXW2712YVr1TVxI6SgdvWOipctyagwMkVvnsT+E2F/KeBv7Ziy3XAixXyfw58EPjvGhGY/0Id71UxP2m72uHfUtE2t6s+drXD7wc9Y0Wwq43/lkK0FdGRtAc1DBOGWkjtUz5hDrB2hfy1qewhre51lbgvML+e92rtN5K0qy3XuF31u1fob9T790NtK4Jdbb2mXvdKq12fICUlyxSlUXwLgK+r6k0tvrsF2FpV9yjL64a5kueo6u+q3PMUbCh6H1VdVJY/GpirqkfX/R/S/BsvqOrg9rp/W0mrXZBe29yuMNyuMIpmVx4HSTwIfFpEymOaXwS6Aw+1ct2/SulRUYaIbATsXbpnexJnjlYSpNUuSK9tblcYblcYhbIrUx6UiGwPbA+sAdwKXAOMAWap6tjSOV2BF4FlwAVYP9dvgEdV9cSye40GUNUDyvKuBY4GfgDMAi4C1gV2UtWl7fuvcxzHccrJmkBdBFxY4auxqrpv2XmbAL8HDsSE6nbgbFVdXHbOGIAW13UHLgW+CvQExgLf1ZXnVDmO4zgdQKYEynEcxykOeeyDchzHcXKAC1SCiMgwEfmniHxQWnl9nIickLRdTtsRkU1EZEbp//QlEXlfRGaKyOEJ2bO9iIwWkcUiMk1ELhaRzknYUmZTqsrIqU1SbVXS86CKzg+xxWh/gM3CPhS4TUT6VRsO3xGIyFhs/pcAb2AL6M5Pyp4soarvi8j92KCc20XkMuBdVW3L/JHVQkT6Ao8C44EjgC2BK7AX0/M72p6INJVRmklZPUymrao1k9eP9juAfhXybgPeSdiuPmWfrwQuSbqsWrH1UWxVkWrHZu3wm08BUyocN5S+fw7YvvT5X8A+CZXNT7AJ6L3L8s4BFpfnJWRbKsooyecohk2pqYdJtVUe4ksQVa20dtWLwPrlGSLyaGmDxmpH5XWs2m7XvNLvdgJ6YRU0raGZs7BG+Bpgj9LxZ2Aa8DlVnVrvH1TVIao6sMJxaqnMtsLeeAF2BV6ptw0xOQR4WFd+674d6AEMTcakT56rtJRRRIc/R7WoVA+TqoNx2qr2aKdcoNLHnlhIppwOrzwi8gAwE9gGuAwsNINtFvkjVd0Ve4O6RJMNzUzBFvN9RFWfUdVngN7AK6r6bAL2bA1MVdVGEVkfaFDV0KWy6kWllf3fxTyoJFfoT1MZRUwhXc8RsGo9TFkdbNlW1b2dcoFKESJyANZXcE2Lr6ZQx8ojIk+JyJQKxw3ROap6KLAhFor5TtnlO9P8tlv3N18R6SMi29Y6yi7ZsZS+3sLG1+ppVwA7A6+W/d1FRI5JyJa2ruzf3qSpjCLS9hwBVethu9bBOFRpq6ZQZ5H3QRJ1RET6AP1rnaeqE1vmichA7G3oH9pifUGqV55/tsVOVR0S87wVInIzcAdwWQeFZkIXEN4R8wjeBhCRHiUbE2lYVPXvwN9Lnz8ENkjCjjIqTXSUKvkdQgrLCFL2HJVTXg9F5NckHB5tpa2qazsFLlD1pk2rs4vIOth6f+9i28u3pMMqT2nkVzdVnVnKOrrsd9o9NKOq1wPXB1yyAzBBm3db3gGLDCTesKSAaiv09yF8hf68k6rnqJV6mGh4tEZbVfd2ykN8dURVr1dVqXWUXyMiPbGYcjfgMC1bSb2Mjqw8fYFRIvKKiLyKPXRnlr5La2jm9RZ/N7FqP14RmUiLviYR2RTrcF/Fiy84aXuOqtXDxOpgjLaq7u2Ue1AJIiJdsFDH1sBepXBHJTqs8qjq20DFZfNTGprZAXig7O9NgEW0UwhLRFb7vi1fUtqRB4GzRWQtVV1QyjsOWIKtM5kItcqwA8unnA59jmrRSj1MpA7GbKvq3k65QCXLH7AJb2cC64jI58q+e1FVl5U+p6rypIVSiGM9Vn5D+y8WQn0e2Knev5lQ49lW/gScAYwUkV8BW2Ar9F+pCU68TlsZJvEcZZA4bVXd2ylfLDZBRGQKUG1uwOaqOqVUeWYCh6rqg6XrDsUGLkxRVa88CVIKmd0EbIS9LY4CztWUVCyxLWp+jw35nYv1712kqisStCnVZeasSq22Cut7qns75QJVMOoRoqpG2t6MOwIR6Q9srKoviO3c/AhwtarenbBpqcXLrDarU0/zVA89xFcw8vTwpgFVnQ5ML31eLiKvAJsma1W68TKrjddTw0fxOSshIpuKrX49QUReF5HLRMQrSwxEZF3gSODhpG3JCl5mtSlynXSBclrSiPUHbAfsBnwW+FKyJqUfsd2Y7wKuUtUJSduTBbzMYlPYOukhPmclPPwSjtj+Sn/FRjNdkbQ9WcDLLD5FrpM+SMKpSin88hJwkL/hVkdErgc6Y/v1eIWKgZdZ2yhanXSBcipSCr88BNzvb7jVEZG9sP2hXgOiodt/VtWrk7Mq3XiZtY0i1kkXKGcVSuGXO7BdTn+YtD2OU3SKWiddoJxV8PCL46SLotZJFyhnJTz84jjposh10gXKcRzHSSU+D8pxHMdJJS5QjuM4TipxgXIcx3FSiQuU4ziOk0pcoBzHcZxU4gLlOI7jpBIXKMdxHCeVuEA5juM4qcQFynEcx0klLlCOk0NEZCsRaRCRn7XI/6OILBCRwUnZ5jhxcYFynByiqm8B1wM/EJF+ACLyv8ApwFGq+kKS9jlOHHwtPsfJKSKyITAZ+AMwERgBnKCqdyZqmOPExLd8d5ycoqozROQq4EdYXT/DxcnJEh7ic5x88ybQHfiPql6TtDGOE4ILlOPkFBHZH7gW+A+wl4jskrBJjhOEC5Tj5BAR2R24FxsosS/wLvCLJG1ynFBcoBwnZ4jIVsCDwL+A76nqcuBnwKEisk+ixjlOAD6Kz3FyRGnk3r8xj+nzqrqslN8Z2zJ8jqrumaCJjhMbFyjHcRwnlXiIz3Ecx0klLlCO4zhOKnGBchzHcVKJC5TjOI6TSlygHMdxnFTiAuU4juOkEhcox3EcJ5W4QDmO4zip5P8BG6PcErKWfLkAAAAASUVORK5CYII=\n",
-      "text/plain": [
-       "<Figure size 432x324 with 1 Axes>"
-      ]
-     },
-     "metadata": {
-      "needs_background": "light"
-     },
-     "output_type": "display_data"
-    }
-   ],
-   "source": [
-    "#Codes for figures\n",
-    "\n",
-    "import numpy as np\n",
-    "import pandas as pd\n",
-    "import matplotlib.pyplot as plt\n",
-    "\n",
-    "def multiple_formatter(denominator=2, number=np.pi, latex='\\pi'):\n",
-    "    def gcd(a, b):\n",
-    "        while b:\n",
-    "            a, b = b, a%b\n",
-    "        return a\n",
-    "    def _multiple_formatter(x, pos):\n",
-    "        den = denominator\n",
-    "        num = np.int(np.rint(den*x/number))\n",
-    "        com = gcd(num,den)\n",
-    "        (num,den) = (int(num/com),int(den/com))\n",
-    "        if den==1:\n",
-    "            if num==0:\n",
-    "                return r'$0$'\n",
-    "            if num==1:\n",
-    "                return r'$%s$'%latex\n",
-    "            elif num==-1:\n",
-    "                return r'$-%s$'%latex\n",
-    "            else:\n",
-    "                return r'$%s%s$'%(num,latex)\n",
-    "        else:\n",
-    "            if num==1:\n",
-    "                return r'$\\frac{%s}{%s}$'%(latex,den)\n",
-    "            elif num==-1:\n",
-    "                return r'$\\frac{-%s}{%s}$'%(latex,den)\n",
-    "            else:\n",
-    "                return r'$\\frac{%s%s}{%s}$'%(num,latex,den)\n",
-    "    return _multiple_formatter\n",
-    "\n",
-    "class Multiple:\n",
-    "    def __init__(self, denominator=2, number=np.pi, latex='\\pi'):\n",
-    "        self.denominator = denominator\n",
-    "        self.number = number\n",
-    "        self.latex = latex\n",
-    "\n",
-    "    def locator(self):\n",
-    "        return plt.MultipleLocator(self.number / self.denominator)\n",
-    "\n",
-    "    def formatter(self):\n",
-    "        return plt.FuncFormatter(multiple_formatter(self.denominator, self.number, self.latex))\n",
-    "\n",
-    "xt=np.linspace(-4*np.pi,4*np.pi,num=500)\n",
-    "##\n",
-    "fig, ax = plt.subplots(figsize=(12, 3))\n",
-    "ax.plot(xt, np.sin(xt), color='b', lw=2, label='$sin(x)$')\n",
-    "ax.plot(xt, np.cos(xt), color='g', lw=2, label='$cos(x)$')\n",
-    "ax.tick_params(axis='both', labelsize= 15)\n",
-    "ax.set_xlabel('$x$', fontsize = 16)\n",
-    "ax.legend(fontsize=14)\n",
-    "\n",
-    "ax.axhline(0, color='black', lw=1)\n",
-    "ax.axvline(0, color='black', lw=1)\n",
-    "\n",
-    "ax.xaxis.set_major_locator(plt.MultipleLocator(np.pi / 2))\n",
-    "ax.xaxis.set_minor_locator(plt.MultipleLocator(np.pi / 12))\n",
-    "ax.xaxis.set_major_formatter(plt.FuncFormatter(multiple_formatter()))\n",
-    "\n",
-    "fig.tight_layout()\n",
-    "fig.savefig('sin_cos.png')\n",
-    "\n",
-    "##\n",
-    "xt=np.linspace(-2*np.pi,2*np.pi,num=500)\n",
-    "fig, ax = plt.subplots(figsize=(6, 4.5))\n",
-    "y = np.tan(xt)\n",
-    "y[:-1][np.diff(y) < 0] = np.nan\n",
-    "\n",
-    "ax.plot(xt,y, color ='r', lw=2)\n",
-    "ax.tick_params(axis='both', labelsize= 15)\n",
-    "\n",
-    "ax.set_xlabel('$x$', fontsize = 16)\n",
-    "ax.set_ylabel(\"$tan(x)$\", fontsize = 16)\n",
-    "\n",
-    "plt.ylim(-10,10)\n",
-    "ax.axhline(0, color='black', lw=1)\n",
-    "ax.axvline(0, color='black', lw=1)\n",
-    "\n",
-    "ax.xaxis.set_major_locator(plt.MultipleLocator(np.pi / 2))\n",
-    "ax.xaxis.set_minor_locator(plt.MultipleLocator(np.pi / 12))\n",
-    "ax.xaxis.set_major_formatter(plt.FuncFormatter(multiple_formatter()))\n",
-    "\n",
-    "fig.tight_layout()\n",
-    "fig.savefig('tan.png')"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 14,
-   "metadata": {
-    "tags": [
-     "remove-cell"
-    ]
-   },
-   "outputs": [
-    {
-     "data": {
-      "image/png": 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\n",
-      "text/plain": [
-       "<Figure size 864x216 with 1 Axes>"
-      ]
-     },
-     "metadata": {
-      "needs_background": "light"
-     },
-     "output_type": "display_data"
-    },
-    {
-     "data": {
-      "image/png": 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\n",
-      "text/plain": [
-       "<Figure size 864x216 with 1 Axes>"
-      ]
-     },
-     "metadata": {
-      "needs_background": "light"
-     },
-     "output_type": "display_data"
-    },
-    {
-     "data": {
-      "image/png": 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\n",
-      "text/plain": [
-       "<Figure size 864x216 with 1 Axes>"
-      ]
-     },
-     "metadata": {
-      "needs_background": "light"
-     },
-     "output_type": "display_data"
-    }
-   ],
-   "source": [
-    "import numpy as np\n",
-    "import pandas as pd\n",
-    "import matplotlib.pyplot as plt\n",
-    "\n",
-    "\n",
-    "#xt=np.linspace(0,15*np.pi,num=500)\n",
-    "xt=np.linspace(0,50,num=1000)\n",
-    "##\n",
-    "fig, ax = plt.subplots(figsize=(12, 3))\n",
-    "\n",
-    "y1=0.62*np.cos(0.52*xt) + 1.09*np.sin(0.52*xt)\n",
-    "y2=3.75 + 4.33*np.cos(0.26*xt) + 2.5*np.sin(0.26*xt)\n",
-    "\n",
-    "ax.plot(xt, y1+y2, color='g', lw=2)\n",
-    "ax.tick_params(axis='both', labelsize= 15)\n",
-    "ax.set_xlabel('$t$', fontsize = 16)\n",
-    "ax.set_ylabel('$f\\,(t)$', fontsize = 16)\n",
-    "\n",
-    "ax.set_xlim(0,50)\n",
-    "fig.tight_layout()\n",
-    "fig.savefig('comp_signal.png')\n",
-    "\n",
-    "fig, ax = plt.subplots(figsize=(12, 3))\n",
-    "\n",
-    "ax.plot(xt, y1, color='b', lw=2)\n",
-    "ax.tick_params(axis='both', labelsize= 15)\n",
-    "ax.set_xlabel('$t$', fontsize = 16)\n",
-    "ax.set_ylabel('$y_1\\,(t)$', fontsize = 16)\n",
-    "\n",
-    "ax.set_xlim(0,50)\n",
-    "fig.tight_layout()\n",
-    "fig.savefig('y1_signal.png')\n",
-    "\n",
-    "fig, ax = plt.subplots(figsize=(12, 3))\n",
-    "\n",
-    "ax.plot(xt, y2, color='y', lw=2)\n",
-    "ax.tick_params(axis='both', labelsize= 15)\n",
-    "ax.set_xlabel('$t$', fontsize = 16)\n",
-    "ax.set_ylabel('$y_2\\,(t)$', fontsize = 16)\n",
-    "\n",
-    "ax.set_xlim(0,50)\n",
-    "fig.tight_layout()\n",
-    "fig.savefig('y2_signal.png')"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 3,
-   "metadata": {
-    "tags": [
-     "remove-cell"
-    ]
-   },
-   "outputs": [
-    {
-     "data": {
-      "application/vnd.jupyter.widget-view+json": {
-       "model_id": "a785b675781e44dcb130fe4aba0b9568",
-       "version_major": 2,
-       "version_minor": 0
-      },
-      "text/plain": [
-       "interactive(children=(FloatSlider(value=0.0, description='c', max=2.0, min=-2.0), Output()), _dom_classes=('wi…"
-      ]
-     },
-     "metadata": {},
-     "output_type": "display_data"
-    },
-    {
-     "data": {
-      "application/vnd.jupyter.widget-view+json": {
-       "model_id": "b9af681dfe384b9fbd3b14793a2feede",
-       "version_major": 2,
-       "version_minor": 0
-      },
-      "text/plain": [
-       "interactive(children=(FloatSlider(value=0.0, description='c', max=2.0, min=-2.0), Output()), _dom_classes=('wi…"
-      ]
-     },
-     "metadata": {},
-     "output_type": "display_data"
-    },
-    {
-     "data": {
-      "text/plain": [
-       "<function __main__.vtrans(c)>"
-      ]
-     },
-     "execution_count": 3,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "from ipywidgets import interact\n",
-    "import matplotlib.pyplot as plt\n",
-    "import numpy as np\n",
-    "\n",
-    "def func(x):\n",
-    "    return x**3\n",
-    "\n",
-    "def htrans(c):\n",
-    "    x = np.linspace(-3,3)\n",
-    "    \n",
-    "    y = func(x+c) #constant c in the argument of the function\n",
-    "    \n",
-    "    plt.plot(x,y, label='$f\\, (x+c)$')\n",
-    "    plt.title('Horizontal translation')\n",
-    "        \n",
-    "    plt.xlabel('x')\n",
-    "    plt.ylabel('y')\n",
-    "    plt.legend(fontsize=12)\n",
-    "    \n",
-    "    plt.ylim(-8, 8) #remember to change the limits for better visualisation if necessary\n",
-    "    \n",
-    "    plt.axhline(0, color='black', lw=1)\n",
-    "    plt.axvline(0, color='black', lw=1)\n",
-    "    \n",
-    "    \n",
-    "\n",
-    "    plt.show()\n",
-    "\n",
-    "def vtrans(c):\n",
-    "    x = np.linspace(-3,3)\n",
-    "    \n",
-    "    y = func(x)+c #constant c summing the whole function\n",
-    "    \n",
-    "    plt.plot(x,y,  label='$f\\, (x)+c$')\n",
-    "    plt.title('Vertical translation')\n",
-    "    \n",
-    "    plt.xlabel('x')\n",
-    "    plt.ylabel('y')\n",
-    "    plt.legend(fontsize=12)\n",
-    "    \n",
-    "    plt.ylim(-8, 8) #remember to change the limits for better visualisation if necessary\n",
-    "        \n",
-    "    plt.axhline(0, color='black', lw=1)\n",
-    "    plt.axvline(0, color='black', lw=1)\n",
-    "    \n",
-    "    plt.show()\n",
-    "    \n",
-    "# create a slider\n",
-    "\n",
-    "interact(htrans, c=(-2.0,2.0))\n",
-    "interact(vtrans, c=(-2.0,2.0))"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 6,
-   "metadata": {
-    "tags": [
-     "remove-cell"
-    ]
-   },
-   "outputs": [
-    {
-     "data": {
-      "application/vnd.jupyter.widget-view+json": {
-       "model_id": "12d5a3f30eab4d5395e543cb666f5e70",
-       "version_major": 2,
-       "version_minor": 0
-      },
-      "text/plain": [
-       "interactive(children=(FloatSlider(value=1.05, description='c', max=2.0, min=0.1), Output()), _dom_classes=('wi…"
-      ]
-     },
-     "metadata": {},
-     "output_type": "display_data"
-    },
-    {
-     "data": {
-      "application/vnd.jupyter.widget-view+json": {
-       "model_id": "bd6164c6fecf4c72b93b1a39f7947633",
-       "version_major": 2,
-       "version_minor": 0
-      },
-      "text/plain": [
-       "interactive(children=(FloatSlider(value=1.05, description='c', max=2.0, min=0.1), Output()), _dom_classes=('wi…"
-      ]
-     },
-     "metadata": {},
-     "output_type": "display_data"
-    },
-    {
-     "data": {
-      "text/plain": [
-       "<function __main__.vstetcom(c)>"
-      ]
-     },
-     "execution_count": 6,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "from ipywidgets import interact\n",
-    "import matplotlib.pyplot as plt\n",
-    "import numpy as np\n",
-    "\n",
-    "def func(x):\n",
-    "    return x**3-2*x\n",
-    "\n",
-    "def hstetcom(c):\n",
-    "    x = np.linspace(-3,3)\n",
-    "    \n",
-    "    y = func(c*x) #constant c multiplying the argument of the function\n",
-    "    \n",
-    "    plt.plot(x,y, label='$f\\, (cx)$')\n",
-    "    plt.title('Horizontal stretching/compression')\n",
-    "        \n",
-    "    plt.xlabel('x')\n",
-    "    plt.ylabel('y')\n",
-    "    plt.legend(fontsize=12)\n",
-    "    \n",
-    "    plt.ylim(-4, 4) #remember to change the limits for better visualisation if necessary\n",
-    "    \n",
-    "    plt.axhline(0, color='black', lw=1)\n",
-    "    plt.axvline(0, color='black', lw=1)\n",
-    "    \n",
-    "    \n",
-    "\n",
-    "    plt.show()\n",
-    "\n",
-    "def vstetcom(c):\n",
-    "    x = np.linspace(-3,3)\n",
-    "    \n",
-    "    y = c*func(x) #constant c multiplying the whole function\n",
-    "    \n",
-    "    plt.plot(x,y,  label='$c\\, f\\, (x)$')\n",
-    "    plt.title('Vertical stretching/compression')\n",
-    "    \n",
-    "    plt.xlabel('x')\n",
-    "    plt.ylabel('y')\n",
-    "    plt.legend(fontsize=12)\n",
-    "    \n",
-    "    plt.ylim(-4, 4) #remember to change the limits for better visualisation if necessary\n",
-    "        \n",
-    "    plt.axhline(0, color='black', lw=1)\n",
-    "    plt.axvline(0, color='black', lw=1)\n",
-    "    \n",
-    "    plt.show()\n",
-    "    \n",
-    "# create a slider\n",
-    "\n",
-    "interact(hstetcom, c=(0.1,2.0))\n",
-    "interact(vstetcom, c=(0.1,2.0))"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 48,
-   "metadata": {
-    "tags": [
-     "remove-cell"
-    ]
-   },
-   "outputs": [
-    {
-     "data": {
-      "image/png": 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\n",
-      "text/plain": [
-       "<Figure size 720x360 with 2 Axes>"
-      ]
-     },
-     "metadata": {
-      "needs_background": "light"
-     },
-     "output_type": "display_data"
-    }
-   ],
-   "source": [
-    "fig, ax = plt.subplots(1,2,figsize=(10, 5))\n",
-    "\n",
-    "xt = np.linspace(-3,5)\n",
-    "\n",
-    "# x-axis reflection\n",
-    "ax[0].plot(xt, (xt-1)**2, color ='r', lw=2, label = '$(x-1)^2$')\n",
-    "ax[0].plot(xt, -(xt-1)**2, color ='b', lw=2, label = '$-(x-1)^2$')\n",
-    "ax[0].tick_params(axis='both', labelsize= 12)\n",
-    "\n",
-    "ax[0].set_xlabel('$x$', fontsize = 14)\n",
-    "ax[0].set_ylabel(\"$y$\", fontsize = 14)\n",
-    "\n",
-    "ax[0].set_title('x-axis reflection')\n",
-    "\n",
-    "ax[0].set_ylim(-10,10)\n",
-    "\n",
-    "ax[0].legend()\n",
-    "\n",
-    "ax[0].axhline(0, color='black', lw=1)\n",
-    "ax[0].axvline(0, color='black', lw=1)\n",
-    "\n",
-    "# y-axis reflection\n",
-    "\n",
-    "xt = np.linspace(-4,4)\n",
-    "\n",
-    "ax[1].plot(xt, (xt-1)**2, color ='r', lw=2, label = '$(x-1)^2$')\n",
-    "ax[1].plot(xt, (-xt-1)**2, color ='b', lw=2,  label = '$[(-x)-1]^2$')\n",
-    "ax[1].tick_params(axis='both', labelsize= 12)\n",
-    "\n",
-    "ax[1].set_xlabel('$x$', fontsize = 14)\n",
-    "#ax[1].set_ylabel(\"$y$\", fontsize = 14)\n",
-    "\n",
-    "ax[1].set_title('y-axis reflection')\n",
-    "\n",
-    "ax[1].set_ylim(-1,10)\n",
-    "\n",
-    "ax[1].legend()\n",
-    "\n",
-    "ax[1].axhline(0, color='black', lw=1)\n",
-    "ax[1].axvline(0, color='black', lw=1)\n",
-    "\n",
-    "fig.tight_layout()\n",
-    "fig.savefig('reflection.png')"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "# 02   - Describing shapes and patterns\n",
-    "  "
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "## Measuring angles  \n",
-    "  \n",
-    "An **angle** is a primitive geometrical element, such as the *point*, the *curve* or the *plane*. It is defined as the figure formed by two line segments that extend from a common point.  \n",
-    "    \n",
-    "<br/>\n",
-    "    \n",
-    "```{image} angle.png  \n",
-    "\n",
-    "```  \n",
-    "  \n",
-    "<br/>\n",
-    "<br/>\n",
-    "      \n",
-    "The two most common units for measuring angles are the *degree* and the *radian*.  \n",
-    "  \n",
-    "  \n",
-    "`````{tabbed} Degrees \n",
-    "\n",
-    "````{panels}  \n",
-    "  \n",
-    "```{image} ./degree.png  \n",
-    "\n",
-    "```  \n",
-    "\n",
-    "---  \n",
-    "  \n",
-    "If we divide a full rotation by 360 equal parts, each of these parts is called a *degree* (noted as $1^{\\circ}$). Thus, the degree corresponds to $1/360$ of one full rotation. Each degree can also be divided in 60 equal parts called *minutes* ($1^{\\circ} = 60'$) and each minute further divided in 60 *seconds* ($1' = 60''$).  \n",
-    "  \n",
-    "+++  \n",
-    "In the figure, one degree (shown in red) and eighty nine degrees (shown in blue).\n",
-    "\n",
-    "  \n",
-    "````  \n",
-    "`````  \n",
-    "  \n",
-    "`````{tabbed} Radians  \n",
-    "\n",
-    "````{panels}  \n",
-    "  \n",
-    "```{image} ./Circle_radians.gif  \n",
-    "\n",
-    "```  \n",
-    "\n",
-    "---  \n",
-    "  \n",
-    "One *radian* is the angle comprised in a circunference when the size of the arc is equal to the radius of the circunference. If $C$ is the size of the arc and $R$ the radius of the circunference, the corresponding angle in radians $\\phi$ will be given by:  \n",
-    "  \n",
-    "$$ \\phi = \\frac{C}{R} $$\n",
-    "  \n",
-    "Thus, if we are considering the arc for the full circunference, $C = 2\\pi R$, and we will have:  \n",
-    "  \n",
-    "$$\\phi = \\frac{2\\pi R}{R} = 2\\pi, $$  \n",
-    "  \n",
-    "and the angle for a full rotation corresponds to $2\\pi$ radians.  \n",
-    "  \n",
-    "```{tip}  \n",
-    "When the angle is written in radians, the unit (radians or rad) is frequently omitted.\n",
-    "\n",
-    "```\n",
-    "  \n",
-    "````  \n",
-    "`````  \n",
-    "  \n",
-    "For a given angle, transforming between units of degrees and radians can be simply done with:  \n",
-    "  \n",
-    "$$\\frac{360^{\\circ}}{\\theta} = \\frac{2\\pi}{\\phi},$$  \n",
-    "  \n",
-    "where $\\theta$ is the angle measured in degrees and $\\phi$, measured in radians. In this lecture, we will consider the angles measured in radians (independent of the symbol used for the variable), unless stated otherwise. \n",
-    "  \n",
-    "```{admonition} Example  \n",
-    "If $\\theta = 90^{\\circ}$, then using the previous expression we have $\\phi = \\displaystyle\\frac{\\pi}{2}$. Since $\\theta$ and $\\phi are directly proportional, we also have:  \n",
-    "  \n",
-    "  $$\\theta = 180^{\\circ} \\longrightarrow \\phi = 2 \\cdot \\frac{\\pi}{2} = \\pi $$  \n",
-    "  $$\\theta = 270^{\\circ} \\longrightarrow \\phi = 3 \\cdot \\frac{\\pi}{2} = \\frac{3\\pi}{2} $$  \n",
-    "  $$\\theta = 360^{\\circ} \\longrightarrow \\phi = 4 \\cdot \\frac{\\pi}{2} = 2\\pi $$\n",
-    "  \n",
-    "```"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "## Trigonometric circle  \n",
-    "  \n",
-    "  \n",
-    "We obtain the trigonometric circle by superimposing a circle of radius $1$ to the cartesian system of coordinates:   \n",
-    "\n",
-    "```{image} trig_circ.png\n",
-    "```  \n",
-    "\n",
-    "  \n",
-    "The projections on the $x$ and $y$-axes have special meanings, if we consider the angle $\\theta$ between the line segment in the radial direction and the $x$-axis.  \n",
-    "  \n",
-    "From the geometric relations on the right triangle (triangle in which one internal angle is a right angle):  \n",
-    "  \n",
-    "````{panels}  \n",
-    "  \n",
-    "```{image} ./right_trig.png\n",
-    "\n",
-    "```  \n",
-    "\n",
-    "---  \n",
-    "  \n",
-    "The larger side of the triangle, opposing the right angle, is called *hypothenuse*, while the other two smaller side are called *catheti*. We define the *sine* and *cosine* of the angle $\\alpha$ as the ratios of the opposing and adjacent catheti, respectively, with the hypothenuse:  \n",
-    "  \n",
-    "$$\\mbox{sin}(\\alpha) = \\frac{a}{c}$$  \n",
-    "$$\\mbox{cos}(\\alpha) = \\frac{b}{c}$$\n",
-    "  \n",
-    "````  \n",
-    "  \n",
-    "Back to the trigonometric circle, it is possible to construct a right triangle where the hypothenuse corresponds to the radius of the superimposed circle. In this case, since this radius equals to $1$, we have:  \n",
-    "  \n",
-    "$$\\mbox{sin}(\\theta) = y$$  \n",
-    "$$\\mbox{cos}(\\theta) = x.$$  \n",
-    "  \n",
-    "In other words, the projections of the end of the radial line segment on the $y$ and $x$-axes correspond to the sine and cosine of the angle $\\theta$. Sine and cosine for this angle can be obtained by the projections even if $\\theta > \\displaystyle\\frac{\\pi}{2}$.  \n",
-    "  \n",
-    "### Trigonometric identities  \n",
-    "  \n",
-    "  With the aid of the projections on the trigonometric circle, some trigonometric identities become imediately clear. Consider $\\theta > 0$ for counterclockwise angle measurements starting from the $x$-axis and $\\theta < 0$ for clockwise measurements. We have:  \n",
-    "    \n",
-    "$$\\mbox{sin}(0) = 0,\\ \\ \\ \\mbox{sin}\\left(\\frac{\\pi}{2}\\right) = 1,\\ \\ \\ \\mbox{sin}(\\pi) = 0,\\ \\ \\ \\mbox{sin}\\left(\\frac{3\\pi}{2}\\right) = -1,\\ \\ \\ \\mbox{sin}(2\\pi) = 0$$  \n",
-    "$$\\mbox{cos}(0) = 1,\\ \\ \\ \\mbox{cos}\\left(\\frac{\\pi}{2}\\right) = 0,\\ \\ \\ \\mbox{cos}(\\pi) = -1,\\ \\ \\ \\mbox{cos}\\left(\\frac{3\\pi}{2}\\right) = 0,\\ \\ \\ \\mbox{cos}(2\\pi) = 1$$\n",
-    "  \n",
-    "From the fact that a full rotation brings you back to the same point:  \n",
-    "  \n",
-    "$$\\mbox{sin}(\\theta) = \\mbox{sin}(\\theta + 2\\pi)$$  \n",
-    "$$\\mbox{cos}(\\theta) = \\mbox{cos}(\\theta + 2\\pi)$$  \n",
-    "  \n",
-    "or from Pythagoras' theorem on the right triangle in the trigonometric circle:  \n",
-    "  \n",
-    "$$\\begin{align}\n",
-    "[\\mbox{sin}(\\theta)]^2 + [\\mbox{cos}(\\theta)]^2 =& 1^2 \\\\\n",
-    "\\mbox{sin}^2(\\theta) + \\mbox{cos}^2(\\theta) =& 1\n",
-    "\\end{align}$$  \n",
-    "  \n",
-    "```{tip}\n",
-    "The notation $\\mbox{sin}^2(\\theta)$ is frequently used to express the square of $\\mbox{sin}(\\theta)$, or:  \n",
-    "  \n",
-    "$$ \\mbox{sin}^2(\\theta) = [\\mbox{sin}(\\theta)]^2 = [\\mbox{sin}(\\theta)]\\cdot[\\mbox{sin}(\\theta)],$$  \n",
-    "  \n",
-    "which is different from $\\mbox{sin}(\\theta^2)$.\n",
-    "```  \n",
-    "  \n",
-    "  \n",
-    "Now watch this [video](https://web.microsoftstream.com/video/b3023026-7d45-4378-a8ee-c166a43eb2e7) for a different application of the same trigonometric construct.\n",
-    "  \n",
-    "  "
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "## Trigonometric functions  \n",
-    "  \n",
-    "Since sine and cosine are defined for any real value of a given angle $x$ ($x \\in {\\rm I\\!R}$), we can define the functions:  \n",
-    "  \n",
-    "$$f(x) = \\mbox{sin}(x)$$  \n",
-    "$$f(x) = \\mbox{cos}(x).$$  \n",
-    "  \n",
-    "Thus, the association of sine and cosine with pure geometric ratios gives place for a more generic definition of functions of real variables. The graphs of these functions are given by:  \n",
-    "  \n",
-    "```{image} ./sin_cos.png\n",
-    "\n",
-    "```  \n",
-    "  \n",
-    "All the other known trigonometric relations can also be defined as functions in a similar way. Take for example the tangent of an angle $\\theta$, which is:  \n",
-    "  \n",
-    "$$tan(\\theta) = \\displaystyle\\frac{\\mbox{sin}(\\theta)}{\\mbox{cos}(\\theta)}.$$\n",
-    "\n",
-    "If $x \\in {\\rm I\\!R}$, than we can define a function  \n",
-    "  \n",
-    "$$ f(x) = \\mbox{tan}(x) = \\displaystyle\\frac{\\mbox{sin}(x)}{cos(x)}, \\ \\ \\ (\\mbox{for }\\mbox{cos}(x) \\ne 0).$$  \n",
-    "  \n",
-    "The graph from the tangent function will be given by:  \n",
-    "  \n",
-    "```{image} ./tan.png\n",
-    "\n",
-    "```  \n",
-    "  \n",
-    "Note by the above graphs that the functions sine and cosine have patterns that repeat themselves every interval of $2\\pi$, while the tangent repeats itself every interval of $\\pi$. We say that these three functions are **periodic**.\n",
-    "\n",
-    "\n",
-    "A function $f(x)$ is periodic if there is a positive constant $a$, so that:  \n",
-    "  \n",
-    "$$f(x + a) = f(x),$$  \n",
-    "  \n",
-    "for all values of $x$ for which the function f(x) is defined.  \n",
-    "  \n",
-    "If $a$ is the smallest number with this property, we call it the *period* of f(x).\n",
-    "\n",
-    "```{admonition} Example  \n",
-    "  \n",
-    "Since $\\mbox{sin}(x + 2\\pi) = \\mbox{sin}(x)$ for any value of $x$ and since $2\\pi$ is the smallest number for which this property holds, then sine is a periodic function with period $2\\pi$. With the same reasoning, it can be shown that the tangent is periodic with period $\\pi$.\n",
-    "```  \n",
-    "```{admonition} Example  \n",
-    "  \n",
-    "Many natural processes can be modelled as periodic. Think for example of a pendulum, the tides, the seasons. Often periodicity can be a good first order approximation, even if it does not strictly hold. For example when describimg the shape of the cloud cover in the picture below.\n",
-    "```\n",
-    "```{image} ./cloud.png\n",
-    "```  \n",
-    "  \n",
-    "### Amplitude and Period  \n",
-    "  \n",
-    "  \n",
-    "From the basic functions $\\mbox{sin}(x)$ or $\\mbox{cos}(x)$ we can write a more general function $f(x)$ given by  \n",
-    "  \n",
-    "$$f(x) = A\\mbox{sin}(\\omega x),\\ \\ \\ \\ \\ \\ A, \\omega \\in {\\rm I\\!R}\\ \\ (A, \\omega \\ne 0)$$  \n",
-    "  \n",
-    "Since the sine ranges from $-1$ to $1$ (see the graph for $sin(x)$ for reference), by its definition, $f(x)$ will range from $-A$ to $A$. We call $A$ the **amplitude** of $f(x)$ and it gives the extent of the variation of the oscillation of the function.  \n",
-    "  \n",
-    "Being sine a periodic function, $f(x)$ will also be, possibly with a different period. The period $T$ of $f(x)$, as discussed before, will be determined by the equation $f(x + T) = f(x)$, that should be valid for every $x$. Thus:  \n",
-    "  \n",
-    "$$\\begin{align}\n",
-    "    f(x + T) &= f(x) \\\\\n",
-    "    A\\mbox{sin}(\\omega(x+T)) &= A\\mbox{sin}(\\omega x) \\\\\n",
-    "    \\mbox{sin}(\\omega(x+T)) &= \\mbox{sin}(\\omega x) \\\\\n",
-    "    \\mbox{sin}(\\omega x+\\omega T)) &= \\mbox{sin}(\\omega x) \\\\\n",
-    "    \\mbox{sin}(z+\\omega T)) &= \\mbox{sin}(z)\\ \\ \\ \\ \\ \\ (\\mbox{where we change the variable to } z=\\omega x)\n",
-    "  \\end{align}$$\n",
-    "  \n",
-    "But since we know that sine is periodic with period $2\\pi$, we should have $|\\omega|T = 2\\pi$. Thus, the **period** of the function $f(x)$ will be:  \n",
-    "  \n",
-    "$$T = \\frac{2\\pi}{\\omega}.$$  \n",
-    "  \n",
-    "The constant $\\omega = \\displaystyle\\frac{2\\pi}{T}$ is called **angular frequency** and measures the number of full rotations the function $f(x)$ oscillates in one interval of $T$.  \n",
-    "  \n",
-    "```{admonition} Example  \n",
-    "  \n",
-    "The function $f(x) = 3.6\\, \\mbox{cos}\\left(\\displaystyle\\frac{\\pi}{3}x\\right)$ has an amplitude $A=3.6$, an angular frequency $\\omega = \\displaystyle\\frac{\\pi}{3}$, and a period $T = \\displaystyle\\frac{2\\pi}{\\omega} = \\displaystyle\\frac{2 \\pi}{\\frac{\\pi}{3}} = 6$.  \n",
-    "```"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "## Modelling complex periodic phenomena  \n",
-    "  \n",
-    "  \n",
-    "Watch this [video](https://web.microsoftstream.com/video/02f373a2-621b-4313-a18a-265f5fb65656) about periodic functions decomposition."
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "## Transformation on graphs  \n",
-    "  \n",
-    "  \n",
-    "Starting from the elementary functions we have discussed  \n",
-    "      \n",
-    "$$ax+b,\\ \\ ax^2+bx+c,\\ \\ \\sqrt{x},\\ \\ \\mbox{e}^x,\\ \\ \\mbox{log}(x),\\ \\ \\mbox{sin}(x),\\ \\ \\mbox{cos}(x)$$  \n",
-    "  \n",
-    "we can introduce several transformations to obtain new functions, by adding or multiplying constants to the argument (function variable) or to the whole function. These are the simplest cases of what is know as *function composition* with the resulting function called *composite function*.  \n",
-    "  \n",
-    "When working with mathematical models, this will help you to understand how the shape of a given function might change by changing one of the parameters, or how to construct a set of functions (that might be useful for your particular model) with small modifications of basic functions.  \n",
-    "  \n",
-    "  \n",
-    "### Translations  \n",
-    "  \n",
-    "A given function can be dislocated, either vertically or horiontally, by *adding* a constant $c$ to the whole function or to its argument. In general, starting from a function $f(x)$, translations are built following the rules:  \n",
-    "  \n",
-    "- **Horizontal** (Constant $c$ summed to the argument of the function)  \n",
-    "\n",
-    "  $y = f(x + c)\\ \\ \\ \\  -  \\left\\{\n",
-    "\\begin{array}{ll}\n",
-    "      c > 0 & \\rightarrow \\ \\  \\mbox{function moves to the left} \\\\\n",
-    "      c < 0 & \\rightarrow \\ \\  \\mbox{function moves to the right} \\\\\n",
-    "\\end{array} \n",
-    "\\right. $  \n",
-    "  \n",
-    "- **Vertical** (Constant $c$ summed to the whole function)  \n",
-    "\n",
-    "  $y = f(x) + c\\ \\ \\ \\  -  \\left\\{\n",
-    "\\begin{array}{ll}\n",
-    "      c > 0 & \\rightarrow \\ \\  \\mbox{function moves up} \\\\\n",
-    "      c < 0 & \\rightarrow \\ \\  \\mbox{function moves down} \\\\\n",
-    "\\end{array} \n",
-    "\\right. $  \n",
-    "  \n",
-    "```{admonition} Example  \n",
-    "If we start with the function $f(x) = x^3$, the function $y = f(x + 2) = (x+2)^3$ will represent a horizontal translation, while the function $y = f(x) + 2 = x^3+2$ will represent a vertical translation.  \n",
-    "```  \n",
-    "  \n",
-    "To see how this works, try to implement the following code in your own jupyter notebook. Try to use different functions $f(x)$ (defined in 'func(x)') to see how different graphs behave to horizontal and vertical translations.\n",
-    "  \n",
-    "```{code}\n",
-    "from ipywidgets import interact\n",
-    "import matplotlib.pyplot as plt\n",
-    "import numpy as np\n",
-    "\n",
-    "def func(x):\n",
-    "    return x**3\n",
-    "\n",
-    "def htrans(c):\n",
-    "    x = np.linspace(-3,3)\n",
-    "    \n",
-    "    y = func(x+c) #constant c in the argument of the function\n",
-    "    \n",
-    "    plt.plot(x,y, label='$f(x+c)$')\n",
-    "    plt.title('Horizontal translation')\n",
-    "    plt.legend(fontsize=12)\n",
-    "    \n",
-    "    plt.ylim(-8, 8) #remember to change the limits for better visualisation if necessary\n",
-    "    \n",
-    "    plt.axhline(0, color='black', lw=1)\n",
-    "    plt.axvline(0, color='black', lw=1)\n",
-    "    \n",
-    "    \n",
-    "\n",
-    "    plt.show()\n",
-    "\n",
-    "def vtrans(c):\n",
-    "    x = np.linspace(-3,3)\n",
-    "    \n",
-    "    y = func(x)+c #constant c summing the whole function\n",
-    "    \n",
-    "    plt.plot(x,y,  label='$f(x)+c$')\n",
-    "    plt.title('Vertical translation')\n",
-    "    \n",
-    "    plt.xlabel('x')\n",
-    "    plt.ylabel('y')\n",
-    "    plt.legend(fontsize=12)\n",
-    "    \n",
-    "    plt.ylim(-8, 8) #remember to change the limits for better visualisation if necessary\n",
-    "        \n",
-    "    plt.axhline(0, color='black', lw=1)\n",
-    "    plt.axvline(0, color='black', lw=1)\n",
-    "    \n",
-    "    plt.show()\n",
-    "    \n",
-    "# create a slider\n",
-    "\n",
-    "interact(htrans, c=(-2.0,2.0))\n",
-    "interact(vtrans, c=(-2.0,2.0))\n",
-    "```  \n",
-    "  \n",
-    "  \n",
-    "### Stretching/ Compression  \n",
-    "  \n",
-    "A given function can also be stretched or compressed, either vertically or horiontally, by *multiplying* a constant $c$ to the whole function or to its argument. Starting from a function $f(x)$, stretchings/compressions are built following the rules:  \n",
-    "  \n",
-    "- **Horizontal** (Constant $c$ multiplied to the argument of the function)  \n",
-    "\n",
-    "  $y = f(cx)\\ \\ \\ \\  -  \\left\\{\n",
-    "\\begin{array}{ll}\n",
-    "      c > 1 & \\rightarrow \\ \\  \\mbox{function is horizontally compressed} \\\\\n",
-    "      0 < c < 1 & \\rightarrow \\ \\  \\mbox{function is horizontally stretched} \\\\\n",
-    "\\end{array} \n",
-    "\\right. $  \n",
-    "  \n",
-    "- **Vertical** (Constant $c$ multiplied to the whole function)  \n",
-    "\n",
-    "  $y = cf(x)\\ \\ \\ \\  -  \\left\\{\n",
-    "\\begin{array}{ll}\n",
-    "      c > 1 & \\rightarrow \\ \\  \\mbox{function is vertically stretched} \\\\\n",
-    "      0 < c < 1 & \\rightarrow \\ \\  \\mbox{function is vertically compressed} \\\\\n",
-    "\\end{array} \n",
-    "\\right. $  \n",
-    "  \n",
-    "```{admonition} Example  \n",
-    "For the function $f(x) = x^3-2x$, the function $y = f(2x) = (2x)^3-2(2x) = 8x^3-4x$ will represent a horizontal compression, while the function $y = 2f(x) = 2x^3 - 4x$ will represent a vertical stretching.  \n",
-    "```  \n",
-    "```{admonition} Example  \n",
-    "One example where we stretch or compress functions comes from chemical processes. Many chemical processes happen faster when the temperature is higher. If we describe the concentration of a certain chemical as a function of time $c(t)$, increasing the temperature will lead to a horizontal compression of the function.\n",
-    "```   \n",
-    "\n",
-    "Now play with the code below to see how different graphs behave to horizontal and vertical stretching/compression.\n",
-    "  \n",
-    "```{code}  \n",
-    "from ipywidgets import interact\n",
-    "import matplotlib.pyplot as plt\n",
-    "import numpy as np\n",
-    "\n",
-    "def func(x):\n",
-    "    return x**3-2*x\n",
-    "\n",
-    "def hstetcom(c):\n",
-    "    x = np.linspace(-3,3)\n",
-    "    \n",
-    "    y = func(c*x) #constant c multiplying the argument of the function\n",
-    "    \n",
-    "    plt.plot(x,y, label='$f\\, (cx)$')\n",
-    "    plt.title('Horizontal stretching/compression')\n",
-    "        \n",
-    "    plt.xlabel('x')\n",
-    "    plt.ylabel('y')\n",
-    "    plt.legend(fontsize=12)\n",
-    "    \n",
-    "    plt.ylim(-4, 4) #remember to change the limits for better visualisation if necessary\n",
-    "    \n",
-    "    plt.axhline(0, color='black', lw=1)\n",
-    "    plt.axvline(0, color='black', lw=1)\n",
-    "    \n",
-    "    \n",
-    "\n",
-    "    plt.show()\n",
-    "\n",
-    "def vstetcom(c):\n",
-    "    x = np.linspace(-3,3)\n",
-    "    \n",
-    "    y = c*func(x) #constant c multiplying the whole function\n",
-    "    \n",
-    "    plt.plot(x,y,  label='$c\\, f\\, (x)$')\n",
-    "    plt.title('Vertical stretching/compression')\n",
-    "    \n",
-    "    plt.xlabel('x')\n",
-    "    plt.ylabel('y')\n",
-    "    plt.legend(fontsize=12)\n",
-    "    \n",
-    "    plt.ylim(-4, 4) #remember to change the limits for better visualisation if necessary\n",
-    "        \n",
-    "    plt.axhline(0, color='black', lw=1)\n",
-    "    plt.axvline(0, color='black', lw=1)\n",
-    "    \n",
-    "    plt.show()\n",
-    "    \n",
-    "# create a slider\n",
-    "\n",
-    "interact(hstetcom, c=(0.1,3.0))\n",
-    "interact(vstetcom, c=(0.1,3.0))\n",
-    "```  \n",
-    "  \n",
-    "  \n",
-    "### Reflection  \n",
-    "  \n",
-    "If the constant $c$ multiplying the whole function or its argument has the particular value $c = -1$, we will have a *reflection*. Reflections can be done in relation to the $x$ or $y$-axes and they are built, starting from a function $f(x)$, following the rules:  \n",
-    "  \n",
-    "  \n",
-    "- **About the x-axis** - $y = -f(x)$  \n",
-    "  \n",
-    "- **About the y-axis** - $y = f(-x)$  \n",
-    "\n",
-    "The figure below shows these two reflections using the function $f(x) = (x-1)^2$.  \n",
-    "  \n",
-    "```{image} ./reflection.png\n",
-    "\n",
-    "```  "
-   ]
-  }
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- "nbformat_minor": 4
-}
diff --git a/_sources/mathscourse/lecture03/03_Analysing_change,_one_step_at_a_time.ipynb b/_sources/mathscourse/lecture03/03_Analysing_change,_one_step_at_a_time.ipynb
deleted file mode 100644
index a3d7bc80..00000000
--- a/_sources/mathscourse/lecture03/03_Analysing_change,_one_step_at_a_time.ipynb
+++ /dev/null
@@ -1,1004 +0,0 @@
-{
- "cells": [
-  {
-   "cell_type": "code",
-   "execution_count": 2,
-   "metadata": {
-    "tags": [
-     "remove-cell"
-    ]
-   },
-   "outputs": [
-    {
-     "data": {
-      "image/png": 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\n",
-      "text/plain": [
-       "<Figure size 864x216 with 1 Axes>"
-      ]
-     },
-     "metadata": {
-      "needs_background": "light"
-     },
-     "output_type": "display_data"
-    },
-    {
-     "data": {
-      "image/png": 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s+82aTE2N6j776E5XonvuUd22reg/wikLF9ocnWhCYBS5PP+8a2XNs3276pQpNtpiFwnVvn1Vb7wxtU26HWzapHr99bkpDKB6+OH296S932XFCtXLL7d+xd69ix7rYiYjtk9piMiJwO+BfYCNwBvAq8DsbP6WqmZKLjABRGQbZiY3N9q+DLhH7TnZjb8zAYiWFnt2G/tAwDmzVHVUUx/GvZ/M7VjfxreBG4EPsVGl32GGs7k8jRWnkHNKE9tR1UmqOkpVR40cObL5aGnTJjSTcR6xNZveeQcdP94u/YC2a4d+4xvovHnutTWXMhl06lT0iCNy2vv3R2+7Da2ri/5X6Ux1deiNN6K9e+e0jx5tf0/a68uKFeiFF1o9AVQEPfNM9JVXSvp+8bMxhhjMYE4rsL0HtobpMtcHDGsuXV1g+2Za2lxKO+vW2US6tm1VwZpHl16qumSJa2XFmTNH9YQTdEfTol8/W9S4ZcuOXSjnVg9Jk8moTp6cG4oG1aOPtqH1tA8A1NaqXnWV6u67m+42bVTPPTf2AAAVHl16FTgjzneqnbCO3/sabRtApTp+00gmY8Poe+2Vqyzf/KYf5lJXp/q979nwOajuuad16BaYIZ06k3n/fVsuEpnLoYf6Mbqoqvr006qDB+e0n3GG6ltvlVVUpU3mq8Cjcb5T7YQNYa8DuuZt+28qPYSdFmpqVL/2tVxlOe641N8aYAfz51tnKNhEwIsuana9TqpMZsoU1Z49TXuPHrbWK+2jRao2SPH97+fqy4gRLR4AqLTJTALexW7rMDTOd6uVsMl4K4EngZOxDt3NJDUZzyVLl9rVEyzknTgx/SMXEU8+mVsGcMABqi+9VPQrqTCZTMbusBedpKedprpqlWtVpfHRR6qf/rTpbtvWJmJWYNpFpU3mTWAbNrt3O/A+8CBwJTCWlCyQxJYVPA3UZQ3npyS9rKDaLFyoOmiQ7pgEOGeOa0WlM3lyrnl01lklr9VxbjKZTG7xaJs2NpTuQ9NI1SZgjhypO+YYVXBuVEVNxsqjIzAKOB8bbXoR2BQZT9zy0pa8MJk1a2wCIKiOHu3NLQFU1foCOnQw7ZddFivycm4yP/yh6e7USfXhh91qiUNdXW4JxgEHVHzJS8VNpmAhNjw8DBhfifJcptSbTH296vHH6441L2mfTp/PokW5pQwXXxw7CnBqMn/8o+5oZjzyiDsdcclkVM85x7QPHJjIDbyqYjKtKaXeZK65xv5te+/t1x3ftm3LXU1PP72sviNnJvPuu7mlGL/+tRsN5fKHP5juzp1VX3stkZ8oZjKxnlYgIm2By4Fjs52pc8nO9lXVVXHKCpTBW2/Btdfa63vvhf793eqJw6RJ8NJL9pSGu++2h5X5wqWXwubNcNZZcMEFrtWUztq18J3v2Otbb4XDDnOjozkHapyAX2L9Lw+R6/yN0krg73HKS2NKdSTzxS/aVen8810ricf69bn7HE+ZUnYxuIhkpk0z3V272gJTn/jud037pz+daAc1RSKZuJeT8dhIUvSY2hOz2xZg81DatsjxAk0ze7Y9Q6hTJ7jmGtdq4jFxol1Vjz/ev+c5/+xnll9xhV/P0F69Gu64w15ff73TZ2XFNZne2KzfaMHCFlWdjC0qrAFuqqC2QD7RI2K/9S3o18+tljg0NMDtt9vrK67w68Fwr74Kzz4LXbvCRRe5VhOPiROhrg4+/3k48kinUuKazIdAZ7WV1h8CvQBUtQZrSl1dWXkBADZtggcesNe+VfaHH4Zly2DoUDjlFNdq4hGZ43nn2RMofUEV7rnHXl9yiVstxDeZ2cDQ7OsFWAdwxErAUc9SK2fyZKitheOOg8GDXauJx333WX7BBX519tbXw5Qp9vr8891qicsLL8CiRTYw8KlPuVYT+1nYt5AzkknALSIyB3gHWzO0ooLaAhFRFHPuuW51xKWmBqZOtddnneVWS1ymT4cNG+CQQ+Cgg1yricf991v+la9AW/fdpLFMRlWnYzfoBvgDMA64D+uj2Qb8R0XVBWDLFnjmGXv9uc85lRKbGTOsX+Coo2DAANdq4vH445b71lENMG2a5SnRHit+FZG/iMgAAFXdrqr/BhyB3bhqsKo+mIDGXZvnnrMT9fDD/RrdAPjHPyw/+WS3Osrhqacs9037e+/BwoXQvTuMavJmdVUlbiP5LGCnoQ1VfV1VHwFqROSYiikLGC+8YPlJJ7nVUQ6RyaSgXyAWq1fbxMfdd4fRo12riceMGZZ/6lPQLm5vSDIUVSEiQ7P7zSuy6xDgOcJcmcryyiuWjxnjVkdc6urgtdess/cYz649M2dafvTR0KGDWy1xibSnqL6UEsn8OzAHW0agwBUicomIHCciXfL264496C1QKVRzJnP00W61xOXNN2H7dhg2DDp3dq0mHq++avlID+8pH5nMUUe51ZFHKfHUjdgtLY/E5sIcDHwWe9BbRkTeBd7CRp3eSEjnrsn778OHH0Lv3jBwoGs18XjtNcsPP9ytjnKITMbxJLbYbNuWO+4p0l40klHVTar6D1W9AZiPLSPoipnOBOAJoBs2h+a8BLXuesyfb/mIEX7NlIVcZXe1KK8lzJ5t+RFHuNURl0WLYOtW2Hdf6NHDtZodxB3CHp739rVsCiTFO+9YPnRo8/ulkXnZLrxDDnGrIy5bt8KSJTa/xLeJjwsWWD5kiFsdjfBoCuYuyNtvW56ySlMSixZZfsABbnXEZfFi6wsbOBDat3etJh6RyaTMHIPJpJkokvHNZLZtg6VLrYk3aJBrNfF4913LfTNHyJnMgQe61dGIYDJpJqXhb1GWLrWRpf79oWNH12ri4bPJLFxoeTCZQEmoworsUrB99nGrJS7vvWf5fvu51VEOPptMdNz339+tjka0OpMRkWdERAukTq61xeLDD63Z0b27zTz1iZRW9pJYutRy35p5AKuyd8Dde2+3OhqRjnnHlecfwBWNtm11IaRsVq60PGUVpiR8jcDAlhQA7LWXWx1x2bzZUqdOqbv3TWs1mXWq+pJrES0iMhmf7oIXsWaN5b17u9VRDh98YLlvJhOZY9++qZtT1eqaS62GKBrwMZKJTlQfTSY6Wfv0casjLlFTKYUr9VuryXxGRGqzaZqIHOpaUGxCJFN9tmyBjRtt9XLPnq7VxCOYTFWZAVwKnIotexgIPCci+zb1BRGZICIzRWTmmugEcY3PfTLRMfQtGogisD59UtfkKEqKTSb1fTIi0p1G97AphKrOz+b5NzN/TkSmY2uuvpNNhb47CbudKKNGjdJC+1QdX6MB8Fe7r52+EEymhZwN/LaE/QpeelR1lYi8gC3o9IcNGyzv3t2tjrhkMvaMJYBevdxqiYvPJpNiY099c0lV71RVKZZKKSpxsZXEV5P56COb7du9u383fErxiVqUjRstT2F9Sb3JtBQR2Qt7dMss11pi4avJ+DyyFJ2oKbpNQslE2lM2Rwb8aC6VTHYU6efAA8D7WKfvD7Dndt/sUFp8fDWZSLdvozNgD9EDe2KkbwSTqRofYn0zPwf2BDYBzwBnqOoSh7ri46vJbN5seZcuze+XRiKT8VF7MJnqoKrLsWdB+c327f5eVVuDyfh2zCHVJtPq+2S8JL+yp+AJgLGoqbHcR5OJDNJnk0mh9mAyacTXphKESMYVkfYQyQRKojWYjG+PQQF/TWbrVqivt9uFpvAmYcFk0khrMJkQyVSP/P6YFC6HCCaTRnw2mdbQJ+Ob9hR3+kIwmXTis8n4eqJC64hkUkgwmTTSGqKB0CdTPVI8sgTBZNJJXZ3lu+3mVkc5+BrJqPpvMiGSCZSMzybjaxS2ZYtNguzY0b+HukXHPKXRYzCZNFJba7mPJuNrJOOrbjCDhNTWl2AyacTnSMbXk9VX3ZD6+hJMJo2kvNI0i68dv9Ex9+0ZV5CLZDql89FiwWTSiM8m42ufzNbsY7lSOGO2KMFkArHx+aoamYxv2qMT1WeTSelFKZhMGvE5kokigpReVZvEV92Qqy8p1R5MJo34ajKqtlAP/BsGDs2lxAgmk0ZSXmmaZNs2y9u3hzaeVa3W0FxKaX3xrCbsIkTRgG8VPooGfHtKAfjdXAp9MoHYRCbj28naGpocPmoPfTKB2PhqMr7qhmCQCRJMJo2EztPqk/J+jWZJefPaK5MRkfEi8qCIrBQRFZGvN7FffxH5q4hsFpG1InKbiPgzccPXiMBnk/FZe36HewrxymSAs4B9gUeb2kFE2gHTgEHAeOBS7Hnak6qgrzL4ajK+6ga/O35Tftx9e+7SeFXNiEgX4Lwm9jkbOAgYrKrvAYjINuB+EblGVRdUSWv5pLzSNInP0UDUXPLtmEOIZCqJqmZK2G0s8K/IYLI8BNQDn01EWKXx3WR80w25E9Vn7cFkqsYwYH7+BlWtB97NfpZ+fO34TXkHZLM0NFjezrfgntSbjKiqaw2xyTaXNgHfUNXfNfpsAfB3Vf1Oo+3PA0tU9T8KlDcBmJB9OxRYBWzI26V7o/e9gLXNfN7S9/nlJ1l23O8nWXbc8tN0XBq/d3lcimlJ4rj0UNXeNIWqOktZgcOKpQLf6wIo8PUCny0Abiqw/QXg3hJ1TSryfmbM/eO+n1mNsuN+PxyXso9T1Y5LGf+zih6XQsl1bHg28NsS9ovzxKr1QI8C23sAH5VYxiNF3sfdP+77apUd9/vhuJT23uVxKaYl6ePycYq5UBoTzUcy9wD/bLStA1AHXFyh359ZiXJclO9r2T5r39WPS2vs+H0MOEpEBuVt+wLQEXi8Qr+R9JybJMv3teyky/e17KTLb3HZXnX8ishwYDjQCfgDcDvwDLBGVWdk92kPzAa2Aldh/T43AdNV9asOZAcCuzS+mcyPgasLfDRDVU/M228f4DbgZMxs7gcuV9XaKsgMBAJ5eGUygUDAP1pjn0wgEEgRwWRKRETOFpG/icjy7OruWSLyZde60oaI7CMiq7LH6TURWSYiq0Xk82WWN1xEnhKRWhFZISI/EZG2adTqI9Wo167nyfjEZcB7wH9hMyDHAX8SkV6qemslfkBEZmDzeQR4B/imqm6sRNnVQlWXicijWEf7/SJyPTbTuvh8ikaISE9gOvAWcDpwAHADdnG8Mk1aq0FC9SPxep3Y2H1rS0CvAtv+BLxXwd/onvf6RuCnCf4907G5Rk2lQc1893lgcYF0V/bzV4Dh2ddPAMeXqfEH2OTKbnnbvgfU5m9r4XGoiNZKHt9q1o9q1OvQXCoRVV1bYPNsoE/+BhGZnr2hVlNpUIFyot/YkC2jDdAZq4xJhfX/jZ3AtwOfyKa7gRXAGFV9vxmdx6nqvgXSf2a1D8autACHA2+UqXEsME13vlrfD+wGnFBmmTuosNbGlH18m6JQ/Whp3SilXrekTkc/ElL5V4G/YreVyN92OLAOG0Ifk013AcuB0SWUORVYg83/6ZK3/U7g37Ovrwe+3ULtPTATOz1v2wPAYy0sdygwO/u6D7C8BWV9APy4wPYabEpCS/9/FdNaxeP7sfqRQN3YqV63tE6HSKZMROTTWD/B7Y0+Wgz0BJ5U1ZdU9SWgG3aFvEFEFhdId0VfVtVxQF8sjL8or9xDyV1lP3bFFZHuIjKsWMr7ysHZ/M1GvzE3/tHYiUOBOXnv24nIWWWW1ZPC683WZz9rKZXU2phEjm8T9aPZuhGHJur1Ypqo06r6cimid8lEmSvAs9/dF1gN/LXAZ8dhV7DBedveBn4ZU98IYG72dRvsStIu+/4DbHl9/v7n0XwfgNq/e8f+F2IRQZvs+92A7cC5rv83eRq3AZcW2L4c+JlrfUW0J3p8o/pRSt2IUWbBet3SOr0rjy6VtQJcRPbA1kctAQotUzgY65hclN1/N6zd3+wVLDuS0kFVV2c3fSnvOwcC76tqg4j0Abap6k5XeFW9EwubS2UEME9zdxscgVXYlkYylaSpFfXdKX1FvSsqenybqR9F60aJ5TdXr8uq0xG7bHNJVe9UVSmW8r+TfeLBo9iq7tNUtaZA0eVWrp7A30XkDRGZg/1jL81+lkRYfzA7h/IHAxlsuDgtzKfR3QxFZADW6Tm/4DfSQ6WPb1P1o8V1o4R63SLD3JUjmVhkn4LwAHblOFZVP2hi17Iql6ouAt/9byoAAANUSURBVEY18dkD2d8m+7t7xRJfmBFYJ2LEPlh43+w6ExFp8TqUxubdDI8Bl4tIV1XdlN02Hrttx4yW6ij2t8TQWYiyjm8zWpqqHy2qGyXW6xYZZjCZ0rkDm6h0KbCHiIzJ+2y2qmbvol3ZypUE2bC6NztfiV7Fmob/Ag5p6rstPPHiMhG4BHhQRH4B7A/8GLhRKzBJMam/pSXH1wGl1OuW1WnXHWS+JKyHvakO1X2z+/TJvh+b971x2P2I57j+GxI6LgOAp4B52NXuerILbytU/nDgaSx6WQn8FGjrg3YfUrF6XYk6HVZhV4lKNDOaQqsbXeyEiPQD+qvqTBHpADwJ/EpVp7jSVCo+aW9J/XFZPyA0l6qG6390UqjqSizCQFXrReQNLEJIPT5p97n+7LKjS2lDRAZkVxvPE5E3ReR6EfGqYonInsAZ2GOCvcIn7b7VlWAy6aEB+L6qHgQcAYwGznQrqXREpCMwGbhZVee51hMHD7V7VVdCcykl+BS6N0bs/i73YqMRN7jWEwcftftWV0LHbwrJhu6vAZ/x4coqIncCbbH7m3hVoXzWDn7UlWAyKSMbuj8OPOrDlVVEjsXuLzMXW5sDcLeq/sqdqtLwWTv4U1eCyaSIbOj+Z+zubJe51hNILz7VlWAyKcL30D1QPXyqK8FkUoLvoXugevhWV4LJBAKBRAnzZAKBQKIEkwkEAokSTCYQCCRKMJlAIJAowWQCgUCiBJMJBAKJEkwmEAgkSjCZQCCQKMFkAoFAogSTCaQWERksIttE5JpG238tIptEpOAjZALpIphMILWo6kLsqZj/JSK9AETkR8A3gS+q6kyX+gKlEdYuBVKNiPQF3sWeDzQfmAR8WVX/4lRYoGTC7TcDqUZVV4nIzcB3sfp6STAYvwjNpYAPLAA6Av9U1dtdiwnEI5hMINWIyEnAb4B/AseKyGGOJQViEkwmkFpE5EjgIazz90RgCfD/XGoKxCeYTCCViMhg4DHgCeBiVa0HrgHGicjxTsUFYhFGlwKpIzui9CIWuZyqqluz29tit5xcr6rHOJQYiEEwmUAgkCihuRQIBBIlmEwgEEiUYDKBQCBRgskEAoFECSYTCAQSJZhMIBBIlGAygUAgUYLJBAKBRPk/8lj8aEr9mNIAAAAASUVORK5CYII=\n",
-      "text/plain": [
-       "<Figure size 288x216 with 1 Axes>"
-      ]
-     },
-     "metadata": {
-      "needs_background": "light"
-     },
-     "output_type": "display_data"
-    }
-   ],
-   "source": [
-    "#Codes for figures\n",
-    "\n",
-    "import numpy as np\n",
-    "import pandas as pd\n",
-    "import matplotlib.pyplot as plt\n",
-    "\n",
-    "def multiple_formatter(denominator=2, number=np.pi, latex='\\pi'):\n",
-    "    def gcd(a, b):\n",
-    "        while b:\n",
-    "            a, b = b, a%b\n",
-    "        return a\n",
-    "    def _multiple_formatter(x, pos):\n",
-    "        den = denominator\n",
-    "        num = np.int(np.rint(den*x/number))\n",
-    "        com = gcd(num,den)\n",
-    "        (num,den) = (int(num/com),int(den/com))\n",
-    "        if den==1:\n",
-    "            if num==0:\n",
-    "                return r'$0$'\n",
-    "            if num==1:\n",
-    "                return r'$%s$'%latex\n",
-    "            elif num==-1:\n",
-    "                return r'$-%s$'%latex\n",
-    "            else:\n",
-    "                return r'$%s%s$'%(num,latex)\n",
-    "        else:\n",
-    "            if num==1:\n",
-    "                return r'$\\frac{%s}{%s}$'%(latex,den)\n",
-    "            elif num==-1:\n",
-    "                return r'$\\frac{-%s}{%s}$'%(latex,den)\n",
-    "            else:\n",
-    "                return r'$\\frac{%s%s}{%s}$'%(num,latex,den)\n",
-    "    return _multiple_formatter\n",
-    "\n",
-    "class Multiple:\n",
-    "    def __init__(self, denominator=2, number=np.pi, latex='\\pi'):\n",
-    "        self.denominator = denominator\n",
-    "        self.number = number\n",
-    "        self.latex = latex\n",
-    "\n",
-    "    def locator(self):\n",
-    "        return plt.MultipleLocator(self.number / self.denominator)\n",
-    "\n",
-    "    def formatter(self):\n",
-    "        return plt.FuncFormatter(multiple_formatter(self.denominator, self.number, self.latex))\n",
-    "\n",
-    "xt=np.linspace(-4*np.pi,4*np.pi,num=500)\n",
-    "##\n",
-    "fig, ax = plt.subplots(figsize=(12, 3))\n",
-    "ax.plot(xt, np.sin(xt), color='b', lw=2, label='$sin(x)$')\n",
-    "ax.plot(xt, np.cos(xt), color='g', lw=2, label='$cos(x)$')\n",
-    "ax.tick_params(axis='both', labelsize= 15)\n",
-    "ax.set_xlabel('$x$', fontsize = 16)\n",
-    "ax.legend(fontsize=14)\n",
-    "\n",
-    "ax.axhline(0, color='black', lw=1)\n",
-    "ax.axvline(0, color='black', lw=1)\n",
-    "\n",
-    "ax.xaxis.set_major_locator(plt.MultipleLocator(np.pi / 2))\n",
-    "ax.xaxis.set_minor_locator(plt.MultipleLocator(np.pi / 12))\n",
-    "ax.xaxis.set_major_formatter(plt.FuncFormatter(multiple_formatter()))\n",
-    "\n",
-    "fig.tight_layout()\n",
-    "fig.savefig('sin_cos.png')\n",
-    "\n",
-    "##\n",
-    "xt=np.linspace(-2*np.pi,2*np.pi,num=500)\n",
-    "fig, ax = plt.subplots(figsize=(4, 3))\n",
-    "y = np.tan(xt)\n",
-    "y[:-1][np.diff(y) < 0] = np.nan\n",
-    "\n",
-    "ax.plot(xt,y, color ='r', lw=2)\n",
-    "ax.tick_params(axis='both', labelsize= 15)\n",
-    "\n",
-    "ax.set_xlabel('$x$', fontsize = 16)\n",
-    "ax.set_ylabel(\"$tan(x)$\", fontsize = 16)\n",
-    "\n",
-    "plt.ylim(-10,10)\n",
-    "ax.axhline(0, color='black', lw=1)\n",
-    "ax.axvline(0, color='black', lw=1)\n",
-    "\n",
-    "ax.xaxis.set_major_locator(plt.MultipleLocator(np.pi / 2))\n",
-    "ax.xaxis.set_minor_locator(plt.MultipleLocator(np.pi / 12))\n",
-    "ax.xaxis.set_major_formatter(plt.FuncFormatter(multiple_formatter()))\n",
-    "\n",
-    "fig.tight_layout()\n",
-    "fig.savefig('tan.png')"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 3,
-   "metadata": {
-    "tags": [
-     "remove-cell"
-    ]
-   },
-   "outputs": [
-    {
-     "data": {
-      "application/vnd.jupyter.widget-view+json": {
-       "model_id": "a785b675781e44dcb130fe4aba0b9568",
-       "version_major": 2,
-       "version_minor": 0
-      },
-      "text/plain": [
-       "interactive(children=(FloatSlider(value=0.0, description='c', max=2.0, min=-2.0), Output()), _dom_classes=('wi…"
-      ]
-     },
-     "metadata": {},
-     "output_type": "display_data"
-    },
-    {
-     "data": {
-      "application/vnd.jupyter.widget-view+json": {
-       "model_id": "b9af681dfe384b9fbd3b14793a2feede",
-       "version_major": 2,
-       "version_minor": 0
-      },
-      "text/plain": [
-       "interactive(children=(FloatSlider(value=0.0, description='c', max=2.0, min=-2.0), Output()), _dom_classes=('wi…"
-      ]
-     },
-     "metadata": {},
-     "output_type": "display_data"
-    },
-    {
-     "data": {
-      "text/plain": [
-       "<function __main__.vtrans(c)>"
-      ]
-     },
-     "execution_count": 3,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "from ipywidgets import interact\n",
-    "import matplotlib.pyplot as plt\n",
-    "import numpy as np\n",
-    "\n",
-    "def func(x):\n",
-    "    return x**3\n",
-    "\n",
-    "def htrans(c):\n",
-    "    x = np.linspace(-3,3)\n",
-    "    \n",
-    "    y = func(x+c) #constant c in the argument of the function\n",
-    "    \n",
-    "    plt.plot(x,y, label='$f\\, (x+c)$')\n",
-    "    plt.title('Horizontal translation')\n",
-    "        \n",
-    "    plt.xlabel('x')\n",
-    "    plt.ylabel('y')\n",
-    "    plt.legend(fontsize=12)\n",
-    "    \n",
-    "    plt.ylim(-8, 8) #remember to change the limits for better visualisation if necessary\n",
-    "    \n",
-    "    plt.axhline(0, color='black', lw=1)\n",
-    "    plt.axvline(0, color='black', lw=1)\n",
-    "    \n",
-    "    \n",
-    "\n",
-    "    plt.show()\n",
-    "\n",
-    "def vtrans(c):\n",
-    "    x = np.linspace(-3,3)\n",
-    "    \n",
-    "    y = func(x)+c #constant c summing the whole function\n",
-    "    \n",
-    "    plt.plot(x,y,  label='$f\\, (x)+c$')\n",
-    "    plt.title('Vertical translation')\n",
-    "    \n",
-    "    plt.xlabel('x')\n",
-    "    plt.ylabel('y')\n",
-    "    plt.legend(fontsize=12)\n",
-    "    \n",
-    "    plt.ylim(-8, 8) #remember to change the limits for better visualisation if necessary\n",
-    "        \n",
-    "    plt.axhline(0, color='black', lw=1)\n",
-    "    plt.axvline(0, color='black', lw=1)\n",
-    "    \n",
-    "    plt.show()\n",
-    "    \n",
-    "# create a slider\n",
-    "\n",
-    "interact(htrans, c=(-2.0,2.0))\n",
-    "interact(vtrans, c=(-2.0,2.0))"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 6,
-   "metadata": {
-    "tags": [
-     "remove-cell"
-    ]
-   },
-   "outputs": [
-    {
-     "data": {
-      "application/vnd.jupyter.widget-view+json": {
-       "model_id": "12d5a3f30eab4d5395e543cb666f5e70",
-       "version_major": 2,
-       "version_minor": 0
-      },
-      "text/plain": [
-       "interactive(children=(FloatSlider(value=1.05, description='c', max=2.0, min=0.1), Output()), _dom_classes=('wi…"
-      ]
-     },
-     "metadata": {},
-     "output_type": "display_data"
-    },
-    {
-     "data": {
-      "application/vnd.jupyter.widget-view+json": {
-       "model_id": "bd6164c6fecf4c72b93b1a39f7947633",
-       "version_major": 2,
-       "version_minor": 0
-      },
-      "text/plain": [
-       "interactive(children=(FloatSlider(value=1.05, description='c', max=2.0, min=0.1), Output()), _dom_classes=('wi…"
-      ]
-     },
-     "metadata": {},
-     "output_type": "display_data"
-    },
-    {
-     "data": {
-      "text/plain": [
-       "<function __main__.vstetcom(c)>"
-      ]
-     },
-     "execution_count": 6,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "from ipywidgets import interact\n",
-    "import matplotlib.pyplot as plt\n",
-    "import numpy as np\n",
-    "\n",
-    "def func(x):\n",
-    "    return x**3-2*x\n",
-    "\n",
-    "def hstetcom(c):\n",
-    "    x = np.linspace(-3,3)\n",
-    "    \n",
-    "    y = func(c*x) #constant c multiplying the argument of the function\n",
-    "    \n",
-    "    plt.plot(x,y, label='$f\\, (cx)$')\n",
-    "    plt.title('Horizontal stretching/compression')\n",
-    "        \n",
-    "    plt.xlabel('x')\n",
-    "    plt.ylabel('y')\n",
-    "    plt.legend(fontsize=12)\n",
-    "    \n",
-    "    plt.ylim(-4, 4) #remember to change the limits for better visualisation if necessary\n",
-    "    \n",
-    "    plt.axhline(0, color='black', lw=1)\n",
-    "    plt.axvline(0, color='black', lw=1)\n",
-    "    \n",
-    "    \n",
-    "\n",
-    "    plt.show()\n",
-    "\n",
-    "def vstetcom(c):\n",
-    "    x = np.linspace(-3,3)\n",
-    "    \n",
-    "    y = c*func(x) #constant c multiplying the whole function\n",
-    "    \n",
-    "    plt.plot(x,y,  label='$c\\, f\\, (x)$')\n",
-    "    plt.title('Vertical stretching/compression')\n",
-    "    \n",
-    "    plt.xlabel('x')\n",
-    "    plt.ylabel('y')\n",
-    "    plt.legend(fontsize=12)\n",
-    "    \n",
-    "    plt.ylim(-4, 4) #remember to change the limits for better visualisation if necessary\n",
-    "        \n",
-    "    plt.axhline(0, color='black', lw=1)\n",
-    "    plt.axvline(0, color='black', lw=1)\n",
-    "    \n",
-    "    plt.show()\n",
-    "    \n",
-    "# create a slider\n",
-    "\n",
-    "interact(hstetcom, c=(0.1,2.0))\n",
-    "interact(vstetcom, c=(0.1,2.0))"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 48,
-   "metadata": {
-    "scrolled": true,
-    "tags": [
-     "remove-cell"
-    ]
-   },
-   "outputs": [
-    {
-     "data": {
-      "image/png": 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\n",
-      "text/plain": [
-       "<Figure size 720x360 with 2 Axes>"
-      ]
-     },
-     "metadata": {
-      "needs_background": "light"
-     },
-     "output_type": "display_data"
-    }
-   ],
-   "source": [
-    "fig, ax = plt.subplots(1,2,figsize=(10, 5))\n",
-    "\n",
-    "xt = np.linspace(-3,5)\n",
-    "\n",
-    "# x-axis reflection\n",
-    "ax[0].plot(xt, (xt-1)**2, color ='r', lw=2, label = '$(x-1)^2$')\n",
-    "ax[0].plot(xt, -(xt-1)**2, color ='b', lw=2, label = '$-(x-1)^2$')\n",
-    "ax[0].tick_params(axis='both', labelsize= 12)\n",
-    "\n",
-    "ax[0].set_xlabel('$x$', fontsize = 14)\n",
-    "ax[0].set_ylabel(\"$y$\", fontsize = 14)\n",
-    "\n",
-    "ax[0].set_title('x-axis reflection')\n",
-    "\n",
-    "ax[0].set_ylim(-10,10)\n",
-    "\n",
-    "ax[0].legend()\n",
-    "\n",
-    "ax[0].axhline(0, color='black', lw=1)\n",
-    "ax[0].axvline(0, color='black', lw=1)\n",
-    "\n",
-    "# y-axis reflection\n",
-    "\n",
-    "xt = np.linspace(-4,4)\n",
-    "\n",
-    "ax[1].plot(xt, (xt-1)**2, color ='r', lw=2, label = '$(x-1)^2$')\n",
-    "ax[1].plot(xt, (-xt-1)**2, color ='b', lw=2,  label = '$[(-x)-1]^2$')\n",
-    "ax[1].tick_params(axis='both', labelsize= 12)\n",
-    "\n",
-    "ax[1].set_xlabel('$x$', fontsize = 14)\n",
-    "#ax[1].set_ylabel(\"$y$\", fontsize = 14)\n",
-    "\n",
-    "ax[1].set_title('y-axis reflection')\n",
-    "\n",
-    "ax[1].set_ylim(-1,10)\n",
-    "\n",
-    "ax[1].legend()\n",
-    "\n",
-    "ax[1].axhline(0, color='black', lw=1)\n",
-    "ax[1].axvline(0, color='black', lw=1)\n",
-    "\n",
-    "fig.tight_layout()\n",
-    "fig.savefig('reflection.png')"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 91,
-   "metadata": {
-    "tags": [
-     "remove-cell"
-    ]
-   },
-   "outputs": [
-    {
-     "data": {
-      "image/png": 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ufMIYcyLzc0TSy1rLwXjb5VPCXz/jRenu2fn8u2WzWLGwkMljcl3PDJzS0lLXE8QH2RaoLcB04KpAAZszvEUkLay1HIifT0apmWPJKJXPyedz93pRmpSnKIlcKasCZa39znWe+1Ymt4gMhLWW/SfPUxuLsyMW53hrB0OHGMpn5/OFe2ezYuEU8hWltNmzZw/V1dWuZ0iaZVWgRILMWsu+t5JR2hfnjWSUls7J54v3zWH5AkXJL+vWrXM9QXygQIkMgLWW2FvnkkdKzbx59t0o/eV9c1i+sJCJo3Nczwy9zZs388gjj7ieIWk24EAZY9ZZazelYYtIIFhr2fvHc973lPbFOXH2IsOGGJbOncSX7p/D8gWFTFCUMio3V0emYZSOI6ivAptSHzDGfNVa+78G+sLGmJ3W2gcG+joiA2Wt5dVLUYrF+ePbXpSWzZ3EV+6fx0MLpihKDq1atcr1BPHBTQfKGPM1YCVQYIz5JLDHWnsk+fQXgQEHCpiYhtcQuSnWWl458c7ls+/eeseL0j3zJvHVB+axfMEUxo9SlLLBli1bdB1UCA3kCKoGeAXv1PDlwN8aY6YBp4F4GraBbjArGWat5eUT71C3N86OfV6Uhg813DN3El97cB7LFxQybtRw1zPlCuXl5a4niA8+MFDGmKHW2sSVj1trO4AGY8xKa+0ryc8dDswA3rjZQcaYY3hhMkDEGPN68vfWWnu9WyGJ3JS+vmSUkqeEnzzXyfChhnvnTeZvHirmoQVTGDdSUcpm7e3trieID/pzBNVhjDkAvArsvfRPa+1pgEtxSv6+Bzg6kEHW2lmXfm+Medlau2QgrydyLV6U3qZ2bzM79sWJn+skZ+gQPlw8iW+sKOHBBVMYO0JRCopYLMbq1atdz5A060+g/g5YBHwI+BQwFLDGmBa8WKWG69C1jrYGQG/xSdr09Vn+8Obb1MbiPL6vOSVKk/nmwyU8MF9RCqr169e7niA++MBAWWv/8dLvjTG5wG3A7Sm/1gPjkp/SBYxK474DaXwtGYT6+iyNb7ztvX23L07L+S5yhg3hPkUpVGpqanSSRAjd0EkS1touYE/yFwDGmDuALwOfBtJ6MYK19tPpfD0ZHBJ9lsbjZ5NRauZUmxeliuLJVC2K8JFbCxijKIVKfn6+6wnigxs+i88YkwM8AFQDq4BpwB+BnwPb0rpOpJ8SfZaXUqJ0uq2L3GFDqCiZTGU0wgPzp5CXqxunhFVFRYXrCeKDfv2NNcYU4MWoGngQGAnsBn4CbLfW7k3nKN2dQvoj0WfZfewsO/a9N0r3lxRQmTxSUpQGh61btxKNRl3PkDTrz2nmvwPKgDbgSeBLQJ219oyPu3y7O4UEW6LP8vtjrdTF4jy+r4Uz7e+N0gO3FjBaURp0dAQVTv35m3wn0In39l0j8CaQzjP1LsvQ3SkkYHoTfexOvn33+L5mzrR3M2L4ED5yawGV0Qj3lyhKg93JkyddTxAf9Odv9d8DUaAc7zRzAIwxb/HeU8xfBZqstQM5NTwTd6eQAOhN9LH72Fm2x+I8sa+Z1gvdjBw+9N0o3TqZUTmKkniamppcTxAf9Oc08w2Xfm+MGY0Xq0V4p5gvwnvLb2zyUzqAvOu93vvdmSL5tXy5O4UEQ2+ij98fO0vtlVGaX0BV8khpZM5Q1zMlC+k6qHC60dPMLwC/S/66zBhThBes/nyX8rp3pkh+nbTenUKyV2+ij12vt1IXa+aJ/c2cvdDNqBzvSKkqGqFCUZJ+0HVQ4ZSW90istceB48Bv+vHpLu9MIVmgJ9HHrqPeiQ5P7G/m7Y4eRuUM5YH5U6iKFlJRUsCI4YqS9F8kEnE9QXyQ8TfxHd+ZQhzpSfTx4tFW6vbGeeJAM+909DA6GaXKaISKksmKkty0srIy1xPEB06/y5zpO1NIZvUk+njhyBnqYnHqD7TwTkcPebnDeGC+d6LDfcWKkqTHtm3bKC0tdT1D0sz5aVC6M0W4dPf28cLRM9Tt9aJ07qIXpQeTUfqwoiQ+WL58uesJ4gMngcr0nSnEX9293pFSbSxO/f5mznf2MiZ3GA8u8N6+u3feJEVJfNXU1MTSpUtdz5A0y3igjDG/B0rJ7J0pJM26e/t4/shpavc28+SBZJRGDOOhBVOovC3CvcWTyB2mKElmHD9+3PUE8YGLI6gPkaE7U0h6dfUmeP4170jpyQMttCWjtHxBIVWLClk2V1ESN3QdVDi5CFQm70whA9TVm+C3Td6JDk8e9KI0dsQwViwspCoaYdncSeQMG+J6pgxyug4qnFycZr7h0u/TcWcKSb/OngTPNZ1mx75mnjrQQltXL+NGDufhhYVULoqwbI6iJNmlqKjI9QTxQSYDZa58IE13ppA06OxJ8GzTaepicXYePEV7Mkoro4VURiMsVZQkixUXF7ueID7IWKCstf3+v9sN3plCblJnT4KGw5ei1MKF7gTjRw2nKhqhclGEpXPyGT5UUZLsV19fr7P4Qsj5dVCSWV6UTlEba+bpZJQmjBpO9e1TqYxGKFeUJICqq6tdTxAfKFCDwMXuS1GK8/ShU3R0J5g4OoePLvaidPdsRUmCrbGxUXeSCCEFKqQudid4JhmlZ1Ki9LHF06iKRrh79kSGKUoSEvG4flxcGClQIdLR3cszh7zvKT196BQXexLkj87h40u8KN01S1GScNJ1UOGkQAVcR3cvTx86dTlKnT195I/O4RN3eFG6U1GSQUDXQYWTAhVAF7rejdIzh70oTcrL5U9Lb6EyGuGuWfkMHXLVWf0ioaXTzMNJgQqIC1297Dx0irq9cRqa3o3SvymdTmXySElRksFq6tSprieIDxSoLNbe1cvOgy3UxeI0HD5NV28fk8fk8mdl06mKRigrUpREABoaGqioqHA9Q9JMgcoybZ097DzonX33bNNpunv7KBiTy5o7Z1AZjVA6c4KiJHKF1atXu54gPlCgskBbZw9PHWyhdm8zz73mRWnK2FzW3jmDqkURSmdMYIiiJPK+GhoaiEZ1Z7SwUaAcOd/Zw85LUWo6TXeij8KxI/jUXTOoika4Q1ES6bfW1lbXE8QHClQGnbvYw1MHvO8p/fa1M+9G6e4ZrFoUYcl0RUnkZug6qHBSoHx2KUq1sTi/fe00PQnL1HEj+Ez5TCqjhYqSSBroOqhwCmWgjDEfBr6B96PlpwKftdZuusbn/QS4aK39m+THTwP3J5/uxbuj+j9Za392I1//XEcP9QeaqYvFef7IGXoSlmnjR/Jvy4uoXBRh8S3jFSWRNNL3n8IplIHC+yGH+4B/Tv66ijHGANWk/FRfYAnwCPC/gVzg88BPjTGN1tqXr/cFE32WRxtPUBeL80JKlNYtLaIyGmHx9PF4X1JE0i0vTz/XNIxCGShrbR1QB2CM2fQ+n/YhYATwfPLz5gDjgcettc3Jx34KfBdYCFw3UAfj5/nmv+xl2viRfHbZLCqjEW6/ZZyiJJIBu3btYsWKFa5nSJqFMlD99HGg1lrbm/y4FDgP7AUwxhQC/wPoA/5wrRcwxqwH1gNMmDaL33xpGYsUJZGMW7NmjesJ4oPBfBfRjwGPpXxcivfW4DljTAcQBz4BfN1ae+BaL2CtrbHWlllry2YXTuR2vY0n4sT27dtdTxAfDMpAGWPmArOBJ1IeLgV+DiwG7kk+9zNr7fczv1BEbkRXV5frCeKDQRkovLf3dlprL6Q8tgR40Vp7xFr7B+CLwF8ZY3R6kEiWW7t2resJ4oPBGqj3vL1njJkFTARilx6z1r6Bd2LEZzK+TkRuyKZNm1xPEB+EMlDGmDxjzGJjzGK8P+OM5MczjDGTgbuBbSn/SineyRAHr3ipJ4E/ychoEblppaWlrieID0IZKKAM7+jnZWAksDH5+7/Hu/bpJWttS8rnlwJHrbUXr3idJ4G5xpiF/k8WEZFUoQyUtbbBWmuu8WsdV5+9h7X229baq34kp7V2Z/Lf25+h6SJyE/bs2eN6gvgglIH6AC8AW1yPEJH0WbdunesJ4oNBFyhr7T9Za0+43iEi6bN582bXE8QHgy5QIhI+ubm5rieIDxQoEQm8VatWuZ4gPlCgRCTwtmzRt5XDSIESkcArLy93PUF8oECJSOC1t7e7niA+UKBEJPBisdgHf5IEjgIlIoG3fv161xPEBwqUiAReTU2N6wniAwVKRAIvPz/f9QTxgQIlIoFXUVHheoL4QIESkcDbunWr6wniAwVKRAJPR1DhpECJSOCdPHnS9QTxgQIlIoHX1NTkeoL4QIESkcDTdVDhpECJSODpOqhwUqBEJPAikYjrCeIDBUpEAq+srMz1BPGBAiUigbdt2zbXE8QHCpSIBN7y5ctdTxAfKFAiEng6zTycFCgRCbzjx4+7niA+UKBEJPB0HVQ4KVAiEni6DiqcFCgRCbyioiLXE8QHCpSIBF5xcbHrCeIDBUpEAq++vt71BPGBAiUigVddXe16gvhAgRKRwGtsbHQ9QXygQIlI4MXjcdcTxAcKlIgEnq6DCicFSkQCT9dBhZMCJSKBp9PMw0mBEpHAmzp1qusJ4gMFSkQCr6GhwfUE8YECJSKBt3r1atcTxAcKlIgEno6gwkmBEpHAa21tdT1BfKBAiUjg6TqocFKgRCTwdB1UOClQIhJ40WjU9QTxgQIlIoGXl5fneoL4QIESkcDbtWuX6wniAwVKRAJvzZo1rieIDxQoEQm87du3u54gPlCgRCTwurq6XE8QHyhQIhJ4a9eudT1BfKBAiUjgbdq0yfUE8YECJSKBV1pa6nqC+ECBEhGRrKRAiUjg7dmzx/UE8YECJSKBt27dOtcTxAcKlIgE3ubNm11PEB8oUCISeLm5ua4niA8UKBEJvFWrVrmeID5QoEQk8LZs2eJ6gvhAgRKRwCsvL3c9QXygQIlI4LW3t7ueID5QoEQk8GKxmOsJ4gMFSkQCb/369a4niA8UKBEJvJqaGtcTxAcKlIgEXn5+vusJ4gMFSkQCr6KiwvUE8YECJSKBt3XrVtcTxAfGWut6QygYY9qAw653iAxSk4AzrkfITZtprZ185YPDXCwJqcPW2jLXI0QGI2NMo/7+hY/e4hMRkaykQImISFZSoNJHF2KIuKO/fyGkkyRERCQr6QhKRESykgIlIiJZSYESEZGspEClgTHmr4wxx4wxncaYPcaYe11vEgk7Y8wmY8z2Kx5bZYzpMMb8F1e7JH0UqAEyxnwS+AHwX4ElwIvADmPMDKfDRAYZY8xngK3At62133G9RwZOgRq4fw9sstb+zFp70Fr7FSAO/KXjXSKDhjHmr4GfA5+31v7A9R5JDwVqAIwxOUApUH/FU/XA0swvEhl8jDH/APw34BPW2v/jeo+kj+7FNzCTgKFAyxWPtwAPZn6OyKDzEFAFrLLW1roeI+mlI6j0uPJqZ3ONx0Qk/fYBR4G/M8aMdz1G0kuBGpgzQAIovOLxAq4+qhKR9IsD9wHjgKeMMRMc75E0UqAGwFrbDezBe5sh1UN4Z/OJiM+stW8BFcBoYKcxRj//PSQUqIH7n8A6Y8znjTHzjTE/AKYCP3G8S2TQsNbG8SKVAzxtjJnkdpGkgwI1QNbaXwNfA74LvALcA1Raa99wOkxkkLHWtgD3Jz98xhhT4HKPDJzuZi4iIllJR1AiIpKVFCgREclKCpSIiGQlBUpERLKSAiUiIllJgRIRkaykQImISFZSoEREJCspUCIhYoz5gjHGGmPajDEjrniuMPnc113tE7kRCpRIuCwGuoA8rv6ZZEuS/3w5o4tEbpICJRIui4HdwKvAx6947lKgXsnoIpGbpECJhIQxxgCL8AL0GFBtjEn9O74YeNNae9bFPpEbpUCJhMc8vLf2XgZ+g/eDM5emPL8Evb0nAaJAiYTH4uQ/X7HWvgy8QfJtPmPMGGAOCpQEiAIlEh5LgB5gf/Ljf+Xd70PdDhiS338yxgwxxnw3+bagSFZSoETCYzFw0Frbnfz4MWCOMSbK1WfwRYFPWv1AOMliCpRIeCzmvW/hPQe8jXcUtQQ4a6190xizAKgDphhjXjHG/GPmp4p8sGGuB4jIwBljpgCFpJxCbq3tNcbU4gXq8tt71toDxpj/Bxy11n7fxV6R/tARlEg4vN81Tr8B7sB7S2qseAYAAACJSURBVC/16KoUaMzALpGbpkCJhMPlM/iuePxxoBPv3ZJLJ0gMBW5DZ/RJljP6HqnI4GKMuQX4g7W2wPUWkevREZTI4BMHGo0xh40x33M9RuT96AhKRESyko6gREQkKylQIiKSlRQoERHJSgqUiIhkJQVKRESykgIlIiJZSYESEZGspECJiEhW+v/DItPKXsV5vgAAAABJRU5ErkJggg==\n",
-      "text/plain": [
-       "<Figure size 432x288 with 1 Axes>"
-      ]
-     },
-     "metadata": {
-      "needs_background": "light"
-     },
-     "output_type": "display_data"
-    }
-   ],
-   "source": [
-    "import matplotlib.pyplot as plt\n",
-    "import numpy as np\n",
-    "    \n",
-    "time = np.linspace(0,15)\n",
-    "rat = 0.5 + ((1-0.5)/10)*time\n",
-    "\n",
-    "fig, ax = plt.subplots(figsize=(6, 4))\n",
-    "\n",
-    "\n",
-    "ax.plot(time,rat)\n",
-    "    \n",
-    "ax.set_xlabel('$N_t$',fontsize = 16)\n",
-    "ax.set_ylabel(r'$\\frac{N_t}{N_{t+1}}$',fontsize = 22)\n",
-    "\n",
-    "ax.set_yticks([0.5,1])\n",
-    "ax.set_yticklabels(['$1/R$','1'],fontsize=14)\n",
-    "\n",
-    "ax.set_xticks([0,10])\n",
-    "ax.set_xticklabels(['0','K'],fontsize=14)\n",
-    "\n",
-    "ax.axhline(1, ls='--', color='black', lw=0.5)\n",
-    "ax.axvline(10, ls='--', color='black', lw=0.5)\n",
-    "\n",
-    "ax.set_ylim(0, 1.5)\n",
-    "ax.set_xlim(0, 15)\n",
-    "\n",
-    "fig.tight_layout()\n",
-    "fig.savefig('parent-offspring.png')"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 1,
-   "metadata": {
-    "tags": [
-     "remove-cell"
-    ]
-   },
-   "outputs": [
-    {
-     "data": {
-      "image/png": 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",
-      "text/plain": [
-       "<Figure size 600x400 with 1 Axes>"
-      ]
-     },
-     "metadata": {},
-     "output_type": "display_data"
-    }
-   ],
-   "source": [
-    "import matplotlib.pyplot as plt\n",
-    "import numpy as np\n",
-    "\n",
-    "lisg1=[]\n",
-    "lisg2=[]\n",
-    "lisg3=[]\n",
-    "\n",
-    "n0g1=1\n",
-    "n0g2=1\n",
-    "n0g3=1\n",
-    "\n",
-    "lisg1.append(n0g1)\n",
-    "lisg2.append(n0g2)\n",
-    "lisg3.append(n0g3)\n",
-    "\n",
-    "vect = range(1,25)\n",
-    "for t in vect:\n",
-    "    n0g1=1.1*n0g1\n",
-    "    lisg1.append(n0g1)\n",
-    "    \n",
-    "    n0g2=1*n0g2\n",
-    "    lisg2.append(n0g2)\n",
-    "    \n",
-    "    n0g3=0.9*n0g3\n",
-    "    lisg3.append(n0g3)\n",
-    "\n",
-    "fig, ax = plt.subplots(figsize=(6, 4))\n",
-    "time = range(25)\n",
-    "\n",
-    "ax.scatter(time,lisg1,  label='$R>1$')\n",
-    "ax.scatter(time,lisg2,  label='$R=1$')\n",
-    "ax.scatter(time,lisg3,  label='$0<R<1$')\n",
-    "    \n",
-    "ax.set_xlabel('t',fontsize = 16)\n",
-    "ax.set_ylabel('$N_t$',fontsize = 16)\n",
-    "\n",
-    "ax.tick_params(\n",
-    "    axis='both',          \n",
-    "    which='both',      \n",
-    "    bottom=False,      \n",
-    "    left=False,\n",
-    "    labelbottom=False,labelleft=False)\n",
-    "\n",
-    "\n",
-    "\n",
-    "#plt.yticks([0,1],['0','K'],fontsize = 13)\n",
-    "#plt.axhline(1, ls='--', color='black', lw=0.5)\n",
-    "\n",
-    "ax.set_ylim(0, 5)\n",
-    "ax.grid()\n",
-    "ax.legend(fontsize=12)\n",
-    "\n",
-    "fig.tight_layout()\n",
-    "fig.savefig('expgrowth.png')"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 3,
-   "metadata": {
-    "tags": [
-     "remove-cell"
-    ]
-   },
-   "outputs": [
-    {
-     "data": {
-      "image/png": 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",
-      "text/plain": [
-       "<Figure size 600x400 with 1 Axes>"
-      ]
-     },
-     "metadata": {},
-     "output_type": "display_data"
-    }
-   ],
-   "source": [
-    "import matplotlib.pyplot as plt\n",
-    "import numpy as np\n",
-    "\n",
-    "lisg1=[]\n",
-    "lisg2=[]\n",
-    "lisg3=[]\n",
-    "\n",
-    "n0g1=0.1\n",
-    "n0g2=0.8\n",
-    "n0g3=1.4\n",
-    "\n",
-    "lisg1.append(n0g1)\n",
-    "lisg2.append(n0g2)\n",
-    "lisg3.append(n0g3)\n",
-    "\n",
-    "vect = range(1,16)\n",
-    "for t in vect:\n",
-    "    n0g1=1.5*n0g1/(1+((1.5-1)/1)*n0g1)\n",
-    "    n0g2=1.5*n0g2/(1+((1.5-1)/1)*n0g2)\n",
-    "    n0g3=1.5*n0g3/(1+((1.5-1)/1)*n0g3)\n",
-    "    \n",
-    "    lisg1.append(n0g1)\n",
-    "    lisg2.append(n0g2)\n",
-    "    lisg3.append(n0g3)\n",
-    "    \n",
-    "time = range(16)\n",
-    "\n",
-    "fig, ax = plt.subplots(figsize=(6, 4))\n",
-    "\n",
-    "ax.scatter(time,lisg1)\n",
-    "ax.scatter(time,lisg2)\n",
-    "ax.scatter(time,lisg3)\n",
-    "    \n",
-    "ax.set_xlabel('t',fontsize = 16)\n",
-    "ax.set_ylabel('$N_t$',fontsize = 16)\n",
-    "\n",
-    "ax.tick_params(\n",
-    "    axis='x',          \n",
-    "    which='both',      \n",
-    "    bottom=False,      \n",
-    "    top=False,\n",
-    "    labelbottom=False)\n",
-    "\n",
-    "\n",
-    "ax.set_yticks([0,1])\n",
-    "ax.set_yticklabels(['0','K'],fontsize=12)\n",
-    "ax.axhline(1, ls='--', color='black', lw=0.5)\n",
-    "\n",
-    "ax.set_ylim(0, 1.5)\n",
-    "\n",
-    "fig.tight_layout()\n",
-    "fig.savefig('logistic.png')"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": null,
-   "metadata": {},
-   "outputs": [],
-   "source": []
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "# 03   - Analysing change, one step at a time\n",
-    "  "
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "Imagine a colony of wasps is founded by a queen with the following pattern of eggs laid:  \n",
-    "\n",
-    "<br />  \n",
-    "    \n",
-    "| Days     |  Eggs  |\n",
-    "|----------|:------:|\n",
-    "| $0$      |  $100$ |\n",
-    "| $1$      |  $135$ |\n",
-    "| $2$      |  $170$ |\n",
-    "| $3$      |  $205$ |\n",
-    "| $4$      |  $240$ |  \n",
-    "    \n",
-    "\n",
-    "Since there is an additive pattern for the increase in the total number of eggs laid, with the number of eggs in any given day, we can determine the number of eggs the colony will have in the following day. If we index the days by a sequence of natural numbers *n*, we have for the above colony:  \n",
-    "  \n",
-    "$$a_{n+1} = a_{n} + 35,$$  \n",
-    "  \n",
-    "where $a_n$ represents the total amount of eggs in a given day $n$ and $a_{n+1}$ the same variable on the day following day $n$. The constant $35$ represents the daily increment (or decrement) on this total number. A sequence of number built with constant additive increments receives the name of *arithmetic progression*.  \n",
-    "  \n",
-    "The above expression defines an iterative process (also known as a *recursion*) with which we can construct the whole sequence starting from a single initial value. If we want to know how many eggs will have been laid on a given day $n$, this shall be given by the expression:  \n",
-    "  \n",
-    "$$a_n = 100 + 35n.$$  \n",
-    "  \n",
-    "Thus, the total amount of eggs laid on day $n=100$ is $a_{100} = 100 + 35 \\cdot 100 = 3600$.  \n",
-    "  \n",
-    "<br />\n",
-    "<br />\n",
-    "Now imagine a single *Escherichia coli* cell put on a Petri dish with a favorable growth medium. If given the right conditions, *E. coli* cells divide, generating two other cells, in a time of approximately $30$ minutes. For our experiment, we would have:  \n",
-    "\n",
-    "<br />  \n",
-    "    \n",
-    "| Minutes  |  # cells  |\n",
-    "|----------|:------:|\n",
-    "| $0$      |  $1$ |\n",
-    "| $30$      |  $2$ |\n",
-    "| $60$      |  $4$ |\n",
-    "| $90$      |  $8$ |\n",
-    "| $120$      |  $16$ |  \n",
-    "  \n",
-    "    \n",
-    "If $t$ represents the number of reproduction intervals since the start of the experiment (assuming values $t=0, 1, 2, \\ldots$), then we can construct the recursion:  \n",
-    "  \n",
-    "$$N_{t+1} = 2N_t,$$  \n",
-    "  \n",
-    "where $N_t$ represents the number of *E. coli* cells after $t$ intervals of reproduction (each interval corresponding to $30$ minutes) and $N_{t+1}$ represents the number of cells on the interval following $t$. See that this time the constant $2$ provides a multiplicative increment, so that the sequence formed by this multiplicative iteration is called *geometric progression*.  \n",
-    "  \n",
-    "If we want $N_t$ for a given interval $t$, we then have:  \n",
-    "  \n",
-    "$$N_t = 2^t,$$\n",
-    "    \n",
-    "leading to an exponential growth in discrete steps.  \n",
-    "  \n",
-    "In general, if we have a population growth described by a geometric progression, starting with a number of individuals $N_0$, the number of individuals on the subsequent generations can be described by the recursion:  \n",
-    "  \n",
-    "$$N_{t+1} = RN_t,$$  \n",
-    "  \n",
-    "where $t$ indicates the number of generations and $R$ is a positive constant ($R>0$).  \n",
-    "  \n",
-    "With this recursion, starting from $N_0$, we can obtain the full sequence by construction:  \n",
-    "  \n",
-    "$$\\begin{align}  \n",
-    "N_1 &= RN_0 \\\\\n",
-    "N_2 &= RN_1 = R(RN_0) = R^2N_0 \\\\\n",
-    "N_3 &= RN_2 = R(RN_1) = R[R(RN_0)] = R^3N_0 \\\\\n",
-    "    &\\vdots \\end{align}$$  \n",
-    "  \n",
-    "From this process of construction, we can obtain what we call the *solution* of the above recursion, given by:  \n",
-    "  \n",
-    "$$N_t = N_0R^t.$$  \n",
-    "  \n",
-    "In problems of population dynamics, $R$ is called **growth constant** and its value determines the long-term behaviour of the function:  \n",
-    "  \n",
-    "```{image} expgrowth.png\n",
-    "```"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "## Sequences  \n",
-    "  \n",
-    "The two last problems are examples of *sequences* for which a formal definition is given as:  \n",
-    "  \n",
-    "$$\\begin{align}\n",
-    "    f\\!:\\ &{\\rm I\\!N} \\rightarrow {\\rm I\\!R} \\\\\n",
-    "       & n \\rightarrow f(n),\\end{align}$$  \n",
-    "         \n",
-    "in other words, a sequence is a function $f$ from the set of natural numbers to the set of real numbers. For each value of the independent variable $n$ (a natural number), the sequence determines its image $f(n)$.  \n",
-    "  \n",
-    "```{admonition} Example  \n",
-    "The sequence defined by the rule $f(n) = \\displaystyle\\frac{1}{n+1}$ defines the sequence of numbers (starting from $n=0$):  \n",
-    "  \n",
-    "$$1,\\frac{1}{2},\\frac{1}{3},\\frac{1}{4},\\frac{1}{5},\\ldots,$$  \n",
-    "  \n",
-    "corresponding to the set of values $n=0,1,2,3,4,\\ldots$.\n",
-    "```  \n",
-    "  \n",
-    "  \n",
-    "\n",
-    "Sequences are usually written as ordered lists of numbers $a_0, a_1, a_2, a_3, \\ldots$ where $a_n=f(n)$ and referred to with the notation $\\{a_n\\}$ (which is short for $\\{a_n: n \\in {\\rm I\\!N}\\}$, making it explicit that $n$ is a natural number).  \n",
-    "  \n",
-    "As we have seen before, two forms of representation are possible:  \n",
-    "  \n",
-    "````{panels}\n",
-    "**Explicit description**\n",
-    "^^^\n",
-    "$$a_n=f(n)$$  \n",
-    "  \n",
-    "```{admonition} Examples  \n",
-    "$$a_n = 100 + 35n$$  \n",
-    "$$a_n = 100\\cdot2^n$$  \n",
-    "$$a_n = \\frac{4n^2-1}{n^2}$$  \n",
-    "$$a_n = \\mbox{cos}\\left(\\displaystyle\\frac{3\\pi n}{2}\\right)$$\n",
-    "```\n",
-    "+++\n",
-    "In this case, any term $a_n$ of arbitrary index $n$ can be obtained by direct calculation if we know the function $f(n)$. \n",
-    "---\n",
-    "\n",
-    "**Recursive description**\n",
-    "^^^\n",
-    "$$a_{n+1}=g(a_n)$$  \n",
-    "  \n",
-    "```{admonition} Examples  \n",
-    "$$a_{n+1} = a_n + 35,\\ \\ \\ \\mbox{with}\\ a_0 = 100$$  \n",
-    "$$a_{n+1} = 2a_n,\\ \\ \\ \\mbox{with}\\ a_0=100$$  \n",
-    "$$a_{n+1} = \\frac{3}{a_n},\\ \\ \\ \\mbox{with}\\ a_0=2$$ \n",
-    "```  \n",
-    "  \n",
-    "The recursive description is also called *difference equation*.\n",
-    "+++\n",
-    "Note that with the form $a_{n+1}=g(a_n)$ the following term $a_{n+1}$ can be determined only with the term one step back $a_n$, knowing the function $g$. In general, recursions can be dependent on several other backward terms. When it depends only on the immediate previous term it is called a **first-order** recursion. \n",
-    "````"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "## Limits  \n",
-    "  \n",
-    "  \n",
-    "In many applications, we are interested to know what is the *long-term behaviour* of a given sequence $\\{a_n\\}$. For example, is a given population tending towards a stable number of individuals as time passes? In other words, we want to know if there is a number $L$ for which the sequence tends to as $n$ increases. To represent this we use the following notation:  \n",
-    "  \n",
-    "$$\\lim_{n\\rightarrow \\infty} a_n = L\\ \\ (\\mbox{or simply}\\ \\lim a_n =L)$$  \n",
-    "  \n",
-    "If such a number exists, we say that the sequence $\\{a_n\\}$ is **convergent**.  \n",
-    "  \n",
-    "```{admonition} Examples  \n",
-    "1. $a_n = \\displaystyle\\frac{1}{n+1}: \\ \\ \\ \\left\\{1,\\displaystyle\\frac{1}{2},\\displaystyle\\frac{1}{3},\\displaystyle\\frac{1}{4},\\displaystyle\\frac{1}{5},\\ldots\\right\\}$  \n",
-    "    <br />\n",
-    "    Note that each term is smaller than the previous and they are all positive. We can conclude that:  \n",
-    "  \n",
-    "    $$\\lim a_n = 0.$$  \n",
-    "\n",
-    "2. $b_n = (-1)^n: \\ \\ \\ \\left\\{1,-1,1,-1,1,-1,\\ldots\\right\\}$  \n",
-    "    <br />\n",
-    "    The terms in this sequence are alternating between $1$ and $-1$, never convergeing for a specific value. Thus, the limit of sequence $\\{b_n\\}$ does not exist.  \n",
-    "    <br />\n",
-    "3. $c_n = 2^n: \\ \\ \\ \\left\\{1,2,4,8,16,32,\\ldots\\right\\}$  \n",
-    "    <br />\n",
-    "    For this sequence, the terms are ever increasing. We frequently say that the sequence tends to infinity (or tends to $\\infty$). However, $\\infty$ represents an abstraction (\"ever increasing\"), not a specific number. Therefore, the limit for this sequence also does not exist.  \n",
-    "    <br />\n",
-    "4. $d_n = \\displaystyle\\frac{n+2}{n+1}: \\ \\ \\ \\left\\{2,\\displaystyle\\frac{3}{2},\\displaystyle\\frac{4}{3},\\displaystyle\\frac{5}{4},\\displaystyle\\frac{6}{5},\\ldots\\right\\}$  \n",
-    "    <br />\n",
-    "    For this sequence, each term is smaller than the previous (look at the decimal forms of the fractions: $2, 1.5, 1.333\\cdots, 1.25, 1.2 \\ldots$) and they are greater than $1$. We can conclude that:  \n",
-    "  \n",
-    "    $$\\lim a_n = 1.$$  \n",
-    "  \n",
-    "```  \n",
-    "  \n",
-    "Although there is a formal way to *prove* that a sequence $\\{a_n\\}$ converges to a certain limit, in practice we use a set of rules to apply limits to sequences and composition of sequences.  \n",
-    "  \n",
-    "```{tip}  \n",
-    "Consider two sequences $\\{a_n\\}$ and $\\{b_n\\}$, and $C$ as a real constant. If these two sequences are convergent with  $\\lim a_n = L$ and $\\lim b_n = M$, we have:  \n",
-    "  \n",
-    "$$\\lim (a_n + b_n) = \\lim a_n + \\lim b_n = L + M$$  \n",
-    "$$\\lim (Ca_n) = C\\lim a_n = CL$$  \n",
-    "$$\\lim (a_nb_n) = (\\lim a_n)(\\lim b_n) = LM$$  \n",
-    "$$\\lim \\left(\\frac{a_n}{b_n}\\right) = \\frac{\\lim a_n}{\\lim b_n} = \\frac{L}{M},\\ \\ (\\mbox{if}\\ M \\ne 0)$$\n",
-    "  \n",
-    "In other other words, if you can decompose a given sequence into the sum, product or division of two other convergent sequences, the limit of this sequence will be the composition of the limits of the two others.\n",
-    "```  \n",
-    "  \n",
-    "```{admonition} Examples  \n",
-    "If $a_n = \\displaystyle\\frac{4n^2-1}{n^2}:$  \n",
-    "  \n",
-    "$$\\lim\\left(\\displaystyle\\frac{4n^2-1}{n^2}\\right) = \\lim\\left(4-\\displaystyle\\frac{1}{n^2}\\right) = \\lim 4 - \\left(\\lim\\displaystyle\\frac{1}{n}\\right)\\left(\\lim\\displaystyle\\frac{1}{n}\\right) = 4,$$  \n",
-    "  \n",
-    "since as we know: $\\ \\lim 4 = 4\\ $ and $\\ \\lim \\displaystyle\\frac{1}{n} = 0$.\n",
-    "\n",
-    "```  \n",
-    "  \n",
-    "For recursions, one way to find the limit is to find the solution (corresponding explicit description) and then calculate the limit of the sequence with the limit rules. However, finding the explicit form of a given sequence is not always possible.  \n",
-    "  \n",
-    "There is a procedure to find *candidates* for the long-term behaviour of a given sequence. We do this by calculating **fixed points**.  \n",
-    "  \n",
-    "Fixed points are specific values on a given sequence for which all subsequent values equal the first. That is, if $a$ is a fixed point of the sequence $\\{a_n\\}$, and if $a_0 = a$, then $a_1 = a$, $a_2 = a$, and all the other values will be $a$. Thus, given the recursion $a_{n+1} = g(a_n)$, if $a$ is a fixed point we will have:  \n",
-    "  \n",
-    "$$a = g(a)$$  \n",
-    "  \n",
-    "and the last equation provides a way of find possible values of $a$.  \n",
-    "  \n",
-    "```{admonition} Example  \n",
-    "  \n",
-    "Consider the following recursion  \n",
-    "  \n",
-    "$$a_{n+1} = \\sqrt{3a_n},\\ \\ \\ a_0 = 2.$$  \n",
-    "  \n",
-    "To obtain the fixed points we calculate:  \n",
-    "  \n",
-    "$$\\begin{align}\n",
-    "  a &= \\sqrt{3a} \\\\\n",
-    "  a^2 &= 3a \\ \\ \\longrightarrow a = 0\\ \\mbox{ or }\\ a = 3 \\end{align}$$  \n",
-    "  \n",
-    "Starting with $a_0 = 2$, we can use the recursion to construct the sequence as:  \n",
-    "  \n",
-    "  $$\\left\\{\\sqrt{6},\\sqrt{3\\sqrt{6}},\\sqrt{3\\sqrt{3\\sqrt{6}}},\\sqrt{3\\sqrt{3\\sqrt{3\\sqrt{6}}}},\\ldots \\right\\}$$  \n",
-    "    \n",
-    "or also: $\\{2.449\\cdots,2.711\\cdots,2.852\\cdots,2.925\\cdots, \\ldots\\}$. We see that starting with $a_0=2$, we have $a_n>2$ for all $n$, with increasing values. Since $a=3$ is a fixed point, we conclude that $\\lim a_n = 3$.  \n",
-    "```  \n",
-    "\n",
-    "```{admonition} Example\n",
-    "Whenever we are asking questions like 'where does this process stabilise' or 'how high/much will it be in the end,' we are interested in the limit of some series. This could be a variety of things, like population size, ground water levels, global mean temperature, number of trees, etc.\n",
-    "```  \n",
-    "  \n",
-    "```{warning}  \n",
-    "  \n",
-    "Remember that fixed points are only **candidates** for long-term behaviour. See for example the recursion:  \n",
-    "  \n",
-    "$$a_{n+1} = \\displaystyle\\frac{3}{a_n}$$  \n",
-    "  \n",
-    "For the fixed points, we calculate:  \n",
-    "  \n",
-    "$$\\begin{align}\n",
-    "  a &= \\displaystyle\\frac{3}{a} \\\\\n",
-    "  a^2 &= 3 \\\\\n",
-    "  &\\longrightarrow a = -\\sqrt{3}\\ \\mbox{ or }\\ a = \\sqrt{3}. \\end{align}$$  \n",
-    "    \n",
-    "Indeed, if we construct the sequence from the recursion starting with any of these two values, all the subsequent elements of the sequence will have the same value. However:  \n",
-    "  \n",
-    "$$\\mbox{if } a_0 = 2 \\ \\mbox{the sequence will be given by: }\\ \\left\\{2,\\displaystyle\\frac{3}{2},2,\\displaystyle\\frac{3}{2},\\ldots \\right\\}$$  \n",
-    "  \n",
-    "or  \n",
-    "  \n",
-    "$$\\mbox{If } a_0 = -3 \\ \\mbox{the sequence will be given by: }\\ \\left\\{-3,-1,-3,-1,\\ldots \\right\\}$$  \n",
-    "  \n",
-    "showing that for any initial value chosen (different from the fixed points), the sequence will oscillate between two values, without a clear tendency to any limit.\n",
-    "```\n"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "## Discrete-time population models  \n",
-    "  \n",
-    "Sequences are generally used in many applications in Biology. Important examples are models for **seasonally breeding** populations.  \n",
-    "  \n",
-    "  \n",
-    "### Density-dependent population growth  \n",
-    "  \n",
-    "As we have seen before, the recursion relation for the exponential growth model is given by $N_{t+1} = RN_t$. Rearranging the terms of this equation we have:  \n",
-    "  \n",
-    "$$\\displaystyle\\frac{N_t}{N_{t+1}}=\\frac{1}{R}.$$  \n",
-    "  \n",
-    "The left-hand term of the last equation is known as *parent-offspring ratio* as it denotes the ratio between the population size at time $t$ (parents) and the population size at the following time $t+1$ (offspring). Here we are considering a population that breeds without overlapping generations. Each generation breeds, dies, and is replaced by their offspring on the following generation.  \n",
-    "  \n",
-    "On the equation we also see that the parent-offspring ratio is equal to $\\displaystyle\\frac{1}{R}$, which is a constant. Therefore, we say that this growth is *density-independent* as the parent-offspring ratio does not depend on the density of individuals at any time step. Relating the parent-offspring ratio to the value of $R$ allows us to predict the behaviour of this population in terms of growth or decline:  \n",
-    "  \n",
-    "$$\\begin{align}\n",
-    "R>1 \\ &- \\ \\mbox{there will be more offspring than parents (population grows)} \\\\\n",
-    "R<1 \\ &- \\ \\mbox{there will be more parents than offspring (population declines)} \\\\\n",
-    "R=1 \\ &- \\ \\mbox{there will be no change in population size} \\end{align}$$\n",
-    "  \n",
-    "We know, however, than in real scenarios, that the growth factor of a given population should depend on the current size of that population, due to space or resources limitation. These are called *density-dependent* effects. One of the simplest methods to include density-dependent effects in this population model is to consider the parent-offspring ratio as a linear function of $N_t$, instead of a constant. Let us assume that the parent-offspring ratio show the following linear behaviour:  \n",
-    "  \n",
-    "```{image} parent-offspring.png\n",
-    "```  \n",
-    "  \n",
-    "This means that the parent-offspring ratio is smaller than $1$ (population growth) for values of $N_t$ smaller than a given population threshold $K$, and greater than $1$ (population decline) if the population size is greater than this threshold. The graph above leads to the following expression:  \n",
-    "  \n",
-    "$$\\begin{align}\n",
-    "\\displaystyle\\frac{N_t}{N_{t+1}} = \\frac{1}{R} + \\frac{1-\\frac{1}{R}}{K}N_t \\\\\n",
-    "N_{t+1} = \\frac{N_t}{\\frac{1}{R} + \\frac{1-\\frac{1}{R}}{K}N_t}, \\end{align}$$  \n",
-    "  \n",
-    "which leads, after rearranging the terms, to:  \n",
-    "  \n",
-    "$$N_{t+1} = \\frac{RN_t}{1 + \\frac{(R-1)}{K}N_t}.$$  \n",
-    "  \n",
-    "The previous expression is known as **Beverton-Holt recruitment curve** (or Beverton-Holt model) and it describes a type of population growth called *logistic growth*.  \n",
-    "  \n",
-    "The fixed points $\\bar{N}$ for this recursion can be calculated as:  \n",
-    "  \n",
-    "$$\\bar{N} = \\frac{R\\bar{N}}{1 + \\frac{(R-1)}{K}\\bar{N}}\\ \\longrightarrow \\ \\bar{N}\\left(1 + \\frac{(R-1)}{K}\\bar{N}\\right) = R\\bar{N}$$  \n",
-    "  \n",
-    "From the last term we can show (try it as an exercise) that $\\bar{N}=0$ and $\\bar{N}=K$ are the two fixed points. Starting from different values for the initial population size $N_0$ it can be shown that the population tends to $K$ as time increases as:  \n",
-    "  \n",
-    "```{image} logistic.png\n",
-    "```  \n",
-    "  \n",
-    "The constant $K$ is know as *carrying capacity* and represents the maximum population size a given population can reach at a given place, due to intra-specific competition for resources or space.  \n",
-    "  \n",
-    "  \n",
-    "Watch this [video](https://www.youtube.com/watch?v=Kas0tIxDvrg) about application of growth models to epidemics.  \n",
-    "  \n",
-    "  \n",
-    "### Annual plants  \n",
-    "  \n",
-    "  \n",
-    "Suppose a given population of plants produces, on average $f$ seeds per individual in the reproductive period. Seeds survive winter with a probability $\\sigma$ and germinate on the next season with probability $\\alpha$ (these seeds reach reproductive maturity on the same season). If $p_n$ is the plant population size on season $n$ and $p_{n+1}$ the population size on the season following $n$, we shall have:  \n",
-    "  \n",
-    "$$p_{n+1} = (\\alpha\\sigma f)p_n$$  \n",
-    "  \n",
-    "which correspond to the density-independent model since $R=\\alpha\\sigma f$ is a constant.  \n",
-    "  \n",
-    "Now suppose that some of the seeds that were produced on the previous season and survived the previous winter, but did not germinate on the current season, might have a second chance of germinating. They might do so on the following season, provided they survive the winter and germinate on the following season. From the average number of seeds produced last season $fp_{n-1}$, a number $\\sigma fp_{n-1}$ of them survived last winter. From this number, a fraction $(1-\\alpha)$ did not germinate this season (1 minus the probability of germinating). These seeds will have a second chance provided they survive next winter (with probability $\\sigma$) and germinate on the next season (with probability $\\alpha$). Thus, the number plant population size on the season following $n$ will be:  \n",
-    "  \n",
-    "$$\\begin{align}\n",
-    "p_{n+1} = \\alpha\\sigma fp_n + \\alpha\\sigma(1-\\alpha)\\sigma fp_{n-1} \\\\  \n",
-    "p_{n+1} = (\\alpha\\sigma f)p_n + [\\alpha(1-\\alpha)\\sigma^2 f]p_{n-1} \\\\  \n",
-    "p_{n+1} = \\beta p_n + \\gamma p_{n-1}\\end{align}$$  \n",
-    "  \n",
-    "where $\\beta=\\alpha\\sigma f$ and $\\gamma = \\alpha(1-\\alpha)\\sigma^2 f$.  \n",
-    "  \n",
-    "The previous model is an example of a discrete population model with overlapping populations. Since the term $n+1$ depends not only on the term $n$ but also on $n-1$ this is called a *second-order difference equation* or *delay (or lagged) difference equation*.  \n",
-    "  \n",
-    "  \n",
-    "### Fibonacci sequence  \n",
-    "  \n",
-    "  \n",
-    "The Fibonacci sequence is a sequence defined by the following rule: each number is equal to the sum of the previous two numbers (starting with $0$ and $1$). Thus the following sequence might be formed:  \n",
-    "  \n",
-    "$$\\left\\{ 0,1,1,2,3,5,8,13,21,\\ldots \\right\\}. $$  \n",
-    "  \n",
-    "It can also be defined by the second-order difference equation:  \n",
-    "  \n",
-    "$$F_{n+1} = F_{n} + F_{n-1}.$$  \n",
-    "  \n",
-    "Although this is clearly not a convergent sequence (with sustained growth for each element), if we define a new sequence given by the ratio between successive elements of the Fibonacci sequence as $b_n = \\displaystyle\\frac{F_{n+1}}{F_n}$, it can be show that this sequence is convergent.  \n",
-    "  \n",
-    "Assuming that $\\lim \\displaystyle\\frac{F_{n+1}}{F_n} = \\lambda$, we have:  \n",
-    "  \n",
-    "$$\\begin{align}\n",
-    "F_{n+1} &= F_{n} + F_{n-1}\\quad \\mbox{(Fibonacci seq.)} \\\\  \n",
-    "\\displaystyle\\frac{F_{n+1}}{F_{n}} &= 1 + \\frac{F_{n-1}}{F_{n}}\\quad \\mbox{(divide both sides by}\\ F_{n}\\mbox{)} \\\\\n",
-    "\\lim \\left(\\displaystyle\\frac{F_{n+1}}{F_{n}}\\right) &= \\lim\\left(1 + \\frac{F_{n-1}}{F_{n}}\\right) \\\\\n",
-    "\\lambda &= 1 + \\frac{1}{\\lambda} \\\\\n",
-    "\\lambda^2 &- \\lambda -1 = 0 \\end{align}$$\n",
-    "  \n",
-    "The previous equation is solved finding the roots of the second-degree polynomial, which gives $\\lambda = \\displaystyle\\frac{1+\\sqrt{5}}{2}$ or $\\lambda = \\displaystyle\\frac{1-\\sqrt{5}}{2}$. Since the solution $\\displaystyle\\frac{1-\\sqrt{5}}{2}$ is negative, it can be the limit of the ratio of Fibonacci terms (all of them positive). Hence we conclude:  \n",
-    "  \n",
-    "$$\\lim \\left(\\displaystyle\\frac{F_{n+1}}{F_{n}}\\right) = \\displaystyle\\frac{1+\\sqrt{5}}{2} \\approx 1.618.$$  \n",
-    "  \n",
-    "The previous number is known as the *Golden ratio* and it has fascinated mathematicians, phylosophers, and artists since ancient times. Fibonacci numbers and the Golden ratio can be found in several interesting natural patterns, as you can see in this [video](https://www.youtube.com/watch?v=ahXIMUkSXX0)."
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diff --git a/_sources/mathscourse/lecture04/04_From_table_arrangements_to_powerful_tools.ipynb b/_sources/mathscourse/lecture04/04_From_table_arrangements_to_powerful_tools.ipynb
deleted file mode 100644
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--- a/_sources/mathscourse/lecture04/04_From_table_arrangements_to_powerful_tools.ipynb
+++ /dev/null
@@ -1,384 +0,0 @@
-{
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-   "metadata": {},
-   "source": [
-    "# 04 - From table arrangements to powerful tools\n",
-    "  "
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "## Systems of linear equations  \n",
-    "  \n",
-    "As we saw previously, the equation $ax + by = c$, with $a$, $b$, and $c$ constants, defines a straight line on the $xy$-plane (note that it can also be written as $y = mx + k$, $m$ and $k$ constants, by a suitable change of the coeffients). Since the coefficients do not depend on the variables $x$ and $y$, we call it a *linear equation*. A possible plot for this equation would be:  \n",
-    "  \n",
-    "```{image} lineq.png\n",
-    "```  \n",
-    "  \n",
-    "Note that all the points $(x,y)$ on the blue line *satisfy* or *solve* the lienar equation $ax + by = c$.  \n",
-    "  \n",
-    "Now consider that we have the following:  \n",
-    "  \n",
-    "$$\\begin{cases} ax + by = c \\\\ dx +ey = f \\end{cases}$$  \n",
-    "  \n",
-    "where $a, b, c, d, e,\\ \\mbox{and}\\ f$ are all constants.In this case we have a *system of linear equations*. Solving this system means to find all possible pairs $(x,y)$ that solve both equations **simultaneously**. For the previous system, we would have 3 options:\n",
-    "  \n",
-    "```{image} syslineq.png\n",
-    "```  \n",
-    "  \n",
-    "On the left panel, the two lines intersect at a *single point*. This point is the only one that solves both linear equations at the same time, and, therefore, constitutes the **only solution** for the system of two equations. On the central panel, the two lines are *parallel*, so there is no point of intersection. In this case there is **no solution** and the system of equations is said to be *inconsistent*. On the right panel, the two lines coincide. Thus, every point in those lines correspond to a possible solution for the system, so that it has **infinite solutions**.  \n",
-    "  \n",
-    "   \n",
-    "```{admonition} Example  \n",
-    "  \n",
-    "Given the system of linear equations  \n",
-    "  \n",
-    "$$\\begin{cases} 2x + 3y = 6 \\\\ 2x + y = 4 \\end{cases}$$\n",
-    "\n",
-    "we can obtain the solution by isolating one of the variables in one of the equations and substituting on the other equation. Isolating $x$ on the first equation gives:  \n",
-    "  \n",
-    "$$x = 3 - \\displaystyle\\frac{3}{2}y$$  \n",
-    "  \n",
-    "Then by substituting on the second we have:  \n",
-    "  \n",
-    "$$\\begin{align} \n",
-    "2\\left(3 - \\displaystyle\\frac{3}{2}y \\right) + y &= 4 \\\\\n",
-    "6 - 3y + y &= 4 \\\\\n",
-    "-2y = -2 \\\\\n",
-    "y = 1 \\end{align}$$  \n",
-    "  \n",
-    "The value of $x$ shall then be given by:  \n",
-    "  \n",
-    "$$x = 3 - \\displaystyle\\frac{3}{2}\\cdot 1 = 3 - \\frac{3}{2} = \\frac{3}{2}$$  \n",
-    "  \n",
-    "We see that the system has a single solution, given by $\\left(\\displaystyle\\frac{3}{2},1\\right)$.  \n",
-    "  \n",
-    "```  \n",
-    "  \n",
-    "````{admonition} Example  \n",
-    "  \n",
-    "Now consider the sytem  \n",
-    "  \n",
-    "$$\\begin{cases} 2x + 3y &= 6 \\\\ -4x + -6y &= -12 \\end{cases}$$\n",
-    "\n",
-    "Isolating $x$ on the first equation gives the same as before:  \n",
-    "  \n",
-    "$$x = 3 - \\displaystyle\\frac{3}{2}y$$  \n",
-    "  \n",
-    "Then by substituting on the second we have:  \n",
-    "  \n",
-    "$$\\begin{align} \n",
-    "-4\\left(3 - \\displaystyle\\frac{3}{2}y \\right) - 6y &= -12 \\\\\n",
-    "-12 + 6y - 6y &= -12 \\\\\n",
-    "0 = 0 \\end{align}$$  \n",
-    "  \n",
-    "which is just a trivial equation.  \n",
-    "  \n",
-    "When we look at the equations in our system we see that the second equation is simply a constant multiplied by the first $(\\text{Eqn.} 2) = -2\\cdot(\\text{Eqn.} 1)$, and thus they should represent the same straight line.  \n",
-    "  \n",
-    "This system has then infinite solutions which are represented by the set of points $\\left(3 - \\displaystyle\\frac{3}{2}y,y\\right)$\n",
-    "  \n",
-    "```{tip}\n",
-    "Note that writing the solution as $\\left(3 - \\displaystyle\\frac{3}{2}y,y\\right)$ represents an infinite set of points, since for every real value of $y$ we would have a different ordered pair. This set of points forms the straight line corresponding to $2x+3y = 6$.\n",
-    "\n",
-    "```\n",
-    "  \n",
-    "````  \n",
-    "  \n",
-    "Solving the system by this method of substitutions, however, can get much more complicated if we have more than two variables. For these cases, we need a simpler method.  \n",
-    "  \n",
-    "One strategy is to find equivalent systems, which have the same solution as the original one, but are much easier to solve. We can apply the following set of operations without changing the solutions of a given system:  \n",
-    "  \n",
-    "- Multiplication of an entire row by a non-zero constant  \n",
-    "- Adding/subtracting one row to another  \n",
-    "- Rearranging the order of the rows  \n",
-    "  \n",
-    "The examples below show this procedure applied to two systems.  \n",
-    "  \n",
-    "```{admonition} Examples  \n",
-    "  \n",
-    "(i) $\\begin{cases} 3x + 5y - z = 10 \\\\ 2x –  y + 3z = 9 \\\\ 4x + 2y - 3z = -1 \\end{cases}$  \n",
-    "  \n",
-    "See the solution on this [video](https://web.microsoftstream.com/video/194b0152-903a-4614-b608-306ce71d9a39)\n",
-    "  \n",
-    "(ii) $\\begin{cases} x - 3y + z = 4 \\\\ x – 2y + 3z = 6 \\\\ 2x - 6y + 2z = 8 \\end{cases}$  \n",
-    "  \n",
-    "See the solution on this [video](https://web.microsoftstream.com/video/067c8938-5e45-4f08-bf90-783cb125c6f5)\n",
-    "```  \n",
-    "  \n",
-    "Note that by applying the rules of matrix multiplication, the system *(ii)* can also be written in the form:  \n",
-    "  \n",
-    "$$\\begin{pmatrix} 1 & -3 & 1\\\\ 1 & -2 & 3 \\\\ 2 & -6 & 2  \\end{pmatrix} \\begin{pmatrix} x \\\\ y \\\\ z   \\end{pmatrix} = \\begin{pmatrix} 4 \\\\ 6 \\\\ 8   \\end{pmatrix} \\quad \\longrightarrow \\quad A\\mathbf{x}=\\mathbf{b}$$  \n",
-    "  \n",
-    "where $A$ is the matrix of coefficients and $\\mathbf{x}$ is the vector of variables.  \n",
-    "  \n",
-    "```{tip}  \n",
-    "We will be using the common notation of uppercase letters ($A$) to represent matrices and lowercase bold letters ($\\mathbf{x}$) to represent vectors.\n",
-    "```\n",
-    "\n",
-    "In general, the system with $n$ variables $\\{x_1, x_2, \\ldots, x_n \\}$ and $m$ equations can be written as  \n",
-    "  \n",
-    "$$\\begin{cases} a_{11}x_1 + a_{12}x_2 + \\ldots + a_{1n}x_n = b_1 \\\\ a_{21}x_1 + a_{22}x_2 + \\ldots + a_{2n}x_n = b_2 \\\\ \\dots \\\\ a_{m1}x_1 + a_{m2}x_2 + \\ldots + a_{mn}x_n = b_m \\end{cases}$$  \n",
-    "  \n",
-    "can be written as $A\\mathbf{x}=\\mathbf{b}$, where  \n",
-    "  \n",
-    "$$A=\\begin{pmatrix} a_{11} & a_{12} & \\cdots & a_{1n} \\\\ a_{21} & a_{22} & \\cdots & a_{2n} \\\\ \\vdots & & \\ddots & \\vdots \\\\ a_{m1} & a_{m2} & \\cdots & a_{mn} \\end{pmatrix}$$  \n",
-    "  \n",
-    "and  \n",
-    "  \n",
-    "$$\\mathbf{x} = \\begin{pmatrix} x_1 \\\\ x_2 \\\\ \\vdots \\\\ x_n   \\end{pmatrix}\\quad \\mbox{and} \\quad \\mathbf{b} = \\begin{pmatrix} b_1 \\\\ b_2 \\\\ \\vdots \\\\ b_m   \\end{pmatrix}$$  \n",
-    "  \n",
-    "The matrix $A$ has size $m \\times n$ (since the number of equations can, in principle, be different from the number of variables), while the vectors $\\mathbf{x}$ and $\\mathbf{b}$ have sizes $n \\times 1$ and $m \\times 1$, respectively.  \n"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "## Matrix operations  \n",
-    "  \n",
-    "Let us quickly remind the basic operations that can be performed with matrices. If $A$ and $B$ are both matrices of size $m \\times n$ ($m$ rows and $n$ columns), we have:  \n",
-    "  \n",
-    "- $A=B\\ $ if and only if $a_{ij} = b_{ij}$ for every $1 \\le i \\le m,  1 \\le j \\le n$ (elements at the same position have to be equal).  \n",
-    "  \n",
-    "- $C = A + B$ is the matrix for which $c_{ij} = a_{ij} + b_{ij}$ (sum is performed summing the elements at the same position)  \n",
-    "  \n",
-    "- If $k \\in \\rm I\\!R$ than $kA$ is the $m \\times n$ matrix with elements $ka_{ij}$  \n",
-    "  \n",
-    "- The following properties are valid:  \n",
-    "    \n",
-    "    $$\\begin{align}\n",
-    "    A+B &= B+A \\\\\n",
-    "    (A+B)+C &= A+(B+C) \\\\\n",
-    "    A + 0 &= A\\ (0\\ \\mbox{here represents the matrix with null elements}) \\end{align}$$  \n",
-    "  \n",
-    "    ```{admonition} Example  \n",
-    "    If $A=\\begin{pmatrix} 1 & 0  \\\\ -1 & 2  \\end{pmatrix}$ and $B=\\begin{pmatrix} 2 & 3  \\\\ -1 & 1  \\end{pmatrix}$, then:  \n",
-    "\n",
-    "    $$C = 2A+B = \\begin{pmatrix} 2 & 0  \\\\ -2 & 4  \\end{pmatrix} + \\begin{pmatrix} 2 & 3  \\\\ -1 & 1  \\end{pmatrix} = \\begin{pmatrix} 4 & 3  \\\\ -3 & 5  \\end{pmatrix}$$\n",
-    "    ```  \n",
-    "      \n",
-    "- If $A$ is the $m \\times n$ matrix with elements $a_{ij}$, the transpose of $A$ (denoted by $A'$) is the $n \\times m$ matrix with elements $a_{ij}' = a_{ji}$.  \n",
-    "  \n",
-    "    ```{admonition} Example  \n",
-    "    $$A=\\begin{pmatrix} 1 & 2 & 3 \\\\ 4 & 5 & 6  \\end{pmatrix} \\longrightarrow A'=\\begin{pmatrix} 1 & 4 \\\\ 2 & 5 \\\\ 3 & 6   \\end{pmatrix}$$\n",
-    "      \n",
-    "    $$B=\\begin{pmatrix} 1 & 2 \\end{pmatrix} \\longrightarrow B' = \\begin{pmatrix} 1 \\\\ 2 \\end{pmatrix}$$\n",
-    "    ```  \n",
-    "      \n",
-    "- **(Matrix multiplication)** Consider now that $A$ and $B$ are matrices with sizes $m \\times l$ and $l \\times n$ (number of columns of $A$ is equal to the number of rows of $B$). Then we can define $C = AB$  as the $m \\times n$ matrix with elements $c_{ij}$ formed by multiplying every element of the $i$-th row of $A$ with the corresponding element of the $j$-th column of $B$.  \n",
-    "    \n",
-    "    ```{admonition} Example  \n",
-    "      If $A=\\begin{pmatrix} 1 & 2  \\\\ -1 & 0  \\end{pmatrix}$ and $B=\\begin{pmatrix} 1 & 2 & 3  \\\\ 0 & -1 & 4  \\end{pmatrix}$, then:  \n",
-    "        \n",
-    "      $$C = AB = \\begin{pmatrix} 1 & 0 & 11  \\\\ -1 & -2 & -3  \\end{pmatrix}.$$  \n",
-    "        \n",
-    "      Note that the matrix $BA$ cannot be defined here since the number of columns of $B$ is different from the number of rows of $A$.\n",
-    "    ```  \n",
-    "      \n",
-    "    ```{warning}  \n",
-    "    If matrices $A$ and $B$ are *square matrices* (same number of rows and columns) then we can define both $AB$ and $BA$. But, in general, we **cannot assume** that $AB=BA$. For example, if  $A=\\begin{pmatrix} 2 & -1  \\\\ 4 & -2  \\end{pmatrix}$ and $B=\\begin{pmatrix} 1 & -1  \\\\ 2 & -2  \\end{pmatrix}$, then:  \n",
-    "      \n",
-    "    $$AB = \\begin{pmatrix} 0 & 0  \\\\ 0 & 0  \\end{pmatrix} \\quad and \\quad BA = \\begin{pmatrix} -2 & 1  \\\\ -4 & 2  \\end{pmatrix}$$\n",
-    "    ```  \n",
-    "      \n",
-    "- For matrix multiplication, the following properties are valid:  \n",
-    "    \n",
-    "    $$\\begin{align}\n",
-    "    (A+B)C&=AC+BC \\\\\n",
-    "    A(B+C)&=AB+AC \\\\\n",
-    "    (AB)C &= A(BC) \\\\\n",
-    "    A0 &= 0A = 0 \\end{align}$$  \n",
-    "  \n",
-    "- We can also define powers of $A$. If $k$ is a positive integer:  \n",
-    "    \n",
-    "    $$A^k = A^{k-1}A = AA^{k-1} = A\\cdot A\\cdot A \\cdots A\\ (\\mbox{product of}\\ k\\ \\mbox{matrices})$$\n"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "## Inverse matrix  \n",
-    "  \n",
-    "Suppose that $A$ is a $n \\times n$ matrix. If there exists a $n \\times n$ matrix $B$ so that  \n",
-    "  \n",
-    "$$ AB = BA = I_n,$$  \n",
-    "  \n",
-    "where $I=\\begin{pmatrix} 1 & 0 & \\cdots & 0 \\\\ 0 & 1 & \\cdots & 0 \\\\ \\vdots & & \\ddots & \\vdots \\\\ 0 & 0 & \\cdots & 1 \\end{pmatrix}$ is the $n \\times n$ *identity matrix* (with elements $1$ on the main diagonal and $0$ otherwise). \n",
-    "  \n",
-    "In this case we call the matrix $B$ the *inverse* of $A$ and denote $B=A^{-1}$. If $A$ has an inverse than it is called *invertible* or *non-singular*. If it does not have an inverse, $A$ is *singular*.\n",
-    "\n",
-    "Finding the inverse of $A$ automatically gives us the solution to the system of linear equations $A\\mathbf x = \\mathbf x$\n",
-    "\n",
-    "$$\\begin{align}  \n",
-    "A\\mathbf{x}&=\\mathbf{b} \\\\\n",
-    "A^{-1} A\\mathbf{x} &= A^{-1}\\mathbf{b} \\\\\n",
-    "\\mathbf{x} &= A^{-1}\\mathbf{b} \\end{align}$$  \n",
-    "\n",
-    "Finding ways of inverting matrices is therefore needed for a wide range of applications. A whole research field is dealing with optimising matrix inversions. \n",
-    "  \n",
-    "```{admonition} Example  \n",
-    "  \n",
-    "To find the inverse of matrix $A= \\begin{pmatrix} 2 & 5  \\\\ 1 & 3  \\end{pmatrix}$, we have to find the matrix $B = \\begin{pmatrix} b_{11} & b_{12}  \\\\ b_{21} & b_{22}  \\end{pmatrix}$ so that  \n",
-    "  \n",
-    "$$ AB = I \\quad (\\mbox{then check that }\\ BA = I.$$  \n",
-    "  \n",
-    "Thus  \n",
-    "  \n",
-    "$$\\begin{align}  \n",
-    "  \\begin{pmatrix} 2 & 5  \\\\ 1 & 3  \\end{pmatrix}\\begin{pmatrix} b_{11} & b_{12}  \\\\ b_{21} & b_{22}  \\end{pmatrix} = \\begin{pmatrix} 1 & 0  \\\\ 0 & 1  \\end{pmatrix} \\\\  \n",
-    "  \\begin{pmatrix} 2b_{11} + 5b_{21}  & 2b_{12} + 5b_{22} \\\\ b_{11} + 3b_{21} & b_{12} + 3b_{22}  \\end{pmatrix} = \\begin{pmatrix} 1 & 0  \\\\ 0 & 1  \\end{pmatrix}, \\\\\n",
-    "\\end{align}$$  \n",
-    "  \n",
-    "which leads to two systems of equations:  \n",
-    "\n",
-    "$$\\begin{cases} 2b_{11} + 5b_{21} &= 1 \\\\ b_{11} + 3b_{21} &= 0 \\end{cases}\\quad \\mbox{and}\\quad \\begin{cases} 2b_{12} + 5b_{22} &= 0 \\\\ b_{12} + 3b_{22} &= 1 \\end{cases}$$  \n",
-    "  \n",
-    "The solutions of these two systems, leads to $B= \\begin{pmatrix} 3 & -5  \\\\ -1 & 2  \\end{pmatrix}$\n",
-    "```  \n",
-    "  "
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "## Determinant  \n",
-    "  \n",
-    "If $A = \\begin{pmatrix} a_{11} & a_{12}  \\\\ a_{21} & a_{22}  \\end{pmatrix}$ then the expression:  \n",
-    "  \n",
-    "$$\\mbox{det}A = a_{11}a_{22} - a_{21}a_{12}$$   \n",
-    "  \n",
-    "given by the product of the elements on the main diagonal minus the product of the elements on secondary diagonal. The term $\\mbox{det}A$ is called the *determinant* of $A$. The determinant can be calculated for matrices of any size, but the expression gets increasingly more complicated for larger matrices.  \n",
-    "  \n",
-    "The determinant establishes a clear criterium for the existence of $A^{-1}$. The matrix $A$ will be invertible if and only if $\\mbox{det}A \\ne 0$.  \n",
-    "  \n",
-    "```{note}  \n",
-    "The previous criterium has an important implication. Consider the system of linear equations defined by $A\\mathbf{x}=\\mathbf{b}$ with equal number of equations and variables. If $\\mbox{det}A \\ne 0$ then the inverse of $A$ exists. Thus, if we multiply $A^{-1}$ to the left of each side of the equation defining the linear system, we have:  \n",
-    "  \n",
-    "$$\\begin{align}  \n",
-    "A\\mathbf{x}&=\\mathbf{b} \\\\\n",
-    "A^{-1}(A\\mathbf{x}) &= A^{-1}\\mathbf{b} \\\\\n",
-    "(A^{-1}A)\\mathbf{x} &= A^{-1}\\mathbf{b} \\\\\n",
-    "\\mathbf{x} &= A^{-1}\\mathbf{b} \\end{align}$$  \n",
-    "  \n",
-    "which means that if $\\mbox{det}A \\ne 0$ the system will have a **single solution** with the variables assuming the values $\\mathbf{x} = A^{-1}\\mathbf{b}$.\n",
-    "```"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "## Structured models  \n",
-    "  \n",
-    "In the models we have seen until now, all the individuals in a given population are identical. However, real populations have importance differences between individuals according to their life stages. For example, some insects have stages of eggs, larvae, pupae, and adults, while many plants pass through stages of seeds, seedlings, saplings, and mature individuals. Thus, understanding the differences in the dynamics of each life stage is important to add realism to our models (especially in applied models, such as pest control or harvesting management).  \n",
-    "  \n",
-    "Suppose our population is composed by two classes: *immature* and *mature* individuals. Let us assume non-overlapping breeding seasons and that the timescales for reproduction and maturation are similar. Thus, time can be counted in discrete steps corresponding to this reproduction/maturation interval. At time $t$ we have $X_t$ immature individuals and $Y_t$ mature individuals. Immature individuals survive until maturity with a probability $p$, while mature individuals reproduce and generate, on average, $m$ immature individuals for the next generation. Mature individuals die after reproduction. With these assumptions, we can calculate the sizes of each population class on the step $t+1$ as:\n",
-    "  \n",
-    "$$\\begin{align}\n",
-    "X_{t+1} = m Y_t \\\\\n",
-    "Y_{t+1} = p X_t \\end{align}$$  \n",
-    "  \n",
-    "See that now we have a model with two equations describing the variable class sizes in our population. Moreover, the dynamics of each class depends on the other, so that we have a system of *coupled* difference equations.  \n",
-    "\n",
-    "Interestingly, we can also write this system in matrix form, as:  \n",
-    "  \n",
-    "$$\\begin{pmatrix} X_{t+1}  \\\\ Y_{t+1} \\end{pmatrix} = \\begin{pmatrix} 0 & m  \\\\ p & 0  \\end{pmatrix} \\begin{pmatrix} X_{t}  \\\\ Y_{t} \\end{pmatrix}, \\quad \\mbox{or}$$\n",
-    "\n",
-    "$$\\mathbf{n_{t+1}} = P\\mathbf{n_{t}}$$\n",
-    "  \n",
-    "where the sum of the components of the population vector $\\mathbf{n_{t}}$, gives the total population (immature plus mature individuals) at time $t$: $N_t = X_t + Y_t$. The matrix $P=\\begin{pmatrix} 0 & m  \\\\ p & 0  \\end{pmatrix}$ is called *projection matrix* or *Leslie matrix*. Multiplying matrix $P$ by the population vector at time $t$, $\\mathbf{n_{t}}$, we can project the next population size at time $t+1$. Thus, from an initial population state, say $\\mathbf{n_{0}}$, the projection matrix governs the dynamics of the model.  \n",
-    "  \n",
-    "<br />\n",
-    "\n",
-    "This model can be easily generalised for a larger number of age classes. Suppose we now have $q+1$ classes, $X_0, X_1, X_2, \\ldots, X_q$, from the youngest (meaningful) age $X_0$ until the oldest age $X_q$. At each time step $t$, individuals from class $X_i$ have a chance of surviving to the next age class $X_{i+1}$, with probability $p_{i+1\\ i}$ (imagine an arrow for direction of age class progession $p_{i+1\\leftarrow i}$. We are going to omit the arrow for simplicity). Additionally, suppose that each age class $X_i$ excluding the youngest, will, in principle, be able to reproduce, with average fecundity $m_i$ generating new individuals for class $X_0$. Thus, with the age class sizes at time $t$, and the parameters for survival probabilities and average fecundities, we have:  \n",
-    "  \n",
-    "$$\\begin{align}\n",
-    "X_0(t+1) &= m_1X_1(t) +  m_2X_2(t) + m_3X_3(t) + \\ldots + m_qX_q(t) \\\\\n",
-    "X_1(t+1) &= p_{10}X_0(t) \\\\\n",
-    "X_2(t+1) &= p_{21}X_1(t) \\\\\n",
-    "X_3(t+1) &= p_{32}X_2(t) \\\\\n",
-    "         &\\vdots \\\\\n",
-    "X_q(t+1) &= p_{q\\ q-1}X_{q-1}(t)\n",
-    "    \\end{align}$$  \n",
-    "    \n",
-    "This can also be written in matrix form as:  \n",
-    "  \n",
-    "$$\\begin{pmatrix} X_0(t+1)  \\\\ X_1(t+1) \\\\X_2(t+1)  \\\\X_3(t+1) \\\\ \\vdots \\\\X_q(t+1)  \\\\ \\end{pmatrix} = \\begin{pmatrix} 0 & m_1 & m_2 & m_3 & \\cdots & m_q \\\\ p_{10} & 0 & 0 & 0 & \\cdots &0 \\\\ 0 & p_{21} & 0 & 0 & \\cdots & 0 \\\\ 0 & 0 & p_{32} & 0 & \\cdots & 0 \\\\ \\vdots & \\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\ 0 & 0 & 0 & \\cdots & p_{q\\ q-1} & 0\\end{pmatrix} \\begin{pmatrix} X_0(t)  \\\\ X_1(t) \\\\X_2(t) \\\\X_3(t) \\\\ \\vdots \\\\X_q(t)  \\\\ \\end{pmatrix}$$  \n",
-    "  \n",
-    "$$\\mathbf{n_{t+1}} = P\\mathbf{n_{t}}$$\n",
-    "  \n",
-    "The generalised projection matrix $P$ has then all the surviving probabilities just below the main diagonal, and the fecundities for each class in its $0$-th row.  \n",
-    "  \n",
-    "```{note}  \n",
-    "Two important differences here. First, the rows and columns for the matrix are indexed starting from $0$ until $q$, not starting from $1$ as usual. Second, the time steps are written in parentheses for each age class, instead of a subscript as we have seen before for other population models, so that it does not conflict with the index for the age classes (going from $0$ to $q$).\n",
-    "```  \n",
-    "  \n",
-    "```{admonition} Example  \n",
-    "Consider a population with three age classes $X_0, X_1,\\mbox{ and}\\ X_2$, and the following projection matrix  \n",
-    "  \n",
-    "$$P=\\begin{pmatrix} 0 & 5 & 10\\\\ 0.5 & 0 & 0  \\\\ 0 & 0.2 & 0 \\end{pmatrix}$$\n",
-    "\n",
-    "  \n",
-    "If the population starts with $10$ individuals in the oldest category at time $t=0$, the population vector at time $t=1$ will be:  \n",
-    "  \n",
-    "$$\\mathbf{n_{1}} = P\\mathbf{n_{0}} \\longrightarrow \\begin{pmatrix} X_0(1)  \\\\ X_1(1) \\\\X_2(1) \\end{pmatrix}=\\begin{pmatrix} 0 & 5 & 10\\\\ 0.5 & 0 & 0  \\\\ 0 & 0.2 & 0 \\end{pmatrix}\\begin{pmatrix} 0  \\\\ 0 \\\\ 10 \\end{pmatrix} \\longrightarrow \\begin{pmatrix} 100  \\\\ 0 \\\\ 0 \\end{pmatrix}$$  \n",
-    "  \n",
-    "The population age distribution at each time step will then be given be the successive application of the matrix $P$ to the previous population distribution. For the first 10 time steps we have:  \n",
-    "  \n",
-    "$$\\begin{pmatrix}0 \\\\ 0 \\\\ 10 \\end{pmatrix} \\rightarrow \\begin{pmatrix} 100  \\\\ 0 \\\\ 0 \\end{pmatrix} \\rightarrow \\begin{pmatrix} 0  \\\\ 50 \\\\ 0 \\end{pmatrix} \\rightarrow \\begin{pmatrix} 250  \\\\ 0 \\\\ 10 \\end{pmatrix} \\rightarrow \\begin{pmatrix} 100  \\\\ 125 \\\\ 0 \\end{pmatrix} \\rightarrow \\begin{pmatrix} 625  \\\\ 50 \\\\ 25 \\end{pmatrix} \\rightarrow \\begin{pmatrix} 500  \\\\ 312 \\\\ 10 \\end{pmatrix} \\rightarrow \\begin{pmatrix} 1662  \\\\ 250 \\\\ 62 \\end{pmatrix} \\rightarrow \\begin{pmatrix} 1875  \\\\ 831 \\\\ 50 \\end{pmatrix} \\rightarrow \\begin{pmatrix} 4656  \\\\ 937 \\\\ 166 \\end{pmatrix}$$  \n",
-    "  \n",
-    "Two important things are worth noting here. First, the total population size (sum accros all age classes), follows the sequence $N_t = \\left\\{ 10,100,50,260,225,700,822,1975,2756,5759,\\ldots \\right \\}$. The reproduction process does not always lead to increasing population sizes in the beginning, but is subject to size fluctuations due to the demographic distribution of this population. After a few steps increases in population size become more consistent. Second, as the population size continues to increase, the population age distribution tends to a given proportion of the population classes given by: $X_0 \\approx 76\\%, X_1 \\approx 22\\%, \\mbox{and}\\ X_2 \\approx 2\\%$.  \n",
-    "  \n",
-    "Now consider that, instead of $10$ individuals in the oldest class, we initiate the population with the following distribution $\\mathbf{n_{0}} = \\begin{pmatrix} 7.6  \\\\ 2.2 \\\\ 0.2 \\end{pmatrix}$. The population progression by successive application of the projection matrix will be  \n",
-    "  \n",
-    "$$\\begin{pmatrix} 7.6  \\\\ 2.2 \\\\ 0.2 \\end{pmatrix} \\longrightarrow \\begin{pmatrix} 13  \\\\ 4 \\\\ 0.4 \\end{pmatrix} \\longrightarrow \\begin{pmatrix} 23  \\\\ 6 \\\\ 0.76 \\end{pmatrix} \\longrightarrow \\begin{pmatrix} 40  \\\\ 12 \\\\ 1 \\end{pmatrix} \\longrightarrow \\begin{pmatrix} 72  \\\\ 20 \\\\ 2 \\end{pmatrix} \\longrightarrow \\begin{pmatrix} 124  \\\\ 36 \\\\ 4 \\end{pmatrix} \\longrightarrow \\begin{pmatrix} 219  \\\\ 62 \\\\ 7 \\end{pmatrix} \\longrightarrow \\begin{pmatrix} 381  \\\\ 109 \\\\ 12 \\end{pmatrix} $$  \n",
-    "  \n",
-    "You can check that although each age cathegory is increasing, the percentage contribution of each class to the total population remains nearly constant. To this distribution of classes we give the name of **stable age distribution**.  \n",
-    "  \n",
-    "Moreover, at each time step, each age class increases with approximately the same factor in relation to the previous step, around $\\lambda = 1.75$. If for each age class $X_i(t+1) = \\lambda X_i(t)$ is valid,then we have:  \n",
-    "  \n",
-    "$$\\mathbf{n_{t+1}}=\\lambda\\mathbf{n_{t}},$$  \n",
-    "  \n",
-    "so that we recover the usual model for exponential growth for the total population size $N_{t+1} = \\lambda N_t$.  \n",
-    "\n",
-    "```  "
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": null,
-   "metadata": {},
-   "outputs": [],
-   "source": []
-  }
- ],
- "metadata": {
-  "celltoolbar": "Tags",
-  "kernelspec": {
-   "display_name": "Python 3 (ipykernel)",
-   "language": "python",
-   "name": "python3"
-  },
-  "language_info": {
-   "codemirror_mode": {
-    "name": "ipython",
-    "version": 3
-   },
-   "file_extension": ".py",
-   "mimetype": "text/x-python",
-   "name": "python",
-   "nbconvert_exporter": "python",
-   "pygments_lexer": "ipython3",
-   "version": "3.11.5"
-  }
- },
- "nbformat": 4,
- "nbformat_minor": 2
-}
diff --git a/_sources/mathscourse/lecture05/05_From_table_arrangements_to_powerful_tools_part2.ipynb b/_sources/mathscourse/lecture05/05_From_table_arrangements_to_powerful_tools_part2.ipynb
deleted file mode 100644
index 78a8425c..00000000
--- a/_sources/mathscourse/lecture05/05_From_table_arrangements_to_powerful_tools_part2.ipynb
+++ /dev/null
@@ -1,396 +0,0 @@
-{
- "cells": [
-  {
-   "cell_type": "code",
-   "execution_count": 40,
-   "metadata": {
-    "tags": [
-     "remove-cell"
-    ]
-   },
-   "outputs": [
-    {
-     "data": {
-      "image/png": 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\n",
-      "text/plain": [
-       "<Figure size 360x360 with 1 Axes>"
-      ]
-     },
-     "metadata": {
-      "needs_background": "light"
-     },
-     "output_type": "display_data"
-    }
-   ],
-   "source": [
-    "import matplotlib.pyplot as plt\n",
-    "import matplotlib.ticker as plticker\n",
-    "import numpy as np\n",
-    "\n",
-    "X = np.array((0,0,0,0))\n",
-    "Y= np.array((0,0,0,0))\n",
-    "U = np.array((3,2,1,8))\n",
-    "V = np.array((-1,3,7,12))\n",
-    "\n",
-    "fig, ax = plt.subplots(figsize=(5, 5))\n",
-    "q = ax.quiver(X, Y, U, V,units='xy', scale=1)\n",
-    "\n",
-    "ax.spines['left'].set_position('zero')\n",
-    "ax.spines['bottom'].set_position('zero')\n",
-    "\n",
-    "# Eliminate upper and right axes\n",
-    "ax.spines['right'].set_color('none')\n",
-    "ax.spines['top'].set_color('none')\n",
-    "\n",
-    "# Show ticks in the left and lower axes only\n",
-    "ax.xaxis.set_ticks_position('bottom')\n",
-    "ax.yaxis.set_ticks_position('left')\n",
-    "\n",
-    "ax.set_aspect('equal')\n",
-    "\n",
-    "ax.annotate('$\\mathbf{v_1}$', (3+0.2,-1-0.8),fontsize=14)\n",
-    "ax.annotate('$\\mathbf{v_2}$', (2+0.2,3-0.8),fontsize=14)\n",
-    "ax.annotate('$A\\mathbf{v_1}$', (1+0.2,7-0.8),fontsize=14)\n",
-    "ax.annotate('$A\\mathbf{v_2}$', (8+0.2,12-0.8),fontsize=14)\n",
-    "\n",
-    "ax.tick_params(axis='both', which='major', labelsize=13)\n",
-    "\n",
-    "loc = plticker.MultipleLocator(base=2) # this locator puts ticks at regular intervals\n",
-    "ax.xaxis.set_major_locator(loc)\n",
-    "\n",
-    "ax.set_ylim(-2,14)\n",
-    "ax.set_xlim(-2,10)\n",
-    "\n",
-    "plt.tight_layout()\n",
-    "plt.savefig('lin_trans.png')"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 46,
-   "metadata": {
-    "tags": [
-     "remove-cell"
-    ]
-   },
-   "outputs": [
-    {
-     "data": {
-      "image/png": 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\n",
-      "text/plain": [
-       "<Figure size 360x360 with 1 Axes>"
-      ]
-     },
-     "metadata": {
-      "needs_background": "light"
-     },
-     "output_type": "display_data"
-    }
-   ],
-   "source": [
-    "import matplotlib.pyplot as plt\n",
-    "import matplotlib.ticker as plticker\n",
-    "import numpy as np\n",
-    "\n",
-    "X = np.array((0,0))\n",
-    "Y= np.array((0,0))\n",
-    "U = np.array((1,2/3))\n",
-    "V = np.array((-1,1))\n",
-    "\n",
-    "fig, ax = plt.subplots(figsize=(5, 5))\n",
-    "q = ax.quiver(X, Y, U, V,units='xy', scale=1)\n",
-    "\n",
-    "ax.spines['left'].set_position('zero')\n",
-    "ax.spines['bottom'].set_position('zero')\n",
-    "\n",
-    "# Eliminate upper and right axes\n",
-    "ax.spines['right'].set_color('none')\n",
-    "ax.spines['top'].set_color('none')\n",
-    "\n",
-    "# Show ticks in the left and lower axes only\n",
-    "ax.xaxis.set_ticks_position('bottom')\n",
-    "ax.yaxis.set_ticks_position('left')\n",
-    "\n",
-    "ax.set_aspect('equal')\n",
-    "\n",
-    "ax.annotate('$\\mathbf{v_1}\\ (\\lambda=-1)$', (1+0.1,-1-0.1),fontsize=14)\n",
-    "ax.annotate('$\\mathbf{v_2}\\ (\\lambda=4)$', (2/3+0.1,1+0.1),fontsize=14)\n",
-    "\n",
-    "ax.tick_params(axis='both', which='major', labelsize=13)\n",
-    "\n",
-    "#loc = plticker.MultipleLocator(base=2) # this locator puts ticks at regular intervals\n",
-    "#ax.xaxis.set_major_locator(loc)\n",
-    "\n",
-    "ax.set_ylim(-2,2)\n",
-    "ax.set_xlim(-2,2)\n",
-    "\n",
-    "plt.tight_layout()\n",
-    "plt.savefig('eigenvec_01.png')"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "# 05 - From table arrangements to powerful tools - part 2\n",
-    "  "
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "## Linear transformations\n",
-    "\n",
-    "In general terms, a given matrix $A$ is a representation of what we call a *linear transformation*, a function from a special kind of set called *vector space* into another vector space. Vector spaces are generalised sets with particular properties, but for the moment you can think of two-dimensional and three-dimensional cartesian spaces as usual vector spaces. Then vectors are gonna be represented as arrays of real numbers as before: $\\mathbf{n_{0}} = \\begin{pmatrix} 7.6  \\\\ 2.2  \\end{pmatrix}$ or $\\mathbf{v} = \\begin{pmatrix} x \\\\ y \\\\ z \\end{pmatrix}$.  \n",
-    "  \n",
-    "Back to linear transformations, they are usually denoted as:  \n",
-    "  \n",
-    "$$\\begin{align}\n",
-    "A:V &\\longrightarrow W \\\\\n",
-    "  \\mathbf{v} &\\rightarrow A(\\mathbf{v}) \\end{align}$$\n",
-    "  \n",
-    "where $V$ and $W$ are the domain and codomain vector spaces, and the transformation $A$ is a function that projects vectors $\\mathbf{v}$ into the images $A(\\mathbf{v})$. The property that defines linear transformations is the following:  \n",
-    "  \n",
-    "$$A(a\\mathbf{v}+b\\mathbf{w})=aA(\\mathbf{v})+bA(\\mathbf{w}),$$  \n",
-    "  \n",
-    "where $a$ and $b$ are constants. In other words,for a linear transformation, the image of the transformation $A$ of a linear combination of two vectors ($\\mathbf{v}$ and $\\mathbf{w}$) is equal to the linear combination of the images of this vectors by $A$.  \n",
-    "  \n",
-    "If we represent vectors with their components in a cartesian system, the images of the transformation $A$ will be given be the multiplication of the matrix representation of $A$ on the vectors.  \n",
-    "  \n",
-    "````{admonition} Example\n",
-    "\n",
-    "Given the matrix $A = \\begin{pmatrix} 1 & 2 \\\\ 3 & 2 \\end{pmatrix}$ and the vectors $\\mathbf{v_1} = \\begin{pmatrix} 3  \\\\ -1  \\end{pmatrix}$ and $\\mathbf{v_2} = \\begin{pmatrix} 2  \\\\ 3  \\end{pmatrix}$, then we have:  \n",
-    "  \n",
-    "$$\\begin{align} A\\mathbf{v_1} &= \\begin{pmatrix} 1 & 2 \\\\ 3 & 2 \\end{pmatrix}\\begin{pmatrix} 3  \\\\ -1  \\end{pmatrix} = \\begin{pmatrix} 1  \\\\ 7  \\end{pmatrix} \\\\  \n",
-    "  A\\mathbf{v_2} &= \\begin{pmatrix} 1 & 2 \\\\ 3 & 2 \\end{pmatrix}\\begin{pmatrix} 2  \\\\ 3  \\end{pmatrix} = \\begin{pmatrix} 8  \\\\ 12  \\end{pmatrix} = 4\\begin{pmatrix} 2  \\\\ 3  \\end{pmatrix} \\end{align}$$  \n",
-    "    \n",
-    "The vectors and their respective transformations are given by:  \n",
-    "  \n",
-    "```{image} lin_trans.png\n",
-    "```  \n",
-    "  \n",
-    "We see that under the transformation $A$, vector $\\mathbf{v_1}$ gets rotated (counterclockwise) and stretched, while vector $\\mathbf{v_2}$ only gets stretched without changing the direction.\n",
-    "\n",
-    "````  \n",
-    "  \n",
-    "## Eigenvalues and eigenvectors  \n",
-    "\n",
-    "  \n",
-    "As seen on the previous example, there might be some vectors that upon application of $A$ do not have their direction changed, only their magnitude (or *module* of the vector). For these vectors, we have, in general:  \n",
-    "  \n",
-    "$$A\\mathbf{v} = \\lambda\\mathbf{v}, \\quad \\mbox{for some constant }\\ \\lambda$$  \n",
-    "  \n",
-    "But this is equivalent to:  \n",
-    "  \n",
-    "$$\\begin{align}\n",
-    "A\\mathbf{v} - \\lambda\\mathbf{v} = 0 \\\\  \n",
-    "A\\mathbf{v} - \\lambda I\\mathbf{v} = 0 \\\\\n",
-    "\\left(A - \\lambda I \\right)\\mathbf{v} = 0 \\end{align},$$  \n",
-    "  \n",
-    "where $I$ is the identity matrix. The equation $\\left(A - \\lambda I \\right)\\mathbf{v} = 0$ defines a system of linear equations where $A - \\lambda I$ is the coefficient matrix, $\\mathbf{v}$ is the vector of variables, and $\\mathbf{b}=0$. See that since $\\mathbf{b}=0$ we already know one possible solution for this system, which is $\\mathbf{v}=0$ (all the variables equal to $0$). Therefore, either this system has an unique solution (the null solution) or it has infinite solutions, but definitely it is not inconsistent.  Thus, if we want to know all possible non-zero vectors $\\mathbf{v}$ that satisfy $A\\mathbf{v} = \\lambda\\mathbf{v}$, we have to impose that the system assumes infinite solutions. This criterium should be defined by the determinant of the matrix of coefficients, as:  \n",
-    "  \n",
-    "$$\\mbox{det}(A-\\lambda I) = 0.$$  \n",
-    "  \n",
-    "  \n",
-    "The determinant then defines a polynomial in the variable $\\lambda$, $\\mbox{det}(A-\\lambda I)=p(\\lambda)$, called *characteristic polynomial*. Since we are interested in the condition $p(\\lambda)=0$, what we want to know are the roots $\\lambda$ of this polynomial.  \n",
-    "  \n",
-    "The solutions $\\lambda$ for these equations are called **eigenvalues** and their corresponding vectors are **eigenvectors**. In summary, the eigenvectors are all the vectors that, upon application of $A$, remain on the same direction, only changing magnitude, with the eigenvalue being the stretching/compressing factor. \n",
-    "  \n",
-    "````{admonition} Example  \n",
-    "\n",
-    "Let us calculate the eigenvalues and eigenvectors of $A = \\begin{pmatrix} 1 & 2 \\\\ 3 & 2 \\end{pmatrix}$. First we need to obtain matrix $(A-\\lambda I)$ and to calculate its determinant. We have:  \n",
-    "  \n",
-    "$$A-\\lambda I = \\begin{pmatrix} 1 & 2 \\\\ 3 & 2 \\end{pmatrix} - \\lambda \\begin{pmatrix} 1 & 0 \\\\ 0 & 1 \\end{pmatrix} = \\begin{pmatrix} 1 -\\lambda & 2 \\\\ 3 & 2 -\\lambda  \\end{pmatrix}$$  \n",
-    "  \n",
-    "The characteristic polynomial will be given by $p(\\lambda)=\\mbox{det}(A-\\lambda I)$, and applying the rule for the calculus of determinants of matrices size 2, we have:  \n",
-    "  \n",
-    "$$p(\\lambda)=\\mbox{det}(A-\\lambda I) = \\mbox{det}\\begin{pmatrix} 1 -\\lambda & 2 \\\\ 3 & 2 -\\lambda  \\end{pmatrix} = (1 -\\lambda)(2 -\\lambda) - 3\\cdot 2 $$  \n",
-    "  \n",
-    "$$p(\\lambda) = 2 -\\lambda -2\\lambda +\\lambda^2 - 6 = \\lambda^2 - 3\\lambda -4.$$  \n",
-    "  \n",
-    "Thus, calculating the roots of this 2nd-degree polynomial, we find $\\lambda_1=-1$ and $\\lambda_2=4$. For each of the eigenvalues, to obtain the eigenvectors we have to solve the linear system  $A\\mathbf{v} = \\lambda\\mathbf{v}$ for some general vector $\\mathbf{v} = \\begin{pmatrix} x  \\\\ y  \\end{pmatrix}$ (with coordinates $x$ and $y$ to be specified). Thus:  \n",
-    "  \n",
-    "- $\\mathbf{\\lambda_1 = -1}$:  \n",
-    "    \n",
-    "    $\\begin{align}\n",
-    "A\\mathbf{v} = \\lambda\\mathbf{v} \\longrightarrow \\begin{pmatrix} 1 & 2 \\\\ 3 & 2 \\end{pmatrix}\\begin{pmatrix} x  \\\\ y \\end{pmatrix}= -1\\begin{pmatrix} x  \\\\ y \\end{pmatrix}\\longrightarrow \\cases{x + 2y = -x \\\\ 3x + 2y = -y} \\longrightarrow \\cases{2x + 2y = 0 \\\\ 3x + 3y = 0}\n",
-    "\\end{align}$  \n",
-    "    \n",
-    "    Note that both equations correspond to the same linear equation $x + y = 0$, which is exactly what we would expect since we built the system to be underdetermined when we did $\\mbox{det}(A-\\lambda I) = 0$. The eigenvector corresponding to $\\lambda_1$ will be such that $x + y = 0$ or $x = -y$. Thus, we can choose $\\mathbf{v_1} = \\begin{pmatrix} 1  \\\\ -1  \\end{pmatrix}$.  \n",
-    "    \n",
-    "    ```{note}  \n",
-    "    Remember that infinite solutions are possible for the same system. The important is that the condition $x = -y$ holds. In fact, multiplying a non-zero constant for the vector $\\begin{pmatrix} 1  \\\\ -1  \\end{pmatrix}$ would still give a vector in the same direction. So if $\\mathbf{v_1}$ is an eigenvector, $k\\mathbf{v_1}\\ (k \\in {\\rm I\\!R})$ also is.  \n",
-    "    \n",
-    "    ```  \n",
-    "    <br />  \n",
-    "    \n",
-    "    \n",
-    "- $\\mathbf{\\lambda_1 = 4}$:  \n",
-    "    \n",
-    "    $\\begin{align}\n",
-    "A\\mathbf{v} = \\lambda\\mathbf{v} \\longrightarrow \\begin{pmatrix} 1 & 2 \\\\ 3 & 2 \\end{pmatrix}\\begin{pmatrix} x  \\\\ y \\end{pmatrix}= 4\\begin{pmatrix} x  \\\\ y \\end{pmatrix}\\longrightarrow \\cases{x + 2y = 4x \\\\ 3x + 2y = 4y} \\longrightarrow \\cases{-3x + 2y = 0 \\\\ 3x - 2y = 0}\n",
-    "\\end{align}$  \n",
-    "    \n",
-    "    Since the eigenvector needs to satisfy $3x=2y$, we choose $\\mathbf{v_2} = \\begin{pmatrix} 2/3  \\\\ 1  \\end{pmatrix}$.  \n",
-    "    \n",
-    "    The representation of these vectors on the cartesian plane will be:  \n",
-    "      \n",
-    "```{image} eigenvec_01.png\n",
-    "```\n",
-    "\n",
-    "````  \n",
-    "  \n",
-    "  \n",
-    "  \n",
-    "````{admonition} Example  \n",
-    "  \n",
-    "Let us calculate the eigenvalues and eigenvectors of matrix $A = \\begin{pmatrix} -2 & 1 \\\\ 0 & -1 \\end{pmatrix}$. We have:  \n",
-    "  \n",
-    "$$p(\\lambda)=\\mbox{det}(A-\\lambda I) = \\mbox{det}\\begin{pmatrix} -2 -\\lambda & 1 \\\\ 0 & -1 -\\lambda  \\end{pmatrix} = (-2 -\\lambda)(-1 -\\lambda) - 1\\cdot 0 $$  \n",
-    "  \n",
-    "$$p(\\lambda)=(-2 -\\lambda)(-1 -\\lambda)$$  \n",
-    "  \n",
-    "Since we need $p(\\lambda)=0$ either one of the two factors has to be zero. So, $\\lambda_1=-2$ and $\\lambda_2=-1$.   \n",
-    "For the eigenvectors:  \n",
-    "  \n",
-    "- $\\mathbf{\\lambda_1 = -2}$:  \n",
-    "    \n",
-    "    $\\begin{align}\n",
-    "A\\mathbf{v} = \\lambda\\mathbf{v} \\longrightarrow \\begin{pmatrix} -2 & 1 \\\\ 0 & -1 \\end{pmatrix}\\begin{pmatrix} x  \\\\ y \\end{pmatrix}= -2\\begin{pmatrix} x  \\\\ y \\end{pmatrix}\\longrightarrow \\cases{-2x + y = -2x \\\\ -y = -2y} \\longrightarrow \\cases{y = 0 \\\\ y = 0}\n",
-    "\\end{align}$  \n",
-    "    \n",
-    "    Since $y=0$, we choose $\\mathbf{v_1} = \\begin{pmatrix} 1  \\\\ 0  \\end{pmatrix}$.  \n",
-    "      \n",
-    "- $\\mathbf{\\lambda_2 = -1}$:  \n",
-    "    \n",
-    "    $\\begin{align}\n",
-    "A\\mathbf{v} = \\lambda\\mathbf{v} \\longrightarrow \\begin{pmatrix} -2 & 1 \\\\ 0 & -1 \\end{pmatrix}\\begin{pmatrix} x  \\\\ y \\end{pmatrix}= -1\\begin{pmatrix} x  \\\\ y \\end{pmatrix}\\longrightarrow \\cases{-2x + y = -x \\\\ -y = -y} \\longrightarrow \\cases{-x + y = 0 \\\\ y = y}\n",
-    "\\end{align}$  \n",
-    "    \n",
-    "    The second equation is trivial and provides no information. From the first equation, since $x=y$, we choose $\\mathbf{v_2} = \\begin{pmatrix} 1  \\\\ 1  \\end{pmatrix}$.  \n",
-    "      \n",
-    "```{tip}  \n",
-    "  \n",
-    "Note that the two eigenvalues $\\lambda_1 = -2$ and $\\lambda_2 = -1$ correspond to the elements of the main diagonal of the matrix $A$. Whenever the matrix $A$ is *triangular*, meaning all the elements *below* the main diagonal are $0$, the eigenvalues will correspond to the elements of the main diagonal.  \n",
-    "  \n",
-    "```\n",
-    "  \n",
-    "\n",
-    "````\n",
-    "                             "
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "## Vector decomposition  \n",
-    "  \n",
-    "Now suppose matrix $A$ has eigenvalues $\\lambda_1$ and $\\lambda_2$, with corresponding eigenvectors $\\mathbf{u_1}$ and $\\mathbf{u_2}$. If $\\lambda_1 \\ne \\lambda_2$, then the vectors $\\mathbf{u_1}$ and $\\mathbf{u_2}$ are *linearly independent* meaning one can not be written as a multiple (multiplied by a real constant) of the other. Geometrically, this means that the directions corresponding to this two eigenvectors are not on the same line.  \n",
-    "  \n",
-    "In two dimensions, this is enough to say that any given vector $\\mathbf{v}$ can be written as a linear combination of $\\mathbf{u_1}$ and $\\mathbf{u_2}$ (or that $\\mathbf{u_1}$ and $\\mathbf{u_2}$ consitute a *basis* of this space). We have:  \n",
-    "  \n",
-    "$$\\mathbf{v}= a_1\\mathbf{u_1} + a_2\\mathbf{u_2},$$  \n",
-    "  \n",
-    "for constants $a_1$ and $a_2$ to be determined.  \n",
-    "  \n",
-    "But given that $A$ is a linear transformation, it should hold that:  \n",
-    "  \n",
-    "$$A\\mathbf{v} = A(a_1\\mathbf{u_1} + a_2\\mathbf{u_2}) = a_1A(\\mathbf{u_1}) + a_2A(\\mathbf{u_2}) = a_1\\lambda_1\\mathbf{u_1} + a_2\\lambda_2\\mathbf{u_2},$$  \n",
-    "  \n",
-    "with the last step coming from the fact that $\\mathbf{u_1}$ and $\\mathbf{u_2}$ are eigenvectors of $A$. Since $A\\mathbf{v}$ is also a vector in this space, we can go ahead and apply the transformation $A$ again, and we will have:  \n",
-    "  \n",
-    "$$A(A\\mathbf{v}) = A(a_1\\lambda_1\\mathbf{u_1} + a_2\\lambda_2\\mathbf{u_2})=a_1\\lambda_1 A(\\mathbf{u_1}) + a_2\\lambda_2 A(\\mathbf{u_2}) = a_1\\lambda_1^2\\mathbf{u_1} + a_2\\lambda_2^2\\mathbf{u_2}.$$\n",
-    "  \n",
-    "  \n",
-    "So, in general:  \n",
-    "  \n",
-    "  \n",
-    "$$A(A(A \\cdots A\\mathbf{v}) \\cdots )) = A^n\\mathbf{v} = a_1\\lambda_1^n\\mathbf{u_1} + a_2\\lambda_2^n\\mathbf{u_2},$$  \n",
-    "or, in other words, if we know the eigenvectors and eigenvalues of $A$ (with $\\lambda_1 \\ne \\lambda_2$), and we know the coeffiencients $a_1$ and $a_2$ of the expansion of $\\mathbf{v}$ into the eigenvectors, then it is easy to calculate what is going to be the image ov vector $\\mathbf{v}$ by any number of successive applications of $A$. This is also equivalent of applying the $n$-th power of matrix $A$ to $\\mathbf{v}$.  \n",
-    "  \n",
-    "This fact will have many important applications, as we will se next.  \n",
-    "  \n"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "## Structured models revisited  \n",
-    "  \n",
-    "Let us considered a structured population model defined by the projection matrix $P=\\begin{pmatrix} 1.5 & 2  \\\\ 0.08 & 0  \\end{pmatrix}$ and an initial population vector given by $\\mathbf{n_0} = \\begin{pmatrix} 105  \\\\ 1 \\end{pmatrix}$. If we calculate the eigenvalues and eigenvectors, we will have:  \n",
-    "  \n",
-    "$$p(\\lambda)=\\mbox{det}(P-\\lambda I) = \\mbox{det}\\begin{pmatrix} 1.5 -\\lambda & 2 \\\\ 0.08 & -\\lambda  \\end{pmatrix} = (1.5 -\\lambda)(-\\lambda) - 0.08\\cdot 2 $$  \n",
-    "  \n",
-    "$$p(\\lambda)=\\lambda^2-1.5\\lambda-0.16=(\\lambda-1.6)(\\lambda+0.1)$$  \n",
-    "  \n",
-    "Thus, we have $\\lambda_1=1.6$ and $\\lambda_2=-0.1$.  \n",
-    "  \n",
-    "- $\\mathbf{\\lambda_1 = 1.6}$:  \n",
-    "    \n",
-    "    $\\begin{align}\n",
-    "A\\mathbf{v} = \\lambda\\mathbf{v} \\longrightarrow \\begin{pmatrix} 1.5 & 2  \\\\ 0.08 & 0 \\end{pmatrix}\\begin{pmatrix} x  \\\\ y \\end{pmatrix}= 1.6\\begin{pmatrix} x  \\\\ y \\end{pmatrix}\\longrightarrow \\cases{1.5x + 2y = 1.6x \\\\ 0.08x = 1.6y} \\longrightarrow \\cases{x - 20y = 0 \\\\ x - 20y = 0}\n",
-    "\\end{align}$  \n",
-    "    \n",
-    "    Since $x=20y$, we choose $\\mathbf{u_1} = \\begin{pmatrix} 20  \\\\ 1  \\end{pmatrix}$.  \n",
-    "  \n",
-    "  <br />\n",
-    "  \n",
-    "- $\\mathbf{\\lambda_2 = -0.1}$:  \n",
-    "    \n",
-    "    $\\begin{align}\n",
-    "A\\mathbf{v} = \\lambda\\mathbf{v} \\longrightarrow \\begin{pmatrix} 1.5 & 2  \\\\ 0.08 & 0 \\end{pmatrix}\\begin{pmatrix} x  \\\\ y \\end{pmatrix}= -0.1\\begin{pmatrix} x  \\\\ y \\end{pmatrix}\\longrightarrow \\cases{1.5x + 2y = -0.1x \\\\ 0.08x = -0.1y} \\longrightarrow \\cases{0.8x + y = 0 \\\\ 0.08x + 0.1y = 0}\n",
-    "\\end{align}$  \n",
-    "    \n",
-    "    Since $-\\displaystyle\\frac{4}{5}x=y$, we choose $\\mathbf{u_2} = \\begin{pmatrix} 5  \\\\ -4  \\end{pmatrix}$.  \n",
-    "\n",
-    "<br />\n",
-    "      \n",
-    "We can also note that the initial population vector can be decomposed into the eigenvectors $\\mathbf{u_1}$ and $\\mathbf{u_2}$ of the matrix $P$ according to:  \n",
-    "  \n",
-    "$$\\mathbf{n_0}=\\begin{pmatrix} 105  \\\\ 1 \\end{pmatrix} = 5\\begin{pmatrix} 20  \\\\ 1 \\end{pmatrix} + \\begin{pmatrix} 5  \\\\ -4 \\end{pmatrix} = 5\\mathbf{u_1} + \\mathbf{u_2}$$  \n",
-    "  \n",
-    "We know that to find the population vector for the following step, we have to apply the projection matrix on the current population vector. Thus:  \n",
-    "  \n",
-    "$$\\mathbf{n_1} = P\\mathbf{n_0}= P(5\\mathbf{u_1} + \\mathbf{u_2})=5P(\\mathbf{u_1}) + P(\\mathbf{u_2})=5(1.6)\\begin{pmatrix} 20  \\\\ 1 \\end{pmatrix} + (-0.1)\\begin{pmatrix} 5  \\\\ -4 \\end{pmatrix} = \\begin{pmatrix} 159.5  \\\\ 8.4 \\end{pmatrix}$$  \n",
-    "  \n",
-    "If we perform successive applications of $P$, we will finally have:  \n",
-    "  \n",
-    "$$\\mathbf{n_t} = P^t\\mathbf{n_0} = P^t(5\\mathbf{u_1} + \\mathbf{u_2})=5(1.6)^t\\begin{pmatrix} 20  \\\\ 1 \\end{pmatrix} + (-0.1)^t\\begin{pmatrix} 5  \\\\ -4 \\end{pmatrix}.$$  \n",
-    "  \n",
-    "<br />  \n",
-    "  \n",
-    "By decomposing into eigenvalues and eigenvectors, we know the behaviour of our population for any time $t$, which is much easier than applying $P$ a number $t$ of times, or than calculating $P^t$. Furthermore, as $t\\rightarrow\\infty$ the first term will dominate (with the second term dying out) and the proportions for each age class in this population will be dictated by the vector $\\begin{pmatrix} 20  \\\\ 1 \\end{pmatrix}$ (or, if we divide by the sum of the components: $\\displaystyle\\frac{1}{21}\\begin{pmatrix} 20  \\\\ 1 \\end{pmatrix} = \\begin{pmatrix} 95\\%  \\\\ 5\\% \\end{pmatrix}$.  \n",
-    "  \n",
-    "In general the eigenvector corresponding to the **largest eigenvalue** (also called *dominant eigenvalue*) of the projection matrix, which is always *real* and *positive*, will correspond to the stable age distribution, for which our population vector converges after we wait enough time.  \n",
-    "  \n",
-    "As we have seen, after enough time, the dominant eigenvalue will also correspond to the growth factor for each of the age classes. Thus, if $\\lambda_1>1$ the population size will increase, whereas if $0<\\lambda_1<1$ the population size will decrease.\n",
-    "  \n"
-   ]
-  }
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diff --git a/_sources/mathscourse/lecture06/06_Analysing_change_continuously.ipynb b/_sources/mathscourse/lecture06/06_Analysing_change_continuously.ipynb
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--- a/_sources/mathscourse/lecture06/06_Analysing_change_continuously.ipynb
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@@ -1,507 +0,0 @@
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\n",
-      "text/plain": [
-       "<Figure size 864x216 with 1 Axes>"
-      ]
-     },
-     "metadata": {
-      "needs_background": "light"
-     },
-     "output_type": "display_data"
-    }
-   ],
-   "source": [
-    "import numpy as np\n",
-    "import pandas as pd\n",
-    "import matplotlib.pyplot as plt\n",
-    "\n",
-    "def multiple_formatter(denominator=2, number=np.pi, latex='\\pi'):\n",
-    "    def gcd(a, b):\n",
-    "        while b:\n",
-    "            a, b = b, a%b\n",
-    "        return a\n",
-    "    def _multiple_formatter(x, pos):\n",
-    "        den = denominator\n",
-    "        num = np.int(np.rint(den*x/number))\n",
-    "        com = gcd(num,den)\n",
-    "        (num,den) = (int(num/com),int(den/com))\n",
-    "        if den==1:\n",
-    "            if num==0:\n",
-    "                return r'$0$'\n",
-    "            if num==1:\n",
-    "                return r'$%s$'%latex\n",
-    "            elif num==-1:\n",
-    "                return r'$-%s$'%latex\n",
-    "            else:\n",
-    "                return r'$%s%s$'%(num,latex)\n",
-    "        else:\n",
-    "            if num==1:\n",
-    "                return r'$\\frac{%s}{%s}$'%(latex,den)\n",
-    "            elif num==-1:\n",
-    "                return r'$\\frac{-%s}{%s}$'%(latex,den)\n",
-    "            else:\n",
-    "                return r'$\\frac{%s%s}{%s}$'%(num,latex,den)\n",
-    "    return _multiple_formatter\n",
-    "\n",
-    "class Multiple:\n",
-    "    def __init__(self, denominator=2, number=np.pi, latex='\\pi'):\n",
-    "        self.denominator = denominator\n",
-    "        self.number = number\n",
-    "        self.latex = latex\n",
-    "\n",
-    "    def locator(self):\n",
-    "        return plt.MultipleLocator(self.number / self.denominator)\n",
-    "\n",
-    "    def formatter(self):\n",
-    "        return plt.FuncFormatter(multiple_formatter(self.denominator, self.number, self.latex))\n",
-    "\n",
-    "#xt=np.linspace(-4*np.pi,4*np.pi,num=500)\n",
-    "xt=np.linspace(0,52,num=1000)\n",
-    "##\n",
-    "fig, ax = plt.subplots(figsize=(12, 3))\n",
-    "\n",
-    "fun = 5 + 5*np.cos((np.pi/12)*(xt-3))\n",
-    "dfun = -(5*np.pi/12)*np.sin((np.pi/12)*(xt-3))\n",
-    "\n",
-    "ax.plot(xt, fun, color='b', lw=2, label='$f\\,(t)$')\n",
-    "ax.plot(xt, dfun, color='g', lw=2, label=\"$f\\,'\\,(t)$\")\n",
-    "ax.tick_params(axis='both', labelsize= 15)\n",
-    "ax.set_xlabel('$t$', fontsize = 16)\n",
-    "ax.legend(fontsize=14)  \n",
-    "\n",
-    "ax.axhline(0, color='black', lw=1)\n",
-    "ax.axvline(0, color='black', lw=1)\n",
-    "\n",
-    "ax.set_xlim(0,55)\n",
-    "ax.set_ylim(-2,12)\n",
-    "\n",
-    "#ax.xaxis.set_major_locator(plt.MultipleLocator(np.pi / 2))\n",
-    "#ax.xaxis.set_minor_locator(plt.MultipleLocator(np.pi / 12))\n",
-    "#ax.xaxis.set_major_formatter(plt.FuncFormatter(multiple_formatter()))\n",
-    "\n",
-    "fig.tight_layout()\n",
-    "fig.savefig('func_diff.png')"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "de6827dc",
-   "metadata": {},
-   "source": [
-    "# 06 - Analysing change, continuously\n"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "c8e845ed",
-   "metadata": {},
-   "source": [
-    "## Average rate of change\n",
-    "\n",
-    "Watch this [video](https://web.microsoftstream.com/video/2953e661-e1c1-43ff-ba78-ca52099ed1ec)\n",
-    "for a short introduction to this lecture."
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "330c1e5e",
-   "metadata": {},
-   "source": [
-    "## Instantaneous rate of change  \n",
-    "  \n",
-    "  \n",
-    "\n",
-    "If we have a continuous function $f(x)$, for each two points $x_{0}$ and $x_{1}$, we can define the average rate of change between these points by the ratio between how much $f$ has changed and how much $x$ has changed:  \n",
-    "  \n",
-    "$$\\frac{\\Delta f(x)}{\\Delta x} = \\frac{f(x_{1}) - f(x_{0})}{x_{1} - x_{0}}$$  \n",
-    "  \n",
-    "For points $x_{0}$ and $x_{1}$ this can be graphically represented by:  \n",
-    "  \n",
-    "```{image} sec_tan.png\n",
-    "```  \n",
-    "  \n",
-    "The line that passes through the points $(x_0, f(x_0))$ and $(x_1, f(x_1))$ (yellow line) is named *secant line*. As we have seen before in our treatment of linear functions, the slope of the secant line will be given by the average rate of change between this points (remember that the slope for linear functions is defined as $m=\\displaystyle\\frac{\\Delta y}{\\Delta x})$.\n",
-    "  \n",
-    "Now consider that we change the value of $x_1$ so that it slowly approaches $x_0$ from the right. In this case, the secant lines also change (from yellow to red), as we change the image of $x_1$ the function, $f(x_1)$. In the limit where $x_1$ is very close to $x_0$, the secant line approaches the *tangent* (sark red line) of the function $f(x)$ at the point $x_0$ (the tangent is the straight line that \"just touches\" the curve at that point).  \n",
-    "  \n",
-    "If we represent the slope of the tangent line at the point $x_0$ by the term $f'(x_0)$, we will have:  \n",
-    "  \n",
-    "$$\\lim_{x_{1} \\to x_{0}} \\frac{\\Delta f(x)}{\\Delta x} = \\frac{f(x_{1}) - f(x_{0})}{x_{1} - x_{0}} = f'(x_{0})$$  \n",
-    "  \n",
-    "Since we can do the same process as we change the value $x_0$, we define a new function $f'(x_{0})$ which we call the **derivative** of the function $f$ at the point $x_0$, and its value tells us about the *instantaneous* rate of change of $f$ at the point $x_0$.  \n",
-    "  \n",
-    "If we look instead at the interval between $x_{0}$ and $x_{1}$ and define $h = x_1 - x_0$, if $x_1$ approaches $x_0$, then $h$ approaches $0$. and we would have for the derivative of $f(x)$:  \n",
-    "  \n",
-    "$$\\lim_{h \\to 0} \\frac{f(x + h) - f(x)}{h} = f'(x)$$  \n",
-    "\n",
-    "Visit this [link](https://www.demonstrations.wolfram.com/SecantAndTangentLines/) for an interactive demonstration of the limit of the tangent curve.\n",
-    "  \n",
-    "``````{note}  \n",
-    "Note that in the previous expression the index $0$ was omitted from $x_0$. We will assume that the derivative is defined for all the points in the domain of the function $f(x)$ for which the limit in the expression exists. \n",
-    "\n",
-    "It is good to have a graphical intuition for when the derivative is not well defined. Some cases are shown below.  \n",
-    "  \n",
-    "`````{tabbed} Case 1   \n",
-    "\n",
-    "````{panels}  \n",
-    "  \n",
-    "```{image} ./cases01a.png  \n",
-    "\n",
-    "```  \n",
-    "  \n",
-    "```{image} ./cases01b.png  \n",
-    "\n",
-    "``` \n",
-    "\n",
-    "---  \n",
-    "  \n",
-    "For the points where the function $f(x)$ presents discontinuities, or for the points where it is simply not defined, the derivative will also not be defined.\n",
-    "\n",
-    "  \n",
-    "````  \n",
-    "`````  \n",
-    "  \n",
-    "`````{tabbed} Case 2   \n",
-    "\n",
-    "````{panels}  \n",
-    "  \n",
-    "```{image} ./cases02.png  \n",
-    "\n",
-    "```  \n",
-    "\n",
-    "---  \n",
-    "  \n",
-    "For the points where the curve of the function $f(x)$ presents kinks, note that there are two tangent lines, one coming from the right (red line) and another coming from the left (yellow line). Since the function $f'(x)$ must have only one value for the image of $x$ (in other words, only one value for the slope of the tangent at point $f(x)$), then the derivative is not defined in this case.\n",
-    "  \n",
-    "````  \n",
-    "`````  \n",
-    "  \n",
-    "`````{tabbed} Case 3\n",
-    "\n",
-    "````{panels}  \n",
-    "  \n",
-    "```{image} ./cases03.png  \n",
-    "\n",
-    "```  \n",
-    "\n",
-    "---  \n",
-    "\n",
-    "For the points where the function $f(x)$ is locally vertical (in this case, for $x=0$), the slope of the tangent curve is infinite. Since infinite is not a real number, the derivative is not defined for those points.\n",
-    "  \n",
-    "````  \n",
-    "`````  \n",
-    "\n",
-    "``````  \n",
-    "  \n",
-    "  \n",
-    "````{admonition} Example  \n",
-    "  \n",
-    "By the definition of the derivative of $f(x)$:  \n",
-    "  \n",
-    "$$f'(x)=\\lim_{h \\to 0} \\frac{f(x + h) - f(x)}{h}$$  \n",
-    "  \n",
-    "Thus, if $f(x)=x^2$, we will have:  \n",
-    "  \n",
-    "$$f'(x)=\\lim_{h \\to 0} \\frac{(x + h)^{2} - x^{2}}{h} = \\lim_{h \\to 0} \\frac{x^{2} + 2xh + h^{2} - x^{2}}{h} = \\lim_{h \\to 0} (h + 2x) = 2x$$  \n",
-    "  \n",
-    "So that the derivative of $f(x)=x^2$ is the function $f'(x)=2x$.  \n",
-    "  \n",
-    "  \n",
-    "```{note}  \n",
-    "Although limits of functions can be defined in a more rigorous way, in this course we will treat them more intuitively, and use, if required, the known rules for limits. Thus, if $C$ is a constant, $\\lim_{x \\rightarrow a} f(x) = L$, and $\\lim_{x \\rightarrow a} g(x) = M$, it is generally valid that:  \n",
-    "  \n",
-    "$$\\lim_{x \\rightarrow a} [f(x) + g(x)] = \\lim_{x \\rightarrow a} f(x) + \\lim_{x \\rightarrow a} g(x) = L + M$$  \n",
-    "$$\\lim_{x \\rightarrow a} [Cf(x)] = C\\lim_{x \\rightarrow a} f(x) = CL$$  \n",
-    "$$\\lim_{x \\rightarrow a} [f(x)g(x)] = [\\lim_{x \\rightarrow a} f(x)][\\lim_{x \\rightarrow a} g(x)] = LM$$  \n",
-    "$$\\lim_{x \\rightarrow a} \\left[\\frac{f(x)}{g(x)}\\right] = \\frac{\\lim_{x \\rightarrow a} f(x)}{\\lim_{x \\rightarrow a} g(x)} = \\frac{L}{M},\\ \\ (\\mbox{if}\\ M \\ne 0)$$  \n",
-    "$$\\lim_{x \\rightarrow a} [C^{f(x)}] = C^{\\lim_{x \\rightarrow a} f(x)} = C^L$$  \n",
-    "\n",
-    "```\n",
-    "  \n",
-    "````\n",
-    "  \n",
-    "  \n",
-    "Other common notations for the derivative are given by:  \n",
-    "  \n",
-    "$$f'(x) = \\frac{df}{dx} = \\frac{d}{dx}f(x) = \\dot{f}(x)$$  \n",
-    "\n",
-    "```{Example}\n",
-    "Derivatives (or rate of change) are needed everywhere in science. Some common examples are the velocity as the derivative of position and acceleration as the derivative of velocity. When asking yourself 'how fast is the population growing?' or 'how steep is this mountain?' you can find the answer by calculating the derivative of the population number or the height profile.\n",
-    "```\n",
-    "\n",
-    "Visit this [link](https://the-learning-machine.com/article/calculus/derivatives) for examples that provide demonstrations of the intuitive interpretation of the derivative."
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "49c8ba11",
-   "metadata": {},
-   "source": [
-    "## Calculating derivatives  \n",
-    "  \n",
-    "  \n",
-    "It is possible to generalise the previous example for $f(x)=x^2$ and obtain the derivative for any power function $f(x)=x^n$ ($n \\ne 0$):  \n",
-    "  \n",
-    "$$f(x) = x^{n} \\implies f'(x) = nx^{n-1}$$  \n",
-    "  \n",
-    "  \n",
-    "```{admonition} Example  \n",
-    "  \n",
-    "$$x^{2} \\implies 2x^{2-1} = 2x$$\n",
-    "$$\\ x^{3} \\implies 3x^{3-1} = 3x^{2}$$\n",
-    "$$\\ x^{4.2} \\implies 4.2x^{4.2-1} = 4.2x^{3.2}$$  \n",
-    "\n",
-    "```  \n",
-    "  \n",
-    "It is possible to show, from the definition of derivative and the rules for limits, that there are simple rules for derivatives when calculating them for a sum of functions or a product by a constant. If $C$ is a constant, we generally have:  \n",
-    "  \n",
-    "$$\\begin{align}\\frac{d}{dx}[f(x)+g(x)] &= \\frac{df}{dx} + \\frac{dg}{dx} \\\\\n",
-    "   \\frac{d}{dx}[Cf(x)] &= C\\frac{df}{dx}\\end{align}$$  \n",
-    "  \n",
-    "  \n",
-    "Thus, differentiating a given polynomial p(x) stands for differentiating each term separately and summing the result.  \n",
-    "  \n",
-    "```{admonition} Example  \n",
-    "  $$f(x) = 3x^{2} - 4x + 6 \\implies f'(x) = 3(2x) - 4(x^{0}) + 0 = 6x - 4$$  \n",
-    "  $$f(x) = x^{60} - 3x^{20} - 4x \\implies f'(x) = 60x^{59} - 60x^{19} - 4$$\n",
-    "```  \n",
-    "  \n",
-    "For products and quotients of functions, however, we have different rules:  \n",
-    "  \n",
-    "$$\\frac{d}{dx}[f(x)g(x)] = \\left(\\frac{df}{dx}\\right) g(x) + f(x)\\left(\\frac{dg}{dx}\\right)$$  \n",
-    "$$\\frac{d}{dx}\\left[\\frac{f(x)}{g(x)}\\right] = \\frac{\\left(\\displaystyle\\frac{df}{dx}\\right)g(x) - f(x)\\left(\\displaystyle\\frac{dg}{dx}\\right)}{[g(x)]^{2}}$$  \n",
-    "\n",
-    "```{tip}\n",
-    "If you do not want to remember the quotients rule, you can use the product rule on $f(x)j(x)$, with $j(x)=g(x)^{-1}$ and use the chain rule to evaluate $\\frac{d}{dx} j(x)$.\n",
-    "```\n",
-    "  \n",
-    "```{admonition} Examples  \n",
-    "  $$f(x) = (x + 2)(x - 4) \\implies f'(x) = (1)(x-4) + (x+2)(1) = x - 4 + x + 2 = 2x - 2$$  \n",
-    "  $$f(x) = \\frac{\\alpha x}{1 + \\beta x} \\implies f'(x) =  \\frac{(\\alpha)(1 + \\beta x) - (\\alpha x)(\\beta)}{(1 + \\beta x)^{2}} = \\frac{\\alpha + \\alpha\\beta x - \\alpha \\beta x}{(1 + \\beta x)^{2}} =\\frac{\\alpha}{(1 + \\beta x)^{2}}$$  \n",
-    "  \n",
-    "```  \n",
-    "  \n",
-    "Moreover, expressions for the derivatives of the other elementary functions can also be found:\n",
-    "\n",
-    "- $f(x) = a^{x} \\implies f'(x) = a^{x} \\cdot (\\ln a) ,\\ (a > 0, a \\neq 1)$  \n",
-    "    $\\left(\\mbox{in particular: }\\ (e^{x})' = e^{x}\\right)$  \n",
-    "    \n",
-    "- $f(x) = \\log_{a}x \\implies f'(x) = \\displaystyle\\frac{1}{(\\ln a)x}$  \n",
-    "    $\\left(\\mbox{in particular: }\\ (\\ln x)' = \\displaystyle\\frac{1}{x}\\right)$  \n",
-    "    \n",
-    "- $f(x) = \\mbox{sin}(x) \\implies f'(x) = \\mbox{cos}(x)$  \n",
-    "\n",
-    "- $f(x) = \\mbox{cos}(x) \\implies f'(x) = -\\mbox{sin}(x)$\n",
-    "  \n",
-    "  \n",
-    "```{admonition} Example  \n",
-    "If $f(x) = \\mbox{tan}(x) = \\displaystyle\\frac{\\mbox{sin}(x)}{\\mbox{cos}(x)}$, then:  \n",
-    "  \n",
-    "$$f'(x) = \\frac{(\\mbox{cos}(x))\\mbox{cos}(x) - \\mbox{sin}(x)(-\\mbox{sin}(x))}{(\\mbox{cos}(x))^{2}} = \\frac{\\mbox{sin}^{2}(x) + \\mbox{cos}^{2}(x)}{\\mbox{cos}^{2}(x)} = \\frac{1}{\\mbox{cos}^{2}(x)} = \\mbox{sec}^{2}(x)$$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "42cb6b21",
-   "metadata": {},
-   "source": [
-    "## Chain rule  \n",
-    "  \n",
-    "The calculation of derivatives for more complicated functions can generally be made easier by breaking the function into components, for example:  \n",
-    "  \n",
-    "- $f(x) = (3x^{2} - 2x + 1)^{3} = (g(x))^{3}$, where $g(x) = 3x^{2} - 2x + 1$\n",
-    "- $f(x) = \\displaystyle\\frac{\\sqrt{2x - 1}}{1 + \\sqrt{2x - 1}} = \\displaystyle\\frac{g(x)}{1 + g(x)}$, where $g(x) = \\sqrt{h(x)}$, and $h(x) = 2x - 1$  \n",
-    "  \n",
-    "We have seen this before when discussing transformation of graphs.  When one function is nested within the other, we have a composite function, or as for the last example above:\n",
-    "\n",
-    "$$f(x) = f(g(h(x)))$$  \n",
-    "  \n",
-    "The derivative of composite functions in relation to the dependent variable is then given by the *chain rule*:  \n",
-    "  \n",
-    "- $\\displaystyle\\frac{d}{dx}f(x) = \\displaystyle\\frac{d}{dx}f(g(x)) = \\left(\\displaystyle\\frac{df}{dg}\\right)\\left(\\displaystyle\\frac{dg}{dx}\\right)$\n",
-    "- $\\displaystyle\\frac{d}{dx}f(x) = \\displaystyle\\frac{d}{dx}f(g(h(x))) = \\left(\\displaystyle\\frac{df}{dg}\\right) \\left(\\displaystyle\\frac{dg}{dh}\\right) \\left(\\displaystyle\\frac{dh}{dx}\\right)$  \n",
-    "  \n",
-    "The strategy then consists in identifying the nested functions, calculating their derivatives and multiplying the results.  \n",
-    "  \n",
-    "```{admonition} Examples  \n",
-    "  \n",
-    "- If $f(x) = (3x^{2} - 2x + 1)^{3}$, we have:  \n",
-    "      \n",
-    "    $$f(g) = g^{3} \\implies \\displaystyle\\frac{df}{dg} = 3g^2$$\n",
-    "    $$g(x) = 3x^{2} - 2x + 1 \\implies \\displaystyle\\frac{dg}{dx} = 6x - 2$$  \n",
-    "    \n",
-    "    Thus:  \n",
-    "      \n",
-    "    $$\\displaystyle\\frac{df}{dx} = \\left(\\displaystyle\\frac{df}{dg}\\right)\\left(\\displaystyle\\frac{dg}{dx}\\right) = 3g^{2} \\cdot g' = 3(3x^{2} - 2x + 1)^2(6x - 2)$$  \n",
-    "      \n",
-    "    <br />  \n",
-    "      \n",
-    "- If $f(x) = \\displaystyle\\frac{\\sqrt{2x - 1}}{1 + \\sqrt{2x - 1}}$, then:  \n",
-    "      \n",
-    "    $$f(g) = \\displaystyle\\frac{g}{1+g} \\implies \\displaystyle\\frac{df}{dg} = \\frac{(1+g) - g}{(1+g)^2} = \\frac{1}{(1+g)^2}$$  \n",
-    "    $$g(h) = \\sqrt{h} \\implies \\displaystyle\\frac{dg}{dh} = \\frac{1}{2}h^{-1/2}$$  \n",
-    "    $$h(x) = 2x-1 \\implies \\displaystyle\\frac{dh}{dx} = 2$$  \n",
-    "      \n",
-    "    Thus:  \n",
-    "      \n",
-    "    $$\\displaystyle\\frac{df}{dx} =  \\left(\\displaystyle\\frac{df}{dg}\\right) \\left(\\displaystyle\\frac{dg}{dh}\\right) \\left(\\displaystyle\\frac{dh}{dx}\\right) = \\frac{1}{(1+\\sqrt{2x - 1})^2}\\cdot \\frac{1}{2(\\sqrt{2x - 1})}\\cdot 2 = \\frac{1}{(\\sqrt{2x - 1})(1+\\sqrt{2x - 1})^2}$$  \n",
-    "      \n",
-    "    <br />\n",
-    "      \n",
-    "- If $f(t) = 5 + 5\\mbox{cos}\\left[\\displaystyle\\frac{\\pi}{12}(t - 3)\\right]$, we have:  \n",
-    "      \n",
-    "    $$f(g) = 5 + 5g \\implies \\displaystyle\\frac{df}{dg} = 5$$  \n",
-    "    $$g(h) = \\mbox{cos}(h) \\implies \\displaystyle\\frac{dg}{dh} = -\\mbox{sin}(h)$$  \n",
-    "    $$h(t) = \\frac{\\pi}{12}(t - 3) \\implies \\displaystyle\\frac{dh}{dt} = \\frac{\\pi}{12}$$  \n",
-    "      \n",
-    "    Thus:  \n",
-    "      \n",
-    "    $$\\displaystyle\\frac{df}{dx} =  \\left(\\displaystyle\\frac{df}{dg}\\right) \\left(\\displaystyle\\frac{dg}{dh}\\right) \\left(\\displaystyle\\frac{dh}{dx}\\right) = 5 \\cdot \\left(-\\mbox{sin}(h)\\right) \\cdot \\frac{\\pi}{12} = -\\frac{5\\pi}{12}\\mbox{sin}\\left[\\displaystyle\\frac{\\pi}{12}(t - 3)\\right]$$  \n",
-    "      \n",
-    "    For this last example if we plot the function $f(t)=5 + 5\\mbox{cos}\\left[\\displaystyle\\frac{\\pi}{12}(t - 3)\\right]$ together with its derivative $f'(t)=-\\displaystyle\\frac{5\\pi}{12}\\mbox{sin}\\left[\\displaystyle\\frac{\\pi}{12}(t - 3)\\right]$ we will have:  \n",
-    "      \n",
-    "    ```{image} func_diff.png  \n",
-    "    ```  \n",
-    "      \n",
-    "    Try to compare the two graphs. What happens with the function $f(t)$ on the intervals where the derivative $f'(t)$ is positive or negative? What happens at the points where the derivative is zero?"
-   ]
-  },
-  {
-   "cell_type": "markdown",
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-   "source": [
-    "## Higher order derivatives  \n",
-    "  \n",
-    "\n",
-    "\n",
-    "If the derivative of a given function is also a \"well-behaved\" function (without kinks or discontinuities), then we can also calculate the derivative of the derivative. Thus:  \n",
-    "  \n",
-    "$$\\frac{d}{dx}\\left(\\frac{df}{dx}\\right) = \\frac{d^{2}f}{dx^{2}} = f''(x)$$  \n",
-    "  \n",
-    "The function $f''(x)$ is called the *second derivative* with respect to $x$.  \n",
-    "  \n",
-    "We can continue this process and get higher-order derivatives:  \n",
-    "  \n",
-    "$$\\frac{d^{n}f}{dx^{n}} = f^{(n)}(x),$$  \n",
-    "  \n",
-    "assuming again that these functions are also well-behaved. Function $f^{(n)}(x)$ is called the $n$-th derivative of $f$ with respect to $x$ (note the parenthesis on $(n)$ to distinguish from powers of $f$).  \n",
-    "\n",
-    "```{admonition} Examples\n",
-    "The velocity is the derivative of the position, and the acceleration is the derivative of the velocity. Therefore, acceleration is the second-order derivative of the position\n",
-    "```\n",
-    "\n",
-    "```{admonition} Examples  \n",
-    "  \n",
-    "- $f(x) = 3x^{2} - 6x + 2$, then:  \n",
-    "      \n",
-    "    $$f'(x) = 6x - 6$$  \n",
-    "    $$f''(x) = 6$$  \n",
-    "      \n",
-    "    Polynomials are always well-behaved and they are infinitely differentiable. However, if $n$ is the degree of the polynomial (*i.e.* the largest exponent in the polynomial), we will have: $p^{(k)}(x) = 0,\\ \\mbox{for}\\ k \\ge n+1$.  \n",
-    "    \n",
-    "    <br />\n",
-    "      \n",
-    "- $f(x) = e^{-x^{2}}$, then:  \n",
-    "      \n",
-    "    $$f'(x) = e^{-x^{2}} \\cdot (-2x) = -2xe^{-x^{2}}$$  \n",
-    "      \n",
-    "    $$\\begin{align}f''(x)\n",
-    "&= -2[e^{-x^{2}} + x(-2x)(e^{-x^{2}})] \\\\\n",
-    "&= -2e^{-x^{2}} + 4x^{2}e^{-x^{2}} \\\\\n",
-    "&= -2e^{-x^{2}}(1 - 2x^{2})\\end{align}$$  \n",
-    "  \n",
-    "```  \n",
-    "  "
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "c0be95ba",
-   "metadata": {},
-   "source": [
-    "## Functions of several variables  \n",
-    "  \n",
-    "A function $z = f(x,y)$ of two dependent variables $x$ and $y$ defines a surface on the three dimensional $xyz$-space. If this function is well-behave it will be smooth, like if we applied different deformations on a rubber sheet. For smooth functions, we can calculate derivatives with respect to each of the dependent variables. This is written as:  \n",
-    "  \n",
-    "$$ \\frac{\\partial f}{\\partial x} \\text{ or } \\frac{\\partial f}{\\partial x}(x, y) \\text{ or } \\frac{\\partial}{\\partial x}f(x,y)$$  \n",
-    "  \n",
-    "$$\\frac{\\partial f}{\\partial y}, \\frac{\\partial f}{\\partial{y}}(x,y), \\frac{\\partial}{\\partial y}f(x,y)$$  \n",
-    "  \n",
-    "The partial derivative informs us how the function $f$ varies along the $x$ (or the $y$) direction at a specific point.  \n",
-    "  \n",
-    "```{admonition} Example  \n",
-    "Consider $f(x,y) = 4x^2y-2xy^3$. To compute the partial derivative with respect to $x$ we consider all terms in $y$ constants, and calculate the derivative as usual. For the partial derivative with respect to $y$, we consider the $x$ terms constant. We have:  \n",
-    "  \n",
-    "$$\\frac{\\partial f}{\\partial x} = (4y) \\cdot (2x) - 2y^{3} \\cdot (1) = 8xy - 2y^{3}$$  \n",
-    "$$\\frac{\\partial f}{\\partial y} = 4x^{2} - 2x(3y^{2}) = 4x^{2} - 6xy^{2}$$\n",
-    "```  \n",
-    "  \n",
-    "Higher order partial derivatives can also be defined, and represented with the common notations:  \n",
-    "  \n",
-    "$$\\frac{\\partial}{\\partial x}\\left(\\frac{\\partial f}{\\partial x}\\right) = \\frac{\\partial^{2}f}{\\partial x^{2}} = \\partial_{xx} f = f_{xx}(x,y)$$\n",
-    "$$\\frac{\\partial}{\\partial y}\\left(\\frac{\\partial f}{\\partial y}\\right) = \\frac{\\partial^{2}f}{\\partial y^{2}} = \n",
-    "\\partial_{yy} f = f_{yy}(x,y)$$  \n",
-    "  \n",
-    "  \n",
-    "with subscripts representing the variable and the order of differentiation.  \n",
-    "  \n",
-    "\n",
-    "For well-behaved functions, the order of the variables chosen to differentiate does not matter (a special case of what is called Schwarz's theorem):  \n",
-    "  \n",
-    "$$f_{yx}(x,y) = \\frac{\\partial}{\\partial y}\\left(\\frac{\\partial f}{\\partial x}\\right) = \\frac{\\partial^{2}f}{\\partial y \\partial x} = \\frac{\\partial^{2} f}{\\partial x \\partial y}= \\frac{\\partial}{\\partial x}\\left(\\frac{\\partial f}{\\partial y}\\right) = f_{xy}(x,y)$$\n",
-    "\n",
-    "```{admonition} Example \n",
-    "Often, a quantity of interest does not depend only on one variable but on many. The population size of a species depends on various aspects, including the availability of food, the number of predators, and other environmental factors.\n",
-    "```"
-   ]
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diff --git a/_sources/mathscourse/lecture07/07_Analysing_change_continuously_part2.ipynb b/_sources/mathscourse/lecture07/07_Analysing_change_continuously_part2.ipynb
deleted file mode 100644
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--- a/_sources/mathscourse/lecture07/07_Analysing_change_continuously_part2.ipynb
+++ /dev/null
@@ -1,860 +0,0 @@
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\n",
-      "text/plain": [
-       "<Figure size 360x360 with 1 Axes>"
-      ]
-     },
-     "metadata": {
-      "needs_background": "light"
-     },
-     "output_type": "display_data"
-    }
-   ],
-   "source": [
-    "import matplotlib.pyplot as plt\n",
-    "import numpy as np\n",
-    "import matplotlib.ticker as plticker\n",
-    "\n",
-    "# Create figure and add axes\n",
-    "#fig = plt.figure(figsize=(5, 5))\n",
-    "#ax = fig.add_subplot(111)\n",
-    "\n",
-    "fig, ax = plt.subplots(figsize=(5, 5)) \n",
-    "\n",
-    "# Move left y-axis and bottim x-axis to centre, passing through (0,0)\n",
-    "ax.spines['left'].set_position('zero')\n",
-    "ax.spines['bottom'].set_position('zero')\n",
-    "\n",
-    "# Eliminate upper and right axes\n",
-    "ax.spines['right'].set_color('none')\n",
-    "ax.spines['top'].set_color('none')\n",
-    "\n",
-    "# Show ticks in the left and lower axes only\n",
-    "ax.xaxis.set_ticks_position('bottom')\n",
-    "ax.yaxis.set_ticks_position('left')\n",
-    "\n",
-    "\n",
-    "#loc = plticker.MultipleLocator(base=3.0) # this locator puts ticks at regular intervals\n",
-    "#ax.xaxis.set_major_locator(loc)\n",
-    "#loc = plticker.MultipleLocator(base=2.0) # this locator puts ticks at regular intervals\n",
-    "#ax.yaxis.set_major_locator(loc)\n",
-    "\n",
-    "x = np.linspace(-3, 3, 100)\n",
-    "y = x**3\n",
-    "ax.plot(x, y, color='b', linewidth=2.5, label=('$f(x) = x^3$'))\n",
-    "\n",
-    "ylim = ax.get_ylim()\n",
-    "xlim = ax.get_xlim()\n",
-    "\n",
-    "ax.tick_params(axis='both', which='major', labelsize=13)\n",
-    "\n",
-    "\n",
-    "ax.legend(fontsize=13)\n",
-    "# Add legend\n",
-    "\n",
-    "#ax.legend(bbox_to_anchor=(1.05, 0.3),loc='lower left', \n",
-    "#          frameon=False, labelspacing=0.2, fontsize=13)\n",
-    "\n",
-    "plt.tight_layout()\n",
-    "plt.savefig('xcube.png')"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 25,
-   "id": "8401c1b7",
-   "metadata": {
-    "tags": [
-     "remove-cell"
-    ]
-   },
-   "outputs": [
-    {
-     "data": {
-      "image/png": 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4U4h8cIehpyKX7uXnadMJ2GQy8VrT1wC4EH+BpX8vNTgiIeybs7MziYmJRodRoCQmJuZqpY2b2XQCBuhVtxel3UsD8NVfX8m7txD3oHTp0pw/f56EhAT5X7pHSikSEhI4f/48pUuXztMxbKYaWk6KOhXlpYCXGL1lNHsj9rLtzDZaVbTs0BAhCiovLy8ALly4QEpKisHR2D9nZ2fKlClj/rnmls1NRc7O5euXqfB1BW6k3eCJGk/w81M/Gx2SsBEyFVnYCfuYipyd0u6leaaenou94vAKwq9abploIYSwFrtIwACvNn0VAIViwl8yMUMIYf/sJgHXKV2HDlU6ADA7dDZXE68aHJEQQtwbu0nAAMMfGg5AQkoC04KnGRyNEELcG7tKwO3ub0e9MroI87e7v+VG6g2DIxJCiLyzqwRsMpl446E3ALh47SILDyw0OCIhhMg7u0rAAD3r9MTX0xeA8TvHy2ByIYTdsrsEXMSxCMMaDwPg78i/WXd8ncERCSFE3thdAgZ4IeAFPIp4ADDuz3EGRyOEEHljlwnYu6g3zzd4HoBNJzcRciHE4IjEnbz55pvUrl0bLy8vfH19GThwIFevZh1KOG/ePKpUqYKbmxtNmjQhJER+r6Jgs8sEDPDaQ6/haHIEpBVsDxwdHZk/fz5RUVGEhYVx7tw5+vfvb358+/btvPjii0ydOpXo6GiCgoLo3LkzcXFxBkYthGXZRS2InPT5uQ/z98/HweTAsZePcX/x+40OSdyl3377jd69exMbGwtA3759SU9P54cffgB0pamKFSsyZswY+vbtm+NxpBaEsBP2WwsiJyOajQAgXaXz1c6vDI5G5MbGjRupV6+e+X5YWBiNGjUy3zeZTDRo0ICwsLBbnjt9+nQCAgIICAggMjLSKvEKYQl2nYDrlalHp6qdAJi9bzaR1+Wf0R4sW7aMGTNmMGFCZk2P+Ph4ihUrlmU/b2/vbLsgBg0aRHBwMMHBwZQqVcri8QphKXadgAFGNhsJQGJqIpN2TzI4GnEnS5cuZeDAgfz66680bNjQvN3T09PcHZEhJiYmz3VWhbAHdp+A21RqQ4BvAKDXjbuefN3giERO5syZwwsvvMDKlStp27Ztlsf8/f3Zu3ev+b5SitDQUPz9/a0dphBWY/cJ2GQy8WbzNwG4mniVGXtnGByRyM7EiRMZPnw469ato3nz5rc8PnDgQJYvX87GjRtJTk5m/PjxJCUlERgYaEC0QliHXY+CyJCWnkbNyTU5dvUY5b3Kc3zYcYo4FjE6LHETk8mEk5MTLi4uWbZfu3bNfHvevHmMHj2aiIgI6taty9SpU7NcmMuOjIIQdiLbURAFIgEDzNw7k4ErBwIw5/E59Kvfz9iAhFVIAhZ2ouANQ7tZn3p9zEV6Pt/xOekq3eCIhBDi9gpMAnZxcuH1pq8DcPjKYVYcXmFwREIIcXsFJgEDDGo0iOJFiwPw6fZPpVSlEMKmFagE7OniydDGQwHYc2EPG09uNDgiIYTIWYFKwADDmgzDzdkNgI+3fWxwNEIIkbMCl4BLupVkcKPBAGw+tZk/z/5pcERCCJG9ApeAAd5o9oZ5HLC0goUQtqpAJmBfT1+eq/8cAKuPrWZfxD6DIxJCiFsVyAQMMLL5SHPB9k+2f2JwNEIIcasCm4ArF6/M0/WeBmDZP8v4J/IfgyMSQoisCmwCBninxTuYMKFQ0hcshLA5BToBVy9ZnafqPAXA4oOLORp11OCIhBAiU4FOwACjWo4C9LJFn2yTvmAhhO0o8Am4dunaBNUMAmD+/vmciD5hcERCCKEV+AQMMKqVbgWnqTQ+3fapwdEIIYRWKBJw/bL1eaz6YwB8H/Y9p2JOGRuQEEJQSBIwwHut3gMgNT1V+oKFEDah0CTgAN8AulTrAsCc0DnSChZCGK7QJGCA0a1HA9IKFkLYhkKVgBv5NqJrta6AbgWfjD5pcERCiMKsUCVggA9afwBIK1gIYbxCl4Ab+TbKMiJCxgULIYxS6BIwZG0Fj9k6xuBohBCFVaFMwA3LNSSwRiAA88LmSY0IIYQhCmUCBviwzYeYMJGu0vlwy4dGhyOEKIQKbQKuW6auuVLaogOLOHj5oMERCSEKm0KbgEGPC3YwOaBQjN482uhwhBCFTKFOwNVLVueZes8AsOzQMlk7TghhVYU6AQO83+p9nBycABj1xyiDoxFCFCaFPgFX8anC8w2eB/QKytvPbDc4IiFEYVHoEzDoesFFnYoC8PbGt1FKGRyREKIwkAQM+Hn5MfTBoQBsP7OddcfXGRyREKIwkAT8r7davIVnEU8A3tn4Dukq3eCIhBAFnSTgf5VwK8HwZsMB2HdxHz/985PBEQkhCjpJwDd5relrlHQrCcCoTaNISUsxOCIhREEmCfgmni6e5mXsj109xqx9swyOSAhRkEkC/o/BAYOp5F0JgA+3fMj15OvGBiSEKLAkAf+Hi5MLY9rqEpUXr13km7++MTgiIURBJQk4G73r9qZemXoAfPHnF1xJuGJwREKIgkgScDYcTA589vBnAMTdiOPjrR8bHJEQoiCSBJyDTlU70aZSGwAm75nM8avHjQ1ICGEIS86MlQScA5PJxLj24wBISU/h3U3vGhyREMIIr659lX6/9ONc3Ll8P7Yk4NsI8A2gV51eAPz494/sPr/b4IiEENZ05MoRpgRPYW7YXF787cV8P74k4Dv45OFPKOJYBIDh64dLoR4hCpE3N7xJanoqJkyMbTs2348vCfgOKnlX4uXGLwOw7cw2fj3yq8ERCSGsYcupLaw4sgKAfvX74V/WP9/PIQn4LrzT8h28i3oDMHLDSJLTkg2OSAhhSekqnTfWvwGAm7ObeW5AfpMEfBd8XH14v9X7AByNOsq04GkGRySEsKSFBxYSEhECwPCHhuPn5WeR80gCvktDGg+hqk9VAEZvHs3VxKsGRySEsITElETe2fgOAGU9yjKi+QiLnUsS8F0q4ljEPCwtOimaMVss85FECGGs8TvHczbuLABj2o7Bo4iHxc4lCTgXHq/+OK0rtgZg0p5JHI06anBEQoj8dCH+Ap9u/xSAemXq0b9+f4ueTxJwLphMJr7q+BUmTKSmpzLy95FGhySEyEfvbHyHhJQEAL7p+A2ODo4WPZ8k4FxqWK4hz/o/C8CKIyvYcGKDwREJIfJD8IVg5obNBeCJGk/QtnJbi59TEnAefPrwp+Z+oVfWvkJqeqrBEdm+xYsX07JlS7y8vHBycrrl8bVr11K7dm1cXV2pU6cO69evNyBKUVgppXh17asAODs4m6/3WJok4Dwo51mOd1vq2hD/RP4jw9LuQvHixXnppZf45ptb6yufOHGCbt268fbbbxMbG8vbb79NYGAgp06dsn6golD68e8f2XF2BwCvNn3VPOLJ0kx3mFor825zkJSaRO0ptTkRfYLiRYtz7OVjlHArYXRYNm/z5s20a9eO1NTMTw0ffPABmzZtYtu2beZtLVu2pF27dnzwwQe3PV5AQADBwcEWi1cUfNeTr1N9UnXOx5+ntHtpjg49SrGixfL7NKbsNkoLOI+KOhVlfIfxgB6W9v4f7xsckf0KCwujUaNGWbY1bNiQsLCwbPefPn06AQEBBAQEEBkZaY0QRQH26fZPOR9/Xt9++FNLJN8cSQK+B49Xf5yHKz8MwLSQaYRdzD5hiNuLj4+nWLGsf/Te3t7ExcVlu/+gQYMIDg4mODiYUqVKWSNEUUAdv3qccX/q/t4A3wD61e9n1fNLAr4HJpOJCZ0m4GhyJF2l8/Kal6VaWh54enoSGxubZVtMTAxeXl4GRSQKizfWv2Gu7fLtI9/iYLJuSpQEfI9ql67NsCbDAF0tbdHBRQZHZH/8/f3Zu3dvlm379u3D3z//q08JkWFd+DpztbNn/Z+lafmmVo9BEnA++KD1B5RxLwPomsHxN+INjsj2pKWlkZSURHKybm0kJSWRlJSEUopnn32W4OBgFi1aREpKCosWLSIkJIS+ffsaHLUoqG6k3uDlNbrMrEcRD/MakNYmCTgfFCtajC/afwFAxLUIxmyVOhH/9cMPP+Dq6krHjh1JS0vD1dUVV1dXTp8+TZUqVVi+fDljx47Fy8uLsWPH8vPPP1OpUiWjwxYF1Pid4zl29RgAH7X5iHKe5QyJQ4ah5ZN0lU6L2S3YeW4nTg5OhA0Oo1apWkaHVeDJMDSRW6djTlNzck0SUxOpU7oOewftxdnR2dKnlWFoluRgcmBy58k4mBxITU/lpd9ekgtyQtig19e/TmJqIgCTO0+2RvLNkSTgfNSgXAOGPDgEgC2nt7DgwAKDIxJC3Gxt+FqWH1oOwDP1nqFVxVaGxiMJOJ+NaTvGfEHujfVvEJMUY3BEQgjQhdaHrNYNJM8innzR7guDI5IEnO+KFS3GVx2/AuDy9cuM2jTK4IiEEACfbPuEE9EnAPj4fx8bduHtZpKALaBXnV60raRL2U3ZM4U95/cYHJEQhduhyEN8vuNzABqVa8RLD75kcESaJGALMJlMTHl0Cs4OzigUg1YNkpKVQhhEKcWLv71ISnoKJkxM6zLN4oXW75YkYAupUbIGb7d4G4DQi6FM3DXR4IiEKJzmhc1jy+ktAAx5cAgBvgEGR5RJxgFbUFJqEvWm1uPY1WO4Obvxz0v/UNG7otFhFSgyDljcTuT1SGpOrklUYhRlPcpyeMhhq1Y7u4mMA7a2ok5FmdZFF2tPSElg6JqhMjZYCCt6ff3rRCVGAbrYjkHJN0eSgC3sf5X/R19/XdNg1dFV/PTPTwZHJEThsP74eubvnw9A12pdCaoZZHBEt5IEbAVfdviSkm4lARi6ZihXE68aHJEQBdv15OsMXjUY0MV2JneejMmUbS+AoSQBW0FJt5J801GvhXb5+mWGrx9ucERCFGyjN4/mZMxJAD753yfcV+w+gyPKniRgK+ldtzePVH0EgDmhc2Q5eyEsZM/5PXz1l54M1cSvic2M+c2OjIKwojOxZ6g9pTbXkq9R2bsyB148gHsRd4ucSymIiYELF/RXdDQkJemv9HQoWlR/ubtD2bJQrhyUKQPOxtUlyRMZBSFulpyWTKPpjTh4+SDODs7se2EftUvXNjosyGEUhJO1oyjMKhSrwKcPf8rLa17mZMxJ3t30Lt90unWZ9txKT4e//4Zt2yAsDA4c0PdzWFItR46OULky1KgBNWtCo0bQtClUqAA22H0mxC0+2fYJBy8fBOC9Vu/ZSvLNkbSArSxdpdNqTit2nN2BCRNb+m2hZcWWuT5OTAz8+iusWAFbtkBUlAWC/VeZMtCmDbRvr78qVLDcuXJLWsAiw4FLB2g4vSGp6anUK1OP4IHBhpaa/I9smzCSgA1wNOoo/tP8SUpNoqpPVcIGh+Hm7HbH5yUnw88/w9y5sGEDpKTcuk+xYlCnjv6qWhX8/PSXjw+4uuovkwlu3NDdEbGxcPEiRETAmTNw5AgcOgTHjkFqDrOn69aFoCD9Vbu2sa1jScACICUthYdmPURIRAiOJkd2Pb+LRr6NjA7rZtIFYSuqlajG2LZjGf77cMKvhvPepvcY33F8jvufPQuTJ8Ps2RAZmfUxHx/o0EG3UNu0gWrV8ichJiXBvn2waxfs2AEbN+p+ZNBdHAcOwOjRUKsW9O0LzzwDvr73fl4h8uLzHZ8TEhECwIhmI2wt+eZIWsAGSUtPo8WcFvx17i9MmNjWfxvNKzTPss+JE/DZZ/D991lbuz4+8OSTugXaurV1LpylpcHevbBmjW6Fh4ZmfdzBATp1gqFDoWNHfd8apAUswi6G8eCMB0lJT6F2qdqEDArBxcnF6LD+S7ogbM3hK4epP60+N9JuUKV4FcIGh+FexJ1Ll+C993SLNy0tc//mzWHwYOjeXY9gMNLx47B0Kcybp7ssblalik7EAwaAp6dl45AEXLglpyXTZGYTQi+G4mhy5K/n/7KpYjs3kVoQtqZGyRp8/L+PATgefZw31o3kiy/ggQdgxozM5Nu1K/z1F2zfrj/qG518QSfZt97Soy1279ZvDO7/jqg7fhxee01frBs1Ci5fNjZWUXB9vPVjQi/qj2Nvt3jbVpNvjqQFbLC09DTazm3LtjPb9IZ56+FEewAeeQQ++QTq13fhlA8AACAASURBVDcwwFyIjdUt4kmT4OjRzO2urvDSSzByJJQunb/nlBZw4bX7/G6azWpGmkqjXpl67Bm4hyKORYwOKyfSArZFSYmO3L//e0j+t/n4+HNUqxfDmjWwerX9JF/QIzBefll3SSxfDk2a6O2JiTB+vB5j/NZbmRfzhMirhJQE+vzchzSVhrODM3OfmGvLyTdHkoANtG+fTrBzv7kf1v07CqLYORq9P5ROnYyN7V44OEBgIOzcCZs2Qat/F55NSIDPP9fdF+PH65EWQuTFm7+/ydEo/THro7YfUb+sHbVUbiIJ2ABKwZQpepZZeLje1qzoIFqV07UiFh1cwI8HfzQwwvxhMkHbtrB5sx633KyZ3h4dDcOH6xl3S5bon4cQd2v98fVM2jMJgOb3NWdEsxEGR5R3koCt7Pp16NkThgzREyscHeHTT2HrFhM/9p5tLls5+LfBnIs7Z3C0+cNkgocf1hcRf/kFqlfX20+fhqee0kl6/35jYxT2ISohiv4r+gPg7uzO3Cfm2sz6bnkhCdiKzp+Hli11qw+gfHk9jfitt3QiLutRlhldZwAQkxRD31/6kq7SDYw4f5lM8PjjcPAgTJ0KJfV7DVu2QIMGMGxY7utXiMJDKcXAlQO5EH8BgG86fUMVnyoGR3VvJAFbSXAwPPig7vcFaNdO326ede4FT9R4gucbPA/AppOb+GrnV1aO1PKcnPSwtaNH4ZVX9JtPejp8+63ulli6VLolxK1m7p3Jz4d/BiCwRiADGgwwOKJ7JwnYCtas0ReiIiL0/Rdf1CMcMlqA//V1p6+pUly/s7+z8R2CLxTMYVbFi8M33+gKbq1b620REXqWX9euegq2EABHrhzh1XWvAuDr6cuMrjNscoWL3JIEbGFLlsBjj+mhWA4OMHGirutwu+nDHkU8WBi0ECcHJ1LSU+i1rBfxN+KtF7SV1a4Nf/yhp1xnvCn99pvePm2abh2LwutG6g16L+9NQkoCJkzMe2IeJdxKGB1WvpAEbEGzZkGvXrqqWJEi8NNPepzs3bxxN/ZrbJ4lF341nKFrhlo4WmOZTLqoz+HD+jtAfLz+tPDww3DypLHxCeO8teEt9kbsBWB4s+E8fP/DBkeUfyQBW8i0afD887r15u6uW3SBgbk7xvBmw2l3fzsA5oXNM6/wWpCVKKFbwmvXZtYd3rwZ6tWD6dOlb7iwWXlkJd/s0osWPOj7IGP/N9bgiPKXJGALmDNHt9wAvL31GNh27XJ/HAeTA/OemEcpt1IADF41mCNXjuRjpLarY0c9WmKwXtiWa9fghRegc+fMvnRRsJ2NPUu/Ff0A8HLxYnH3xXY52+12JAHns4ULdRUwAC8vnXybNs378cp5lmNe4DwArqdcp8fSHiSmJOZDpLbP01MPV1u3Tg/ZA90yrltXl8QUBVdqeiq9l/fmauJVAGZ0ncH9xe83OKr8Jwk4H/36Kzz7rP6Y7O6uk0WjfKgL3alqJ95u8TYABy4f4JW1r9z7Qe1Ihw66APzTT+v7UVHQrVtmF48oeEZtGsX2M9sBGNRwEE/WftLgiCxDqqHlk1279IyuxERd/WvNmsyhVfkhNT2VtnPbmv8o5wfO5+l6T+ffCezE4sW6eycmRt93cQngr7+C7apokbi9VUdX0XVRVwDqlanHXwP+wtXZ1eCo7plUQ7OU8HDo0iVzqNnSpfmbfAGcHJxYFLTIPFV50KpB/H357/w9iR3o2VNPW874+d64oauuTZwoF+gKglMxp3j252cB8CziyU89fioIyTdHkoDv0ZUrum7vlSv6/tSp8OijljlXea/yzA+cjwkTCSkJdFvSjbgbhW/u7n336TXqxozR95OT9Yy6bt2k1KU9u5F6gx5LexCdpH+Jsx+fzQMlHjA4KsuSBHwPUlL0umwZFc3efRcGDbLsOTtW7cjoNqMBvbrycyue4w7dSAWSo6NebaN69czhar/8Ag0b6hU6hP0ZtmaYedbnsMbD6F6ru8ERWZ4k4Hvw6quwdau+/fTTmS0ySxvVahSPVNWlK5cdWlYg60XcLQ8PXVPjscf0/VOnoEULXVeiEL4v2a2Ze2cyfe90AJqWb8q4DuMMjsg6JAHn0cyZuqYvQECAXsPNWlPTHUwO/BD4AxWLVQTgzQ1vsvHERuuc3Ab5+OjW71df6SneKSm6slrPnno2nbBtu8/vZsjqIQCUdi/NTz1+KnDjfXMiCTgPdu7Ua5yBXuNs+XI98sGaSriVYNmTyyjqVJQ0lcZTPz3FqZhT1g3ChphMeiHQbdt0HzHoOhwPPqgXDhW2KfJ6JN2XdCc5LRlHkyNLeyzFz8vP6LCsRhJwLkVF6WpdKSm6rOKyZZn/8NbWyLcR07voj21RiVEE/hhIQkqCMcHYiCZNYO9ePZMO4MgRaNxYD18TtiU5LZnuS7tzNk6XvRvfYTytKrYyOCrrkgScC0rpQjHn/l2o4ptvdH+jkfr49+HVJrpMX+jFUAb8OqBQXpS7WcmSutznhx/qlnFCgi6K9Oqr+o1T2IZha4ax9bS+iPJ03acZ1mSYwRFZnyTgXPj6a11UB6B798xuCKON6zCOtpXaArD44GI+2faJwREZz8EB3n9fJ+LixfW2CRN0TY5Ll4yNTcDUPVP5LuQ7QBfZKSj1fXNLZsLdpd279eoVqal6efW9e3WhHVtxJeEKjWc05mSMrtu47MlldKvZzeCoLC8gIIDg4NsXrD95Ug8XzFiNxM9Pdx01aWKFAMUt/jj5Bx3mdyA1PZVyHuXYM3BPYej3lZlweXXtWmZdX2dn+PFH20q+ACXdSrKy10o8i3gC0OfnPuyL2GdwVLahcmXYsUPX6QC9Nl+rVrpes7Cuo1FHCVoSRGp6KkUci7D8qeWFIfnmSBLwXRg+HE6c0Lc//VRfWbdFtUvXZnH3xTiYHEhISeCxxY+ZFzAs7FxddZ3hb7/VF0+Tk3Uxn4zVqYXlRSVE8ejCR80z3WY9Noum5e+hVGABIAn4Dtasge90VxWtW+uhTras8wOdGddeD2I/F3eOLgu7cC35msFR2QaTCYYOhU2b9PBB0GO5H35Y+oUtLTktmaAlQYRf1dNG32v1Hs/Ue8bgqIwnCfg2oqIya/t6euoWlIMd/MRea/oagxrqOdH7Lu6j17JepKWnGRyV7WjZUq9SHRCg72/frm+HhBgbV0GVsZz8ltNbAOhZpycftvnQ4Khsgx2kE+MMGZK5+sKECVCpkqHh3DWTycTkRyfTsYoeDLvq6CpeWftKoR+edrP77tPTyDP6hc+d00MKFy40Nq6C6P0/3mdemF5UoGn5psx5fE6hHPGQHUnAOfj1V32xDXSdgX79DA0n15wcnFjSYwl1S9cFYPKeyXz555cGR2VbMvqFv/5af7JJStI1PUaOhDT5wJAvpodMZ+w2vY5bVZ+q/NrzV4o6FTU4KtshCTgbcXGZY3y9vfUCm/b4hu3l4sXqp1fj6+kLwMgNI/kh7AeDo8pZWloaI0aMoFSpUnh6ehIUFMSVjDqfFmIy6Qka69ZljhceNw66ds0s+i7y5rejv/HSb/ofqaRbSdY8vYZS7qUMjsq2SALOxttv66FKoP8Zy5UzNp57Ud6rPGufXksxl2IAPPfrc6wLX2dwVNn77LPPWLFiBbt27eLcv9MN+/TpY5Vzt2sHe/ZA7dr6/po1epzw4cNWOX2B8+fZP+mxtAdpKg1XJ1dW9lpJVZ+qRodle5RSt/sqdHbsUMpkUgqUatNGqfR0oyPKH5tPblYuY1wUo1HuH7urXed2GR3SLSpUqKBmzpxpvh8eHq4AdfLkyRyf06hRo3yNIS5Oqccf179/UMrLS6nVq/P1FAXegUsHlPdn3orRKIcPHdQvh34xOiRbkG2Ove1MuE6dOql7+QgYGRlJqVL285FDKfjnH90XaDLp1pCLi37M3l5LdqKTojlxVQ9odnRwpHrJ6rg62cZyL2lpaYSGhlKzZk3c3NzM2/ft20flypXxvmnmS2RkpLlr4saNG9S3wIJwFy5kXoAFPXuubNl8P80t7P3vLDktmcNXDpOSpotuVPSuaF5Gy57d6+8lJCRknVKq0y0P5JSZVT60gPO7dWJp48Zltnw+/jjrY/b2WnLyXfB3itEoRqPKfVlOHb963OiQlFJKnTlzRgHqxIkTWbZXqFBB/fDDDzk+z83NzWIxLVmilJtb5t/EM88olZBgsdMppez77+xC3AVVdWJV89+X35N+RoeUb/Lh95JtjpU+4H9duKCrZ4Fe5mb4cGPjsZRBjQbh94+e+hlxLYL2P7S3idlynp56CnVsbGyW7TExMXh5eRkREj166CnMGUsezZ+vJ+NkXB8Qma4kXKHdD+3MEy2GPzScsset8JHBzkkC/tfIkbrmA+jpqkUKcEH+ssfL8lbztwA4EX2C/839H5euGTsVzNvbmwoVKrB3717zthMnThAXF0e9evUMi6t+fX1xrmVLfX/PHj0V/a+/DAvJ5sQkxdDhhw78E/kPAC80eoEv2n9hcFT2waIJeJClV6jMJ1u3woIF+nZQELRvf+s+9vJa7sagQYP45OFPeClADxE6EnWEh+c9TOT1SMPj+vzzzzl58iRxcXG8+eabdOzYkUq3mQFTsqTl+xdLl4YNG+CFF/T9iAjdEp47N//PZW9/Z3E34nhkwSPsu6gLPz1T7xmmPDoFk8lkd6/ldiz2WnLqm1CFZBRESopSdevqPj5XV6VOnzY6IutJS09Tz6943txnV29qPXXl+hXD4klNTVVvvPGGKlGihPLw8FCBgYEqMjLyts+xdp/plClKOTll9gu/+qr+GyqMYpNi1UMzHzL//QT9GKRS0grpD+POcj8KgkJQD3j69MyWzdixemn5wiRdpdN/RX/zVFH/Mv5seHaD3Vy5vpt6wPlt82bdP5wxQKhdOz1r0sfHqmEYKqPl++fZPwF4rPpjLO2xtNAsppkHUg/4v+Lj4b339O2KFeGNN4yNxwgOJgdmPzab3nV7AxB2KYy2c9ty+fplgyOzXW3a6L5gf399f8MG3S988KChYVlNbFJsoUq+N27oWuCWUKgT8BdfwOV/88ynn0LRQjpF3dHBkblPzKVPPT3r7ODlg7T5vg0R8RF3eGbhVamSHiHRo4e+f+IENG2qV8guyKISonh43sPm5Nu1WtcCnXwBRozQff4nT1rg4Dn1Tah86gPu37+/Kl++vPL09FRly5ZV/fv3V1evXs2PQ9+Ts2d1ny8o9eCDSqWl3fk5SUlJatCgQapq1arKw8ND3XfffWr48OEqMTHR8gFbwIQJE1Tjxo2Vq6urqlKlikpNS1X9f+lv7tOrMqGKOhl90ugws5WamqqGDx+unJyclIeHh+rWrdsd+4stIT1djxnPmD0JSr37rlKpqXd/jEWLFqkWLVooT09P5ejoaLlg71FEfISqM6WO+e8jcHGgSkpJyrLPyJEjVa1atZSnp6cqV66cev7551VUVJRBEd+7J5/83vx7dXbeqIKCgtTpvF0oMmYc8Ouvv87hw4eJi4vj0KFDJCQkMGTIEEuf9o5GjYLERH17/Pi7q/ObmppKyZIlWblyJTExMWzbto1Nmzbx5ptvWjZYC/H19WXkyJG8+2/Ht6ODIzMfm8kLjXSn+PHo47SY3YJDkYeMDDNbGXUjatSoYfW6ETczmeCdd2DlSsgYrvzxx7qC3t0W8ylevDgvvfQS33zzjeUCvUenY07T+vvWHLys+1l61enFj91/xMXJJct+jo6OzJ8/n6ioKMLCwjh37hz9+/c3IuR7dukSbNz4NKDrge/b9xAVKlSgZ8+e+XeSnDKzssAoiOjoaNWrVy/VsGHD/D50roSGZrZYAgPv7ViTJk1S9erVy5/ADDJnzhxVpUoV8/309HQ1fN1wc0un5BclVfD5YAMjvFVG3YiMURB3UzfC0g4fVqpGjcyWcNWqSh04cPfP/+OPP2yyBXzw0kHlN97P/PcwYMUAlZp2d038VatWKS8vLwtHmP/S05V65JHM3+W8eUpdu3ZNvfHGG8rHxycvhzRuJtxnn32Gp6cnxYsX55dffjG3uIwyapT+sTo6wuef39uxNm7caOhEAUswmUx80f4LPv7fx4Ce5dRmbhvWH19vcGRabGwsZ86coVGjRuZtVapUwcvLi/379xsWV/XqsGsXPP64vh8ervuFlywxLKR7tvPsTlrOacn5eD3979UmrzK963QcHRzv6vn2+v8xaZKuiAfw0EOnGDKkGB4eHkyYMIHRo0fn34lyyszqDi3gvn37KvQwtWy/3n333Vuec+LECTVq1CgVFhaWl3eQfLFjR+a72qBBmdvz8nq+/vprVapUqbz2CVlMbl/Lf1vAN5u8e7IyjTYpRqOcPnJS80LnWeMl3NbNdSNuHgd8p7oR1pKWptRHH2XtF37jjTuPF7a1FvCKwyuU61hXc8v3s22fqfRclAf86aeflIeHhwoJCbFglPkvJEQpFxf9e6tYUamYGL09IiJCjR07Vv3xxx95OWy2OTbPCTg+Pl5FRkbm+HX9+vVsn7dr1y7l5+en0u7mqlc+S0/XJSZB/4DPns18LLev56uvvlJlypRRB3LzGdNKcvtabpeAlVJq6d9LzaUsGY36ZOsnufpHzG/R0dEKUPv27cuSgL28vNSKFSsMi+u/Vq9Wyts7Mwm3bq1URETO+9tSAv5217fK4UMHc0nJmSEz7/ykmyxZskQVL15cbdq0yUIRWkZUlFKVKunfl6OjUtu2ZX380qVLyt3dPS8XFvM3AefVjh07FKDi4uIscfjb+v33rDOY8uqjjz5Sfn5+6vDhw/kXnIHulICVUmrLqS2q2KfFzEn4uV+eUzdSb1gpwltVqFBBzZo1y5yAjx8/bngfcHbCw5Xy98/8uytX7tZ/6gy2kIBT01LV62tfN/+e3T52UyuPrMzVMWbPnq2KFy+utm/fbqEoLSMtTalHH838XY0bd+s+58+fV0BeGl7WT8CXLl1Sc+fOVdHR0UoppY4cOaKaN2+uWrRoca+HzrX0dD3cDJRyd1fq0qW8HWf48OGqQoUKKjw8PH8DNEBKSopKTExU06dPV/fff79KTEy87ZC6A5cOqApfVzD/c7b5vo2KSjBmiNHYsWNVtWrVVJ06dVRsbKzq3r276tixoyGx3Mn160o9+2zmP7ajo1JffplZ7D81NVUlJiaqdevWKUdHR/PvwdqfMuKS4lTXhV3Nv9+yX5bN9cXXCRMmKB8fH7V7924LRWk5Y8dm/o66dVMqNTVNffvtt+rSv8ni7Nmz6oknnlCVKlVSKbmff279BHz58mXVtm1bVbx4ceXm5qYqVKigXnjhBXXx4sV7PXSurViRdZxmXpw6dUoBqkiRIsrd3d38VatWrfwN1ko++OCDbPuIbyciPkI1mdHE/E9adWJVdfDSQStFnCmjboSjo+Nd140wUnq6Ut99p1SRIpl/h48/rtTVq/oTSHa/B2u25o9fPa5qT65t/r3WmlxLnYo+levjAMrJySnL/4e7u7sFIs5fq1Zl9tk/8IDu901LS1OPPPKIKlWqlHJzc1O+vr6qd+/eeW18ZZtjC0UtCKX0VNGQEChWDE6d0ottirxJTEmk34p+LPlbX973KOLBD4E/8ESNJ6weixG1IO5FcLCePXfqlL5fubIeJREQYFxMm05u4smlTxKVGAVA5wc6syhoEV4uxtRhtrb9+6F5c12O1s1NlxqtWzffT1N4a0GsXauTL+gVcCX53htXZ1cWBS1iTNsxAFxLvkbgj4G8/8f7pKXLeu63ExAAe/fqiRqgp7c2awbffKMbCtaklOLz7Z/T/of25uQ7/KHh/Nrz10KTfC9d0itgX7umJ9UsWGCR5JuznJrGykIX4awtPV2ppk31RwtPT/2RT+SfXw//qjw/8TR/dG03r526dC2PHex5YK9L+KSn637gm0tbdumilLV6UWKTYlXg4kDz763o2KJqbuhc65zcRiQkZOYGUOrzzy16OuMmYhhp48bM1QtefhmKFzc2noKma/Wu7B64mxolawCw4cQGGnzXgG2ntxkcmW0zmXT1ve3bdWEfgFWrdIW1TZsse+7gC8E0+K4BPx/+GYDK3pXZOWAnz/o/a9kT25DUVHjqqczc0L+/LrpjbQU+AX/0kf7u7g6vvWZsLAVVjZI12DNwj7mk5YX4C7Sd25aPtnxEarqF6vgVEE2awL590L27vn/hgq4v/OabkJycv+dSSvH1zq9pNqsZJ6L16tidH+hMyKAQ6pfN/5WlbVV6Ojz/vK7fAbq86LRp+k3R2gp0At6yBbb92xB76SWwwuo1hZZHEQ/mB85n2qPTcHF0IU2l8cHmD2jzfRtOx5w2Ojyb5u2tL8R99x24uuoPxF98AQ89BP/8kz/niIiP4NGFj/L6+tdJSU/BycGJce3HsbLXSoq7Fp6PhUrpBXczlpNq2BBWrDBuDcgCPQrikUf0BThXV32xo0wZoyMqHMIuhtF7eW/zIo1eLl5M6DSBvv59MeVzM8PeRkHcyeHD0Lu3bhUDuLjAZ5/BsGF3V7EvO8v+WcYLq14wX2ir5F2JxUGLaVK+ST5FbR+Ugg8+gDH62jHVqukGWunSVjl94RoFsX+/Tr4AAwZI8rUm/7L+7Bm4hxcDXgT08jX9V/TnscWPcSH+gsHR2bYaNXS/5Jtv6o/EN27orrN27XJfEPxKwhWeXv403Zd2NyffZ+o9w74X9hXK5Pvuu5nJ188P1q+3WvLNWU5X55Sdj4J4+ml9ZdPBQakTJ4yOpvBaeWSlKvtlWfPV9uKfFVez9s7Kt1le9joK4m5s26ZU5cqZV+nd3ZWaPPnOiwekp6erxQcWq1JflDL/3H0+91FLDi6xTuA2Jj1dqZEjM3+O5csrdeyY1cMoPKMgTp+GxYv17Sef1IPdhTG6VOvC3y/9zdN1dWHr6KRoBvw6gDZz23D4ymGDo7NtLVpAWFjmorHXr8OQIfC//8GxY9k/52T0SR5b/Bg9l/UkMiES0MsGHXjxAD1q97BS5LYjLQ2GDtV96gAVKuhrQ1WrGhtXhgKZgL/6Sv/gAUaONDYWAT6uPszvNp+fn/oZX09fALae3or/NH/e2fgO15KvGRyh7fL01Ffof/9dLxwLOoHUrQuffJI5UiIpNYkxW8ZQa0otVh1dBUApt1IsDlrMip4rzD/3wiQxUY8umTJF369YUf/s7r/f2LhuVuAuwkVF6Xe5hARo31738wjbEXcjjnc3vsvkPZNR//55+Xr68kW7L+hdt3euL9IVtItwtxMfD2+9BVOnZs6aq11H8dTo5cw9/ybHo4+b9+1Xvx/j2o+jpFvhHPoTFaVnG/6p1w6lbl1dYN3Pz7CQCsdFuClTdPIFaf3aIi8XL77t/C1/Pf8Xjf0aA3rc8DM/P0PTWU3ZfGqzsQHaME9PmDxZT96oXRvw3cPfD7bm/YPdzcm3bum6bOu/jTmPzym0yTc0VE/5zki+bdvq0Q4GJt8cFagEfONG5seN+vXh4YeNjUfkrLFfY3YO2Mmcx+dQxl0PUdl9fjdt57al84LO7I3Ya3CEtsur6kGqvtsNBjWGiv8OdE8oQdFNE+kdv5cHy7QwNkADLVyoa2tkFDvq3Vu3fIsVMzSsHBWoBLxkCVy8qG+/9poxM1vE3XMwOdCvfj+OvnyUUS1H4ebsBsCa8DU0mt6IxxY9RvCFwtG9cDcOXj5I72W9qTe1HiuO6mnEzg7OVL38Bkw8RtLWl3n7TSdq1oSffrJ+cR8jJSToi21PP637fh0c9IW3+fP1WGpbVWD6gJXKrDRVpoweCWHLP3hxq4j4CD7a8hEz983MMoW5Y5WODG82nIcrP3xLH3Fh6APeeXYnn27/lJVHV5q3OZgc6Ovfl/dbv08l70qsXQuvvw6HDmU+r0kTfaHuf/8zIGgr2rtXJ97D/w6q8fHRo6Datzc2rv/IvjmY0/g0ZWfjgLdtyxznN3q00dGIe3Hi6gk18NeByukjJ/M4Vkaj/Kf6qzn75qiE5ATzvgV1HPCN1Btqfth81XhG4yw/A4cPHVTPn3qqQ5GHbnlOcrJSkyYpVaJE5v8CKNWunVJ//mnAi7CwpCS9+OnNFeWaN1fKxlalylCwC7J37w7Lluk53WfOyMy3guB0zGnG/TmOOaFzSEhJMG8vXrQ4ff37MqjRIPo80qdAtYCPRh1l9r7ZfB/6PZeuXzJvL+JYhL7+fRnZfCRVfW4/iDUmBr78Er7+OvOCNOhrIu+9B61bWyp66/njD3jxRThyRN93coLRo/UMQicnQ0PLSbYt4AKRgE+dgipVdJWjfv1gzhyjIxL5KSohimnB05i0ZxIXr13M8pjbbDc++fETetbpSRkP+3zXjbweybJDy1h4YCHbzmQt41nKrRSDAwYzOGBwrsfyXrwIH38M06dnrazWpInurujWzWaTVY7Cw/WbSMZEK4A6dWD2bL3qjQ0ruAl4xAj9jg+6iEn9wlNZr1BJSUthxZEVTAuexsaTG/XG74AXdJ9oiwotCKwRyBM1nqCSdyUjQ72jM7FnWHV0Fb8c/oVNJzeRprKuJNL8vuYMbDiQp+o8RVGnovd0rvPnYdw4XW0tKSlze4UKukpgv362/4nx/Hldx2HWLF3LF/TyQR9+CK+8As7OxsZ3FwpmAk5M1OP7oqOhVSs900UUfOFXw/kh7Ac+6/sZyQNuLZxbvUR1OlTpQIcqHWh2XzN8XH0MiDJT3I04tp3exh+n/uD3E7+z/9L+W/bx8/Sjd93ePNfgOXOB+/x06RJ8+62eWRcVlbndyQkef1wXJe/QwbaSWWio7kpZtAhSUjK39+wJn3+u30TsRMFMwN9/r/9wQA9D61H4prsXagEBAUz8eSLL/lnG8sPLORVzKtv9apWqRbPyzWhYriH+Zf2pV6YeHkU8LBLTteRrHIo8ROjFUPZc2MOeC3s4cOnALa1cgHIe5ehWsxtPzek3SgAAC+dJREFU1X6K5hWa42Cy/MjQhAQ9PGviRPj776yP+fjo6ylPPgktWxpTJzcuTl/P+f572Lo162OdO+tuFTv8lFswE3CTJrB7N5Qtqy++2dK7t7C8m4ehKaUIvRjKmvA1rD++nh1nd9x2RQ4/Tz+q+lSlqk9VynuVx9fTl3Ie5SjuWhwvFy88i3hSxLEIjg6OOJocSUlPISk1icSURGJvxHIl4QpXEq5wPu48p2NPczr2NMeijnE6NucC9I4mRwJ8A+j8QGe6VOtC/bL1rZJ0s6MU7NwJM2bAjz/qT5M38/KCjh2hUyd94e7++y03tv7SJV0+9rff9EoVN3eVODhAUJDut27a1DLnt4KCl4CDgzM73t97L3P5IVF43G4ccPyNeHae28mOMzvYcXYHu8/vJj453qrxeRTxIMA3gMa+jWldqTUtK7TE08XTqjHcjfh4vTLEwoW6fkpaNotb+/rqWWYNGugWaJ06uvvP0TF350pM1BfTQkJ042nnTt3V8F9lysAzz+gJFhnr5tmxgpeABwzQVz8dHfVIiPLljY5IWFtuJmKkq3ROxZwi9GIoBy4dIDw6nGNRxzgefZwrCVfuKY7S7qWpWKwilYtXplbJWtQuXZvapWpTrUQ1HB1ymaEMdvWqbo2uWqW/R0fnvK+zs64ydt99uvuieHHdcs5YvSMtTQ+Li46GK1d0Ufnz53M+nre37mbo00cXobe3URq3UbAScHS0fvdNTITAQFi+3OiIhBHyayZccloyl65dIuJaBLFJscTdiCPuRhwp6SmkpaeRptJwdnDG1dmVok5FKeZSjJJuJSnhVoIy7mVwdXbNh1dje9LS9OoyW7fqC9zBwXD2bP4d380NGjXS/c2PPKK7GApQ0r1ZtgnYbl/q3LmZfVYvvmhsLML+FXEswn3F7uO+YvcZHYpNcXTUXQ4NGujhXqBHUISF6UkQJ0/qT5/nz+tGUXS07s7IaNeZTLpVW7y4/qpYURdDr1IF6tWDWrUKbMK9K3bZAlYKatbUfwAPPKDngOd1wUJh3wpDLQhRIBScesDbt2dOQRw0SJKvEMI+2WXqmjlTf3d2hmefNTYWIYTIK7tLwDExsHSpvv344zawrLQQQuSR3SXgRYsyL74NHGhsLEIIcS/sLgHPmKG/V6yoxwkKIYS9sqsEvHevrnYG8NxzcvFNCGHf7CqFZVx8c3DILMAjhBD2ym4ScGKinqcOujjIfTJeXghh5+wmAa9YAbGx+ra0foUQBYHdJOC5c/X34sWha1djYxFCiPxgFwn4wgVdIg90JXxZbl4IURDYRQJesEAvuAnQt6+xsQghRH6x+QSsVGb3Q40a0LixsfEIIUR+sfkEvHdv5rpVfftabkkUYTkTJ06kSZMmuLm5UbVq1Wz3mTdvHlWqVMHNzY0mTZoQEhJi5SiFsD6bT8Dff6+/m0x6eRJhf3x9fRk5ciTvvvtuto9v376dF198kalTpxIdHU1QUBCdO3cmLi7OypEKYV02nYCTk3XtB9DTjmXJIfvUvXt3goKC8PPzy/bxGTNm0K1bNzp06ICLiwsjRozAxcWFn3/+2cqRCmFdNp2A16/X1fdByk4WZGFhYTRq1Mh832Qy0aBBA8LCwrLdf/r06QQEBBAQEEBkZKS1whQi39l0Al6wQH93c4MnnjA2FnGrfv36YTKZcvwaNWrUXR0nPj6eYsWKZdnm7e2dYxfEoEGDCA4OJjg4mFKlSt3z6xDCKDa7GtO1a3r2G+i6vx4exsYjbjVp0iS+/PLLHB93c3O7q+N4enoSmzHN8V8xMTFUqVLlnuITwtbZbAL+5ZfMur+9exsbi8ieh4cHHvnwzujv78/evXvN95VShIaG0q1bt3s+thC2zGa7IDIK75QoAR07GhuLuDepqakkJSWRkpKCUoqkpCSSkpLMjw8cOJDly5ezceNGkpOTGT9+PElJSQQGBhoYtRCWZ5Mt4MjIzKnHPXrotd+E/Ro7diwffvih+b6rqyugW7oALVq0YMqUKQwcOJCIiAjq1q3L6tWr8fLyMiReIazFJpelnzwZhg7Vt7dtgxYtjIhC2ANZll7YCftZlj6j+6FCBWjWzNhYhBDCUmwuAZ86BX/+qW/37i3LDgkhCi6bS28ZS86DLj0phBAFlc0l4B9/1N+rV4d69YyNRQghLMmmEvDx45BRBOvJJ6XymRCiYLOpBHxz98OTTxoXhxBCWINNJeAlS/T3WrWgTh1jYxFCCEuzmQR87Bjs26dvS+tXCFEY2EwCzmj9gp79JoQQBZ3NJeA6dXQXhBBCFHQ2kYCPHIH9+/Xtp54yNhYhhLAWm0jAy5Zl3u7e3bg4hBDCmmwiAS9frr/XqqWXnhdCiMLA8AR8+nTm5IugIGNjEUIIazI8Ad+88K0sgCCEKEwMT8AZ/b+VK4O/v7GxCCGENRmagC9ehB079O1u3aT2gxCicDE0Aa9YARkLckj/rxCisDE0AWeMfihXDpo0MTISIYSwPsMScHQ0bNqkbwcGysoXQojCx7C0t2oVpKbq2zL6QQhRGBmWgFes0N+9vaFVK6OiEEII4xiSgJOSYO1affvRR8HZ2YgohBDCWIYk4D/+gOvX9e3HHzciAiHE/9u7f5e21jiO459ga4sm6iAutlppRbBUB7PYimKRthAUoVNxcYp/QXDJFhzEgA46KK7+A4bW26KC6FKw2iBOgsXBdghI/AFq0eYOhyQ3t7H34u3znOT6foH4TXI838flw+HLyXngPlcCOD1+uH1bevnSjRUAgPusB/CPH1Is5tTPn0sVFbZXAACFwXoAf/okff3q1IwfANxk1gN4fj5b9/ba7g4AhcN6AKfnv21t0r17trsDQOGwGsBfvkhbW07d12ezMwAUHqsB/NfxA/NfADed1QB++9b5ff++1NJiszMAFB5rAXxyIq2sOHUgwLN/AcBaAC8uSt+/O3UgYKsrABQuawGcHj/cvet8AQMAbjorAZxKSe/eOXV3t1RWZqMrABQ2KwH8+XP222+MHwDAYSWA0+MHiQAGgDSrAfz4sfTggY2OAFD4jAdwIiF9/OjUXP0CQJbxAF5YyG49TwADQJaVAJakykrp6VPT3QCgeBgN4MtL6cMHp37xQrp1y2Q3ACguRgN4fV06OHDqV69MdgKA4mM0gN+/z9bs/QYAuYwGcHrr+SdPpNpak50AoPgYC+CDg+ztZ4wfAOBnxgJ4cdHZAVkigAEgH2MBnB4/lJdLz56Z6gIAxctIAKdS2QDu7pbu3DHRBQCKm5EA3tqSvn1zasYPAJCfkQBOX/1KBDAAXMVoAD96JD18aKIDisX5+bmGhobU2Ngon8+nuro6hUIhnZ2d5Rw3Njam2tpalZeXq6enR7u7uy6tGLDHSAC/eSP19kqvX5s4O4rJxcWFqqurFYvFlEwmtbq6quXlZQ0PD2eOmZub09jYmGKxmBKJhJqbm9XX16fLy0sXVw6Y50mlH1WW3y8/BK5jampKMzMzisfjkqSuri51dnYqEolIkk5OTlRTU6OFhQV1dXX98lx+v1/r6+vG1wz8R3n3gbe2KSeQtrS0pJaWlszreDyutra2zGuv16vGxsZMQAP/VwQwrm1wcFAej+fKn3A4/NPfTExMaG1tTSMjI5n3jo+PVVlZmXNcVVWVjo6O8vadmZmR3++X3+9XIpH4vf8UYBEPiMS1TU5OKhqNXvl52d+2vx4fH9fo6KiWl5dVV1eXed/n8+nw8DDn2GQyqYqKirznDQaDCgaDkpwRBFCsCGBcm9frldfr/VfHRiIRTU9Pa2VlRU1NTTmftba2amNjQ/39/ZKcGfDOzo5aW1t/+5qBQsIIAsaFQiHNzs7mDV/JuaKdnp7W5uamTk9PFQ6H1dDQoI6ODhdWC9jDFTCM2tvbUzQaVWlpac4VbX19vba3tyVJAwMD2t/fVyAQUDKZVHt7u+bn51VSUuLWsgEruA0NRY3b0FAkuA0NAAoJAQwALiGAAcAlBDAAuIQABgCXEMAA4JJ/ug0NKGgej+ePVCrFY/9RlAhgAHAJIwgAcAkBDAAuIYABwCUEMAC4hAAGAJf8CUkFhu4lo5cBAAAAAElFTkSuQmCC\n",
-      "text/plain": [
-       "<Figure size 360x360 with 1 Axes>"
-      ]
-     },
-     "metadata": {
-      "needs_background": "light"
-     },
-     "output_type": "display_data"
-    }
-   ],
-   "source": [
-    "import matplotlib.pyplot as plt\n",
-    "import numpy as np\n",
-    "import matplotlib.ticker as plticker\n",
-    "\n",
-    "# Create figure and add axes\n",
-    "#fig = plt.figure(figsize=(5, 5))\n",
-    "#ax = fig.add_subplot(111)\n",
-    "\n",
-    "fig, ax = plt.subplots(figsize=(5, 5)) \n",
-    "\n",
-    "# Move left y-axis and bottim x-axis to centre, passing through (0,0)\n",
-    "ax.spines['left'].set_position('zero')\n",
-    "ax.spines['bottom'].set_position('zero')\n",
-    "\n",
-    "# Eliminate upper and right axes\n",
-    "ax.spines['right'].set_color('none')\n",
-    "ax.spines['top'].set_color('none')\n",
-    "\n",
-    "# Show ticks in the left and lower axes only\n",
-    "ax.xaxis.set_ticks_position('bottom')\n",
-    "ax.yaxis.set_ticks_position('left')\n",
-    "\n",
-    "\n",
-    "#loc = plticker.MultipleLocator(base=3.0) # this locator puts ticks at regular intervals\n",
-    "#ax.xaxis.set_major_locator(loc)\n",
-    "#loc = plticker.MultipleLocator(base=2.0) # this locator puts ticks at regular intervals\n",
-    "#ax.yaxis.set_major_locator(loc)\n",
-    "\n",
-    "x = np.linspace(-3, 3, 100)\n",
-    "y = x**3-(3/2)*x**2-6*x+3\n",
-    "ax.plot(x, y, color='b', linewidth=2.5, label=('$f\\,(x)$'))\n",
-    "\n",
-    "x = np.linspace(-3, 3, 100)\n",
-    "y = 3*x**2-3*x-6\n",
-    "ax.plot(x, y, color='g', linewidth=2.5, label=('$f\\,\\'(x)$'))\n",
-    "\n",
-    "ylim = ax.get_ylim()\n",
-    "xlim = ax.get_xlim()\n",
-    "\n",
-    "ax.tick_params(axis='both', which='major', labelsize=13)\n",
-    "\n",
-    "\n",
-    "ax.legend(fontsize=15)\n",
-    "# Add legend\n",
-    "\n",
-    "#ax.legend(bbox_to_anchor=(1.05, 0.3),loc='lower left', \n",
-    "#          frameon=False, labelspacing=0.2, fontsize=13)\n",
-    "\n",
-    "plt.tight_layout()\n",
-    "plt.savefig('func_diff.png')"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 31,
-   "id": "cb32140f",
-   "metadata": {
-    "tags": [
-     "remove-cell"
-    ]
-   },
-   "outputs": [
-    {
-     "data": {
-      "image/png": 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\n",
-      "text/plain": [
-       "<Figure size 360x360 with 1 Axes>"
-      ]
-     },
-     "metadata": {
-      "needs_background": "light"
-     },
-     "output_type": "display_data"
-    }
-   ],
-   "source": [
-    "import matplotlib.pyplot as plt\n",
-    "import numpy as np\n",
-    "import matplotlib.ticker as plticker\n",
-    "\n",
-    "# Create figure and add axes\n",
-    "#fig = plt.figure(figsize=(5, 5))\n",
-    "#ax = fig.add_subplot(111)\n",
-    "\n",
-    "fig, ax = plt.subplots(figsize=(5, 5)) \n",
-    "\n",
-    "# Move left y-axis and bottim x-axis to centre, passing through (0,0)\n",
-    "ax.spines['left'].set_position('zero')\n",
-    "ax.spines['bottom'].set_position('zero')\n",
-    "\n",
-    "# Eliminate upper and right axes\n",
-    "ax.spines['right'].set_color('none')\n",
-    "ax.spines['top'].set_color('none')\n",
-    "\n",
-    "# Show ticks in the left and lower axes only\n",
-    "ax.xaxis.set_ticks_position('bottom')\n",
-    "ax.yaxis.set_ticks_position('left')\n",
-    "\n",
-    "\n",
-    "#loc = plticker.MultipleLocator(base=3.0) # this locator puts ticks at regular intervals\n",
-    "#ax.xaxis.set_major_locator(loc)\n",
-    "#loc = plticker.MultipleLocator(base=2.0) # this locator puts ticks at regular intervals\n",
-    "#ax.yaxis.set_major_locator(loc)\n",
-    "\n",
-    "x = np.linspace(-3, 3, 100)\n",
-    "y = x**3-(3/2)*x**2-6*x+3\n",
-    "ax.plot(x, y, color='b', linewidth=2.5, label=('$f\\,(x)$'))\n",
-    "\n",
-    "x = np.linspace(-3, 3, 100)\n",
-    "y = 6*x-3\n",
-    "ax.plot(x, y, color='orange', linewidth=2.5, label=('$f\\,\\'\\'(x)$'))\n",
-    "\n",
-    "ylim = ax.get_ylim()\n",
-    "xlim = ax.get_xlim()\n",
-    "\n",
-    "ax.tick_params(axis='both', which='major', labelsize=13)\n",
-    "\n",
-    "\n",
-    "ax.legend(fontsize=15)\n",
-    "# Add legend\n",
-    "\n",
-    "#ax.legend(bbox_to_anchor=(1.05, 0.3),loc='lower left', \n",
-    "#          frameon=False, labelspacing=0.2, fontsize=13)\n",
-    "\n",
-    "plt.tight_layout()\n",
-    "plt.savefig('plot_concav.png')"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 37,
-   "id": "26e54118",
-   "metadata": {
-    "tags": [
-     "remove-cell"
-    ]
-   },
-   "outputs": [
-    {
-     "data": {
-      "image/png": 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\n",
-      "text/plain": [
-       "<Figure size 360x360 with 1 Axes>"
-      ]
-     },
-     "metadata": {
-      "needs_background": "light"
-     },
-     "output_type": "display_data"
-    }
-   ],
-   "source": [
-    "import matplotlib.pyplot as plt\n",
-    "import numpy as np\n",
-    "import matplotlib.ticker as plticker\n",
-    "\n",
-    "# Create figure and add axes\n",
-    "#fig = plt.figure(figsize=(5, 5))\n",
-    "#ax = fig.add_subplot(111)\n",
-    "\n",
-    "fig, ax = plt.subplots(figsize=(5, 5)) \n",
-    "\n",
-    "# Move left y-axis and bottim x-axis to centre, passing through (0,0)\n",
-    "ax.spines['left'].set_position('zero')\n",
-    "ax.spines['bottom'].set_position('zero')\n",
-    "\n",
-    "# Eliminate upper and right axes\n",
-    "ax.spines['right'].set_color('none')\n",
-    "ax.spines['top'].set_color('none')\n",
-    "\n",
-    "# Show ticks in the left and lower axes only\n",
-    "ax.xaxis.set_ticks_position('bottom')\n",
-    "ax.yaxis.set_ticks_position('left')\n",
-    "\n",
-    "\n",
-    "#loc = plticker.MultipleLocator(base=3.0) # this locator puts ticks at regular intervals\n",
-    "#ax.xaxis.set_major_locator(loc)\n",
-    "#loc = plticker.MultipleLocator(base=2.0) # this locator puts ticks at regular intervals\n",
-    "#ax.yaxis.set_major_locator(loc)\n",
-    "\n",
-    "x = np.linspace(-3, 4, 100)\n",
-    "\n",
-    "y = (3/2)*x**4-2*x**3-6*x**2+2\n",
-    "ax.plot(x, y, color='b', linewidth=2.5, label=('$f\\,(x)$'))\n",
-    "\n",
-    "y = 6*x**3-6*x**2-12*x\n",
-    "ax.plot(x, y, color='g', linewidth=2.5, label=('$f\\,\\'(x)$'))\n",
-    "\n",
-    "y = 18*x**2-12*x-12\n",
-    "ax.plot(x, y, color='orange', linewidth=2.5, label=('$f\\,\\'\\'(x)$'))\n",
-    "\n",
-    "#ylim = ax.get_ylim()\n",
-    "#xlim = ax.get_xlim()\n",
-    "ax.set_ylim(-50,50)\n",
-    "\n",
-    "ax.tick_params(axis='both', which='major', labelsize=13)\n",
-    "\n",
-    "\n",
-    "ax.legend(fontsize=15)\n",
-    "# Add legend\n",
-    "\n",
-    "#ax.legend(bbox_to_anchor=(1.05, 0.3),loc='lower left', \n",
-    "#          frameon=False, labelspacing=0.2, fontsize=13)\n",
-    "\n",
-    "plt.tight_layout()\n",
-    "plt.savefig('plot_extrema.png')"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 51,
-   "id": "797a95f5",
-   "metadata": {
-    "tags": [
-     "remove-cell"
-    ]
-   },
-   "outputs": [
-    {
-     "data": {
-      "image/png": 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\n",
-      "text/plain": [
-       "<Figure size 432x288 with 1 Axes>"
-      ]
-     },
-     "metadata": {
-      "needs_background": "light"
-     },
-     "output_type": "display_data"
-    }
-   ],
-   "source": [
-    "import matplotlib.pyplot as plt\n",
-    "import numpy as np\n",
-    "    \n",
-    "dens = np.linspace(0,4)\n",
-    "\n",
-    "fig, ax = plt.subplots(figsize=(6, 4))\n",
-    "\n",
-    "\n",
-    "ax.plot(dens,2*dens**2/(1+dens**2))\n",
-    "    \n",
-    "ax.set_xlabel('$N$',fontsize = 16)\n",
-    "ax.set_ylabel('$f\\,(N)$',fontsize = 16)\n",
-    "\n",
-    "ax.set_yticks([])\n",
-    "ax.set_yticklabels([])\n",
-    "\n",
-    "ax.set_xticks([np.sqrt(1/3)])\n",
-    "ax.set_xticklabels([r'$\\sqrt{\\frac{1}{3ah}}$'],fontsize=16)\n",
-    "\n",
-    "ax.axvline(np.sqrt(1/3), ls='--', color='black', lw=0.5)\n",
-    "\n",
-    "ax.set_ylim(0, 2)\n",
-    "ax.set_xlim(0, 4)\n",
-    "\n",
-    "fig.tight_layout()\n",
-    "fig.savefig('hollingIII.png')"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 57,
-   "id": "1c07acfd",
-   "metadata": {
-    "tags": [
-     "remove-cell"
-    ]
-   },
-   "outputs": [
-    {
-     "data": {
-      "image/png": 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AEjDuv/9+brnlFq688soDfb169XKwIqngdtsH4M47D+LinK0nVGgIQgJCQUEBP/30E7GxsfTq1Yvk5GT69etHRkbGYc9NT08nLS2NtLQ0srOzHag2/Hz/PezebdoafvAeBbAEhJycHDweD6+88gqvv/4627Zt47zzzmPQoEHs3bv3kOcOHz6cjIwMMjIyqF+/vkMVh5f33jO3kZEKYG9SAEtAqFWrFgBDhw6lW7duxMTEcP/991NWVsYPP/zgcHXhzeOB99837f79oV49Z+sJJQpgCQi1a9cmNTUVVyVn9q6sT/znf/8zK+AArrjC2VpCjQJYAsbIkSOZMWMGq1atwu128/TTTxMbG0ufPn2cLi2szZ5tbl0uuOwyZ2sJNZoFIQHj3nvvZd++fZx99tkUFxfTs2dPPv30U2rreueOsSw7gPv2hYYNna0n1Lgsyzra40d9UMRpaWlplc6UEO/IyLCXHD//PNxxh7P1BLFKx9E0BCEiR1Qx+wHg8sudqyNUKYBFpFIHDz+cdho0bepsPaFIASwilVq2DNatM23NfvANBbCIVGrWLLutAPYNBbCIHMay7AA+9VRITXW0nJClABaRw/zvf/aVj6+91tFSQpoCWEQOU7H3GxEBV1/tbC2hTAEsIocoL4d//9u0+/c3lx8S31AAi8ghvvkGduww7WuucbSUkKcAFpFDVAw/REdr9oOvKYBF5IDSUnvxxQUXQJ06ztYT6hTAInLAZ59BTo5pa/aD7ymAReSAt94yt/Hx5tLz4lsKYBEBzJ7vRx+Z9hVXQEKCs/WEAwWwiABm6llpqWn/5S/O1hIuFMAiAsCbb5rbZs2gXz9HSwkbCmARYd06qLj26f/9n7n6sfieAlhEDhx8A7j+eufqCDcKYJEw5/HYww8nnQQdOzpbTzhRAIuEuQUL7DOfae/XvxTAImFuxgxzGxWlcz/4mwJYJIzl5tpnPrvkEkhOdraecKMAFgljs2ZBYaFp33STs7WEIwWwSBibPt3cNm8O557rbC3hSAEsEqaWLIGMDNMeNkxzf52gABYJUxV7vy4XDB3qbC3hSgEsEoaKimDmTNM+7zxo0cLZesKVAlgkDM2ebWZAgA6+OUkBLBKGXnrJ3CYnw8UXO1tLOFMAi4SZpUvh++9N+8YbISbG2XrCmQJYJMz84x/m1uWCESOcrSXcKYBFwkhurn3w7cILoWVLZ+sJdwpgkTDy5ptQUGDaI0c6W4sogEXChmXBtGmm3aoVnH++s/WIAlgkbMybB7/9Ztq33goR+tvvOP0KRMLEiy+a29hYrXwLFApgkTCQmQkffmja114L9eo5W48YCmCRMPD88+bSQwB//auztYhNASwS4vLy4NVXTfucc6BrV2frEZsCWCTEvfoq7Ntn2tr7DSwKYJEQ5nbDc8+Zdvv2MHCgs/XIoRTAIiHsww9h40bTvusuTT0LNPp1iIQoy4JnnjHtOnV0yflApAAWCVELFsCPP5r2LbdAzZrO1iOHUwBLwPF4PPTp0weXy8WWLVucLidoPfGEua1Rwww/SOBRAEvAmTp1KvHx8U6XEdSWLYNPPjHtYcOgYUNn6wlq7kL4/TUzpuNlCmAJKGvWrGHatGn87W9/c7qUoPbkk+Y2IgLuvdfZWoKWpwzW/gM+bgMLb4QtH3n9IxTAEjA8Hg/Dhg3j6aefJikp6YjPS09PJy0tjbS0NLKzs/1YYXBYvx5mzTLtP/3JnPlMToDlgQ3/hLkd4OeRULTd9G+Y6fWPUgBLwHjuuedo1KgRl19++VGfN3z4cDIyMsjIyKB+/fp+qi54TJliLzseM8bZWoKKZcHW/8CnPeGHP0P+etNfMxVOexNO/5fXPzLK6+8oUgXr1q1jypQpZGRkOF1KUNu2zV52PHAgdO/ubD1BI+s7WHo/ZH9v98U2gM4PQJvhEOmbC+cpgCUgLFiwgOzsbLp06QKY4QiAbt268eijjzJSl284Lk8+CSUlpj1+vLO1BIWcpbB0HGz7xO6LToSO90H7uyA6wacf77KOfmTP+4f9RCpRWFjInj17DtzfsmULp512Gj///DMdOnQgIaHyvwhpaWnaa95v2zYz3ltSAgMGwBdfOF1RANu3DpY9CBvfsfsiY6HdHdBpLNSo6+1PdFXWqT1gCQjx8fGHTD1zu90ANGrU6IjhK4d66il773fiRGdrCViF22DFI/D7dLDMnzFckdD6JujyAMQ39Ws5CmAJSKmpqRzjf2dykO3b4eWXTfucc+CMM5ytJ+CU7IFVT8Kav0N5kd3f4hro+jAktnWkLAWwSAh46ikoLjZt7f0exF0Aq5+DVU9BWa7d33ggdJ8MdXs6VxsKYJGgt3UrvPSSaZ9zDvTt62w9AaG8FH5/xQw3FO+0++ufDt0fhwaB8UNSAIsEuYcftvd+H3rI0VKc5yk3B9aWPQgFmXZ/Ulfo/hg0uRBclR4Pc4QCWCSIrV1rz/u98MIwHvu1LNg6F5aNh73L7f6EVmaMN/VacAXeujMFsEgQe+ABKC837cmTna3FMTvnm0UUu360+2IbmVkNrW/y2SIKb1AAiwSpX36Bf+1fHTtkSBiuetuz2Cyi2P6Z3RedBJ3ug/Z3QlTgnwBZASwSpMaNM7dRUWYcOGzkrYFlD8Cmf9t9kXFm5Vqn+yCmjnO1nSAFsEgQ+uor+O9/Tfvmm6F1a2fr8YvCLbD8YVj/Glj7x11cUdDmZjPcENfY2fqqQAEsEmTKy2HUKNOuWdOMA4e0kt2w6glY/Xfw7F/qhwtaXAvdHoZawfuvjwJYJMi88QYsXWraY8dC4+Db8Ts+Zfmw+ln49Wkoy7P7m1xoFlHUCf5BbwWwSBDJz7fPctasGdxzj7P1+ER5CaxLh5WPQnGW3V//jP2LKEJnrp0CWCSIPPUU7Nhh2o8/DiF16TxPubnqxPKJULDR7k/qDj0eh8YXBNQiCm9QAIsEic2boeJSeWlpZupZSLAsc721ZeMhd5Xdn9Aauj0CLf4UkIsovEEBLBIk7rkHivafyGvKFHPBzaC3cx4suR92L7T74hpDl4nQehhERDtXmx8ogEWCwJdfwnvvmfY118CZZzpbT7XtzjCLKHYcdNb4mDrmZOjtboeoUBpbOTIFsEiAKy2F22837YQEexgiKOWthqUTYPN7dl9k/EGLKI58NexQpAAWCXDPPgurV5v2xInQ1L8XbfCOgs2w4mFYP8NeRBERDa2HQ5cJENfI2focogAWCWBbttjLjDt2hLvucraeE1a8C1Y9DmtePHQRReqfodskc7ayMKYAFglgd9wBBQWm/cILEB0sx6TK9sFvz8CvU8C9z+5vejF0f9Scn1cUwCKB6v334cMPTXvIEDj7bGfrOS7lJbD2JbOIomSX3d/gLLOIov5pztUWgBTAIgFo7177wFvdujB1qrP1HJPHDZlvwfKHoHCT3V+np7kSRePzQ24RhTcogEUC0P33mysdAzzzDDRo4Gw9R2RZsOUDM7Mh71e7v1Zb6PYopFwZsosovEEBLBJgvvvOvsjmgAFw/fXO1nNEO74yiyj2/Gz3xTWFrhOh1Q0hv4jCGxTAIgGkoACGDjXt2FgTxAH3P/ddP5lFFDu/svti6kLncdB2JETFOVdbkFEAiwSQsWPh999N+7HHAuxE67m/wrIJsPl9uy+qJnS4BzqMgpjaztUWpBTAIgHi66/NVDMwVze+805n6zmgYKM5uJb5Jlge0xcRA21ugS7jITZQB6gDnwJYJADk5cGwYaYdHw+vvw6RkY6WBMXZsPIxWDsNPKWmzxUBqddB14cgIdXJ6kKCAlgkANx9N2zcfwrcp55yeOihLM8soPjtGXDn2/3NLjUzG5I6O1dbiFEAizjs3XdhxgzTPvtsuPVWhwopL4Y102DVY+Y6bBUa9DMnRE8+1aHCQpcCWMRBmzbB8OGmXbeuud6b38/z63FD5hv7F1Fssfvr9jar1xoNCMCpGKFBASzikPJy+L//M6veAKZPN9d58xvLA5tnw7IHzGkiKyS2N0MNza9Q8PqYAljEIY89ZhZdgNkLvuwyP32wZZkToS8dB3sW2f3xzaDrJGh5PUQoGvxBP2URB3z9NTz0kGm3b2+WG/vFroWw9H5zKaAKNepB5/HQ9laIjPVTIQIKYBG/27YNrr0WPB6z2u3f/4aaNX38oXtXmkUUWz60+6ISzCKKjqMgOtHHBUhlFMAifuR2m/DNyjL3p02Dbt18+IH5G8xl3jPfAizTFxFjlgx3Hgex9X344XIsCmARPxo/Hr791rSHDrXP++B1RTth5WRY9xJ4ykyfKwJa3mBOllMzxUcfLCdCASziJ7NmmUUWAF272suOvao0F379G6yeCu4Cu7/5FdDtEajd0QcfKlWlABbxg19+sZcaJyWZq13Ee/PK6+4iWPMCrHoCSvfY/Y0GmBOi1zvJix8m3qIAFvGx7Gy49FIoKjKLLP71L2jTxktv7nGbKw0vnwRFW+3+uieZ1WuNzvHSB4kvKIBFfKikBK680qx4AzMEcd55XnhjywOb3jMzG/attfsTO5g93maXahFFEFAAi/iIZcFNN9kH3f78Z7jnHi+86fbPzCKKnF/s/vgUc5n31OsgwunTqMnxUgCL+MjDD8PMmabdp49ZalytndLsH8wiiqxv7b4a9aHLBGgzAiJrVKte8T8FsIgPzJxpr3Rr1cpcXj62qovM9i6HpeNh68d2X1Qt6DgaOtwN0bWqW644RAEs4mWff27PeKhTBz75BOpXZb1D/npYNhE2vI29iKIGtLsNOt0PscneKlkcoutFS0AYM2YMnTt3JjExkSZNmnDzzTezZ8+eY78wwPz0E1x+OZSVQUyMmW7Wvv0JvknRDvj5dpjbATbMBCyziKL1TTB4LfSaovANEQpgCQiRkZHMnDmT3bt3s3TpUrZs2cJQny0T843ffoNBg8yVjV0uePtt6NfvBN6gdC8sGQdzWsPaF+0VbClXwYWr4JRXoGZzX5QuDnFZlnW0x4/6oIiv/Oc//2HIkCHk5uYe9XlpaWlkZGT4qaoj27gR+vaFzZvN/X/8A2655Thf7C6ENX+HVU9CaY7d3+g86PGYOTG6BLtKD79qDFgC0ldffUW3I5ylJj09nfT0dACys7P9WValtmyB/v3t8J006TjD11MGv78KKx6Gou12f71TzSKKhv18Ua4EEO0BS8CZPXs2N9xwA/Pnz6dXr15Hfa7Te8Dbt8NZZ8Ha/WshRo2Cp58+xnQzywMbZ8GyByH/d7u/dmfoPhmaXqxFFKFHe8AS+N59911GjBjBnDlzjhm+TtuxA845xw7fO+44RvhaFmz7xEwp27vU7q+Zaq5EkfpnLaIIMwpgCRgzZsxg1KhRfPzxx5x++ulOl3NUW7eaKxivWWPuDx8Ozz13lPDNWmAWUWQvsPtiG0DnCdBmuBZRhCkFsASE559/nkmTJvHZZ59x0kmBfeaujRtN+K5fb+7fdJM56FZp+OYsNXu82/5j90UnmkUU7e+G6AS/1CyBSWPAEhBcLhdRUVHUqHHonmB+fv5RX+fvMeA1a+Dcc+2T69x+u9nzPexS8vt+N2O8G9/hwF+jyFhodwd0GmOuwybhRGPAEriOsSMQEBYtgoEDzekl4QgH3Aq3wYpH4PfpYLlNnysSWt8IXR6E+KZ+r1sClwJY5Dh89ZU5p2/FDvmDD5pzPRwI39IcM4939fNQXmS/sMU10PVhSGzr75IlCCiARY7hnXfghhugtNQE7t//Drfdtv9Bd4EJ3VVPQtlBi0YaX2DOy1u3pxMlS5BQAIscgWXB5MnwwAPmfnS0OcvZ1VcD5aVmmGHFw1C8035Rch+ziKLBmY7ULMFFASxSidJSM7XsjTfM/aQkmD0bzu5XDpnvmEu956+3X5DUFbpNhqYXaRGFHDcFsMgf7NhhLiP0/ffmfsuW8J+5Fh0T58Kn4yB3hf3kmi3N1YZbXKNFFHLCFMAiB/n5Z7jsMrPQAuDUU+GT17+lzsaxsOtH+4mxjaDLA+YUkZExzhQrQU8BLLLfjBlw663mQpoAE+/8hQcuHkfkov/aT4pOMvN4298BUTWdKVRChgJYwl5hoZnV8Prr5n77JmuZ8+iDtKsxCyqOr0XGQfs7oeN9UG1SBAEAABUUSURBVKOuU6VKiFEAS1hbvRquugqWL4cmdbby2JBHuL7PdFyUmye4oswwQ5cHIL6Js8VKyFEAS1iyLHj1VbjrLoiN2M2T1z7JXRf8nRpRxfuf4YIW15pLvddq42itEroUwBJ29uyBm2+Gz/6Tz18HPsvoC5+mdnye/YQmF5rz8tbp7lyREhYUwBJW5s6F228tYXCndH6f+igNa2fZD9Y/A7o/Dg3OcK5ACSsKYAkLubnw17vLca97m3mjJtKywQb7waTuZtlwk4FaRCF+pQCWkPfB+xYfv/QR95wzgS7nrjzQbyW0xtXtEWjxJ3PZdxE/UwBLyNq6Ff4x6RsuajaW125YeKC/PKYxkd0n4mo9DCKiHaxQwp0CWEJOWRnMmraYRlnjeLTfZwf6S6w6xPQYQ2T7OyAq3sEKRQwFsISU/32+mr3fPsB1nd6F+qav2B1PeZu7qZk2GmKSnC1Q5CAKYAkJ61dsYd0Hkzg7dQZRncwiitLyaHbUHE7KwAkQ18jhCkUOpwCWoOYuc/P1lHvpU+8FWrU2J3HweFysKvoz7a6YRErdVg5XKHJkCmAJSnl79pExcyqu3OWc3Xjpgf5fsgfTdOBkurTr6mB1IsdHASxBZd/eEha+/RLdoyZzdnI2kftnjy3b0ZfYU56g55A+zhYocgIUwBIUcveW8+2bb9EjYiID6m460F9cFsfi2u/T85rzcUVoEYUEFwWwBLQd2y2+fP0D0mImMLjxrwf6N+xpy84GjxDb8Gl6XXiBgxWKVJ3LsqyjPX7UB0V8ZcUK+O8bX3Fm7XGc3OqnA/079zVhS52H6Hn5DURERZOWlkZGRoaDlYocl0r/e6Y9YAkY5eXw6afw37d/5tKW93Nvz68OPJZbXJette+nw5W30TAmzsEqRbxHASyOy8qC116DL2b/yq2nP8ALg2cfeKyorCbZ9e4m5crR1I6p7WCVIt6nABZHlJfDF1/A9Omw+LtNTLjkIT6/6w0iIzwAuD3R5De+haQ+40mJa+hwtSK+oQAWv1q5Et56C2bOhJK8bMZd8hhvPzWNGtGlAFiWC3fz64ju9RBJCS0drlbEtxTA4nObNsG//gXvvAO//AK14vK4Z+AzjBo0hVpx+QeeZzW9BFf3R4lO6uJgtSL+owAWn9i4EWbPhvfegx9/NH01oou5+4J/MP7SySTX2m0/uUE/6PE4ruRTHalVxCkKYPEKyzJXFv7oI7MtWmQ/Fhnh5i993+CRP02iSe3N9gN1ekGPx6HRuboShYQlBbBUWUEBzJsHn3xito0b//gMi9sGz+bBSyfQIHa13V2rnbnoZfMrFLwS1hTActw8HliyxMxe+PxzWLAASksPf16XLhaj/vwFV7cbR3zxQbvC8c2g60PQ8i8QoT96IvpbIEfk8ZgVafPnmz3d+fPNJd3/KCYGzjwTLrwQruq/kKa77oed86B4/xNq1INO46DdSIiM9et3EAlkCmA5oLAQMjLghx/g++/NHu7evZU/t00bOO88OP98OPtsSHCvhGUTYOWH9pOiEqDDKOh4D0Qn+udLiAQRBXCYKi+H1avh55/hp5/MtmQJuN2VP79RI+jf32wDBkDLiim6+Rtg+UOQ+SYHTh0SEQNtb4XO4yG2vu+/jEiQUgCHgdJSWLXKzMFdsgQWLzbtgoIjv6ZFCzOs0Lev2dq3/8PxsuIsWDEZ1v0DPGWmzxUBLa8347w1W/jyK4mEBAVwCCkvh8xME7YrV5ppYcuXw2+/HXnPFiAuDnr3hpNPhj594LTToEmTIzy5NBd+/RusngrugxK8+eXQ7RGo3cmr30kklCmAg1BuLqxda7bVq03A/vabaRcXH/21sbHQtasJ3N69IS0NOneG6OhjfKi7CNa+CCsfh9KDjsQ1PAe6PwbJJ1f7e4mEGwVwACovh23bYMMGWL/e7NWuXw/r1pktO/v43qdJE+jWzQRut27Qs6cZSog6kd+6xw3rZ8DySVC01e6vm7Z/EcWAE/lqInIQBbCfWRbs3g1bt8KWLWbbvNmcL2HzZrOYYfPmow8Z/FGLFtChg9k6djR7tJ07Q5061SnUA5veMzMb9q21+xM7QvdHodllWkQhUk0KYC8pKjLntc3Kgp07YccOc7t9u2lv3272ardtq3zxwrE0bWqmfrVubbZ27czWpg3Ex3vxi1gWbP8Mlo6DnF/s/viU/YsortMiChEv0d+kP/B4IC8PcnLMtmfPoduuXYdu2dlmO9qMgmOJiDDDBSkpZmvRAlJTzW2rVuY21h/rF7J/hKX3Q9Z8u69GMnSeAG1vgcgafihCJHyETAC73ZCfb4Jw3z6z5eeb27y8Q29zc007N9dse/ea25wcc3v0y+SdmNq1oXFje2va1IRt06bQrJnZGjU6joNgvrR3BSwdD1vn2H1RtaDjKOhwD0TXcq42kRDmkwD2eKCkpPKtuNi+PXgrKjryVlh46FZQYN8WFJigrcp/66uiZk2oV89s9eubLTkZGjaEBg3M/YYN7S0ukC9flr8elk2EDW9jL6KoAe1ug073Q2yyX8spLy9n7NixvP766xQXF3Peeefx8ssvk5zs3zpE/MUnAXzZZTBnzrGf54SICLNXmphotqQkc792bXPQKinJbHXqQN265rZePdOuW9dPQwG+VrQDVjwKv6cfuoii1VDoMhFqNnekrCeeeIKPPvqIhQsXUq9ePYYNG8Z1113Hp59+6kg9Ir7mkwCu4YWhwuhoc3ApLs5sNWva92vWtO8nJNj3a9Wy79eqdeiWmGhu4+PD+OB96V749Wn47VkoL7T7m19pZjYktneuNiA9PZ0HH3yQVq1aAfDUU0/Rpk0bNmzYQGpqqqO1ifiCTwL42muhVy8TxDVqmLNlxcba9+Pi7PsV7T/eRkb6orIw5S6ENS/AqiegNMfub3Sumctbt7dzte2Xm5vLpk2b6N3brqV169YkJiaybNkyBbCEJJd1lCNOF1xwgbVr164qv3l2djb164fGyViC87tYULILirbbQw1AmRVDdGJqQB1cKy0tZfny5XTp0oUaB/0Xavny5TRt2pS6dese6MvOzqbiz2VJSQk9evTwe72+Epx/ziqn72JbtGjRZ5ZlXXDYA5ZlHW2rlt69e1f3LQJGUH0XT7llZf7Tsj5qY1lvY29zO1nWpg+s3r17OV3hYXJycizA+uWXXw7pT0xMtD766KMjvi4+Pt7XpflVUP05OwZ9l0NUmrER1fhHQQKNZcHWT+DTXvDDEMhfZ/prtoBT34CBy6D5pUDgDYInJSWRkpLC4sWLD/StX7+evLw8unXr5mBlIr4TMvOAw17WArOIInuB3VejPnR5ANoMD4pFFMOHD+fJJ5+kf//+1KtXjzFjxnD++edr/FdClk8DePjw4b58e78K2O+Ss8wsG972H7svOhE6job2d0N0wmEvCdTvMnbsWHJycjjppJMoKSnh3HPPZebMmUd9TajNEQ7U301V6Lsc21EPwnFgdr4EnH2/w/KJsOGfHPg1RcZCu9uh01hzHbYwkJaWRkZGhtNliBxLpeN+GoIINkXbYcUjsO4VsPafMs0VCa1vhC4PQnxTZ+sTkeOmAA4WpTmw6ilY/RyUF9n9KX+Cbg9DYjvnahORKvH5LIhhw4bRvHlzEhMTady4McOGDSMnJ+fYLwxAJSUljBgxgrZt21KrVi1SUlIYPXo0xce6DEV1uAth5RPwUSuzkKIifBufDxcshjNmVTl8n3/+eU455RTi4+Np06aNF4v2vfLyckaPHs3SpUupVasWV1xxBdWZs+6kWbNm0bdvXxITE4k6obPlB54xY8bQuXNnEhMTadKkCTfffDN79uw59gsD1Pjx42nZsiWJiYk0aNCAK6+8kk2bNnnt/X0ewPfccw+//fYbeXl5/PrrrxQWFnLbbbf5+mN9wu12k5yczMcff8zevXv57rvv+PrrrxkzZoz3P6y8FNZMgzmtzeyGsv3Xh08+Dc75Bvr/F+r2rNZHNGnShPvuu4/x48dXv14/qzhvRIcOHdiyZQsA1113ncNVVU2dOnUYOXIkzz77rNOlVFtkZCQzZ85k9+7dLF26lC1btjB06FCny6qy6667jiVLlpCXl8eGDRtISUnhmmuu8d4HHGmCsOWFhRh/lJOTY1177bVWr16BtxCgql544QWrW7du3ntDT7llrZ9pWR+1+sMiii6WtXmOZXk83vus/WbMmGG1bt3a6+/rSykpKdb06dMPTJBft26dBViZmZnOFlYN8+bNsyIjI50uw6vmzp1rJSYmOl2GV+Tn51ujRo2y6tatW5WXV5qxfvn/zhNPPMHkyZPJz88nLi7umFOLgslXX33lnYUClmWmki0dD3uX2f01U83VhltcCxE6QQbovBHBxGt/Pxz0z3/+k1tvvZW8vDyioqJ45plnvPbeVR6CuOGGG3C5XEfcJkyYcOC5Y8eOZd++faxfv55Ro0YF5HjjiXyfCs8++ywLFixg8uTJ1fvwrG/hizNg/mA7fGMbQu+/w0WroeX/nVD4VuW7BJO8vDwAateufUh/UlLSgcfEebNnz+aVV17hueeec7qUahkyZAi5ubls376dhx56iK5du3rtvau8B/zCCy/wt7/97YiPx1dyobKWLVsyePBgBg0axKZNm4iICJyV0Cf6faZOncqTTz7J119/TUpKStU+NGcJLBkH2w863210beh0H7S/C6JqVultq/K7CSa1apmTCOXm5h7Sv3fvXhITE50oSf7g3XffZcSIEcyZM4devXo5XY5XNGrUiJtvvplWrVqxadOmQ04QVVVVDuCEhAQSEg5fZXUsbrebrVu3UlBQcOAvUiA4ke/zyCOP8PLLLzN//nzat6/COXTz1sLyB2HjLLsvMhba3QmdxkCN6v1iq/q7CRY6b0RgmzFjBqNGjeLjjz/m9NNPd7ocr3K73RQUFLBt2zavBLBPd0GzsrJ488032bvXHMFfs2YN9913H2eccUZAhe+JGD16NNOnT69a+BZuhZ9ugf90tMPXFQVtRsDg36Hnk9UO3xPhdrspLi6mrKwMy7IoLi727ZQ6L6o4b0RJSQl5eXlBfd6I8vJyiouLKd1/Xa2K34PlzYsT+snzzz/Pvffey2effRb04evxeHjhhRfIysoCYMuWLdx2222kpqbSoUMH73zIkY7OWV6YBZGVlWX179/fqlOnjhUfH2+lpKRYI0aMsHbs2FHdt3bEhg0bLMCKiYmxataseWDr1KnT0V9YvMuyFo+2rFmxh85sWHCNZeWt9U/xlZg4caKFWcd8yBYM3G63NWrUKCsyMtJKSEiwLrvsMis7O9vpsqpkxowZlf4egnFGB2BFRUUd8vejZs2aTpdVJeXl5dbAgQOt+vXrW/Hx8VaTJk2sIUOGWOvWravK21WasToXhC+V5cPqZ81lgMoOOjjUZBB0nwx1QudE4k7RuSAkSOhcEH5TXgLr0mHlo1CcZffXPx26Pw4N+jpXm4gEDAWwN3nKzSXel0+Egg12f1I36P6Y2fMN2yuCisgfKYC9wbJg6xyziCJ3pd2f0Gr/IoprzGXfRUQOogCurp3fwJL7Yff/7L7YRtD1QWh1I0TGOFaaiAQ2BXBV7VlsrkSx/TO7LzoJOo+FdndAVHAvdhAR31MAn6i8NbDsAdj0b7svMt6sXOs0GmLqOFebiAQVBfDxKtwCyyfB+hlglZu+ikUUXSZAXCNn6xORoKMAPpbiXeZE6GteAE/J/k4XpA6BrpOgVmtHyxOR4KUAPpKyfPhtqllE4d5n9ze5CHo8BkneOyOSiIQnBfAflZfAupdhxaNQkm331+8LPR43iylERLxAAVzBUw4bZu5fRLHR7k/qvn8RxUAtohARr1IAWxZs+RCWTYDcVXZ/rbZmEUXKVVpEISI+Ed4BvONrc8HL3T/ZfXFNoOtEaDUUIqKdq01EQl54BvDun80iih1f2n0xdaDT/dDudoiKc642EQkb4RXAub+ZoYbNs+2+yHjo8FfoeC/EJDlXm4iEnfAI4IJNZhFF5utgeUxfRLRZRNF5AsQ1dLQ8EQlPoR3Axdmw8nFY+yJ4Svd3uqDlddD1IUho6WR1IhLmQjOAy/Lg12fgtyngzrf7m10C3R6FpC7O1SYisl9oBXB5Maz9B6x8DEp22f0NzoIeT0Dyqc7VJiLyB6ERwB43ZL4Jyx+Cws12f51eZvVao3O1iEJEAk5wB7BlmRkNyyZA3mq7v1Y76P4oNL9CiyhEJGAFZwBblpnDu3Qc7DnoirjxzczBtZZ/gYjg/GoiEj6CL6V2/WRWr+382u6rUQ86jYO2t2oRhYgEjeAJ4NxVsHQCbPnA7ouqCR3ugQ6jIKa2c7WJiFRB4Adw/gZzcG3DWwctoogxe7udx0FsAyerExGpssAN4OIsWDEZ1r1kL6JwRUDL6804b80WjpYnIlJdgRfApblmAcVvz4C7wO5vdil0nwy1OzlXm4iIFwVOALuLzJLhlY9D6R67v+HZ5oToyac4V5uIiA84H8AeN6x/3YzzFm21++um7V9EMcCpykREfMq5ALY8ZhHF0gmwb43dn9jBnK+h+eVavSYiIc3/AWxZsP1zs4giZ7HdH998/yKK67WIQkTCgn+TLvtHs4gia77dVyPZTCdreytExvq1HBERJ/kngPeugKXjYeucgz65FnQcZa5GEZ3olzJERAKJb89Uk58JP1wPn3Szwzeihlm9dvF6c/FLhW/YKykpYcSIEbRt25ZatWqRkpLC6NGjKS4udro0EZ/yzR5w0U5Y+Sisexk8ZabPFWGuNNxlItRs7pOPleDkdrtJTk7m448/pm3btmzZsoXLL7+c0tJSnnvuOafLE/EZl2VZR3v8qA9WyuOGOa0OPS9v8yuh2yNQu8MJv52EpxdffJH09HSWLl161OelpaWRkZFx1OeIBIBKp3QdK4BFHOFyud4HCizLuq6Sx4YDw/ffjbUsS9eYkqCkABafcrlcrwN/OcpTJluWNeEPr7kbGAekWZa1yYfliThKASw+5XK5EoCjzS8stCyr8KDn/xUYAwywLGuFr+sTcZICWAKGy+V6ABgBnGNZ1upjPV8k2GnJmQQEl8v1NHA1cJZlWb87XY+IP2gPWBzncrlaABuAUqDsoIc2WpbV2ZGiRPxAASwi4hBds11ExCEKYBERhyiARUQcogAWEXGIAlhExCEKYBERhyiARUQcogAWEXHI/wPXV9PF1ilhCgAAAABJRU5ErkJggg==\n",
-      "text/plain": [
-       "<Figure size 360x360 with 1 Axes>"
-      ]
-     },
-     "metadata": {
-      "needs_background": "light"
-     },
-     "output_type": "display_data"
-    }
-   ],
-   "source": [
-    "import matplotlib.pyplot as plt\n",
-    "import numpy as np\n",
-    "import matplotlib.ticker as plticker\n",
-    "\n",
-    "# Create figure and add axes\n",
-    "#fig = plt.figure(figsize=(5, 5))\n",
-    "#ax = fig.add_subplot(111)\n",
-    "\n",
-    "fig, ax = plt.subplots(figsize=(5, 5)) \n",
-    "\n",
-    "# Move left y-axis and bottim x-axis to centre, passing through (0,0)\n",
-    "ax.spines['left'].set_position('zero')\n",
-    "ax.spines['bottom'].set_position('zero')\n",
-    "\n",
-    "# Eliminate upper and right axes\n",
-    "ax.spines['right'].set_color('none')\n",
-    "ax.spines['top'].set_color('none')\n",
-    "\n",
-    "# Show ticks in the left and lower axes only\n",
-    "ax.xaxis.set_ticks_position('bottom')\n",
-    "ax.yaxis.set_ticks_position('left')\n",
-    "\n",
-    "\n",
-    "#loc = plticker.MultipleLocator(base=3.0) # this locator puts ticks at regular intervals\n",
-    "#ax.xaxis.set_major_locator(loc)\n",
-    "#loc = plticker.MultipleLocator(base=2.0) # this locator puts ticks at regular intervals\n",
-    "#ax.yaxis.set_major_locator(loc)\n",
-    "\n",
-    "x = np.linspace(-3, 3, 100)\n",
-    "y = np.exp(x)\n",
-    "ax.plot(x, y, color='b', linewidth=2.5, label=('$e^x$'))\n",
-    "\n",
-    "y = 1+x\n",
-    "ax.plot(x, y, color='orange', linewidth=2.5, label=('$1+x$'))\n",
-    "\n",
-    "\n",
-    "ax.tick_params(axis='both', which='major', labelsize=13)\n",
-    "\n",
-    "\n",
-    "ax.legend(fontsize=15)\n",
-    "ax.set_ylim(-2,10)\n",
-    "# Add legend\n",
-    "\n",
-    "#ax.legend(bbox_to_anchor=(1.05, 0.3),loc='lower left', \n",
-    "#          frameon=False, labelspacing=0.2, fontsize=13)\n",
-    "\n",
-    "plt.tight_layout()\n",
-    "plt.savefig('exp_linapp.png')"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "f7510b82",
-   "metadata": {},
-   "source": [
-    "# 07 - Analysing change, continuously (Part 2)"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "4d5884a5",
-   "metadata": {},
-   "source": [
-    "In many different contexts we may be interested in finding the maximum or minimum values a given function may assume. This process is referred to as *optimization*.  \n",
-    "  \n",
-    "In Ecology, some of the most common uses for optimization methods are:\n",
-    "\n",
-    "1 - *Statistical estimation and inference*: optimizing the likelihood of a model and its parameters being true, for a given set of observations.\n",
-    "\n",
-    "2 - *Resources management*: finding a balance between different wildlife and economic priorities (market, legal, natural constraints)\n",
-    "\n",
-    "3 - *Optimality theory*: how can the behavioural or life-history trade-offs result in optimal strategies for different organisms"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "1733c93f",
-   "metadata": {},
-   "source": [
-    "## Extreme values of functions  \n",
-    "  \n",
-    "For a given function $f(x)$ that is **continuous** in an interval $a \\le x \\le b$, there will always be a global maximum and a global minimum, points at which the function assumes its maximum and minimum values in the interval.  \n",
-    "```{image} extrema.png  \n",
-    "  \n",
-    "```\n",
-    "  \n",
-    "A function $f(x)$ presents a *local maximum* value if for a given vicinity of the point $x_0$, $f(x_0)$ is greater than all the other $f(x)$ for $x$ in this vicinity. The analogous definition can be made for a *local minimum* value when $f(x_0)$ is smaller than all the other $f(x)$ in the vicinity. Local minima and local maxima constitue the *local extrema* for $f(x)$, and with the set of all extrema we can find the global maximum and the global minimum.  \n",
-    "  \n",
-    "An important result is that, if the function $f$ has a local extremum at a given point $x=c$ in that interval ($a<c<b$) and the derivative of the function exists at that point (in other words, if $f'(c)$ is well defined), then we will have:  \n",
-    "  \n",
-    "$f'(c)=0$  \n",
-    "    \n",
-    "  \n",
-    "`````{warning}  \n",
-    "\n",
-    "````{panels}  \n",
-    "  \n",
-    "```{image} ./xcube.png  \n",
-    "\n",
-    "```  \n",
-    "\n",
-    "---  \n",
-    "  \n",
-    "Note, however, that the contrary is not always true. If $f'(c)=0$, not necessarily $f(c)$ will be an extremum point. On the left plot, $f(x)=x^3 \\implies f'(x)=2x^2$, thus $f'(0)=0$ but there is no extremum at $x=0$ .\n",
-    "      \n",
-    "```` \n",
-    "`````  \n",
-    "  \n",
-    "Thus the points $x=c$, for which $f'(c)$, will constitute **candidates** for local extrema, and will have to be evaluated under other sets of conditions. Also important to have in mind that point for which the derivative is not definided (kinks or discontinuities) may also constitute local extrema (see on the first figure).  \n",
-    "  \n",
-    "  \n",
-    "In general, when searching for local extrema, we have to adopt the following rules:  \n",
-    "\n",
-    "- Do not assume points $x$ for which $f'(x) = 0$ are extrema. They are candidates.\n",
-    "- Check points where derivative is not defined.\n",
-    "- Check endpoints of domain.\n",
-    "  \n",
-    "```{tip}\n",
-    "If possible, create a plot. Visuals are powerful tools and can help verify whether calculations have been done correctly.\n",
-    "```"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "fd6ac48c",
-   "metadata": {},
-   "source": [
-    "## Monotonicity and concavity  \n",
-    "  \n",
-    "\n",
-    "\n",
-    "Imagine a function $f(x)$ defined in a given interval $a \\le x \\le b$. For every $x_1$ and $x_2$ in this interval, if $x_{2} > x_{1}$ implies that $f(x_{2}) > f(x_{1})$ then the function is **increasing** in this interval. If on the other hand, $x_{2} > x_{1}$ implies that $f(x_{2}) < f(x_{1})$ then the function is **decreasing** in this interval.  This can be graphically shown as:  \n",
-    "  \n",
-    "```{image} incr_decr.png  \n",
-    "```  \n",
-    "  \n",
-    "\n",
-    "  \n",
-    "```{note}  \n",
-    "Since we are using the stric inequalities (\"$>$\" or \"$<$\") it can be added that the function is **monotonic**, or monotonically increasing or decreasing. If we had $x_{2} > x_{1}$ implying that $f(x_{2}) \\ge f(x_{1})$ (or $f(x_{2}) \\le f(x_{1})$), the function would be *non-decreasing* (or *non-increasing*).  \n",
-    "  \n",
-    "```\n",
-    "  \n",
-    "Now suppose that $f$ is well-behaved in a given interval $a \\le x \\le b$ (in mathematical terms, we say: $f$ is continuous and differentiable in $(a,b)$), then an important results is that\n",
-    "\n",
-    "$$\\mbox{If }\\ f'(x) > 0\\ \\mbox{ for}\\ a \\le x \\le b \\implies f(x)\\ \\mbox{is increasing for}\\ a \\le x \\le b$$\n",
-    "\n",
-    "$$\\mbox{If }\\ f'(x) < 0\\ \\mbox{ for}\\ a \\le x \\le b \\implies f(x)\\ \\mbox{is decreasing for}\\ a \\le x \\le b$$  \n",
-    "  \n",
-    "  \n",
-    "In other words, if the derivative of function $f$ is positive (negative) for every point $x$ in a given interval then the function is increasing (decreasing) in this interval.  \n",
-    "  \n",
-    "````{admonition} Example  \n",
-    "Let us calculate the intervals where the function $f(x) = x^{3} - \\displaystyle\\frac{3}{2}x^{2} - 6x + 3$ is increasing and decreasing. Since $f(x)$ is continuos and differentiable for all real values of $x$ (remember that all polynomial functions are well-behved), we can use the first derivative test:  \n",
-    "  \n",
-    "$$f'(x) = 3x^{2}-3x-6 = 3(x^{2}-x-2) = 3(x+1)(x-2)$$  \n",
-    "  \n",
-    "and we see that $f'(x)$ has roots in $x=-1$ and $x=2$. Since $f'(x)$ is a polynomial of degree 2 with the coefficient of the leading term positive, it is *concave up*. From this, we can conclude that:  \n",
-    "  \n",
-    "$$x<-1\\ \\mbox{or}\\ x>2 \\implies f'(x) > 0 \\implies f(x)\\ \\mbox{is}\\ increasing$$  \n",
-    "$$-1<x<2 \\implies f'(x) < 0 \\implies f(x)\\ \\mbox{is}\\ decreasing$$  \n",
-    "  \n",
-    "The following plot shows the function $f(x)$ and its derivative. Compare the intervals where the derivative is positive (negative) with the intervals where the function $f(x)$ is increasing (decreasing).  \n",
-    "  \n",
-    "```{image} func_diff.png  \n",
-    "```  \n",
-    "\n",
-    "````  \n",
-    "  \n",
-    "When we discussed polynomials of degree 2, depending on the sign of the coefficient of the leading term, we had the two cases:  \n",
-    "  \n",
-    "```{image}  concav.png  \n",
-    "```\n",
-    "  \n",
-    "Let us now generalise the notion of concavity for any arbitrary function. Suppose $f(x)$ is differentiable in a given interval $a \\le x \\le b$ (in other words, the derivative $f'(x)$ is well-defined for every point in this interval). Then we have the following results:  \n",
-    "  \n",
-    "$$\\mbox{If }\\ f'(x)\\ \\mbox{is increasing for}\\ a \\le x \\le b \\implies f(x)\\ \\mbox{is}\\ concave\\ up$$\n",
-    "\n",
-    "$$\\mbox{If }\\ f'(x)\\ \\mbox{is decreasing for}\\ a \\le x \\le b \\implies f(x)\\ \\mbox{is}\\ concave\\ down$$  \n",
-    "  \n",
-    "This means that if for a given function $f(x)$ the slopes of the tangent lines are increasing in a given interval of $x$, in this interval the function should resemble an \"upwards U-shape\". If on the other hand, the slopes of the tangent lines are decreasing in this interval, the function should resemble a \"downwards U-shape\". This can be visualised on the plots below.  \n",
-    "  \n",
-    "```{image}  slopes_inc_dec.png  \n",
-    "```\n",
-    "  \n",
-    "If the function $f$ is twice differentiable in $a \\le x \\le b$ ($f''(x)$ is well-define for $a \\le x \\le b$), we can use the last two results to find a general criterium for concavity. This will be given by:  \n",
-    "  \n",
-    "$$\\mbox{If }\\ f''(x) > 0\\ \\mbox{ for}\\ a \\le x \\le b \\implies f'(x)\\ \\mbox{is increasing for}\\ a \\le x \\le b \\implies f(x)\\ \\mbox{is}\\ concave\\ up$$\n",
-    "\n",
-    "$$\\mbox{If }\\ f''(x) < 0\\ \\mbox{ for}\\ a \\le x \\le b \\implies f'(x)\\ \\mbox{is decreasing for}\\ a \\le x \\le b \\implies f(x)\\ \\mbox{is}\\ concave\\ down$$  \n",
-    " \n",
-    "```{admonition} Example\n",
-    "If you have a population curve, these tools will help you analyze questions such as 'How fast is the population growing?' 'Is the growth accelerating?' or 'Are any of these factors changing over different periods?\n",
-    "```\n",
-    "  \n",
-    "````{admonition} Example  \n",
-    "Let us analyse the function $f(x) = x^{3}-\\displaystyle\\frac{3}{2}x^{2}-6x+3$ again. We have:  \n",
-    "  \n",
-    "$$\\begin{align} \n",
-    "f'(x) = 3x^{2}-3x-6 \\\\\n",
-    "f''(x) = 6x -3 \\end{align}$$  \n",
-    "  \n",
-    "The function f''(x) has a root in $x=\\displaystyle\\frac{1}{2}$. Since its slope is positive, we have:  \n",
-    "  \n",
-    "$$x<\\displaystyle\\frac{1}{2} \\implies f''(x) < 0 \\implies f(x)\\ \\mbox{is}\\ concave\\ down$$  \n",
-    "$$x>\\displaystyle\\frac{1}{2} \\implies f''(x) > 0 \\implies f(x)\\ \\mbox{is}\\ concave\\ up$$  \n",
-    "  \n",
-    "On the plots below, compare the intervals where the second derivative is positive (negative) with the intervals where the function $f(x)$ is concave up (concave down).  \n",
-    "  \n",
-    "```{image}  plot_concav.png  \n",
-    "```  \n",
-    "\n",
-    "````"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "8d0f05bc",
-   "metadata": {},
-   "source": [
-    "## Finding extrema  \n",
-    "  \n",
-    "  \n",
-    "Back to our strategy for finding maxima or minima of functions, we first determine the set:  \n",
-    "  \n",
-    "- find all points $c$ for which $f'(c) = 0$ or all point $c$ where $f'(c)$ does not exist. These are called **critical points**.\n",
-    "- find endpoints of domain of $f$.  \n",
-    "  \n",
-    "Additionally, if $f$ is twice differentiable (in some interval containing $c$):  \n",
-    "  \n",
-    "- If $f'(c) = 0$ and $f''(c) < 0 \\implies c$ is a local maximum (since the function is concave down)\n",
-    "- If $f'(c) = 0$ and $f''(c) > 0 \\implies c$ is a local minimum (since the function is concave up)  \n",
-    "  \n",
-    "````{admonition} Example  \n",
-    "Let us analyse the function $f(x) = \\displaystyle\\frac{3}{2}x^{4}-2x^{3}-6x^{2}+2$, and find its local extrema. We have:  \n",
-    "  \n",
-    "$$\\begin{align}\n",
-    "f'(x)&= 6x^{3}-6x^{2}-12x=6x(x-2)(x+1) \\\\\n",
-    "f''(x)&=18x^{2}-12x-12=6(3x^{2}-2x-2)\\end{align}$$  \n",
-    "  \n",
-    "Since this is a polynomial function, the derivative is well-defined for all points. Also, for the critical points (roots of $f'(x)$) we have $x=0$, $x=-1$, and $x=2$. Calculating the value of the second derivative at the critical points:  \n",
-    "  \n",
-    "$$\\begin{align}\n",
-    "&x=0 \\implies f''(0)=-12 \\text{ (local maximum)} \\\\\n",
-    "&x=-1 \\implies f''(-1)=18 \\text{ (local minimum)} \\\\\n",
-    "&x=2 \\implies f''(2)=36 \\text{ (local minimum)}\\end{align}$$  \n",
-    "  \n",
-    "Now we can check on the plot of $f(x)$ that we have successfully determined the local extrema.  \n",
-    "  \n",
-    "```{image} plot_extrema.png  \n",
-    "```\n",
-    "\n",
-    "````"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "15f830da",
-   "metadata": {},
-   "source": [
-    "## Inflection points  \n",
-    "  \n",
-    "  \n",
-    "As we saw before, given a function $f(x)$, for some values of $x$, there might be a change in the concavity of the function:  \n",
-    "  \n",
-    "```{image} inflection.png  \n",
-    "```\n",
-    "    \n",
-    "The points $x$ for where the function changes its concavity are called **inflection points**. If $f$ is twice differentiable and $c$ is an inflection point, then $f''(c)=0$.  \n",
-    "\n",
-    "Inflection points are the points where the function is (locally) steepest. It therefore helps us answer questions like when the growth/decline was strongest.\n",
-    "  \n",
-    "```{note}  \n",
-    "Note again that the points $x=c$ where $f''(c)=0$ are only candidates for inflection points. See that if $f(x)=x^{4}$, $f''(x)=12x^{2}$. Then, $f''(0)=0$, but $x=0$ is not an inflection point of $f$.  \n",
-    "  \n",
-    "After calculating the candidate points $x=c$, if the concavities of the function (given by the sign of the second derivative) to the left and to the right of $c$ change, then $c$ is an inflection point.  \n",
-    "\n",
-    "```  \n",
-    "\n",
-    "````{admonition} Example  \n",
-    "\n",
-    "Consider the Holling Type III functional response   \n",
-    "  \n",
-    "$$f(N)=\\frac{aN^{2}}{1+ahN^{2}},$$  \n",
-    "  \n",
-    "where $f(N)$ is prey consumption rate per predator, $N$ is the prey population density, $a$ is the attack rate, and $h$ is handling time. Let us check if this function presents any inflection point. We have:  \n",
-    "  \n",
-    "$\\begin{align}f'(N) \n",
-    "&= \\frac{2aN(1+ahN^{2})-aN^{2}(2ahN)}{(1+ahN^{2})^{2}} \\\\\n",
-    "&= \\frac{2aN}{(1+ahN^{2})^{2}}\\end{align}$  \n",
-    "  \n",
-    "<br />\n",
-    "  \n",
-    "$\\begin{align}f''(N)\n",
-    "&=\\frac{2a(1+ahN^{2})^{2}-2aN[2(1+ahN^{2})\\cdot2ahN]}{(1+ahN^{2})^{4}} \\\\\n",
-    "&=\\frac{2a(1+2ahN^{2}+a^{2}h^{2}N^{4})-2a[4ahN^{2}+4a^{2}h^{2}N^{4}]}{(1+ahN^{2})^{4}} \\\\\n",
-    "&=\\frac{2a(1-3ahN^{2})}{(1+ahN^{2})^{3}}\\end{align}$  \n",
-    "  \n",
-    "See that the function $f''(N)$ has two roots: $N=-\\sqrt{\\displaystyle\\frac{1}{3ah}}$ and $N=\\sqrt{\\displaystyle\\frac{1}{3ah}}$. Since $N$ is a population density, it only makes sense to analyse $N \\ge 0$. It is also possible to show that:\n",
-    "  \n",
-    "$$N<\\sqrt{\\displaystyle\\frac{1}{3ah}} \\implies f''(N) > 0$$  \n",
-    "$$N>\\sqrt{\\displaystyle\\frac{1}{3ah}} \\implies f''(N) < 0$$  \n",
-    "  \n",
-    "This means that we have a change of concavity at the point $N=\\sqrt{\\displaystyle\\frac{1}{3ah}}$ and thus this is and inflection point of this curve.  \n",
-    "  \n",
-    "Analyse how the function $f(N)$ changes with $N$.  \n",
-    "  \n",
-    "```{image} hollingIII.png  \n",
-    "```\n",
-    "  \n",
-    "How does this function compares with the Holling type II functional response?\n",
-    "\n",
-    "````  \n",
-    "  \n",
-    "```{tip}  \n",
-    "  \n",
-    "For the intervals where the function is concave up you can also think of it as representing \"accelerated change\" of that function (since the first derivative, the \"velocity\" is increasing), while the intervals where it is concave down represent \"deccelerated change\" of the function. Understanding this can be particularly important if you function has a specific biological meaning, as in the previous example.  \n",
-    "  \n",
-    "```"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "ed45757b",
-   "metadata": {},
-   "source": [
-    "## Series expansions \n",
-    "  \n",
-    "From the formal definition of a derivative  \n",
-    "  \n",
-    "$$f'(a)= \\lim_{x \\to a} \\frac{f(x)-f(a)}{(x-a)} \\xrightarrow{\\text{$x$ very close to $a$}} f'(a) \\approx \\frac{f(x)-f(a)}{(x-a)},$$  \n",
-    "  \n",
-    "which means that if $x$ is sufficiently close to $a$, the average rate of change is sufficiently close to the value of the derivative at that point. From the previous approximation, we have:  \n",
-    "  \n",
-    "$$f(x) = f(a)+f'(a)(x-a),$$  \n",
-    "  \n",
-    "which provides a linear approximation of $f(x)$ around $a$ (see that if $\\alpha= f(a)$ and $\\beta=f'(a)$, both constants, then $f(x) = \\alpha+\\beta(x-a)$, which can easily be arranged on the form $y=mx+b$).  \n",
-    "  \n",
-    "`````{admonition} Example  \n",
-    "  \n",
-    "````{panels}  \n",
-    "  \n",
-    "```{image} ./exp_linapp.png\n",
-    "\n",
-    "```  \n",
-    "\n",
-    "---  \n",
-    "  \n",
-    "Say we have $f(x)=e^{x}$ and we want to use this linear expression to approximate the function close to $a=0$: \n",
-    "\n",
-    "$$f(x)=f(0)+f'(0)(x-0)=1+x,$$  \n",
-    "  \n",
-    "since $f'(x) = e^{x}$ and thus $f(0)=f'(0)=1$.  \n",
-    "  \n",
-    "The approximation is reasonable if $x$ very close to $0$, but fails when $x$ increasingly departs from $0$. \n",
-    "  \n",
-    "````  \n",
-    "  \n",
-    "`````"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "dcb07889",
-   "metadata": {},
-   "source": [
-    "One way of trying to make the approximation better is to add terms of higher order of $x$. Suppose we want to approximate the function $f(x)$ at point $a$ by a given polynomial of degree $n$  \n",
-    "  \n",
-    "$$P(x)=a_{0}+a_{1}(x-a)+a_{2}(x-a)^{2}+\\cdots+a_n(x-a)^{n}$$  \n",
-    "  \n",
-    "so that the derivatives of $f(x)$ and $P(x)$ are the same at $x=a$:\n",
-    "\n",
-    "$$\\begin{align}f(a)&=P(a) \\\\\n",
-    "f'(a)&=P'(a)\\\\\n",
-    "&\\vdots \\\\\n",
-    "f^{(n)}(a)&=P^{(n)}(a)\\end{align}.$$  \n",
-    "  \n",
-    "But for the polynomial, we have:  \n",
-    "  \n",
-    "$$\\begin{align}  \n",
-    "P(a)&=a_{0} \\\\\n",
-    "P'(a)&=1 \\cdot a_{1} \\\\\n",
-    "P''(a)&=2 \\cdot 1 \\cdot a_{2} \\\\\n",
-    "P'''(a)&=3 \\cdot 2 \\cdot 1 \\cdot a_{3} \\\\  \n",
-    "P''''(a)&=4 \\cdot 3 \\cdot 2 \\cdot 1 \\cdot a_{4} \\\\\n",
-    "&\\vdots \\\\\n",
-    "P^{(n)}(a)&= \\underbrace{n(n-1)(n-2) \\cdots 3 \\cdot 2 \\cdot 1}_{\\displaystyle\\text{$$n!$$}}\\cdot a_{n} \\end{align}$$  \n",
-    "  \n",
-    "where $n! = n\\cdot(n-1)\\cdot(n-2) \\cdots 3 \\cdot 2 \\cdot 1$ is the factorial of $n$.  \n",
-    "  \n",
-    "Thus, by making derivatives of $f(x)$ and $P(x)$ equal at $x=a$, we obtain: \n",
-    "  \n",
-    "$$f^{(n)}(a) = P^{(n)}(a) \\implies f^{(n)}(a) = n!\\, a_n \\implies a_{n} = \\frac{f^{(n)}(a)}{n!}$$  \n",
-    "  \n",
-    "The polynomial with these coefficients is called **Taylor polynomial** of degree $n$ at $x = a$:  \n",
-    "  \n",
-    "$$P(x)=f(a)+f'(a)(x-a)+\\frac{f''(a)}{2!}(x-a)^{2}+\\cdots+\\frac{f^{(n)}(a)}{n!}(x-a)^{n}.$$  \n",
-    "  \n",
-    "````{admonition} Example  \n",
-    "  \n",
-    "To find approximation of degree 3 for $f(x)=e^{x}$, at $x=0$, we have:  \n",
-    "  \n",
-    "$$f(0)=1 \\\\\n",
-    "f'(0)=1 \\\\\n",
-    "f''(0)=1 \\\\\n",
-    "f'''(0)=1$$  \n",
-    "  \n",
-    "Thus, \n",
-    "  \n",
-    "$$P_{3}(x)=1+(x-0)+\\frac{1}{2!}(x-0)^2+\\frac{1}{3!}(x-0)^3=1+x+\\frac{x^2}{2}+\\frac{x^3}{6}$$  \n",
-    "  \n",
-    "See how the approximation to the function $f(x)=e^{x}$  at $x=0$ gets increasingly better when more terms are added to the polynomial.  \n",
-    "  \n",
-    "```{image} exp_series.gif  \n",
-    "```\n",
-    "  \n",
-    "````  \n",
-    "  \n",
-    "  \n",
-    "```{admonition} Example  \n",
-    "For the function $f(x)=\\displaystyle\\frac{1}{1-x}\\ (x\\neq1)$ at $x=0$, we have:  \n",
-    "  \n",
-    "$$f(x)=\\frac{1}{1-x} \\implies f(0)=1 \\\\\n",
-    "f'(x)=\\frac{1}{(1-x)^{2}} \\implies f'(0)=1 \\\\\n",
-    "f''(x)=\\frac{2}{(1-x)^{3}} \\implies f''(0)=2 \\\\\n",
-    "f'''(x)=\\frac{3\\cdot2}{(1-x)^{4}} \\implies f'''(0)=3! \\\\\n",
-    "f^{(n)}(x)=\\frac{n!}{(1-x)^{n+1}} \\implies f^{(n)}(0)= n!$$  \n",
-    "  \n",
-    "Thus the Taylor polynomial of degree $n$ will be:  \n",
-    "  \n",
-    "$$\\begin{align}P(x)\n",
-    "&=1+1(x-0)+\\frac{2}{2}(x-0)^{2}+\\frac{3!}{3!}(x-0)^{3}+\\cdots+\\frac{n!}{n!}(x-0)^{n} \\\\\n",
-    "&=1+x+x^{2}+x^{3}+\\cdots+x^{n}\\end{align}$$\n",
-    "\n",
-    "  \n",
-    "```  \n",
-    "  \n",
-    "For some functions, the approximation $f(x)=P_n(x)$ at a given reference point $x=a$ can be made increasingly better by adding more terms to $P(x)$, so that in the limit $n\\rightarrow \\infty$ the Taylor polynomials (called **Taylor series** in this limit) for the functions $e^{x}$, $\\mbox{sin}(x)$, $\\mbox{cos}(x)$ converge exactly for all values of $x$. For other functions, such as $\\mbox{ln}(1+x)$ and $\\displaystyle\\frac{1}{1-x}$, there is a restricted range of values of $x$ for which the approximation can be made better by adding terms.  \n",
-    "  \n",
-    "We have the following approximations by Taylor series, with ranges in $x$, at $x = 0$:\n",
-    "\n",
-    "$$e^x=\\sum_{n=0}^\\infty \\frac{x^n}{n!} =1+x+\\frac{x^2}{2}+\\frac{x^3}{3!}+\\cdots,\\ -\\infty<x<\\infty$$\n",
-    "\n",
-    "$$\\mbox{sin}(x)=\\sum_{n=0}^\\infty \\frac{(-1)^nx^{1+2n}}{(1+2n)!} = x-\\frac{x^3}{3}+\\frac{x^5}{5!}-\\frac{x^7}{7!}+\\cdots,\\ -\\infty<x<\\infty$$\n",
-    "\n",
-    "$$\\mbox{cos}(x)=\\sum_{n=0}^\\infty \\frac{(-1)^nx^{2n}}{(2n)!} = 1-\\frac{x^2}{2}+\\frac{x^4}{4!}-\\frac{x^6}{6!}+\\cdots,\\ -\\infty<x<\\infty$$\n",
-    "\n",
-    "$$\\mbox{ln}(1+x)=-\\sum_{n=1}^\\infty \\frac{(-1)^n x^n}{n} = \\frac{x^2}{2!}+\\frac{x^3}{3!}-\\frac{x^4}{4!}+\\frac{x^5}{5!}+\\cdots,\\ -1 < x \\leq 1$$\n",
-    "\n",
-    "$$\\frac{1}{1-x} = \\sum_{n=0}^\\infty x^n = 1+x+x^2+x^3+\\cdots,\\ -1 \\leq x < 1$$  \n",
-    "  \n",
-    "Knowing the Taylor series often gives insight into the properties of a function."
-   ]
-  }
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diff --git a/_sources/mathscourse/lecture08/08_Calculating_accumulated_change.ipynb b/_sources/mathscourse/lecture08/08_Calculating_accumulated_change.ipynb
deleted file mode 100644
index 0768e2ed..00000000
--- a/_sources/mathscourse/lecture08/08_Calculating_accumulated_change.ipynb
+++ /dev/null
@@ -1,336 +0,0 @@
-{
- "cells": [
-  {
-   "cell_type": "markdown",
-   "id": "b869064b",
-   "metadata": {},
-   "source": [
-    "# 08 - Calculating accumulated change"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "c400afba",
-   "metadata": {},
-   "source": [
-    "## Series  \n",
-    "  \n",
-    "For every sequence $\\{a_n\\}$ of real numbers, it is possible to define another sequence $\\{S_n\\}$ for which the elements are given by the sum of the first $n$ terms of the sequence $\\{a_n\\}$:\n",
-    "\n",
-    "$$\\begin{align}\n",
-    "S_0 &= a_0 \\\\\n",
-    "S_1 &= a_0 + a_1 \\\\\n",
-    "S_2 &= a_0 + a_1 + a_2 \\\\\n",
-    "&\\vdots \\\\\n",
-    "S_n &= a_0 + a_1 + \\cdots + a_{n-1} + a_n \\\\\n",
-    "S_n &= \\sum_{i = 0}^n a_i \\end{align}$$  \n",
-    "  \n",
-    "The sequence $\\{S_n\\}$ is called a *series* (or an infinite series, to emphasize the sum over increasing number of terms $\\{a_n\\}$).  \n",
-    "  \n",
-    "The same way as we asked about $\\lim a_n$, we can ask if $\\lim S_n$ exists. If it exists, or, in other words, if the sum of infinite terms $\\{a_n\\}$ is equal to a finite number, we say that the series $S_n$ is convergent.  \n",
-    "  \n",
-    "  \n",
-    "```{admonition} Examples  \n",
-    "  \n",
-    "The following expressions are example of convergent series:  \n",
-    "  \n",
-    "$$\\begin{align}&\\sum_{n=0}^\\infty \\frac{x^n}{n!} \\hspace{0.1cm}, \\hspace{0.5cm} x \\in \\mathbb{R} \\\\\n",
-    "&\\sum_{n=0}^\\infty x^n \\hspace{0.1cm}, \\hspace{0.5cm} -1<x<1 \\\\\n",
-    "&\\sum_{n=0}^\\infty (-1)^n \\frac{x^{2n+1}}{(2n+1)!} \\hspace{0.1cm}, \\hspace{0.5cm} x \\in \\mathbb{R}\\end{align}$$  \n",
-    "  \n",
-    "```  \n",
-    "  "
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "99112c46",
-   "metadata": {},
-   "source": [
-    "## Calculating areas  \n",
-    "  \n",
-    "  \n",
-    "The idea of summing infinite quantities can be used to solve the problem of determining the area below a given curve. Let $f(x)$ be a function defined on the interval $a \\le x \\le b$ as:\n",
-    "\n",
-    "```{image}  sum_areas.png\n",
-    "```\n",
-    "\n",
-    "The area below the curve of $f(x)$ can be found dividing the interval $a \\le x \\le b$ into $n$ subintervals:\n",
-    "\n",
-    "$$a=x_0<x_1<x_2<\\cdots<x_{n-1}<x_n=b$$  \n",
-    "  \n",
-    "where each interval defines a rectangle of height $f(x_i)$ and width $(x_{i+1}-x_i) = \\Delta x_i = \\displaystyle\\frac{(b-a)}{n}$. The area below the curve will then be approximately the sum of the areas of the rectangles\n",
-    "\n",
-    "$$A \\approx \\sum_{i=0}^n f(x_i)\\Delta x_i.$$  \n",
-    "  \n",
-    "As we increasethe number of intervals $n$, while decreasing the widths of each rectangle, the sum of the areas of the rectangles gives a better approximation of the area below the curve. The exact result is given by the limit $n \\rightarrow \\infty$. In this limit, the sum is represented by the following symbol:\n",
-    "\n",
-    "$$\\lim_{n \\to \\infty} \\sum_{i=0}^n f(x_i) \\Delta x_i = \\int_a^b f(x)dx$$  \n",
-    "  \n",
-    "If this limit exists, we say that the function $f$ is **integrable** on the interval $a \\lt x \\lt b$ and denote the limit by the **definite integral** $\\int_a^b f(x)dx$.  \n",
-    "  \n",
-    "The interpretation of the integral as area can be generalized for when $f(x)$ assumes negative values  \n",
-    "  \n",
-    "```{image}  defin_int.png\n",
-    "``` \n",
-    "  \n",
-    "In the case of the previous figure, we will have:  \n",
-    "  \n",
-    "$$\\begin{align} \\int_a^b f(x)dx \n",
-    "&= A_1 - A_2 + A_3 \\hspace{0.1cm}, \\hspace{0.5cm} (A_i \\geq 0) \\\\\n",
-    "&= [\\text{area above x-axis}] - [\\text{area below x-axis}] \\end{align}$$  \n",
-    "  \n",
-    "Other intuitive results related to the calculation of the area are:\n",
-    "\n",
-    "$$\\int_a^a f(x)dx = 0\\quad \\mbox{(size of the interval equal to zero)}$$\n",
-    "\n",
-    "$$\\int_a^b f(x)dx = -\\int_b^a f(x)dx\\quad  \\mbox{(orientation of the integral)}$$   \n",
-    "  \n",
-    "Furthermore, some of the properties of the integral follows directly from the properties of limits:\n",
-    "\n",
-    "$$\\begin{align}\n",
-    "\\int_a^b kf(x)dx &= k \\int_a^b f(x)dx \\\\\n",
-    "\\int_a^b [f(x) + g(x)]dx &= \\int_a^b f(x)dx + \\int_a^b g(x)dx \\\\\n",
-    "\\int_a^b f(x)dx &= \\int_a^c f(x)dx + \\int_c^b f(x)dx ,\\ \\  (a \\le c \\le b) \\end{align}$$"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "9697057a",
-   "metadata": {},
-   "source": [
-    "## Fundamental theorem of Calculus  \n",
-    "  \n",
-    "  \n",
-    "The problem of calculating integrals as limits of sums requires specific methods for each function and becomes quite impractical. Fortunately, this problem can be solved with the very important result that the area problem is intrinsically related to the tangent problem.\n",
-    "\n",
-    "Suppose we define a function $F(x)$ that gives the area below the curve $f(u)$ from $a$ until a variable value $x$. By definition:  \n",
-    "  \n",
-    "$$F(x) = \\int_a^x f(u)du$$  \n",
-    "  \n",
-    "```{image} fund_calc.png  \n",
-    "```\n",
-    "  \n",
-    "If we consider a small increment $h$ to $x$, the new area below the curve shall be given by:  \n",
-    "  \n",
-    "$$F(x+h) = \\int_a^{x+h} f(u)du$$  \n",
-    "  \n",
-    "  \n",
-    "The difference between these two areas shall be approximately given by the area of the retangle with width $h$ and height $f(x)$:  \n",
-    "  \n",
-    "$$F(x+h) - F(x) \\approx hf(x) \\implies \\displaystyle\\frac{F(x+h) - F(x)}{h} \\approx f(x)$$  \n",
-    "  \n",
-    "But see that in the limit where the increment $h$ gets very small, the approximation gets exactly equal to the increment in area, and we obtain the definition of the derivative of $F(x)$:  \n",
-    "  \n",
-    "$$\\lim_{h\\rightarrow 0} \\displaystyle\\frac{F(x+h) - F(x)}{h} = \\displaystyle\\frac{dF}{dx} = f(x).$$  \n",
-    "  \n",
-    "This results is known as the *Fundamental theorem of Calculus*, and it say that the instantaneous rate of change of the area below a given curve $f$ at a point $x$ is given by the value of the function at that point. In other words, the area $F(x)$ is a function that, if differentiated, gives the function $f(x)$.  \n",
-    "  \n",
-    "The function $F(x)$ is then called an **antiderivative** of $f(x)$. In general, there is a *family* of antiderivates of $f(x)$, since $\\displaystyle\\frac{d(F(x) + C)}{dx}=\\displaystyle\\frac{dF}{dx}=f(x)$, independent of the value of the constant $C$. We therefore call **indefinite integral**:\n",
-    "\n",
-    "$$\\int f(x)dx = F(x) + C$$  \n",
-    "   \n",
-    "It also directly follows that\n",
-    "\n",
-    "$$\\int_b^c f(x)dx = F(c) - F(b)$$  \n",
-    "  \n",
-    "Thus the problem of calculating the area below the curve $f(x)$ from $b$ to $c$, which, by definition, is given by the infinite sum of terms $f(x_i)\\Delta x_i$ (with $\\Delta x_i \\rightarrow 0$), can be reduced to obtaining the antiderivative of $f$, $F(x)$, and subtracting the values of this function at points $c$ and $b$. The main problem then becomes how to calculate the antiderivates of a given function $f(x)$, for which different methods can be used."
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "02bc5c04",
-   "metadata": {},
-   "source": [
-    "## Calculating integrals  \n",
-    "  \n",
-    "  \n",
-    "The simplest integrals can be calculated by inverting the differentiation of fundamental functions, as\n",
-    "\n",
-    "$$\\int x^n \\,dx = \\frac{x^{n+1}}{n+1} + C$$\n",
-    "\n",
-    "$$\\int e^x \\,dx = e^x + C$$\n",
-    "\n",
-    "$$\\int \\frac{1}{x}\\,dx = \\ln |x| + C$$\n",
-    "\n",
-    "$$\\int \\mbox{cos}(x)\\, dx = \\mbox{sin}(x) + C$$\n",
-    "\n",
-    "$$\\int \\mbox{sin}(x)\\, dx = -\\mbox{cos}(x) + C$$  \n",
-    "  \n",
-    "This are obtained simply because we know that the derivative of the functions on the righ-hand sides of the equations give exactly the function that is being integrated. When the integrals are not so simple, we must apply other methods to turn them into integrals we know. \n",
-    "  \n",
-    "  \n",
-    "### Integration by substitution  \n",
-    "  \n",
-    "  \n",
-    "This method consists in finding an appropriate substitution of the variable of integration so that we end up with a simpler integral to calculate.  \n",
-    "  \n",
-    "  \n",
-    "````{admonition} Examples  \n",
-    "  \n",
-    "- $\\int\\frac{1}{\\sqrt{x+4}}dx$  \n",
-    "    \n",
-    "    Note that if $u=x+4 \\implies \\displaystyle\\frac{du}{dx}=1 \\implies du=dx$. Thus, the integral can be given by:  \n",
-    "    \n",
-    "    $$\\int u^{-\\frac{1}{2}}du = \\frac{1}{\\left(-\\frac{1}{2}+1\\right)} \\cdot u^{\\frac{1}{2}} + C = 2\\sqrt{x+4} + C.$$  \n",
-    "      \n",
-    "      \n",
-    "    \n",
-    "    ```{note}  \n",
-    "    Note that three mathematicians die everytime we operate with $du$ and $dx$ as if they were numbers on a fraction. Formally, the correct substitution on the integral would be given by a function $u=u(x)$, so that $ \\int f(u(x)) u'(x)dx = \\int f(u)du$. But we can procede with the heresy above as long as we have this in mind.  \n",
-    "    ```  \n",
-    "    <br />\n",
-    "      \n",
-    "- $\\int xe^{(x^2-1)}dx$  \n",
-    "      \n",
-    "    See that $u=x^2-1 \\implies \\displaystyle\\frac{du}{dx}=2x \\implies  dx=\\displaystyle\\frac{du}{2x}$. Then:  \n",
-    "    \n",
-    "    $$\\int \\frac{1}{2}e^{u}du = \\frac{1}{2}e^u + C = \\frac{e^{(x^2-1)}}{2} + C $$  \n",
-    "      \n",
-    "````\n",
-    "\n",
-    "### Integration by parts  \n",
-    "  \n",
-    "When integrating by parts, we have to find an appropriate association $(u=u(x), v=v(x))$, so that:  \n",
-    "  \n",
-    "$$\\int uv'dx=uv-\\int u'vdx$$  \n",
-    "  \n",
-    "```{admonition} Example  \n",
-    "  \n",
-    "- $\\int x\\ln x\\, dx $  \n",
-    "      \n",
-    "    Note that  \n",
-    "      \n",
-    "    $$\\begin{align}\n",
-    "    u=\\ln x \\implies u' = \\displaystyle\\frac{1}{x} \\\\\n",
-    "    v'=x \\implies v=\\displaystyle\\frac{x^2}{2} \\end{align}$$  \n",
-    "      \n",
-    "    Thus:  \n",
-    "      \n",
-    "    $$\\begin{align}\\int x\\ln x\\, dx \n",
-    "&= \\frac{x^2}{2}\\ln x - \\int \\frac{x^2}{2} \\cdot \\frac{1}{x}dx \\\\\n",
-    "&= \\frac{x^2 \\ln x}{2} - \\frac{1}{2}\\int xdx \\\\\n",
-    "&= \\frac{x^2 \\ln x}{2} - \\frac{1}{4}x^2 + C \\\\\n",
-    "&= \\frac{1}{4} x^2 (2\\ln x - 1) + C \\end{align}$$  \n",
-    "  \n",
-    "- $\\int xe^x \\,dx$  \n",
-    "      \n",
-    "    Note that  \n",
-    "      \n",
-    "    $$\\begin{align}\n",
-    "    u=x \\implies u'=1 \\\\\n",
-    "    v'=e^x \\implies v=e^x \\end{align}$$  \n",
-    "      \n",
-    "    Thus:  \n",
-    "      \n",
-    "    $$\\begin{align}\\int xe^xdx  \n",
-    "    &= xe^x - \\int e^x dx \\hspace{0.1cm} \\\\\n",
-    "    &= xe^x-e^x + C \\\\\n",
-    "    &= e^x (x-1) + C \\end{align}$$  \n",
-    "      \n",
-    "```\n",
-    "\n",
-    "```{tip}\n",
-    "For complicated integrals, computer algebra programs like Mathematica or Sympy can be helpful. An especially user-friendly option is the website WolframAlpha.\n",
-    "```\n",
-    "  \n",
-    "### Integration by partial fractions  \n",
-    "  \n",
-    "If the integral has the form \n",
-    "\n",
-    "$$\\int \\frac{P(x)}{Q(x)}dx$$ \n",
-    "\n",
-    "where the degree of $P(x)$ is smaller than the degree of $Q(x)$, and $Q(x)=a(x-x_1)(x-x_2)\\cdots(x-x_n)$, we can write\n",
-    "\n",
-    "$$\\frac{P(x)}{Q(x)} = \\frac{1}{a}\\left[\\frac{A_1}{x-x_1}+\\frac{A_2}{x-x_2}+\\cdots+\\frac{A_n}{x-x_n}\\right]$$ \n",
-    "\n",
-    "with $\\{A_n\\}$ to be determined.  \n",
-    "  \n",
-    "```{admonition} Example  \n",
-    "  \n",
-    "- $\\int \\frac{1}{x(x-1)}\\, dx$  \n",
-    "      \n",
-    "    First, note that $\\displaystyle\\frac{1}{x(x-1)} =  \\displaystyle\\frac{A}{x} + \\displaystyle\\frac{B}{(x-1)} \\implies 1 = A(x-1) + Bx \\implies 1 = x(A+B) - A$. By comparing the coeffients of the terms in $x$ and the terms independent of $x$, we obtain:  \n",
-    "      \n",
-    "    $$\\begin{cases}\n",
-    "    A + B &= 0\\\\\n",
-    "       -A &= 1\n",
-    "    \\end{cases} \\implies  \n",
-    "    \\begin{cases}\n",
-    "    A &= -1 \\\\\n",
-    "    B &= 1\\end{cases}$$  \n",
-    "      \n",
-    "    Thus:  \n",
-    "      \n",
-    "    $$\\frac{1}{x(x-1)} = -\\frac{1}{x} + \\frac{1}{x-1}$$\n",
-    "      \n",
-    "    Then we will have for the integral:  \n",
-    "      \n",
-    "    $$\\begin{align} \\int\\frac{1}{x(x-1)}dx  \n",
-    "    &= \\int\\left(-\\frac{1}{x}+\\frac{1}{x-1}\\right)dx \\\\\n",
-    "    &= -\\int \\frac{1}{x}dx + \\int\\frac{1}{x-1}dx \\\\\n",
-    "    &= - \\ln|x| + \\ln|x-1| + C \\\\\n",
-    "    &= \\ln \\left|\\frac{x-1}{x}\\right| + C\\end{align}$$  \n",
-    "      \n",
-    "```\n",
-    "    "
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "a24228ef",
-   "metadata": {
-    "tags": [
-     "remove-cell"
-    ]
-   },
-   "source": [
-    "## Fourier coefficients  \n",
-    "  \n",
-    "For a given periodic function $f(t)$ (period $P$), we have shown\n",
-    "\n",
-    "$$f(t)=a+\\sum_{n=1}^N[\\beta_ncos(\\omega nt) + \\gamma_n sin(\\omega nt)]$$  \n",
-    "  \n",
-    "For well-behaved functions $N \\rightarrow \\infty$ gives the exact correspondence between the function $f(t)$ and the series.  \n",
-    "  \n",
-    "If $$\\omega_n=\\frac{2\\pi}{P} \\cdot n$$\n",
-    "\n",
-    "then:\n",
-    "\n",
-    "$$f(t) = a + \\sum_{n=1}^\\infty \\beta_n cos\\left[\\frac{2\\pi nt}{P}\\right] + \\gamma_n sin\\left[\\frac{2\\pi nt}{P}\\right]$$  \n",
-    "  \n",
-    "Multiplying both sides by $cos\\left[\\frac{2\\pi nt}{P}\\right]$ and integrating over one period $P$:\n",
-    "\n",
-    "$$\\beta_n = \\frac{2}{P} \\int_{P} f(t) cos\\left[\\frac{2\\pi nt}{P}\\right]dt$$\n",
-    "\n",
-    "since \n",
-    "\n",
-    "$$\\begin{align} \\int_{-\\pi}^\\pi cos(nx)cos(mx)\n",
-    "&=\\pi\\gamma_{mn} \\\\\n",
-    "\\int_{-\\pi}^\\pi sin(mx)cos(nx)&=0\\end{align}$$"
-   ]
-  }
- ],
- "metadata": {
-  "celltoolbar": "Tags",
-  "kernelspec": {
-   "display_name": "Python 3 (ipykernel)",
-   "language": "python",
-   "name": "python3"
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-  "language_info": {
-   "codemirror_mode": {
-    "name": "ipython",
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-   "mimetype": "text/x-python",
-   "name": "python",
-   "nbconvert_exporter": "python",
-   "pygments_lexer": "ipython3",
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diff --git a/_sources/mathscourse/tutorials/lecture01/Tutorial_01.ipynb b/_sources/mathscourse/tutorials/lecture01/Tutorial_01.ipynb
deleted file mode 100644
index 4b53a044..00000000
--- a/_sources/mathscourse/tutorials/lecture01/Tutorial_01.ipynb
+++ /dev/null
@@ -1,203 +0,0 @@
-{
- "cells": [
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "# Tutorial 01"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "1.  An autocatalytic reaction uses its resulting product for the formation of a new product, as in the reaction  \n",
-    "  \n",
-    "    $$A + X \\rightarrow X$$  \n",
-    "    \n",
-    "    If we assume that this reaction occurs in a closed vessel, then the reaction rate is given by\n",
-    "    \n",
-    "    $$\n",
-    "       R(x) = kx(a - x)\n",
-    "    $$\n",
-    "    \n",
-    "    for $0 \\le x \\le a$, where $a$ is the initial concentration of $A$ and $x$ is the concentration of $X$.\n",
-    "    \n",
-    "    <br/>\n",
-    "    \n",
-    "    (a) Show that $R(x)$ is a polynomial and determine its degree.\n",
-    "    <br/>\n",
-    "    (b) Without using any graphic visualiser, graph $R(x)$ for $k = 2$ and $a = 6$. Find the value of $x$ at which the reaction rate is maximal.  \n",
-    "    \n",
-    "    <br/>\n",
-    "      \n",
-    "2. Find the maximum/minimum value of the following polynomials  \n",
-    "  \n",
-    "    <br/>\n",
-    "    \n",
-    "    (a) $p(x) = x^2 - 2x - 3$\n",
-    "    \n",
-    "    (b) $p(x) = -x^2 - 6x + 8$\n",
-    "    \n",
-    "    (*Hint:* Calculate the roots and consider the symmetry of the parabola)  \n",
-    "    \n",
-    "    <br/>\n",
-    "      \n",
-    "3. There is a function that is frequently used to describe the per capita growth rate of organisms when the rate depends on the concentration of some nutrient and becomes saturated for large enough nutrient concentrations. If we denote the concentration of the nutrient by $N$, then the per capita growth rate $r(N)$ is given by the *Monod growth function*\n",
-    "\n",
-    "    $$\n",
-    "        r(N)=\\frac{aN}{k+N},\\ \\ \\ N \\ge 0\n",
-    "    $$\n",
-    "    \n",
-    "    where $a$ and $k$ are positive constants.\n",
-    "    \n",
-    "    <br/>\n",
-    "    \n",
-    "    (a) Use Python or R to investigate the Monod growth function. Graph $r(N)$ for (i) $a = 5$ and $k = 1$, (ii) $a = 5$ and $k = 3$, and (iii) $a = 8$ and k = 1. Place all three graphs in the same coordinate system.\n",
-    "    \n",
-    "    (b) On the basis of the graphs in (a), describe in words what happens when you change $a$.\n",
-    "    \n",
-    "    (c) On the basis of the graphs in (a), describe in words what happens when you change $k$.  \n",
-    "    \n",
-    "    <br/>\n",
-    "      \n",
-    "      \n",
-    "4. The function \n",
-    "\n",
-    "    $$\n",
-    "        f(x) = \\frac{x^n}{b^n + x^n},\\ \\ \\ x \\ge 0\n",
-    "    $$ \n",
-    "    \n",
-    "    where $n$ is a positive integer and $b$ is a positive real number, is used in biochemistry to model reaction rates as a function of the concentration of some reactants    \n",
-    "    \n",
-    "    <br/>\n",
-    "    \n",
-    "    (a) Use Python or R to graph $f(x)$ for $n = 1, 2,\\ \\mbox{and}\\ 3$ in the same coordinate system when $b = 2$.\n",
-    "    \n",
-    "    (b) Where do the three graphs in (a) intersect?\n",
-    "    \n",
-    "    (c) What happens to $f(x)$ as $x$ gets larger?\n",
-    "\n",
-    "    (d) For an arbitrary positive value of $b$, show that $f (b) = 1/2$. On the basis of this demonstration and your answer in C. ,  explain why $b$ is called the half-saturation constant.  \n",
-    "    \n",
-    "    <br/>\n",
-    "  \n",
-    "  \n",
-    "5. The file [words_moby_dick.csv](https://imperialcollegelondon.box.com/s/940y521dq1owarx8pt4d4mu2d0f2cz85) contains data on the cumulative distribution of the number of times specific words occur in the text of the novel *Moby Dick*, by Herman Melville. Assume that this distribution is a power law of the form \n",
-    "\n",
-    "    $$\n",
-    "        y = x^D\n",
-    "    $$\n",
-    "    \n",
-    "    (a) Using the provided data, and R/Python, estimate the exponent of this power law. \n",
-    "    (*Hint:* If you plot this data in log-log scale, what should correspond to the exponent $D$?)\n",
-    "    \n",
-    "    (b) Can you think of a better way to estimate D?  \n",
-    "    \n",
-    "    <br/>\n",
-    "  \n",
-    "  \n",
-    "6. Assume that a population size at time $t$ is $N(t)$ and that\n",
-    "\n",
-    "    $$\n",
-    "        N(t) = 40 \\cdot 2^t,\\ \\ \\ t \\ge 0\n",
-    "    $$    \n",
-    "\n",
-    "    (a) Find the population size at time t = 0.\n",
-    "\n",
-    "    (b) Show that\n",
-    "\n",
-    "    $$\n",
-    "        N(t) = 40\\, \\mbox{e}^{t\\, ln 2},\\ \\ \\ t \\ge 0\n",
-    "    $$    \n",
-    "\n",
-    "    (c) How long will it take until the population size reaches 1000?\n",
-    "    \n",
-    "   (*Hint*: Find $t$ so that $N(t) = 1000$.)  \n",
-    "   \n",
-    "   <br/>\n",
-    "     \n",
-    "7. (*Adapted from Moss, 1980*) Hall (1964) investigated the change in population size of the zooplankton species *Daphnia galeata mendota* in Base Line Lake, Michigan. The population size $N(t)$ at time $t$ was modeled by the equation\n",
-    "\n",
-    "    $$ \n",
-    "        N(t) = N_0\\mbox{e}^{rt}\n",
-    "    $$\n",
-    "    \n",
-    "    where $N_0$ denotes the population size at time $0$. The constant $r$ is called the **intrinsic rate of growth**.\n",
-    "    \n",
-    "    <br/>\n",
-    "    \n",
-    "    (a) Plot $N(t)$ as a function of $t$ if $N_0 = 100$ and $r = 2$. Compare your graph against the graph of $N(t)$ when $N_0 = 100$ and $r = 3$. Which population grows faster?\n",
-    "    \n",
-    "    (b) The constant $r$ is an important quantity because it describes how quickly the population changes. Suppose that you determine the size of the population at the beginning and at the end of a period of length $1$, and you find that at the beginning there were $200$ individuals and after one unit of time there were $250$ individuals. Determine $r$. (*Hint*: Consider the ratio $N(t+1)/N(t)$.)  \n",
-    "    \n",
-    "    <br/>\n",
-    "      \n",
-    "8. Fish are indeterminate growers; that is, they grow throughout their lifetime. The growth of fish can be described by the von Bertalanffy function\n",
-    "\n",
-    "    $$\n",
-    "        L(x) = L_{\\infty}(1-\\textrm{e}^{-kx} )\n",
-    "    $$\n",
-    "    \n",
-    "    for $x \\ge 0$, where $L(x)$ is the length of the fish at age $x$ and $k$ and $L_{\\infty}$ are positive\n",
-    "    \n",
-    "    <br/>\n",
-    "    \n",
-    "    (a) Using Python or R, graph $L(x)$ for $L_{\\infty} = 20$, for (i) $k = 1$ and (ii) $k = 0.1$.\n",
-    "    \n",
-    "    (b) For $k = 1$, find $x$ so that the length is 90\\% of $L_{\\infty}$. Repeat for 99\\% of $L_{\\infty}$. Can the fish ever attain length $L_{\\infty}$? Interpret the meaning of $L_{\\infty}$.\n",
-    "    \n",
-    "    (c) Compare the graphs obtained in A. Which growth curve reaches 90\\% of $L_{\\infty}$ faster? Can you explain what happens to the curve of $L(x)$ when you vary $k$ (for fixed $L_{\\infty}$)?  \n",
-    "    \n",
-    "    <br/>\n",
-    "    \n",
-    "9. For the following pairs of functions, plot the ratio (quotient) between the two using R or Python. Based on the behaviour of the ratio when $x \\rightarrow \\infty$ (very large values of $x$), how does each of these functions compare to the other in the velocity that they grow?\n",
-    "\n",
-    "    <br/>  \n",
-    "    \n",
-    "    (a) $ f(x) = e^x \\quad \\text{ and } \\quad g(x) = x^2 $\n",
-    "\n",
-    "    (b) $ f(x) = e^x \\quad \\text{ and } \\quad g(x) = x^{20} $\n",
-    "    \n",
-    "    (c) $ f(x) = x \\quad \\text{ and } \\quad g(x) = \\log(x) $  \n",
-    "    \n",
-    "    <br/>\n",
-    "      \n",
-    "10. A community measure that takes both species abundance and species richness into account is the Shannon diversity index $H$. To calculate $H$, the proportion $p_i$ of species $i$ in the community is used. Assume that the community consists of $S$ species. Then\n",
-    "\n",
-    "    $$\n",
-    "        H = -\\sum_{i=1}^S p_i\\, \\textrm{ln}\\, p_i = -( p_1 \\textrm{ln}\\, p_1 + p_2 \\textrm{ln}\\, p_2 + \\cdots + p_S \\textrm{ln}\\, p_S )\n",
-    "    $$\n",
-    "    \n",
-    "    <br/>\n",
-    "    \n",
-    "    (a) Assume that $S = 5$ and that all species are equally abundant; that is, $p_1 = p_2 = \\cdots = p_5$. Compute $H$.\n",
-    "    \n",
-    "    (b) Assume that $S = 10$ and that all species are equally abundant; that is, $p_1 = p_2 = \\cdots = p_{10}$. Compute $H$.\n",
-    "    \n",
-    "    (c) A measure of equitability (or evenness) of the species distribution can be measured by dividing the diversity index $H$ by $\\textrm{ln}\\, S$. Compute $H/\\textrm{ln}\\, S$ for $S = 5$ and $S = 10$."
-   ]
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diff --git a/_sources/mathscourse/tutorials/lecture05/Tutorial_05.ipynb b/_sources/mathscourse/tutorials/lecture05/Tutorial_05.ipynb
deleted file mode 100644
index 7d975283..00000000
--- a/_sources/mathscourse/tutorials/lecture05/Tutorial_05.ipynb
+++ /dev/null
@@ -1,275 +0,0 @@
-{
- "cells": [
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "# Tutorial 05\n",
-    "\n",
-    "\n",
-    "1. Solve the following systems of linear equations\n",
-    "    <br/>\n",
-    "    \n",
-    "    $$\n",
-    "        \\mbox{(a)}  \\ \n",
-    "        \\begin{cases}\n",
-    "            5x - y + 2z = 6 \\\\\n",
-    "            x + 2y - z = -1 \\\\\n",
-    "            3x + 2y - 2z = 1\\\\\n",
-    "        \\end{cases}\n",
-    "    $$\n",
-    "    \n",
-    "    <br/>\n",
-    "    \n",
-    "    $$\n",
-    "        \\mbox{(b)}  \\ \n",
-    "        \\begin{cases}\n",
-    "            2x - y + 3z = 3 \\\\\n",
-    "            2x + y + 4z = 4 \\\\\n",
-    "            2x - 3y + 2z = 2\\\\\n",
-    "        \\end{cases}\n",
-    "    $$\n",
-    "    \n",
-    "    <br/>\n",
-    "    \n",
-    "2. Three different species of insects are reared together in a laboratory cage. They are supplied with two different types of food each day. Each individual of species 1 consumes 3 units of food $A$ and 5 units of food $B$, each individual of species 2 consumes 2 units of food $A$ and 3 units of food $B$, and individual of species 3 consumes 1 unit of food $A$ and 2 units of food $B$. Each day, 500 units of food $A$ and 900 units of food $B$ are supplied. How many individuals of each species can be reared together? Is there more than one solution? What happens if we add 550 units of a third type of food, called $C$, and each individual of species 1 consumes 2 units of food $C$, each individual of species 2 consumes 4 units of food $C$, and each individual of species 3 consumes 1 unit of food $C$?\n",
-    "\n",
-    "    <br/>\n",
-    "\n",
-    "3. Networks are abstractions of real systems used to identify and study patterns of connectivity between the individual units that compose those systems. Networks (often referred to as *graphs*) are mathematically defined as a set of *nodes*, or vertices, (representing the individual units) and a set of *links*, or connections, between these nodes, which account for the interaction patterns within the studied systems. In Ecology, graphs have been widely used in studies investigating food-webs, plant-pollinator interactions, and disease transmission in populations.  \n",
-    "    \n",
-    "    <br />\n",
-    "  \n",
-    "    If the nodes of a network of size $N$ are indexed with positive integers such as $i = 1, 2, \\ldots, N$, an useful method for representing the structure of a network is to construct a matrix $A$ for which the elements $a_{ij}$ are given by:  \n",
-    "      \n",
-    "    $$a_{ij} = \\begin{cases}1,\\ \\mbox{if nodes}\\ i \\ \\mbox{and}\\ j \\ \\mbox{are linked} \\\\ 0,\\ \\mbox{otherwise}  \\end{cases}$$  \n",
-    "    \n",
-    "    Matrix $A$ is called the *adjacency matrix* of the network. One example of a network with $6$ nodes is shown below.\n",
-    "\n",
-    "    <br/>  \n",
-    "    \n",
-    "    ```{image} graph_example.png  \n",
-    "    ```  \n",
-    "          \n",
-    "    (a) Based on the graphical representation of this network, write down its adjacency matrix.  \n",
-    "        \n",
-    "    <br/>\n",
-    "    \n",
-    "    (b) By the definition of the adjacency matrix, the sum of the matrix elements on row $i$, $k_i = \\sum_{j=1}^N a_{ij}$, corresponds to the number of links of node $i$. The quantity $k_i$ is called the *degree* of node $i$. Calculate the corresponding degrees for each node on the previous network.\n",
-    "        \n",
-    "    <br/>\n",
-    "    \n",
-    "    (c) The Laplacian matrix is a mathematical object that can be used to find many useful properties of a network. By definition, $L$ is constructed as  \n",
-    "    \n",
-    "    <br/>  \n",
-    "    \n",
-    "    $$ L = D - A $$  \n",
-    "    \n",
-    "    <br/>  \n",
-    "    \n",
-    "    Where $A$ is the adjacency matrix, and $D$ is the degree matrix. The degree matrix $D$ is a diagonal matrix which contains information about the degree of each node. With your answer for itens (a) and (b), calculate $L$.\n",
-    "        \n",
-    "    <br/>\n",
-    "    \n",
-    "    (d) Use R/Python to numerically calculate the eigenvalues of the Laplacian matrix.\n",
-    "    \n",
-    "    <br/>\n",
-    "   \n",
-    "4. Write each system in matrix form\n",
-    "   <br/>\n",
-    "    \n",
-    "    $$\n",
-    "        \\mbox{(a)}  \\ \n",
-    "        \\begin{cases}\n",
-    "            2x_2-x_1 = x_3 \\\\\n",
-    "            4x_1 + x_3 = 7x_2 \\\\\n",
-    "            x_2 - x_1 = x_3\\\\\n",
-    "        \\end{cases}\n",
-    "    $$\n",
-    "    \n",
-    "    <br/>\n",
-    "    \n",
-    "    $$\n",
-    "        \\mbox{(b)}  \\ \n",
-    "        \\begin{cases}\n",
-    "            2x_1 - 3x_2 = 4 \\\\\n",
-    "            -x_1 + x_2 = 3 \\\\\n",
-    "            3x_1 = 4\\\\\n",
-    "        \\end{cases}\n",
-    "    $$\n",
-    "    \n",
-    "    <br/>\n",
-    "    \n",
-    "5. Suppose that \n",
-    "\n",
-    "    <br/>\n",
-    "    \n",
-    "    $$A = \\begin{pmatrix}\n",
-    "            2 & 4 \\\\\n",
-    "            3 & 6 \n",
-    "            \\end{pmatrix}\n",
-    "        \\ \\ \\ \\text{,}\\ \\ \\\n",
-    "        X = \\begin{pmatrix}\n",
-    "            x \\\\\n",
-    "            y \n",
-    "            \\end{pmatrix} \\ \\ \\ \\text{and}\\ \\ \\\n",
-    "            B = \\begin{pmatrix}\n",
-    "            b_1 \\\\\n",
-    "            b_2 \\\\\n",
-    "            \\end{pmatrix}$$\n",
-    "\n",
-    "    (a) Write $AX = B$ as a system of linear equation.\n",
-    "        \n",
-    "    <br/>\n",
-    "    \n",
-    "    (b) Show that if \n",
-    "   \n",
-    "    $$\n",
-    "        B = \\begin{pmatrix}\n",
-    "            3 \\\\\n",
-    "            \\frac92 \\\\\n",
-    "            \\end{pmatrix}\n",
-    "    $$\n",
-    "    \n",
-    "    then $AX = B$ has infinitely many solutions. Graph the two straight lines associated with the corresponding system of linear equations, and explain why the system has infinitely many solutions.\n",
-    "        \n",
-    "    <br/>\n",
-    "    \n",
-    "    (c) Find a column vector\n",
-    "    \n",
-    "    <br/>\n",
-    "    \n",
-    "    $$B = \\begin{pmatrix}\n",
-    "            b_1 \\\\\n",
-    "            b_2 \\\\\n",
-    "            \\end{pmatrix}$$\n",
-    "\n",
-    "    so that $AX = B$ has no solutions.\n",
-    "    \n",
-    "    <br/>\n",
-    "    \n",
-    "6. Assume that a population is divided into three age classes and that 80\\% of the individuals age 0 and 10\\% of the individuals age 1 survive until the end of the next breeding season. Assume further that individuals age 1 have an average of 1.6 offspring and individuals age 2 have an average of 3.9 offspring. If, at time 0, the population consists of 1000 individuals age 0, 100 individuals age 1, and 20 individuals age 2, find the projection matrix and the age distribution at time 3.\n",
-    "\n",
-    "    <br/>\n",
-    "\n",
-    "7. Consider the following projection matrix representing a population with five age classes:\n",
-    "\n",
-    "    <br/>  \n",
-    "    \n",
-    "    $$\n",
-    "        P = \\begin{pmatrix}\n",
-    "        0 & 5 & 3 & 2 & 1 \\\\\n",
-    "        0.9 & 0 & 0 & 0 & 0 \\\\\n",
-    "        0 & 0.3 & 0 & 0 & 0 \\\\\n",
-    "        0 & 0 & 0.1 & 0 & 0 \\\\\n",
-    "        0 & 0 & 0 & 0.05 & 0 \\\\\n",
-    "    \\end{pmatrix}\n",
-    "    $$\n",
-    "    \n",
-    "    <br/>\n",
-    "    \n",
-    "    (a) Making use of Python or R code, begin with a population distributed as $(0,0,0,0,10)$ individuals at time $t=0$, project the population at time $t=20$, and plot the total number over time. Plot the logarithm of the total population over time.\n",
-    "        \n",
-    "    <br/>\n",
-    "    \n",
-    "    (b) Repeat the above but begin with $(80,16,5,1,1)$ individuals. How do these results compare to the results in (a) ?\n",
-    "    \n",
-    "    <br/>\n",
-    "    \n",
-    "8. In the problems below, find the eigenvalues $\\lambda_1$ and $\\lambda_2$ and corresponding eigenvectors $\\mathbf{v_1}$ and $\\mathbf{v_2}$ for each matrix $B$. Determine the equations of the lines through the origin in the direction of the eigenvectors $\\mathbf{v_1}$ and $\\mathbf{v_2}$, and graph the lines together with the eigenvectors $\\mathbf{v_1}$ and $\\mathbf{v_2}$ and the vectors $A\\mathbf{v_1}$ and $A\\mathbf{v_2}$.\n",
-    "\n",
-    "    <br/>\n",
-    "    \n",
-    "    $$\n",
-    "    \\text{(a)} \\ \\ B = \\begin{pmatrix}\n",
-    "    2 & 3\\\\\n",
-    "    0 & -1\n",
-    "    \\end{pmatrix}\n",
-    "    $$\n",
-    "\n",
-    "    <br/>\n",
-    "\n",
-    "    $$\n",
-    "    \\text{(b)} \\ \\ B = \\begin{pmatrix}\n",
-    "    3 & 6\\\\\n",
-    "    -1 & -4\n",
-    "    \\end{pmatrix}  \n",
-    "    $$\n",
-    "    \n",
-    "    <br/>\n",
-    "    \n",
-    "9. Suppse that \n",
-    "\n",
-    "    <br/>\n",
-    "    \n",
-    "    $$\n",
-    "    P = \\begin{pmatrix}\n",
-    "    0.5 & 2.3\\\\\n",
-    "    a & 0\n",
-    "    \\end{pmatrix}\n",
-    "    $$\n",
-    "    \n",
-    "    <br/>\n",
-    "    \n",
-    "    is the projection matrix of a population with two age classes. For which values of $a$ does this population grow?\n",
-    "    \n",
-    "    <br/>\n",
-    "    \n",
-    "10. Suppose that \n",
-    "\n",
-    "    <br/>\n",
-    "    \n",
-    "    $$\n",
-    "    P = \\begin{pmatrix}\n",
-    "    0.5 & 2.0\\\\\n",
-    "    0.1 & 0\n",
-    "    \\end{pmatrix}\n",
-    "    $$\n",
-    "    \n",
-    "    <br/>\n",
-    "    \n",
-    "    is the projection matrix of a population with two age classes.\n",
-    "        \n",
-    "    <br/>\n",
-    "    \n",
-    "    (a) If you were to manage this population, would you need to be concerned about its long-term survival?\n",
-    "    \n",
-    "    <br/>\n",
-    "    \n",
-    "    (b) Suppose that you can improve either the fecundity or the survival of the zero-year-olds, but due to physiological and environmental constraints, the fecundity of zero-year-olds will not exceed 1.5 and the survival of zero-year-olds will not exceed 0.4. Investigate how the growth rate of the population is affected by changing either the survival or the fecundity of zero-year-olds, or both. What would be the maximum achievable growth rate?\n",
-    "    \n",
-    "    <br/>\n",
-    "    \n",
-    "    (c) In real situations, what other factors might you need to consider when you decide on management strategies?"
-   ]
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diff --git a/_sources/mathscourse/tutorials/lecture06/Tutorial_06.ipynb b/_sources/mathscourse/tutorials/lecture06/Tutorial_06.ipynb
deleted file mode 100644
index 39d7af68..00000000
--- a/_sources/mathscourse/tutorials/lecture06/Tutorial_06.ipynb
+++ /dev/null
@@ -1,224 +0,0 @@
-{
- "cells": [
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "# Tutorial 06"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "1. **Island Model** Assume that a species lives in a habitat that consists of many islands close to a mainland. The species occupies both the mainland and the islands, but, although it is present on the mainland at all times, it frequently goes extinct on the islands. Islands can be recolonized by migrants from the mainland. The following model keeps track of the fraction of islands occupied: Denote the fraction of islands occupied at time $t$ by $p(t)$. Assume that each island experiences a constant risk of extinction and that vacant islands (the fraction $1-p$) are colonized from the mainland at a constant rate. Then\n",
-    "\n",
-    "    $$\n",
-    "        \\frac{dp}{dt}=c(1-p)-ep\n",
-    "    $$\n",
-    "\n",
-    "     where $c$ and $e$ are positive constants.\n",
-    "\n",
-    "    <br/>\n",
-    "    \n",
-    "    (a) The gain from colonization is $f (p) = c(1 - p)$ and the loss from extinction is $g(p) = ep$. Graph (without graph visualisers) $f(p)$ and $g(p)$ for $0 \\le p \\le 1$ in the same coordinate system. Explain why the two graphs intersect whenever $e$ and $c$ are both positive. Compute the point of intersection and interpret its biological meaning.\n",
-    "    \n",
-    "     <br/>\n",
-    "     \n",
-    "    (b) The parameter $c$ measures how quickly a vacant island becomes colonized from the mainland. The closer the islands, the larger is the value of $c$. Use your graph in (a) to explain what happens to the point of intersection of the two lines as c increases. Interpret your result in biological terms.  \n",
-    "      \n",
-    "    <br />  \n",
-    "      \n",
-    "2. **Biotic Diversity** (Adapted from Valentine, 1985.) Walker and Valentine (1984) suggested a model for species diversity which assumes that species extinction rates are independent of diversity but speciation rates are regulated by competition. Denoting the number of species at time $t$ by $S(t)$, the speciation rate by $b$, and the extinction rate by $a$, they used the model\n",
-    "\n",
-    "    $$\n",
-    "        \\frac{dS}{dt} = S\\left[b\\left(1-\\frac{S}{K}\\right)-a \\right]\n",
-    "    $$\n",
-    "      \n",
-    "    where $K$ denotes the number of \"niches\", or potential places for species in the ecosystem.\n",
-    "    \n",
-    "    <br/>\n",
-    "    \n",
-    "    (a) Find possible equilibria ($dS/dt = 0$) under the condition $a < b$.\n",
-    "    \n",
-    "    <br/>\n",
-    "    \n",
-    "    (b) Use your result in (a) to explain the following statement by Valentine (1985):\n",
-    "    \n",
-    "     ```{admonition} Quotation \n",
-    "     \"In this situation, ecosystems are never 'full', with all potential niches occupied by species so long as the extinction rate is above zero.\"\n",
-    "     ```\n",
-    "    \n",
-    "    <br/>\n",
-    "    \n",
-    "    (c) What happens when $a \\ge b$?  \n",
-    "      \n",
-    "    <br />  \n",
-    "      \n",
-    "3. Find a point on the curve\n",
-    "    \n",
-    "    $$y = 1 - 3x^3$$\n",
-    "    \n",
-    "     whose tangent line is parallel to the line $y = -x$. Is there more than one such point? If so, find all other points with this property.  \n",
-    "       \n",
-    "    <br />  \n",
-    "      \n",
-    "4. Find a point on the curve\n",
-    "\n",
-    "    $$y = 2x^3 - 4x + 1$$\n",
-    "    \n",
-    "    whose tangent line is parallel to the line $y-2x = 1$. Is there more than one such point? If so, find all other points with this property.  \n",
-    "      \n",
-    "    <br />  \n",
-    "      \n",
-    "5. In the following, use the product and quotient rules to find the derivative with respect to the independent variable.\n",
-    "\n",
-    "    <br/>\n",
-    "    \n",
-    "    (a) $f(x) = (2x^3 - 1)(3 + 2x^2)$\n",
-    "    \n",
-    "    <br/>\n",
-    "    \n",
-    "    (b) $h(s) = (4 - 3s^2 + 4s^3)^2$\n",
-    "    \n",
-    "    <br/>\n",
-    "    \n",
-    "    (c) $f(x) =\\sqrt{3x}(x^2 - 1)$\n",
-    "    \n",
-    "    <br/>\n",
-    "    \n",
-    "    (d) $g(t) = 4(3t^2 - 1)(2t + 1)$\n",
-    "    \n",
-    "    <br/>\n",
-    "    \n",
-    "    (e) $f(x) = 3(x^2 + 2)(4x^2 - 5x^4) - 3$\n",
-    "    \n",
-    "    <br/>\n",
-    "    \n",
-    "    (f) $g(x) = \\displaystyle\\frac{1 - 4x^3}{1 - x}$\n",
-    "    \n",
-    "    <br/>\n",
-    "    \n",
-    "    (g) $h(x) = \\displaystyle\\frac{1 + 2x^2 - 4x^4}{3x^3 - 5x^5}$\n",
-    "    \n",
-    "    <br/>\n",
-    "    \n",
-    "    (h) $f(t) =\\displaystyle\\frac{3 - t^2}{(t + 1)^2}$\n",
-    "    \n",
-    "    <br/>\n",
-    "    \n",
-    "    (i) $h(s) = \\displaystyle\\frac{2s^3 - 4s^2 + 5s - 7}{(s^2 - 3)^2}$\n",
-    "    \n",
-    "    <br/>\n",
-    "    \n",
-    "    (j) $g(s) = \\displaystyle\\frac{s^{1/7} - s^{2/7}}{s^{3/7} + s^{4/7}}$\n",
-    "      \n",
-    "    <br />  \n",
-    "      \n",
-    "6. **Logistic Growth**\n",
-    "\n",
-    "    <br/>\n",
-    "    \n",
-    "    (a) Find the derivative of the logistic growth curve  \n",
-    "      \n",
-    "    $$N(t)=\\frac{K}{1+\\left(\\frac{K}{N(0)}-1\\right)e^{-rt}}$$  \n",
-    "    \n",
-    "    where $r$ and $K$ are positive constants and $N(0)$ is the population size at time $0$.\n",
-    "        \n",
-    "    <br/>\n",
-    "            \n",
-    "    (b) Show that $N(t)$ satisfies the equation  \n",
-    "      \n",
-    "    $$\\frac{dN}{dt}=rN\\left(1-\\frac{N}{K}\\right)$$  \n",
-    "      \n",
-    "    [*Hint:* Use the function $N(t)$ given in (a) for the right-hand side, and simplify until you obtain the derivative of N(t) that you computed in A.]\n",
-    "        \n",
-    "    <br/>\n",
-    "        \n",
-    "    (c) What happens to the *per-capita* rate of growth $\\frac{1}{N}\\frac{dN}{dt}$ as you increase N? What biological principle and what type of interaction is your model describing?  \n",
-    "      \n",
-    "    <br />  \n",
-    "      \n",
-    "7. In the following, find the derivative with respect to the independent variable (*Hint*: use the chain rule).\n",
-    "    \n",
-    "    <br/>\n",
-    "    \n",
-    "    (a) $f(x) = \\mbox{sin}(e^{2x} + x)$\n",
-    "    \n",
-    "    <br/>\n",
-    "    \n",
-    "    (b) $h(x) = \\mbox{exp}[x^2 - 2 \\mbox{cos}\\, x]$\n",
-    "    \n",
-    "    <br/>\n",
-    "    \n",
-    "    (c) $f(x) =\\displaystyle\\frac{x-2^{−x}}\n",
-    "{1+x2^{−x}}$\n",
-    "    \n",
-    "    <br/>\n",
-    "    \n",
-    "    (d) $g(t) = \\mbox{log}\\left(\\displaystyle\\frac{1 - t}{1 + 2t}\\right)$\n",
-    "    \n",
-    "    <br/>\n",
-    "    \n",
-    "    (e) $f(t) = \\mbox{sin}(\\mbox{ln}(3t))$\n",
-    "    \n",
-    "    <br/>\n",
-    "    \n",
-    "    (f) $g(x) = \\mbox{ln}(\\mbox{cos}(1 - x))$\n",
-    "    \n",
-    "    <br/>\n",
-    "    \n",
-    "    (g) $h(x) = \\displaystyle\\frac{\\sqrt{x^2 - 1}}{2+\\sqrt{x^2+1}}$ \n",
-    "    \n",
-    "    <br/>\n",
-    "    \n",
-    "    (h) $f(t) =\\sqrt[3]{t^2+\\sqrt{1-t}}$\n",
-    "    \n",
-    "    <br/>\n",
-    "      \n",
-    "8. The functional responses of some predators are sigmoidal; that is, the number of prey attacked per predator as a function of prey density has a sigmoidal shape. If we denote the prey density by $N$, the predator density by $P$, the time available for searching for prey by $T$, and the handling time of each prey item per predator by $T_h$, then the number of prey encounters per predator as a function of $N$, $T$, and $T_h$ can be expressed as\n",
-    "\n",
-    "    $$\n",
-    "        f (N, T, T_h) = \\frac{b^2N^2T}{1 + cN + bT_hN^2}\n",
-    "    $$\n",
-    "    \n",
-    "    where $b$ and $c$ are positive constants.\n",
-    "    \n",
-    "    <br/>\n",
-    "    \n",
-    "    (a) Calculate the first-order partial derivative of $f(N, T, T_h)$ with respect to the prey density $N$. If $N$ increases, what happens to $f(N, T, T_h)$?\n",
-    "     \n",
-    "    <br/>\n",
-    "    \n",
-    "    (b) Calculate the first-order partial derivative of $f(N, T, T_h)$ with respect to the the time $T$ available for search. If $T$ increases, what happens to $f(N, T, T_h)$?\n",
-    "     \n",
-    "    <br/>\n",
-    "    \n",
-    "    (c) Calculate the first-order partial derivative of $f(N, T, T_h)$ with respect to the handling time $T_h$. If $T_h$ increases, what happens to $f(N, T, T_h)$?\n",
-    "    \n",
-    "    [*Hint:* The function $f(N, T, T_h)$ will be an increasing (decreasing) function of the independent variable if the first-order partial derivative of $f$ with respect to this variable is positive (negative).]"
-   ]
-  }
- ],
- "metadata": {
-  "kernelspec": {
-   "display_name": "Python 3",
-   "language": "python",
-   "name": "python3"
-  },
-  "language_info": {
-   "codemirror_mode": {
-    "name": "ipython",
-    "version": 3
-   },
-   "file_extension": ".py",
-   "mimetype": "text/x-python",
-   "name": "python",
-   "nbconvert_exporter": "python",
-   "pygments_lexer": "ipython3",
-   "version": "3.7.4"
-  }
- },
- "nbformat": 4,
- "nbformat_minor": 2
-}
diff --git a/_sources/notebooks/Appendix-Maths.ipynb b/_sources/notebooks/Appendix-Maths.ipynb
index 4fe30957..819a74f0 100644
--- a/_sources/notebooks/Appendix-Maths.ipynb
+++ b/_sources/notebooks/Appendix-Maths.ipynb
@@ -37,7 +37,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 1,
+   "execution_count": 3,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -3090,7 +3090,7 @@
  ],
  "metadata": {
   "kernelspec": {
-   "display_name": "Python 3",
+   "display_name": "Python 3 (ipykernel)",
    "language": "python",
    "name": "python3"
   },
@@ -3104,7 +3104,7 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.8.10"
+   "version": "3.12.4"
   },
   "latex_envs": {
    "LaTeX_envs_menu_present": true,
diff --git a/_sources/notebooks/Appendix-MathsForBiologists/MathsForBiologists.ipynb b/_sources/notebooks/Appendix-MathsForBiologists/MathsForBiologists.ipynb
new file mode 100644
index 00000000..bf41a0bb
--- /dev/null
+++ b/_sources/notebooks/Appendix-MathsForBiologists/MathsForBiologists.ipynb
@@ -0,0 +1,5883 @@
+{
+ "cells": [
+  {
+   "cell_type": "code",
+   "execution_count": 3,
+   "metadata": {
+    "tags": [
+     "remove-cell"
+    ]
+   },
+   "outputs": [],
+   "source": [
+    "from ipywidgets import interact\n",
+    "import matplotlib.pyplot as plt\n",
+    "import numpy as np\n",
+    "\n",
+    "def fun(a,b):\n",
+    "    x = np.linspace(-2,2)\n",
+    "    plt.plot(x,a*x+b)\n",
+    "    plt.ylim(-8, 8)\n",
+    "    plt.show()\n",
+    "    \n",
+    "# create a slider\n",
+    "\n",
+    "interact(fun, a=(-6.0,4.0),b=(-1.0,1.0))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 62,
+   "metadata": {
+    "tags": [
+     "remove-cell"
+    ]
+   },
+   "outputs": [
+    {
+     "data": {
+      "image/png": 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",
+      "text/plain": [
+       "<Figure size 432x432 with 1 Axes>"
+      ]
+     },
+     "metadata": {
+      "needs_background": "light"
+     },
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "import matplotlib as mpl\n",
+    "import matplotlib.pyplot as plt\n",
+    "import numpy as np\n",
+    "\n",
+    "\n",
+    "# Create figure and add axes\n",
+    "fig = plt.figure(figsize=(6, 6))\n",
+    "ax = fig.add_subplot(111)\n",
+    "\n",
+    "\n",
+    "m = np.linspace(-2, 2, 11)\n",
+    "\n",
+    "# Get colors from coolwarm colormap\n",
+    "colors = plt.get_cmap('coolwarm', 11)\n",
+    "# Create variable reference to plot\n",
+    "f_d, = ax.plot([], [], linewidth=2.5)\n",
+    "# Add text annotation and create variable reference\n",
+    "temp = ax.text(1, 1, '', ha='right', va='top', fontsize=24)\n",
+    "\n",
+    "# Move left y-axis and bottim x-axis to centre, passing through (0,0)\n",
+    "ax.spines['left'].set_position('center')\n",
+    "ax.spines['bottom'].set_position('center')\n",
+    "\n",
+    "# Eliminate upper and right axes\n",
+    "ax.spines['right'].set_color('none')\n",
+    "ax.spines['top'].set_color('none')\n",
+    "\n",
+    "# Show ticks in the left and lower axes only\n",
+    "ax.xaxis.set_ticks_position('bottom')\n",
+    "ax.yaxis.set_ticks_position('left')\n",
+    "\n",
+    "\n",
+    "\n",
+    "labels = ['-2','-1.6','-1.2','-0.8','-0.4','0.0','0.4','0.8','1.2','1.6','2']\n",
+    "\n",
+    "for i in range(len(m)):\n",
+    "    x = np.linspace(-2, 2, 100)\n",
+    "    y = m[i]*x\n",
+    "    ax.plot(x, y, color=colors(i), linewidth=2.5, label=labels[i])\n",
+    "    \n",
+    "ax.tick_params(axis='both', which='major', labelsize=13)\n",
+    "\n",
+    "    \n",
+    "# Add legend\n",
+    "\n",
+    "leg = ax.legend(bbox_to_anchor=(1.05, 0.3),loc='lower left', \n",
+    "          frameon=True, labelspacing=0.2, fontsize=13)\n",
+    "leg.set_title(r\"$m$\", prop = {'size':'x-large'})\n",
+    "\n",
+    "plt.tight_layout()\n",
+    "plt.savefig('slope_01.png')"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 59,
+   "metadata": {
+    "tags": [
+     "remove-cell"
+    ]
+   },
+   "outputs": [
+    {
+     "data": {
+      "image/png": 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",
+      "text/plain": [
+       "<Figure size 432x360 with 1 Axes>"
+      ]
+     },
+     "metadata": {
+      "needs_background": "light"
+     },
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "import matplotlib.ticker as plticker\n",
+    "\n",
+    "# Create figure and add axes\n",
+    "fig = plt.figure(figsize=(6, 5))\n",
+    "ax = fig.add_subplot(111)\n",
+    "\n",
+    "\n",
+    "# Move left y-axis and bottim x-axis to centre, passing through (0,0)\n",
+    "ax.spines['left'].set_position('zero')\n",
+    "ax.spines['bottom'].set_position('zero')\n",
+    "\n",
+    "# Eliminate upper and right axes\n",
+    "ax.spines['right'].set_color('none')\n",
+    "ax.spines['top'].set_color('none')\n",
+    "\n",
+    "# Show ticks in the left and lower axes only\n",
+    "ax.xaxis.set_ticks_position('bottom')\n",
+    "ax.yaxis.set_ticks_position('left')\n",
+    "\n",
+    "\n",
+    "loc = plticker.MultipleLocator(base=3.0) # this locator puts ticks at regular intervals\n",
+    "ax.xaxis.set_major_locator(loc)\n",
+    "loc = plticker.MultipleLocator(base=2.0) # this locator puts ticks at regular intervals\n",
+    "ax.yaxis.set_major_locator(loc)\n",
+    "\n",
+    "x = np.linspace(3.1, 11, 100)\n",
+    "y = 2 + 1/(x-3)\n",
+    "ax.plot(x, y, color='b', linewidth=2.5)\n",
+    "\n",
+    "x = np.linspace(-6, 2.9, 100)\n",
+    "y = 2 + 1/(x-3)\n",
+    "ax.plot(x, y, color='b', linewidth=2.5)\n",
+    "\n",
+    "ylim = ax.get_ylim()\n",
+    "plt.vlines(3, ylim[0], ylim[1], ls='dashed',alpha=0.3)\n",
+    "\n",
+    "xlim = ax.get_xlim()\n",
+    "plt.hlines(2, xlim[0], xlim[1], ls='dashed',alpha=0.3)\n",
+    "\n",
+    "ax.tick_params(axis='both', which='major', labelsize=13)\n",
+    "    \n",
+    "# Add legend\n",
+    "\n",
+    "#ax.legend(bbox_to_anchor=(1.05, 0.3),loc='lower left', \n",
+    "#          frameon=False, labelspacing=0.2, fontsize=13)\n",
+    "\n",
+    "plt.tight_layout()\n",
+    "plt.savefig('rational_01.png')"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 66,
+   "metadata": {
+    "tags": [
+     "remove-cell"
+    ]
+   },
+   "outputs": [
+    {
+     "data": {
+      "image/png": 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",
+      "text/plain": [
+       "<Figure size 432x360 with 1 Axes>"
+      ]
+     },
+     "metadata": {
+      "needs_background": "light"
+     },
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "import matplotlib as mpl\n",
+    "import matplotlib.pyplot as plt\n",
+    "import numpy as np\n",
+    "\n",
+    "\n",
+    "# Create figure and add axes\n",
+    "fig = plt.figure(figsize=(6, 5))\n",
+    "ax = fig.add_subplot(111)\n",
+    "\n",
+    "alphaset = [-2.5,-2,-1,-0.5, 0.0, 0.5, 1.0, 2.0, 2.5]\n",
+    "colors = plt.get_cmap('coolwarm', 9)\n",
+    "\n",
+    "labels = ['-5/2','-2','-1','-1/2','0','1/2','1','2','5/2']\n",
+    "\n",
+    "plt.ylim(top=10)\n",
+    "\n",
+    "ax.spines['right'].set_color('none')\n",
+    "ax.spines['top'].set_color('none')\n",
+    "\n",
+    "for i in range(len(alphaset)):\n",
+    "    \n",
+    "    x = np.linspace(0.1, 2, 100)\n",
+    "    y = x**alphaset[i]\n",
+    "    ax.plot(x, y, color=colors(i), linewidth=2.5, label=labels[i])\n",
+    "    \n",
+    "ax.tick_params(axis='both', which='major', labelsize=13)\n",
+    "\n",
+    "    \n",
+    "# Add legend\n",
+    "\n",
+    "leg = ax.legend(bbox_to_anchor=(1.05, 0.3),loc='lower left', \n",
+    "          frameon=True, labelspacing=0.2, fontsize=13,)\n",
+    "leg.set_title(r\"${\\alpha}$\", prop = {'size':'x-large'})\n",
+    "\n",
+    "plt.tight_layout()\n",
+    "plt.savefig('powerlaw_01.png')"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 65,
+   "metadata": {
+    "tags": [
+     "remove-cell"
+    ]
+   },
+   "outputs": [
+    {
+     "data": {
+      "image/png": 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",
+      "text/plain": [
+       "<Figure size 432x360 with 1 Axes>"
+      ]
+     },
+     "metadata": {
+      "needs_background": "light"
+     },
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "import matplotlib as mpl\n",
+    "import matplotlib.pyplot as plt\n",
+    "import numpy as np\n",
+    "import matplotlib.ticker as plticker\n",
+    "\n",
+    "\n",
+    "# Create figure and add axes\n",
+    "fig = plt.figure(figsize=(6, 5))\n",
+    "ax = fig.add_subplot(111)\n",
+    "\n",
+    "alphaset = [-5,-2,-1,1, 2, 5]\n",
+    "colors = plt.get_cmap('coolwarm', 6)\n",
+    "\n",
+    "temp1 = colors(1)\n",
+    "temp0 = colors(0)\n",
+    "\n",
+    "labels = ['-5','-2','-1','1','2','5']\n",
+    "\n",
+    "plt.ylim(top=4)\n",
+    "\n",
+    "# Move left y-axis and bottim x-axis to centre, passing through (0,0)\n",
+    "ax.spines['left'].set_position('zero')\n",
+    "ax.spines['bottom'].set_position('zero')\n",
+    "\n",
+    "# Eliminate upper and right axes\n",
+    "ax.spines['right'].set_color('none')\n",
+    "ax.spines['top'].set_color('none')\n",
+    "\n",
+    "# Show ticks in the left and lower axes only\n",
+    "ax.xaxis.set_ticks_position('bottom')\n",
+    "ax.yaxis.set_ticks_position('left')\n",
+    "\n",
+    "loc = plticker.MultipleLocator(base=1.0) # this locator puts ticks at regular intervals\n",
+    "ax.yaxis.set_major_locator(loc)\n",
+    "\n",
+    "for i in range(len(alphaset)):\n",
+    "    \n",
+    "    x = np.linspace(-2, 2, 100)\n",
+    "    y = np.exp(alphaset[i]*x)\n",
+    "    #if i==0: ax.plot(x, y, color=temp1, linewidth=2.5, label=labels[i])\n",
+    "    #if i==1: ax.plot(x, y, color=temp0, linewidth=2.5, label=labels[i])\n",
+    "    #if i==2 or i==3: ax.plot(x, y, color=colors(i), linewidth=2.5, label=labels[i])\n",
+    "    ax.plot(x, y, color=colors(i), linewidth=2.5, label=labels[i])\n",
+    "    \n",
+    "ax.tick_params(axis='both', which='major', labelsize=13)\n",
+    "\n",
+    "    \n",
+    "# Add legend\n",
+    "\n",
+    "leg = ax.legend(bbox_to_anchor=(1.05, 0.3),loc='lower left', \n",
+    "          frameon=True, labelspacing=0.2, fontsize=13,)\n",
+    "leg.set_title(r\"${\\lambda}$\", prop = {'size':'x-large'})\n",
+    "\n",
+    "plt.tight_layout()\n",
+    "plt.savefig('exp_lamb_01.png')"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 55,
+   "metadata": {
+    "tags": [
+     "remove-cell"
+    ]
+   },
+   "outputs": [
+    {
+     "data": {
+      "image/png": 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",
+      "text/plain": [
+       "<Figure size 432x324 with 1 Axes>"
+      ]
+     },
+     "metadata": {
+      "needs_background": "light"
+     },
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "import matplotlib as mpl\n",
+    "import matplotlib.pyplot as plt\n",
+    "import numpy as np\n",
+    "from math import log\n",
+    "import matplotlib.ticker as plticker\n",
+    "\n",
+    "\n",
+    "# Create figure and add axes\n",
+    "fig = plt.figure(figsize=(6, 4.5))\n",
+    "ax = fig.add_subplot(111)\n",
+    "\n",
+    "alphaset = [0.1,0.2,0.5,2,5,10]\n",
+    "colors = plt.get_cmap('coolwarm', 6)\n",
+    "\n",
+    "\n",
+    "labels = ['1/10','1/5','1/2','2','5','10']\n",
+    "\n",
+    "#plt.ylim(top=4)\n",
+    "\n",
+    "# Move left y-axis and bottim x-axis to centre, passing through (0,0)\n",
+    "ax.spines['left'].set_position('zero')\n",
+    "ax.spines['bottom'].set_position('zero')\n",
+    "\n",
+    "# Eliminate upper and right axes\n",
+    "ax.spines['right'].set_color('none')\n",
+    "ax.spines['top'].set_color('none')\n",
+    "\n",
+    "# Show ticks in the left and lower axes only\n",
+    "ax.xaxis.set_ticks_position('bottom')\n",
+    "ax.yaxis.set_ticks_position('left')\n",
+    "\n",
+    "#loc = plticker.MultipleLocator(base=1.0) # this locator puts ticks at regular intervals\n",
+    "#ax.yaxis.set_major_locator(loc)\n",
+    "\n",
+    "\n",
+    "for i in range(len(alphaset)):\n",
+    "    \n",
+    "    x = np.linspace(0.1, 10, 100)\n",
+    "    y = np.log(x) / np.log(alphaset[i])\n",
+    "    #if i==0: ax.plot(x, y, color=temp1, linewidth=2.5, label=labels[i])\n",
+    "    #if i==1: ax.plot(x, y, color=temp0, linewidth=2.5, label=labels[i])\n",
+    "    #if i==2 or i==3: ax.plot(x, y, color=colors(i), linewidth=2.5, label=labels[i])\n",
+    "    ax.plot(x, y, color=colors(i), linewidth=2.5, label=labels[i])\n",
+    "    \n",
+    "ax.tick_params(axis='both', which='major', labelsize=13)\n",
+    "\n",
+    "    \n",
+    "# Add legend\n",
+    "\n",
+    "leg = ax.legend(bbox_to_anchor=(1.05, 0.3),loc='lower left', \n",
+    "          frameon=True, labelspacing=0.2, fontsize=13,)\n",
+    "leg.set_title(r\"$a$\", prop = {'size':'x-large'})\n",
+    "\n",
+    "plt.tight_layout()\n",
+    "plt.savefig('log_base_01.png')"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 53,
+   "metadata": {
+    "tags": [
+     "remove-cell"
+    ]
+   },
+   "outputs": [
+    {
+     "data": {
+      "image/png": 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",
+      "text/plain": [
+       "<Figure size 432x324 with 1 Axes>"
+      ]
+     },
+     "metadata": {
+      "needs_background": "light"
+     },
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "import matplotlib as mpl\n",
+    "import matplotlib.pyplot as plt\n",
+    "import numpy as np\n",
+    "from math import log\n",
+    "import matplotlib.ticker as plticker\n",
+    "\n",
+    "\n",
+    "# Create figure and add axes\n",
+    "fig = plt.figure(figsize=(6, 4.5))\n",
+    "ax = fig.add_subplot(111)\n",
+    "\n",
+    "\n",
+    "colors = plt.get_cmap('coolwarm', 2)\n",
+    "\n",
+    "\n",
+    "labels = [r\"$e^x$\",r\"$ln\\ x$\"]\n",
+    "\n",
+    "plt.ylim(-3,3)\n",
+    "plt.xlim(-3,3)\n",
+    "\n",
+    "ax.spines['left'].set_position('zero')\n",
+    "ax.spines['bottom'].set_position('zero')\n",
+    "\n",
+    "# Eliminate upper and right axes\n",
+    "ax.spines['right'].set_color('none')\n",
+    "ax.spines['top'].set_color('none')\n",
+    "\n",
+    "# Show ticks in the left and lower axes only\n",
+    "ax.xaxis.set_ticks_position('bottom')\n",
+    "ax.yaxis.set_ticks_position('left')\n",
+    "\n",
+    "#loc = plticker.MultipleLocator(base=1.0) # this locator puts ticks at regular intervals\n",
+    "#ax.yaxis.set_major_locator(loc)\n",
+    "\n",
+    "xe = np.linspace(-3,3, 100) \n",
+    "y = np.exp(xe)\n",
+    "ax.plot(xe, y, color=colors(0), linewidth=2.5, label=labels[0])\n",
+    "\n",
+    "xl = np.linspace(0.01, 10, 100)\n",
+    "y = np.log(xl)\n",
+    "ax.plot(xl, y, color=colors(1), linewidth=2.5, label=labels[1])\n",
+    "    \n",
+    "ax.plot(xe,xe,ls='dashed',lw=1.5)\n",
+    "\n",
+    "ax.tick_params(axis='both', which='major', labelsize=13)\n",
+    "\n",
+    "    \n",
+    "# Add legend\n",
+    "\n",
+    "leg = ax.legend(bbox_to_anchor=(1.05, 0.3),loc='lower left', \n",
+    "          frameon=True, labelspacing=0.2, fontsize=13)\n",
+    "\n",
+    "plt.tight_layout()\n",
+    "plt.savefig('log_base_02.png')"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Maths for Biologists\n",
+    "  \n",
+    "## Making mathematical statements"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Sets and relations  \n",
+    "  \n",
+    "Start by watching this video.\n",
+    "\n",
+    "<div style=\"padding:56.25% 0 0 0;position:relative;\"><iframe src=\"https://player.vimeo.com/video/1046665765?title=0&amp;byline=0&amp;portrait=0&amp;badge=0&amp;autopause=0&amp;player_id=0&amp;app_id=58479\" frameborder=\"0\" allow=\"autoplay; fullscreen; picture-in-picture; clipboard-write; encrypted-media\" style=\"position:absolute;top:0;left:0;width:100%;height:100%;\" title=\"Making_Mathematical_Statements_01\"></iframe></div><script src=\"https://player.vimeo.com/api/player.js\"></script>\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Polynomial functions  \n",
+    "\n",
+    "<div style=\"padding:56.25% 0 0 0;position:relative;\"><iframe src=\"https://player.vimeo.com/video/1046665782?title=0&amp;byline=0&amp;portrait=0&amp;badge=0&amp;autopause=0&amp;player_id=0&amp;app_id=58479\" frameborder=\"0\" allow=\"autoplay; fullscreen; picture-in-picture; clipboard-write; encrypted-media\" style=\"position:absolute;top:0;left:0;width:100%;height:100%;\" title=\"Making_Mathematical_Statements_02\"></iframe></div><script src=\"https://player.vimeo.com/api/player.js\"></script>\n",
+    "\n",
+    "One of the simplest elemntary functions is given by:  \n",
+    "  \n",
+    "  $$ f(x) = a_n x^n + a_{n-1} x^{n-1}+ \\cdots + a_1 x^1 + a_0, $$  \n",
+    "  \n",
+    "where $n$ is a *non-negative integer*, and the coefficients $a_n$, $a_{n-1}$, ..., $a_0$. The highest value of $n$ (for which the coefficient $a_n$ is different from $0$) defines the *degree* of the polynomial.  \n",
+    "  \n",
+    "```{admonition} Examples\n",
+    "$f(x) = 3x^5 + 12x^4 - 6x^3 - x^2 + x - 16$ (Polynomial function of degree 5)  \n",
+    "$g(x) = \\frac{3}{4}x^2 - \\frac{5}{2} $ (Polynomial function of degree 2)  \n",
+    "$h(x) = 1.6x^8 - 0.576x^4 + 1.89x^2 - 0.7$ (Polynomial function of degree 8)  \n",
+    "```  \n",
+    "  \n",
+    "\n",
+    "```{tip}\n",
+    "When the domain of the polynomial function is not explicitly given, it is frequently assumed to be all real numbers, ${\\rm I\\!R}$.\n",
+    "```\n",
+    "```{tip}\n",
+    "When no application is specified, the function name \\'$f$\\' and variable name \\'$x$\\' are usually chosen. For application purposes, the function and variable are often named for what they represent; for example, $A(r)=\\pi r^2$ for the area of a circle dependent on its radius.\n",
+    "```\n",
+    "\n",
+    "Let us discuss three of the most common forms of polynomial functions with degrees $0$, $1$, and $2$.\n",
+    "\n",
+    "#### Degree 0  \n",
+    "  \n",
+    "Polynomial functions of degree 0 assume the form:  \n",
+    "  \n",
+    "  $$ f(x) = c, $$  \n",
+    "  \n",
+    "where $c$ is a real number. This function is graphically represented by:  \n",
+    "  \n",
+    "```{image} ./poly_deg0.png\n",
+    "\n",
+    "``` \n",
+    "\n",
+    "```{admonition} Example  \n",
+    "For most mobile phone contracts nowadays a fixed fee is paid for the month with unlimited usage. So the cost as a function of used minutes would be a polynomial of degree zero, since the cost does not depend on the usage.\n",
+    "```\n",
+    "  \n",
+    "  \n",
+    "#### Degree 1\n",
+    "\n",
+    "Polynomial functions of degree 1 are also called *linear functions* and are given by:  \n",
+    "  \n",
+    "$$ f(x) = mx + b,$$  \n",
+    "  \n",
+    "where again $m$ and $b$ are real numbers (with $m \\ne 0$).  \n",
+    "  \n",
+    "The constants $m$ and $b$ are frequently referred to as *slope* and *intercept*. The intercept $b$ shows where the function crosses the $y$-axis, while the slope $m$ is related to the inclination of the function.\n",
+    "  \n",
+    "````{panels}  \n",
+    "  \n",
+    "```{image} ./poly_deg1.png\n",
+    "\n",
+    "```  \n",
+    "\n",
+    "---  \n",
+    "  \n",
+    "If we consider two arbitrary points, $(x_1,y_1)$ and $(x_2,y_2)$ on the line, their projections define two intervals on the $y$ and $x$ axes. These intervals have sizes $\\Delta y = y_2 - y_1$ and $\\Delta x = x_2 - x_1$ respectively.  \n",
+    "  \n",
+    "The slope $m$ will be given by the ratio between the changes in $y$ and $x$, for any chosen two points on the line. Thus:  \n",
+    "\n",
+    "  $$m = \\frac{y_2 - y_1}{x_2 - x_1} = \\frac{\\Delta y}{\\Delta x}. $$  \n",
+    "  \n",
+    "Also note that the triangle defined by the sides $\\Delta y$, $\\Delta x$ and this fragment of the function line forms a right triangle. Thus, there is a relationship between the slope $m$ and the angle $\\theta$ between the function line and the $x$-axis (or any horizontal line parallel to it) which is given by:  \n",
+    "  \n",
+    "  $$m =  \\frac{\\Delta y}{\\Delta x} = tan(\\theta).$$\n",
+    "  \n",
+    "````  \n",
+    "  \n",
+    "The sign of the slope $m$ also determines if the function increases or decreases as a function of $x$. We have:  \n",
+    "  \n",
+    "  $$ \\begin{align}\n",
+    "      m>0 &\\rightarrow \\mbox{Increasing function of } x \\\\\n",
+    "      m<0 &\\rightarrow \\mbox{Decreasing function of } x\n",
+    "  \\end{align} $$  \n",
+    "    \n",
+    "    \n",
+    " \n",
+    "````{panels}  \n",
+    "  \n",
+    "```{image} ./slope_01.png  \n",
+    "\n",
+    "```  \n",
+    "\n",
+    "---  \n",
+    "  \n",
+    "The graph on the left show the functions $y = mx$ (intercept $b=0$), for different values of $m$ (see legend colours).  \n",
+    "  \n",
+    "Note that positive values of $m$ give **increasing** functions, while negative values of $m$ return **decreasing** functions.\n",
+    "      \n",
+    "```` \n",
+    "  \n",
+    "One can also ask what is the value of $x$ where the function $f(x)$ crosses de $x$-axis, or, in mathematical notation, what is $x\\ \\mbox{so that}\\ f(x) = 0$. This value of x is called a **root of $f$** and it can be obtained by:  \n",
+    "\n",
+    "$$\\begin{align}\n",
+    "    f(x) &= 0 \\\\\n",
+    "    mx + b &= 0 \\\\\n",
+    "    mx &= -b \\\\\n",
+    "    x &= -\\frac{b}{m}\n",
+    "\\end{align} $$\n",
+    "    \n",
+    "`````{admonition} Example  \n",
+    "\n",
+    "````{panels}  \n",
+    "  \n",
+    "```{image} ./poly_deg1_02.png\n",
+    "\n",
+    "```  \n",
+    "\n",
+    "---  \n",
+    "  \n",
+    "  Consider the function $y = -2x + 1$.  \n",
+    "  We have:  $m=-2$ and $b=1$. The root of this function will be $x = -\\displaystyle\\frac{b}{m} = -\\displaystyle\\frac{1}{(-2)} = \\displaystyle\\frac{1}{2}$.  \n",
+    "  Additionally, the function crosses the $y$-axis at the intercept $b=1$.\n",
+    "  \n",
+    "````\n",
+    "`````  \n",
+    "\n",
+    "```{admonition} Examples  \n",
+    "The metabolic rate of some organisms depends (approximately) linearly on the body's temperature, with higher rates at higher temperatures.\n",
+    "```\n",
+    "  \n",
+    "#### Degree 2\n",
+    "\n",
+    "Now we look at polynomial functions of degree 2, given by the general form:  \n",
+    "\n",
+    "$$f(x) = ax^2 + bx +c,$$  \n",
+    "  \n",
+    "the coefficients $a$, $b$, and $c$ are real numbers (with $a \\ne 0$). The graph of a polynomial function of degree 2 is a *parabola* and many of its properties can be readily known by analysing the coefficients.\n",
+    "\n",
+    "##### Concavity\n",
+    "\n",
+    "````{panels}  \n",
+    "Concave up (positive concavity)\n",
+    "^^^\n",
+    "  \n",
+    "```{image} ./poly_deg2_01.png  \n",
+    "\n",
+    "```  \n",
+    "\n",
+    "---  \n",
+    "  \n",
+    "Concave down (negative concavity)\n",
+    "^^^\n",
+    "  \n",
+    "```{image} ./poly_deg2_02.png  \n",
+    "\n",
+    "```  \n",
+    "  \n",
+    "````\n",
+    "\n",
+    "##### Intercept & Roots  \n",
+    "\n",
+    "````{panels}  \n",
+    "  \n",
+    "```{image} ./poly_deg2_roots.png  \n",
+    "\n",
+    "```  \n",
+    "\n",
+    "---  \n",
+    "  \n",
+    "For the function $f(x) = ax^2 +bx + c$ the coefficient $c$ represents the **intercept**, the point where the function curve crosses the $y$-axis.  \n",
+    "  \n",
+    "The **roots** of the function $f$ are the values of $x$ for which $ax^2 +bx + c = 0$, where the curve crosses the $x$-axis. A polynomial function of degree 2 has a maximum number of two real roots*, given by:  \n",
+    "\n",
+    "$$\\begin{align}\n",
+    "    x_1 &= \\frac{-b-\\sqrt{b^2-4ac}}{2a} \\\\\n",
+    "    x_2 &= \\frac{-b+\\sqrt{b^2-4ac}}{2a}\n",
+    "\\end{align}$$\n",
+    "\n",
+    "+++\n",
+    "\\* A polynomial function of degree $n$ will always have $n$ roots, but these can be imaginary or multiple roots. It therefore can (but does not have to) have a maximum of $n$ real roots.\n",
+    "  \n",
+    "````  \n",
+    "  \n",
+    "The expression inside the square root distinguish between different properties of this function and therefore receives the name of *discriminant*:  \n",
+    "\n",
+    "$$ \\Delta = b^2 -4ac.$$  \n",
+    "  \n",
+    "Depending on the value of the discriminant a quick conclusion can be reached regarding the roots of the function.\n",
+    "\n",
+    "`````{tabbed} Case 1  -  $\\Delta > 0$  \n",
+    "\n",
+    "````{panels}  \n",
+    "  \n",
+    "```{image} ./poly_deg2_delta01.png  \n",
+    "\n",
+    "```  \n",
+    "\n",
+    "---  \n",
+    "  \n",
+    "When $\\Delta > 0$ there are two real **distinct** roots for the polynomial function.  \n",
+    "  \n",
+    "```{admonition} Examples\n",
+    "Consider the function  $y = 0.5 x^2 - x -1.5$. We have:  \n",
+    "  \n",
+    "  $$\\Delta = (-1)^2 - 4(0.5)(-1.5) = 1 + 3 = 4.$$  \n",
+    "  \n",
+    "  The roots will then be given by:  \n",
+    "    \n",
+    "  $$\\begin{align}\n",
+    "  x_1 &= \\frac{-(-1)-\\sqrt{4}}{2(0.5)} = \\frac{1-2}{1} = -1 \\\\\n",
+    "  x_2 &= \\frac{-(-1)+\\sqrt{4}}{2(0.5)} = \\frac{1+2}{1} = 3 \\end{align}$$  \n",
+    "  \n",
+    "``` \n",
+    "\n",
+    "  \n",
+    "````  \n",
+    "`````  \n",
+    "  \n",
+    "`````{tabbed} Case 2  -  $\\Delta = 0$  \n",
+    "\n",
+    "````{panels}  \n",
+    "  \n",
+    "```{image} ./poly_deg2_delta02.png  \n",
+    "\n",
+    "```  \n",
+    "\n",
+    "---  \n",
+    "  \n",
+    "When $\\Delta = 0$ there are two real **identical** roots for the polynomial function.  \n",
+    "  \n",
+    "```{admonition} Examples\n",
+    "Consider the function  $y = 0.5 x^2 - x +0.5$. We have:  \n",
+    "  \n",
+    "  $$\\Delta = (-1)^2 - 4(0.5)(0.5) = 1 -1 = 0.$$  \n",
+    "  \n",
+    "  The roots will then be given by:  \n",
+    "    \n",
+    "  $$\\begin{align}\n",
+    "  x_1 = x_2 = \\frac{-(-1)\\pm\\sqrt{0}}{2(0.5)} = 1\\end{align}$$  \n",
+    "  \n",
+    "``` \n",
+    "  \n",
+    "````  \n",
+    "`````  \n",
+    "  \n",
+    "`````{tabbed} Case 3  -  $\\Delta < 0$\n",
+    "\n",
+    "````{panels}  \n",
+    "  \n",
+    "```{image} ./poly_deg2_delta03.png  \n",
+    "\n",
+    "```  \n",
+    "\n",
+    "---  \n",
+    "When $\\Delta < 0$ there are no real roots for the polynomial function, since the square root of a negative number is not defined for real numbers.  \n",
+    "  \n",
+    "  ```{admonition} Examples\n",
+    "The graph on the left corresponds to the function  $y = 0.5 x^2 - x + 1$. We see that the curve does not cross the $x$-axis at any point, showing there is no real root for this function.  \n",
+    "\n",
+    "``` \n",
+    "  \n",
+    "````  \n",
+    "`````  \n",
+    "```{admonition} Example\n",
+    "If we throw a ball that is heavy enough so that air resistance is negligible (alternatively we can travel to the moon) the height of the ball as a function of time follows a quadratic equation. Knowing the roots we can calculate when and where the ball will land.\n",
+    "```\n",
+    "\n",
+    "##### Minimum / Maximum \n",
+    "  \n",
+    "  Depending on the concavity of the parabola (up or down), there will be a point where the function assumes a minimum or a maximum value. This point is called the *vertex* of the parabola. This extreme point can be easily determined by the symmetry properties of the function curve.  \n",
+    "   \n",
+    "   ````{panels}  \n",
+    "  \n",
+    "```{image} ./poly_deg2_vertex.png  \n",
+    "\n",
+    "```  \n",
+    "\n",
+    "---  \n",
+    "  \n",
+    "The coordinate $(x_v,y_v)$ of the vertex is given by:  \n",
+    "  \n",
+    "  $$\\begin{align}\n",
+    "  x_v &= -\\frac{b}{2a} \\\\\n",
+    "  y_v &= -\\frac{(b^2-4ac)}{4a} = -\\frac{\\Delta}{4a}\n",
+    "  \\end{align}$$  \n",
+    "    \n",
+    "  ```{tip}  \n",
+    "  Since the parabola is symmetric in relation to a vertical line passing through the vertex, the $x$ coordinate of the vertex is given by the middle point between the two roots $x_v = (x_1 + x_2)/2$. The $y$ coordinate is then given by substituting $x_v$ in the general form of the polynomial of degree 2, as $y_v = f(x_v) = ax_v^2 + bx_v + c$. Try to obtain the expressions above as an exercise.\n",
+    "  ```\n",
+    " ```{tip}  \n",
+    "The extrema is the point of the parabola where the derivative becomes zero $f'(x)=0$.\n",
+    "  ```\n",
+    "  \n",
+    "````\n",
+    "```{admonition} Example\n",
+    "Imagine the population of a specific region having a negative quadratic relationship - it undergoes initial growth, reaches its peak, and then declines. To determine both the maximum population size and the time when it peaked, we need to identify the extrema of this parabolic curve.\n",
+    "```\n",
+    "\n",
+    "  \n",
+    "<!-- Now watch this [video](https://web.microsoftstream.com/video/f64abd8a-9997-4b5e-b11f-0ca09b7603fe) for a short introduction to complex numbers.\n",
+    " -->\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Rational functions  \n",
+    "\n",
+    "<div style=\"padding:56.25% 0 0 0;position:relative;\"><iframe src=\"https://player.vimeo.com/video/1046665803?title=0&amp;byline=0&amp;portrait=0&amp;badge=0&amp;autopause=0&amp;player_id=0&amp;app_id=58479\" frameborder=\"0\" allow=\"autoplay; fullscreen; picture-in-picture; clipboard-write; encrypted-media\" style=\"position:absolute;top:0;left:0;width:100%;height:100%;\" title=\"Making_Mathematical_Statements_03\"></iframe></div><script src=\"https://player.vimeo.com/api/player.js\"></script>\n",
+    "\n",
+    "Rational functions are defined as quotients of polynomial functions:  \n",
+    "  \n",
+    "$$f(x) = \\frac{p(x)}{q(x)},\\ \\ \\ q(x) \\ne 0$$  \n",
+    "    \n",
+    "      \n",
+    "````{panels}  \n",
+    "  \n",
+    "```{image} ./rational_01.png  \n",
+    "\n",
+    "```  \n",
+    "\n",
+    "---  \n",
+    "  \n",
+    "The plot on the left shows the graph for the function $y = \\displaystyle\\frac{2x-5}{x-3}$.  \n",
+    "  \n",
+    "Note that the graph approaches a vertical line at $x=3$ when $x$ is close to $3$, and it approaches a horizontal line at $y=2$ when $x$ approaches large values (positive or negative). These lines are called *vertical asymptote* and *horizontal asymptote*, respectively.  \n",
+    "    \n",
+    "Vertical asymptotes appear for the values of $x$ corresponding to the roots of the denominator function ($q(x)$) of the rational function $f(x)$, while horizontal asymptotes to the limits of the function $f(x)$ (if it exists) for large $x$*.  \n",
+    "  \n",
+    "Asymptotes give us a lot of information about the limiting behaviours of the function, and they are especially relevant when modelling biological phenomena.\n",
+    "\n",
+    "+++\n",
+    "\\* We will be talking more about limits on the following lectures.\n",
+    "      \n",
+    "````  \n",
+    "\n",
+    "```{admonition} Example\n",
+    "Rational functions are often used to express ratios, for example of chemicals. Lets assume chemical A increases quadratically $A(t) = 2 t^2$ and chemical B increases linearly $B(t)=t+5$, the ratio will be:\n",
+    "\n",
+    "\n",
+    "$$r(t) = \\frac{A(t)}{B(t)} = 2\\frac{t^2}{t+5}$$\n",
+    "```\n",
+    "        "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Power functions  \n",
+    "  \n",
+    "Another type of function used in many applications are power functions (sometimes also called power laws). These functions have the functional form:  \n",
+    "  \n",
+    "  $$f(x) = Cx^{\\alpha}$$  \n",
+    "    \n",
+    "where $C$ and $\\alpha$ are real constants (with $C \\ne 0$).\n",
+    "  \n",
+    "  \n",
+    "````{panels}  \n",
+    "  \n",
+    "```{image} ./powerlaw_01.png  \n",
+    "\n",
+    "```  \n",
+    "\n",
+    "---  \n",
+    "  \n",
+    "The plot on the left shows the graphs for the functions $y = x^{\\alpha}$ (with $C=1$) for different values of the exponent $\\alpha$ (see the legend colours).  \n",
+    "  \n",
+    "There are a few interesting aspects to note in this plot. First, the graphs were presented only for positive values of $x$. Some of these functions are also defined (on the set of real numbers) for negative values of $x$ which could be presented on the plot. This is the case for $x^{2}$ or $x^{-2}$, but not the case for $x^{1/2}$ or $x^{-1/2}$.  \n",
+    "  \n",
+    "When analysing positive values of $x$ (as it is frequently the case for models in Biology), for $C > 0$, the exponent $\\alpha$ determines the behaviour of the functions, with **increasing** functions of x for $\\alpha > 0$ and **decreasing** functions of x for $\\alpha < 0$.  \n",
+    "      \n",
+    "````  \n",
+    "Many commonly used functions are examples of power functions, e.g. the square root $\\sqrt{x}=x^{\\frac 12}$ or the inverse function $x^{-1}$.\n",
+    "\n",
+    "<!-- Watch this [video](https://web.microsoftstream.com/video/bf7415e1-823f-449e-ace1-e379fac8fe0a) about additional properties of power law functions. -->"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Exponential functions  \n",
+    "\n",
+    "<div style=\"padding:56.25% 0 0 0;position:relative;\"><iframe src=\"https://player.vimeo.com/video/1046665831?title=0&amp;byline=0&amp;portrait=0&amp;badge=0&amp;autopause=0&amp;player_id=0&amp;app_id=58479\" frameborder=\"0\" allow=\"autoplay; fullscreen; picture-in-picture; clipboard-write; encrypted-media\" style=\"position:absolute;top:0;left:0;width:100%;height:100%;\" title=\"Making_Mathematical_Statements_04\"></iframe></div><script src=\"https://player.vimeo.com/api/player.js\"></script>\n",
+    "\n",
+    "\n",
+    "Similar to power functions, but with very different properties are the exponential functions:  \n",
+    "  \n",
+    "  $$f(x) = a^x, $$  \n",
+    "    \n",
+    "where $a$ is a positive constant. The parameter $a$ is called *base* and the variable $x$ is the exponent (remember that for power functions, the variable $x$ is the base, raised to the power of the parameter $\\alpha$).  \n",
+    "  \n",
+    "  Exponential functions are used to model many different real-world applications, from bacterial growth to radioactive decay.  \n",
+    "    \n",
+    "```{admonition} Example  \n",
+    "\n",
+    "Suppose we start a bacterial colony with a single individual on a Petri dish. At each time step, counted as an average reproduction interval, each individual bacterium duplicates, originating two other individuals. The number of bacteria in the colony as a function of time, $N(t)$, can be counted as:  \n",
+    "    \n",
+    "    \n",
+    "| $t$      |      $N(t)$   |\n",
+    "|----------|:-------------:|\n",
+    "| $0$      |  $1\\ (= 2^0)$ |\n",
+    "| $1$      |  $2\\ (= 2^1)$ |\n",
+    "| $2$      |  $4\\ (= 2^2)$ |\n",
+    "| $3$      |  $8\\ (= 2^3)$ |\n",
+    "| $4$      |  $16\\ (= 2^4)$ |\n",
+    "| $5$      |  $32\\ (= 2^5)$ |  \n",
+    "  \n",
+    "If no limitation (on space or resources) is imposed to the colony, the number of bacteria at each time step $t$ can be well modelled by an exponential function of the form:  \n",
+    "  \n",
+    "$$ N(t) = 2^t. $$\n",
+    "  \n",
+    "```  \n",
+    "  \n",
+    "Exponential functions of the form $f(x) = a^x$ (with $a>0$) can never assume negative values and their behaviour depends on the value of $a$.  \n",
+    "  \n",
+    "```{image} ./exponential_01.png  \n",
+    ":align: center\n",
+    "```  \n",
+    "  \n",
+    "```{tip}  \n",
+    "Note that the function *increases* with $x$ if $a>1$ (growth) and *decreases* with $x$ if $0<a<1$ (decay). To remember this, one can directly apply the properties of exponentiation. If $f(x) = (1/3)^x$ (thus $a = (1/3) < 1$), we have:  \n",
+    "  \n",
+    "$$f(x) = \\left(\\frac{1}{3}\\right)^x = \\frac{1}{3^x},$$  \n",
+    "  \n",
+    "and since $3^x$ is an increasing function of $x$, then $1/3^x$ should be a decreasing function of $x$ (as the denominator increases, the fraction decreases).\n",
+    "```\n",
+    "```{admonition} Example  \n",
+    "In the early stages of a new desease, we can approximate the spread as exponential. In this case $a$ is the number of people every infected person infects themselves. For $a<1$ every person infects less than one other person on average and the desease will slow down, for $a>1$ the desease will spread. At the beginning of the COVID-19 pandemic, an estimate for $a$ (or in this case $R$ for reproductive value) was $a=4$. If we start with 1 infected person, how many do we have after 10 infection cycles?\n",
+    "```\n",
+    "  \n",
+    "Two of the most used bases are $a=10$ and $a=e$, where $e=2.71828$ (called *Euler's number* or *Napier's constant*). Exponential functions with $a=e$ are commonly written as:  \n",
+    "  \n",
+    "$$f(x) = e^x = \\mbox{exp}(x) $$  \n",
+    "\n",
+    "The unique charachteristic of the exponential of base $e$ is, that it is its own derivative\n",
+    "\n",
+    "$$\\frac{\\text{d}}{\\text{d}x} e^x = e^x $$\n",
+    "  \n",
+    "````{panels}  \n",
+    "  \n",
+    "```{image} ./exp_lamb_01.png  \n",
+    "\n",
+    "```  \n",
+    "\n",
+    "---  \n",
+    "  \n",
+    "Another common representation of the exponential function is:  \n",
+    "  \n",
+    "$$ f(x) = e^{\\lambda x}, $$  \n",
+    "  \n",
+    "where $\\lambda$ is a real constant, usually called *rate constant*. The absolute value of the parameter $\\lambda$ controls how fast is the growth (in case $\\lambda > 0$) or the decay (for $\\lambda < 0$) of the function $f(x)$ for increasing values of $x$.  \n",
+    "  \n",
+    "For a fixed value of $x$, which $\\lambda$ corresponds to the highest and the lowest values among these exponential functions? Does it change if $x$ is positive or negative? How would it change if, instead of $a=e$, it was a different base $a<1$?\n",
+    "      \n",
+    "````  \n",
+    "The both formulations can be changed into one another via\n",
+    "\n",
+    "\n",
+    "$$f(x) = e^{\\lambda x} = ({\\underbrace{e^\\lambda}_a})^x.$$\n",
+    "\n",
+    "\n",
+    "We have exponential growth when $a>1$ and therefore when $\\lambda>0$ and decay when $0<a<1$ and therefore $\\lambda<0$."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "\n",
+    "### Logarithmic functions  \n",
+    "  \n",
+    "Let us start this section by understanding what the expression  \n",
+    "  \n",
+    "$$ \\mbox{log}_a b = x $$  \n",
+    "  \n",
+    "actually means. One possible way to read it is: \"$\\mbox{log}_a b$ is the number with which I have to exponentiate the base $a$ so that I find the number $b$. Since \"$\\mbox{log}_a b = x$, the above expression is equivalent to:  \n",
+    "  \n",
+    "$$ a^x = b.$$  \n",
+    "  \n",
+    "The expression $\\mbox{log}_a b$ is called the *logarithm of b in the base a* and it is also possible to define a logarithmic function given by:  \n",
+    "  \n",
+    "$$f(x) = \\mbox{log}_a x,$$  \n",
+    "  \n",
+    "where $a$ is the base of the logarithm and it is assumed $a>0$ and $a \\ne 1$.  \n",
+    "  \n",
+    "By defining a function like this, for each $x$ we will have a number $y = f(x)$ for which $a^y = x$. Therefore, the logarithmic function is only defined for $x>0$. Like the exponential function, the behaviour of the logarithmic function for increasing $x$ will depend on the value of the base $a$.  \n",
+    "  \n",
+    "````{panels}  \n",
+    "  \n",
+    "```{image} ./log_base_01.png  \n",
+    "\n",
+    "```  \n",
+    "\n",
+    "---  \n",
+    "Note that the logarithmic functions are *increasing* if $a>1$ and *decreasing* if $0<a<1$. For a fixed value of $x$ the functions showing the highest and lowest values $f(x)$ will depend both on the value of the base and on which interval $x$ belongs to ($0<x<1$ or $x>1$). Can you determine these values for each case?  \n",
+    "  \n",
+    "        \n",
+    "````\n",
+    "\n",
+    "````{tip}  \n",
+    "The similarities between the graphs for exponential and logarithmic functions are not coincidental. In fact, the graph for a logarithmic function with base $a$ can be obtained by drawing the graph for an exponential function with base $a$ and reflecting it in relation to the line $y = x$.  [MAKE COMMENT ON INVERSE FUNCTIONS]\n",
+    "  \n",
+    "```{image} ./log_base_02.png\n",
+    "```  \n",
+    "    \n",
+    "(here we use the common nomenclature $\\mbox{log}_e x = \\mbox{ln}\\ x$).  \n",
+    "````\n",
+    "\n",
+    "<div style=\"padding:56.25% 0 0 0;position:relative;\"><iframe src=\"https://player.vimeo.com/video/1046665847?badge=0&amp;autopause=0&amp;player_id=0&amp;app_id=58479\" frameborder=\"0\" allow=\"autoplay; fullscreen; picture-in-picture; clipboard-write; encrypted-media\" style=\"position:absolute;top:0;left:0;width:100%;height:100%;\" title=\"Making_Mathematical_Statements_05\"></iframe></div><script src=\"https://player.vimeo.com/api/player.js\"></script>"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 13,
+   "metadata": {
+    "editable": true,
+    "slideshow": {
+     "slide_type": ""
+    },
+    "tags": [
+     "remove-cell"
+    ]
+   },
+   "outputs": [
+    {
+     "data": {
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",
+      "text/plain": [
+       "<Figure size 864x216 with 1 Axes>"
+      ]
+     },
+     "metadata": {
+      "needs_background": "light"
+     },
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "image/png": 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",
+      "text/plain": [
+       "<Figure size 432x324 with 1 Axes>"
+      ]
+     },
+     "metadata": {
+      "needs_background": "light"
+     },
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "#Code for figures\n",
+    "\n",
+    "import numpy as np\n",
+    "import pandas as pd\n",
+    "import matplotlib.pyplot as plt\n",
+    "\n",
+    "def multiple_formatter(denominator=2, number=np.pi, latex='\\pi'):\n",
+    "    def gcd(a, b):\n",
+    "        while b:\n",
+    "            a, b = b, a%b\n",
+    "        return a\n",
+    "    def _multiple_formatter(x, pos):\n",
+    "        den = denominator\n",
+    "        num = np.int(np.rint(den*x/number))\n",
+    "        com = gcd(num,den)\n",
+    "        (num,den) = (int(num/com),int(den/com))\n",
+    "        if den==1:\n",
+    "            if num==0:\n",
+    "                return r'$0$'\n",
+    "            if num==1:\n",
+    "                return r'$%s$'%latex\n",
+    "            elif num==-1:\n",
+    "                return r'$-%s$'%latex\n",
+    "            else:\n",
+    "                return r'$%s%s$'%(num,latex)\n",
+    "        else:\n",
+    "            if num==1:\n",
+    "                return r'$\\frac{%s}{%s}$'%(latex,den)\n",
+    "            elif num==-1:\n",
+    "                return r'$\\frac{-%s}{%s}$'%(latex,den)\n",
+    "            else:\n",
+    "                return r'$\\frac{%s%s}{%s}$'%(num,latex,den)\n",
+    "    return _multiple_formatter\n",
+    "\n",
+    "class Multiple:\n",
+    "    def __init__(self, denominator=2, number=np.pi, latex='\\pi'):\n",
+    "        self.denominator = denominator\n",
+    "        self.number = number\n",
+    "        self.latex = latex\n",
+    "\n",
+    "    def locator(self):\n",
+    "        return plt.MultipleLocator(self.number / self.denominator)\n",
+    "\n",
+    "    def formatter(self):\n",
+    "        return plt.FuncFormatter(multiple_formatter(self.denominator, self.number, self.latex))\n",
+    "\n",
+    "xt=np.linspace(-4*np.pi,4*np.pi,num=500)\n",
+    "##\n",
+    "fig, ax = plt.subplots(figsize=(12, 3))\n",
+    "ax.plot(xt, np.sin(xt), color='b', lw=2, label='$sin(x)$')\n",
+    "ax.plot(xt, np.cos(xt), color='g', lw=2, label='$cos(x)$')\n",
+    "ax.tick_params(axis='both', labelsize= 15)\n",
+    "ax.set_xlabel('$x$', fontsize = 16)\n",
+    "ax.legend(fontsize=14)\n",
+    "\n",
+    "ax.axhline(0, color='black', lw=1)\n",
+    "ax.axvline(0, color='black', lw=1)\n",
+    "\n",
+    "ax.xaxis.set_major_locator(plt.MultipleLocator(np.pi / 2))\n",
+    "ax.xaxis.set_minor_locator(plt.MultipleLocator(np.pi / 12))\n",
+    "ax.xaxis.set_major_formatter(plt.FuncFormatter(multiple_formatter()))\n",
+    "\n",
+    "fig.tight_layout()\n",
+    "fig.savefig('sin_cos.png')\n",
+    "\n",
+    "##\n",
+    "xt=np.linspace(-2*np.pi,2*np.pi,num=500)\n",
+    "fig, ax = plt.subplots(figsize=(6, 4.5))\n",
+    "y = np.tan(xt)\n",
+    "y[:-1][np.diff(y) < 0] = np.nan\n",
+    "\n",
+    "ax.plot(xt,y, color ='r', lw=2)\n",
+    "ax.tick_params(axis='both', labelsize= 15)\n",
+    "\n",
+    "ax.set_xlabel('$x$', fontsize = 16)\n",
+    "ax.set_ylabel(\"$tan(x)$\", fontsize = 16)\n",
+    "\n",
+    "plt.ylim(-10,10)\n",
+    "ax.axhline(0, color='black', lw=1)\n",
+    "ax.axvline(0, color='black', lw=1)\n",
+    "\n",
+    "ax.xaxis.set_major_locator(plt.MultipleLocator(np.pi / 2))\n",
+    "ax.xaxis.set_minor_locator(plt.MultipleLocator(np.pi / 12))\n",
+    "ax.xaxis.set_major_formatter(plt.FuncFormatter(multiple_formatter()))\n",
+    "\n",
+    "fig.tight_layout()\n",
+    "fig.savefig('tan.png')"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 14,
+   "metadata": {
+    "tags": [
+     "remove-cell"
+    ]
+   },
+   "outputs": [
+    {
+     "data": {
+      "image/png": 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",
+      "text/plain": [
+       "<Figure size 864x216 with 1 Axes>"
+      ]
+     },
+     "metadata": {
+      "needs_background": "light"
+     },
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "image/png": 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RF/ToISVxH3tMltG9JH9+ux9ct24yMUJ0NWfOSCNhQI6d3LnNxpNSTZsClStLs+2ePU1HQ17w7bey4pkhg8zku6kacnKMHCn94BYskIkRomvp108yue67T86ZpnlygOWks2elyzMgKYKFCpmNJzVKlbJH7S+9xI2hdHXr1gGzZ8ugfPx47114Adk7duutwL593klvJHMGDZJm1XfcIceO1yglm7XTpZOKWD/+aDoicrOLF6W3FCD7VUuUMBtPahQpYu+X7NBB9iESXcnWrZLdkiaN7NV2QxsCF4Rg1ogRcpN26632QMuL+vSRqlg//WSXcCdKSmtJhwVkRt+LF15AmsJa+dWDBzNPn65s9257ED5hAhAVZTae1CpTxm4G3rUr9x/SlY0fL73ebrhBshW8qmtXmUDescP8fhpyL63txYW2bWUizQ0CPcD66y+7WuDYsXLT5lWZM9vpjX36ME+fLu+996TMeb58dl8Ur6pcWfYgnjvn7ZsICq+uXe0WHPfdZzqa0PTpI/sKVq2SPWVESR09ahfsGjNGUgS9Kn16e2DVvz8LedHlLVwoBbty57bbzrhBoAdYfftKbn6dOnKz5nV16wIPPSQn2MGDTUdDbnPmjL1/Y9AgIGtWs/E4YfhwIGNGYN484LvvTEdDbvPtt7LHNksWaajtddmz2zfPPXpIs2SixN58U/bXVqvmnUqZV1OjBlC7NnD6tH3sE1kuXLB7Xg0Y4K5KmYEdYG3bJql0UVHe6HmVHErZqTDjxjFtiv5r+HDgzz+BO++U2Xw/KFLEXr3q2ZNpU2TT2q6U2aOHrNr6wQsvAOXKAb/9xrQp+q+9e+19tcOHe3N/7eUMGSJ7amJigJ07TUdDbjJlCvDrr0Dp0nJudJPADrB69JBGq23byn+MX5QvL1XhYmOB114zHQ25xT//2IPvUaPcsQHUKd26AXnzSgXNxYtNR0NuMX++FIPIn9+uIOgHadPK3mFAVisOHzYbD7lHnz5S4KJJE/fsQ3FCuXJAixayx8brqe3knJMnJXUUkEF4unRm40nKR7dZybd8OfDpp5Ii1bev6WicN3CgHGizZwObN5uOhtxg0CBJEaxVC6hUyXQ0zsqa1Z5M6NMHiIszGw+Zd+GCfSPWv7/sUfWT6tXl4+RJ/2RgUGjWrZNU6QwZ3LUPxSn9+0s6+IcfAmvWmI6G3GDYMODIEeCBB9zZ5y1wAyyt7XzN3r3NNyILhxIlgHbt5O/K5sO0fz8waZI8tvpH+U3r1nLcb98OzJxpOhoyLSYG2LNHKu+1aGE6mvAYNEg+jx8vqb8UbNa1vnNnSZ32m4IF7abD3bszHTzoDh60C7u5NR02cAOsJUskbSRfPqBjR9PRhM+rrwLZsgGffw6sWGE6GjKpXz+Z0W/UCLjtNtPRhEd0tD1r27cvmw8H2Zkz9rEwZIi3q8NezZ13AvXqSTq4H1csKPm++Uau8zly+LsRdffuMim+Zo1kIVFwDRwo1/mnn3ZvddjADbASpxL5LW0ksdy57Q3er7/O2Z6g+uUXYNYsucn0ewWm+vVl38Gff0q/IwqmiRNlz+E99wBPPGE6mvAaMED2U06bJit2FDxa2/c13brJIMuvsmWzM5B4XxNcv/8u5zylgDfeMB3NlQVqgHXsGLBlizTkbd3adDTh17EjkCuXlK/++mvT0ZAJr78uxVxatZKmk36WJo2dAjl8uKxkULCcPi15+YCs3LoxbcRJZcoAzz8v+w779TMdDZnw1VfA6tVyre/UyXQ04de2rRSu+ekn4OOPTUdDJgwaJFk5DRoAZcuajubKAjXAOnhQPr/6qreb7yVXtmzAyy/LY872BM/WrdKAL316OeaDoEYN4K67ZAVj8mTT0VCkTZggm57vvVeKQARB375S1GjOHGk/QsGhtVzbAamM7IfehteSMaNdwMaaQKTg2L8fePttmVC1jn23CtQA6/x5oHhxoHlz05FEzksvSbrgmjUy00XBYe3LaN1aNggHgVL2TP6wYcDZs0bDoQg6dUpWLgGpOOb31StLsWKyQq0192IFzWefSfXAPHmA9u1NRxM5rVpJJtKWLVJVkIJj4EBpRdCwoazgu1mgBliAzPZFR5uOInKyZuVerCD65RfpAxQdbTfiDYrHHgMqVOAqVtCMHw/8+y9w//3AI4+YjiayevWSVax589iINSgSr1717AlkyWI2nkjKkAF45RV53Lev9Mci//vtN2DGDG+sXgEBG2AVLAg895zpKCKvfXuZ4Vq3Tma8yP8GDpQLcMuWMtMXJFzFCp6TJ+1G2kFavbIUKQI0aya/84MHm46GIuHjj4ENG2Q/0osvmo4m8lq0AIoWlcnEhQtNR0OR8Oabst+0cWPgxhtNR3NtgRpgFSjg35K9V5M5s72KYd14k3/t3AnMnSsz2kHtg1azpqxi/f239EQifxs3Djh6FHjwQaBaNdPRmNGrFxAVBbz7LrB3r+loKJy0tqun9eoFZMpkNh4ToqPtioKDB/O+xu/27pUel1FRdtVMtwvUACvI2rYFcuaUvVjffWc6GgqnQYNk42+zZjLDF0RK2SkEw4axL5afnTkDjBolj4O4emUpUUIyNOLjgaFDTUdD4bRsmaxe5c0bjIrIV9KsmUycb9rE7By/GzZMzm2NGwMlS5qOJnk4wAqILFnsEq6DBpmNhcLn11+lmlhUlD27F1S1awPlywN//SU9M8ifpk2TvVf33gtUqWI6GrP69JEB5owZwIEDpqOhcLGu4Z07S1W9oMqQQXp/AczO8bNDh+ScppS3GmlzgBUgHTpIuuAXX0gPCfKfIUNkluf556ViZpApZZfzHTFCKg+Rv1y4YO+96t07uKtXltKlpeH2xYt2PzDyl7VrgRUrpA1Lu3amozGvTRtm5/jdmDFyrn/ySeCmm0xHk3wcYAVIrlz2ZlhuhPafgweBWbPkJjPoq1eWJ5+Um879+6XCGvnLnDnAH39Is8natU1H4w5WdbWpU2X1lvzFuna/9BKQPbvZWNwgcXaO1Wie/OP4cWDiRHnstT3lHGAFTNeusjl04UJgxw7T0ZCTxoyRmetnngFKlTIdjTukSWMXeBk6lCkkfpJ4r1GvXvJ/TcDNNwNPPQXExgJjx5qOhpy0dSuwZImkxlmDCpLsnCxZgC+/BNavNx0NOWnSJOlxWLUqcPfdpqNJGV6SAqZAASlvqjU3QvvJiRN2z6eg9b26lueeAwoVAn7+Gfj0U9PRkFMWL5aKmddfDzRoYDoad7HOARMnys0J+cOQIfK5VSspcEGC2Tn+dO4cMHq0PPba6hXAAVYgde9ul/Pdv990NOSEmBjpBfTQQ8Bdd5mOxl3Sp5eVW8C+QSFvS9zvqXt3aUlAtnvvBSpWlIkXFnjxh717pf1G2rTAyy+bjsZ9unSRc/2HHwLbt5uOhpwwYwbwzz9SrOrhh01Hk3IcYAVQiRIy4xsXZ88OkHfFxtr/j16qsBNJrVoBOXIAq1bJB3nb8uWSCpQnD9C8uelo3Kl7d/k8ahQLvPjBiBHSfqNx4+C237iaAgWkbDtgt20g74qLA4YPl8e9enmzgBEHWAFlzYBNmyabCMm75syRMqa33AJUr246GnfKmlU2hQNMjfUDq0Je587BbLKaHLVqScWtAwdY4MXrjh4F3nlHHnP16sq6dJEb8VmzpMk8ede8ecC+fcCNN8qeUi/iACugbr8dqFYNOH0amDLFdDSUWpcu2TebPXp4c5YnUjp2lJ4xS5fKfizypm3bgK++koGVte+C/leaNPbN+LBhLPDiZVOnAmfPAo8+CpQrZzoa9ypdGnjiCcnqmDDBdDSUWolrBPToIVtavIgDrACzLr5WjwHyniVLZKN/0aLS/4auLHE6GVNIvMtKh23WTPrf0JU995ykTm3dKhXWyHsuXgTGj5fHXbqYjcULrPuaCRNkUEres3y5nLMKFJCUWK/iACvAqleX2bCDB5lC4lXW6lXXrtzonxydO8sq37vvMoXEiw4fBmbPlscsU31t6dPb/05sPOxNCxdKr7ebbmIKeHI88ABwzz3/Taskb7EmQNu3l3OYV3GAFWBKAd26yeO33mIKidesXi3d63PmBFq2NB2NN5QqJSkkFy7YzQvJO2JiJP2nVi3Jzadra9NGegR9/TWwYYPpaCgltLZvNjt1Ygp4cihlr2KNHCn98sg7du6UdioZMsi5y8s4wAq4Ro2A/PmBLVtkWZa8w7rwtmsnN1CUPFbJ9okTpc8GeUPifRVMlUq+HDnsGxWrKhd5w9q1wA8/SJ+nJk1MR+MdTz0FFC8O7NkDfPyx6WgoJazm6E2aALlzm40lVBxgBVz69NIFHZBVLPKGffuAjz6StMB27UxH4y0VK0pfjSNHJFWQvGHePOCvv6RaZtWqpqPxls6dpX/S/Pnsfegl1iRa27aslpkSUVH2JAzva7wjcVqnH1LAUzTAUkrdq5Tqp5T6XCm1RSm1Wym1Vin1jlKquVKKW449yDp5f/GFbCwk9xs3TioI1q8PFCxoOhpvUcpexRo1Sv4dyd20totbWPvoKPkKFwaefVaOdatgArnb/v3SNDdtWtmLQinTvLmkz69ZIx/kftOm+ataZrIGWEqppkqprQDWAOgMIBOA3QDWATgG4B4A0wD8mTDYKh6meCkMcuUCWrSQxyNHmo2Fru3UKTkRAXKzSSlXrx5QqBCwfbtMLJC7rVwJbNwI5M0rac2Ucta5YupUac9B7sZJtNBkyWK3ceAqlvtdvCjHPOCf+5prDrCUUpsBDAHwKYDyAHJqrStprZ/RWjfWWtfUWt8EIBeAVgDyAtimlGLRaA/p3Fn6plhNa8m93nkHOHnSTnWjlEuXTvpiAZxU8AJr9erFF2XzM6XcXXdJhbUTJ4CZM01HQ1dz6pQMhAHuNwzFSy/JuX7RIuC330xHQ1fz4YdSLbNMGf9Uy0zOCtYMAMW11j211hu1vnytOa31Ca31HK11TQD3ATjuZKAUXjfcANSpI7MIkyebjoauJD5e+pYB/pnlMaV1ayBzZmDZMinyQu60Zw+weDEQHc3GwqGyzhljxjA11s1mzOAkmhMKFAAaNJAUY6bGups1idapk0z2+8E1/xpa69Fa6/MAoJT6QClVJBk/s1lrzcQbj7EuvpMmAefPm42FLu+TT+SGs1gxGRBT6uXIYZe3Z+Nh9xo7Vm6QGjUC8uUzHY23PfkkcP31wO7dUgqZ3Cc+3q6kxtWr0FnFEqZNk5VBcp/vv5ePnDn9VS0zpePEugAKXO4LSqlcSqn7Qw+JTKlYEbj9dmnmOXeu6WjocqxZno4dpVIShcbqLcPUWHc6cQJ4+215zBXb0KVNa1eNtc4l5C5Ll8okWvHi0rOPQlO+vNzbnDzJxsNuZZ2L2rSRrBK/SM4erNJKqXJKqWt9bykA3zkTFpmglH0TM3o0Gw+7zaZNwDffyOZdqygJhaZECemZcvGi3WOJ3GP6dCnIUKUKcNttpqPxh5Yt5SZm+XKmxrqRtZrOSTTnWKtYY8cyNdZtDhwAFizwZ7XM5KxgNQCwFcBpABpAH6VUR6XUg0qpxO1NswNgYpnHNWgglbo2bwa+/dZ0NJSYtfeqRQsge3azsfiJVbJ98mQ2HnaTuDimSoVDjhxSwhqwzynkDhs3ynU3a1ZOojmpTh1Jjf31V6bGus348ZIWW6+etJPwk+QMsEYCqAbgNQAKwM0AhgFYCeC4UmqnUuojAJMBcD7M49KntzeS8+LrHn//Dbz3nqwyWtXvyBn33y8V1v79l42H3WTxYukFVLIkUKuW6Wj8pWNHOzX2n39MR0MWK1XqhReAbNnMxuInadNKRUGAqbFucuYMMGWKPPZjCnhyilyc0lp/o7UeAWAHgPoAsgK4E0BrAF8CyAZgI4AXwhgrRUjbtlKxa/FiYO9e09EQIKsrFy5ITv4NN5iOxl+YGutOVqqUn6pKuUWpUkDt2kBsLKvGusWhQ8D778uxbu2TI+ckTo39+WfT0RAg7SKOHwfuuw+4+27T0TgvRZctrXVZrfUGrfVFrfUmrfXbWusOWutqCX2xtoUrUIqc/PlZ2tRNzp8HJk6Ux36c5XGDevWkmecvvwBffWU6GvrxR2D1akmFbdbMdDT+ZJ1LJhF/m8UAACAASURBVE6UgRaZNXGi7AV98kkpcEHOypnTPpcwO8e8S5fs/we/poB7cl5QKVVWKbVcKXVWKXVQKTVAKcXtoA5KXNr05EmzsQTd3LmSxnP77UDlyqaj8ad06ewUEpZsN89K42nVSoq6kPOqVAFuvVXSj+fNMx1NsJ07Z68k+vVm0w2slcF33wWOHDEbS9B9/jmwaxdQtKgUmvKj5FQRXKyUuiO5L6iUyqCU6qqUahtaaFd8/ZwAlkEKbtQBMABANwD9w/F+QXXnnUClStI3gqVNzdHavtns3FnS2Sg8WrcGMmaUE//27aajCa4//wQ++EAqqDFVKnwSp8aOGsXUWJOsG/4KFYAHHjAdjX+VLg3UrClZIdbeHzLDmsjs0EH2yPlRclawfgfwvVJqXUL1wDuVUv/551BKFVRKPamUmg7gEIAWAH4KQ7wA0BZARgBPa62/0lpPhgyuuiqluC3UQSxtat6KFVLRMV8+Sduk8LnuOuD55+UxU0jMmTBBKgg+84zMblL4NGwoVWM3bWLVWFMST6J16cJJtHCzJhUmTJCUTIq8n38Gli2TPXEv+LhyQ3KKXHQAUBbADwD6AfgRwHml1FGl1CGl1HkABwB8CKAcgM4AbtVa/xCmmB8D8IXWOnHi2lzIoIsJVA6qUwcoVkyaHn7yielogsm68LZrJxUeKbysSYVZs6SqIEXW2bNATIw85n7D8MuQgVVjTfvqK9n7WbAgULeu6Wj87+GHgbJlgYMHpf8SRZ51X9O8ubSN8Ktk7cHSWu9JGGjlB1AVQB8AswAsBjACQDMAxbXW92qtZ2qtw7neUQZSzTBxfL8DOJvwNXJIVJS9L4UX38j79VdgyRKp6Ng2LAm3lNRNNwE1asieCKaQRN6sWcDRo8A990hlKQo/Vo01y0qVeukl+X+g8FLKnkhjamzkHT5st0Pxe8uZlFYRvKC1/lZrPUxr3Vlr3VZr/YrWerbWen+4gkwiJ4Djl3n+WMLXyEGJS5tu3Wo6mmAZN05O/s89J2k8FBnWJvPx45lCEklBqCrlRvnzS6qg1nLOocjZvl32fGbMCLRpYzqa4GjcGMiVS6qVfv+96WiCJSZGqpbWri3tIvzMk1UEIQUuklKXe14p1VoptV4ptf7w4cPhj8xncuSQZVyAq1iRdOIE8Pbb8pipUpH1yCN2Csn8+aajCY4vvgB27AAKFwaeftp0NMFizehPn86qsZFkXVObNpUbfoqMTJmkqBHA+5pIio2VvW9AMCbRvDjAOgbgclmb2XGZlS2t9RStdQWtdYU8efKEPTg/SlzalGPUyJg+HTh9GqhaVUopU+SwupoZiatKpUtnNpagueMOaQFx6pQ9sUPh9e+/khIL2ANcipz27WUbxIIFwIEDpqMJhg8+AP76S+5pqlQxHU34eXGAtQNJ9loppYoAyIwke7PIGTfeCNSqJbMP3JcSfnFxUrkR4OqVKY0bS1XB9euBNWtMR+N/27bJZv9MmaT3FUWeda4ZOxaIjzcbSxDExMhez8ceA8pw93jEFS4sRUXi4+1VFQofre1JtKC0nPHiAOszANWVUlkTPVcfwDkALDQbJtYM28SJwIULZmPxu48/BvbvB264QQa2FHkZM9qFRayKRxQ+1r9xs2ZATu6kNeLxx4ESJYDffpPiOhQ+Fy7IHk8gGKlSbmVNKkyZIhVMKXy++w7YuBHIk0f2fAZBWAZYSqlwbsmfDCAWwIdKqYeVUq0h5eNHJindTg5KXNqU+1LCy7rZ7NQJSOPFKRCfaNdOUtU+/BDYt890NP51+DAwe7Y8ZqqUOVFRdlUvTiqE1wcfAIcOAeXKybWVzLj3XqlYeuyYna5J4WGdU158UdpDBEG4bt/Wh+l1obU+BqAagCgASyBNhkcB6Buu96T/7ksZPZr7UsJlwwaZ6cme3S4uQmYULAjUry/V7VhdLXysqlK1akk6MpnTvDmQNas0Hd640XQ0/hTEVCk3S3xfcymcDYYCbO9eYNEiaUNg9d0LglQPsJRST1zpA0BYx6da61+01lW11hm11gW01q9prZk1HmaJ96WsXm06Gn+yZnleeAHIksVsLGRffKdNkwIA5KygVZVyu2zZpDUHwOpq4bJqFfDTT0Du3NKCg8x65hnZj7Vzp5TMJ+dZLWcaNpS2EEERygrWRwA6A+hymY+sV/k58qjE+1KsGThyzsGDwLx5khZoNXgms8qXBypWlNLVM2aYjsZ/rKpSt9wiFTPJvA4dZFXl/ffl/4acZV07X3xRrqlkVrp0dqVk3tc47+RJqYoMBK9oVygDrN0AWmqtqyT9AHDEofjIZdq3lxPSokWyGZqcM2mSNLZ96imgWDHT0ZCF1dXCg6lS7lSiBFCnjhRimDTJdDT+YqVKpUsnezzJHVq1AjJnBpYtA7ZuNR2Nv0yfLtkfDz0E3H676WgiK5QB1kwAua/wtckhvC65WIECssx76ZJdSpxCd+4cMDnht4apUu5Spw5QvDiwZw+wdKnpaPzD2ueTNy/QqJHpaCgxa1Jh0iTg/HmzsfjJ2LHBTJVyu5w57T3PXMVyTlycnWrctavZWExI0QBLKdXAeqy1Hqy1/vFy36e1HhhqYORe1gBg+nTgxAmzsfjFnDnAkSNAhQrA/febjoYSY3W18Bg5Uj63bx+cqlJeUamSNB8+fFhSBSl0x4/bqVJBvNl0u06dZBV9zhzg779NR+MPixZJy5lSpYLZcialK1izlFJfK6XKhiUa8oTbb5fl3lOn7AsGpZ7W9o07U6XcqUULqa62YgWwaZPpaLxv1y7ptZQ+fbCqSnkFq8Y6b9o04PRpoFo14LbbTEdDSZUsCTzxBFNjnWRNonXuHMyWMyn9K5cHkA7ARqXUW0op1jkLKGsVa+xYWQam1Fu+HNi2TdIv69UzHQ1dTuLqalzFCp31b9ikiTSeJPepXx/Ilw/YskUmFij1Ll60U+q5euVe1n3NxIlMjQ3V2rXykTMn0LSp6WjMSNEAS2u9VWtdEUBrAI0B7FRKBaQnMyVWu7bM+OzfL8vAlHrWzWb79tIngtyJ1dWccfQo8M478jhoVaW8JH16uxADJxVCs3AhcOAAUKYMUKOG6WjoShKnxs6ZYzoab7P2srVpIwVEgihVi3Za65kASgNYBGC2UuobpVQ5RyMjV0uTRnKWAW4KDcWuXcAnn8gelDZtTEdDV1OiBPDkk0whCVVMjBR1qVEDKMerhqu1bSuTPkuWAL/+ajoab9IaGDFCHnfpEsxUKa9Qyl5hHDWKqbGptW+fTCqkTRvsljOp/lXXWp/QWrcHcBekmuBGpdQIpRR7YAVEs2ZAjhzAmjXADz+YjsabrAo7TZpI40lyN1ZXC82FC9J0EmCqlBfkzSvNcLVm1djUWrUKWL9ezu9NmpiOhq7l2WclXX/bNuCrr0xH401jx0ql6QYNgEKFTEdjTooHWEqpdEqpu5VSHZVS7wFYCKAcgLQA2gPYoZR6wuE4yYWyZJH+EQBXsVIjcaqUtRpI7laxInDnnZJC8t57pqPxnnnzgEOHgJtvBh5+2HQ0lBzWpMLbb0slPEoZa6N/u3ZsLOwF0dH2qgvva1Lu5Ekp6AKw5UxKy7SvAXASwFoAIwDcCGAJgPoACgPIC2AugAVKqbbOhkpu1KGDlLGeP19yzCn5Jk4Ezp4FqldnqpRXsLpa6mlt32x26cJqmV5x661A1arAmTOsGptSv/4KLF4sN+1sLOwdbdrIYPjzz4Ht201H4y1WY+HKlWUyMshSuoJ1GsBgAI8CyKG1rqC17qS1nq+1Pqi1Pqm17gbgVQB9nA6W3KdIEal8Fx8PjB9vOhrvOH/eTpXq3t1sLJQy9etLk9CtW4GvvzYdjXdYJe7ZWNh7rJno0aOlIh4lz5gxMrHQuLFUZCRvuO46u/IdC7wkX9AbCyeV0iqCj2qtB2itl2utz1zlW1dCVrQoAKyLb0yMLA/Ttc2aBfzzj8zwVK1qOhpKiehoqfgIMIUkJdhY2Ltq1gTKlgX++IONh5Pr2DFJqwSYKuVFVqbCrFnAkSNmY/GKjz6SytIlS0ql6aALVz2bzQDqhOm1yWXuvlvKm544AUyZYjoa94uPB956Sx53785UKS9q00YGCZ98IpUg6ep27gSWLmVjYa9Kk8ZeaR82jKmxyTFliqSAP/qo7DkkbyldWiYWzp+XdH66Oq3t+5qgNhZOKiz/BFrrc1rrJeF4bXKnnj3l86hRQGys2Vjc7uOPgd27gWLFgLp1TUdDqZEnj6T9AEwhSY7hw+Vz06ZsLOxVjRoBBQtKdbXPPjMdjbvFxrKxsB9Ykwpjx8oeRLqyb7+VatLXXScVpil8K1gUMI89BtxyC3DwIBv0XYt1s9mli/SJIG+yUkhmzJB0T7q8P/+UNBulgJdfNh0NpVZ0tJ3qNmyY2VjcbvZsuRbeequsYJE3Va4M3HMP8O+/dronXd6QIfK5Q4fgNhZOigMscoRSQI8e8njYMOmBQP9r9Wpg7VogVy6gZUvT0VAoypUDHn9cUkjYI+jKrMIIdesCpUqZjoZC0bo1kC2bzFavW2c6GneKj7cn0Xr2ZAq4lyllZ+e89RYLvFzJpk3AF18AmTIFu7FwUhxgkWPq1weKFpX9FosXm47GnayZ33btOMvjB716yecJE1jg5XKOHQMmT5bH1o0KeVe2bPYeOmsQQf+1aJHsyyxWTJrWkrfVqSP7sX7/HZg713Q07mTd17RqJSmCJDjAIsekSwd06yaPhw7lRuikduyQ/Vfp03OWxy/uv1+aDx8/LlU06b8mTQJOn5amwuXLm46GnNCxo6QLfvih7CUlm9Z2qlT37kwB94M0aezJoaFDmZ2T1N690kA+bVruN0yKAyxyVMuWMoOxbh2wcqXpaNzFqrDTtCl7ovhJ797yeeRISRckce6c3ROFq1f+UbAg0KSJDCZGjDAdjbt8/TWwfr0Ucmne3HQ05JTnngMKFZICL598YjoadxkxQgadDRtKBhPZOMAiR2XObK/ODB1qNhY3+f13YOZMmQ3jRn9/qVEDuO024K+/ZHM7iXfekeIf5csD1aqZjoacZLWXeOcdOe5JWKtXnTsDGTOajYWcEx1tZ+cMHszsHMs//9jFP6w9+GTjAIsc99JLstnxs8+ALVtMR+MOQ4dKl/MGDbjR32+UsvdiDRsmm9yDLi7OXrHlRn//KV1a9qbExtoNpINu/Xpg2TIga1bZY0v+0qoVkDOnFKlatcp0NO4wdqxkbdSuzV5vl8MBFjkud27ghRfk8eDBZmNxg4MHgenT5XGfPmZjofCoWxcoUQL49Vdg4ULT0Zj3/vuSm1+yJPD006ajoXB49VX5PGECcPiw2VjcwFq9atsWyJHDbCzkvCxZpAQ5YP9fB9mxY8C4cfLYmmCk/+IAi8Li5Zel6MW8ecD27aajMeutt2Sm95lnpLQ3+U/atHZTyjffDPZG6Lg4+TcAgFdeAaKizMZD4VG+PFCrFnD2rDSYD7KtW2ViJX16uz8e+U+HDpL6+emnwMaNpqMxa8wYqZxbrRrwwAOmo3EnDrAoLIoUkVUsrYE33jAdjTmHD9tlql95xWwsFF7Nm8tG6K1bgY8+Mh2NOXPnSpnqEiWAxo1NR0Ph9Npr8nncOODoUbOxmDRggHxu00aKgJA/5c4tK5QA0L+/2VhMOn5c+hsCQN++ZmNxMw6wKGx69ZJVrLlzpUR5EI0cKdXUatcG7rjDdDQUTunT2ymg/foFcxUrPt5evXr1VZap9rt77gGqV5dS/NYNV9Bs3QosWCC//6yW6X89e8oq1uLFwE8/mY7GjLFjgRMngIcekjYldHkcYFHYFC0KtGgR3FWso0eB8ePlsbVfgfytZUugcGHg55+lT1DQzJsnjcaLF+fqVVC8/rp8HjNGZraDhqtXwZIvn13EpF8/o6EYcfKknRLM1aur4wCLwqp3b3sVa+dO09FE1ujRMrP7yCMy00v+l3gVq3//YK1ixcfbEymvvCK/9+R/998v+zBOnpSZ7SDh6lUwde8uq1hLlkj1yCAZN04mUipVkhUsujIOsCisrr9e9qZcumSnDgXBkSP2LI+1T4GCoUUL2YP488/Bqig4f76kAhcrBjz/vOloKJKsVaxRoyR1KCis1avWrbl6FST58gHt28vjIO3FOnXKbsvA1atr4wCLwq53b9mL8d57wVnFGjpUVq9q1GCOctAEcRUr8epVnz5cvQqaSpWAypVlZtvqf+Z3XL0Ktu7dpd/n0qXAjz+ajiYyrGI2Dz4IVKliOhr34wCLwq5YMXsVKwizHgcP2nuvgrRqRzZrFWvbNlnZ8bt33wV++UVWrJs2NR0NmTBokHweNQr4+2+zsUSCdS1r3Vqqh1Kw5M0LvPSSPA7CXqyjR4Fhw+Rxv35sHp8cHGBRRLz2msz0zZsHbNhgOprwevNN6W7+zDPSK4aCJzraLmzy2mvAxYtm4wmn2Fg7RWzAAPm7U/Dcf79USz1zxh5s+dX330srhowZ2WQ1yLp3BzJnlr5Yq1aZjia8hgyR9N9HHpE9l3RtHGBRRBQpYndB9/MF6bffgKlTZXbHys+nYGreHChVCti9G5g2zXQ04TNpEvD778DNNwPPPWc6GjJp4EA5902eDOzfbzqa8NAa6NFDHnfpwr1XQZY7N9C1qzzu0UOODT/64w9JDwT8P3niJA6wKGJ69wayZweWLZMPP+rbF4iLkxLVZcuajoZMSpfOvhj17y978vzm5Em5qQaAwYOBqCiz8ZBZt94KNGwIXLjg37SpTz8FvvsOyJXLHmhRcHXvDuTJA6xdCyxaZDqa8BgwQLJy6tUDKlQwHY13cIBFEZMrl7161auX/zb/r18PzJ4tKVJ+vbmglHnmGSnR//ffwIgRpqNx3ltvScXMBx8EatUyHQ25wYABUtRo1izZg+gn8fH2NeyVV2TCkIIta1Z7P16vXv5LB9+5E3j7bZk8457ylOEAiyKqY0dJqdiwAfjgA9PROEdroFs3edyxI1CihNl4yB2UkoqSgAxG/LT5/9Ahu2TvkCHc9Ezihhuk8MOlS3JO9FPa1Jw50n6haFG72SxR69ZAyZLArl3A9Ommo3FWr14ysdCyJXDjjaaj8RYOsCiiMmWyV3d69gTOnjUajmMWLQJWrgSuu05mNokslSvL6s7p0/7qmdK7txQ0eOIJ4IEHTEdDbtKvn6zufPGFpNT5wZkz9rn9jTeADBnMxkPukTgdvF8/6RflB8uXy71N5szBqADtNA6wKOJatABuu002xg8fbjqa0F24IHnYgNxA58hhNh5yn6FDJcUiJgbYvNl0NKH74Qdg5kxJh/Vj6iOFJk8e+4asa1d/pE0NHiyb/cuXZzEX+l9169rp4Na+VC+LiwM6d5bHr7zCYi6pwQEWRVxUFDB2rDweMsT71aYmTAD27AHKlJFUAaKkypUD2reXtKkOHbydNnXpkqTBAlJFrWRJs/GQO7VvLylFu3bJOdLL9uyxJwPHjWMxF/pfStn3NSNHyt4lL4uJkXTYEiXkPE8pxwEWGVGpElC/vlSmsVZ/vOjQITvlcfhwSRUgupz+/WVm/7vvgPffNx1N6r37LrBuHVCgANNh6coSr2727y/FULyqa1fJVHj+eeC++0xHQ251992yV+niRZmE8upE2r//Sv9GQH6HmQ6bOhxgkTHDh0ujxvnzgRUrTEeTOl26SKnq2rVZRY2uLkcOWbEFgJdf9mae/qlTdhW1IUOkghbRldSqBTz6KHD8uOy59aLPPwc+/hjIksX+/SW6ksGD5Vz/5ZfA4sWmo0md118Hjh2ThsJ16piOxrs4wCJjihSRjfKAVGSKjTUbT0p98QUwb54MEseNYxU1urZmzWSW89Ah2SjvNa++KrHffbf0eiO6GittKjpaSj1/843piFLm/HmgUyd5/PrrsmpLdDV58tjn9s6dgXPnzMaTUuvWSfP4qChg9Gje14SCAywyqnt3ydPfvt1bG0PPnbPL9PbrBxQrZjIa8oo0aYDx4+WiNXIksGmT6YiS74cf7P0nMTHydyG6ltKlZWAOAG3ayKDFK958U/aQ3XSTPdAiupa2baXp9v793qoce/Ei0KqVpDa+/DJw882mI/I2z10ilVL1lVIfKqUOKaW0UqqZ6Zgo9TJkAKZNk8eDBwNbtpiNJ7kGDQL27pUTEDeAUkrcdZcUuoiPl4qacXGmI7q2ixeBF16w+73dfrvpiMhLevYEypYFdu/2zkTali1S/VMpuUZFR5uOiLwibVqZhFJK+h9u2GA6ouQZMQLYulUKW7z+uulovM9zAywAdQEUA7DUcBzkkIoVZTUoLk42iLr9hnPjRjsXf/JkFraglBs4ELj+ejmWvFDmfNgw+8LLfiiUUtHRwJQp8njIEKlO5mbx8TKhEBcn16b77zcdEXnNvfdKiqA1kXbhgumIrm7nTnu1bfJk6VlKofHiAKu+1vpOAFw38JHBg2VP1vr1kvfrVufPSyWpuDgpQ8wGq5QaWbLYN5x9+wK//GI2nqvZuNGulBkTwwsvpc4DD0jqVFwc0LSpu284hw4FfvwRKFxYrk1EqfHGGzIptWWLu4+jixeBJk3k/qZpU+CRR0xH5A+eG2BprS+ZjoGcly2bzJoAkq/v1lTB11+X2dcbb5RZfaLUevRRmdmMjQUaNXJnkZdz56SYRVwc8NJLwMMPm46IvGzoUNmv+tNP7t2bsn69vUo7fTorZVLqZc5sb4F44w1g7Vqz8VzJoEEyoVC0KDBmjOlo/MNzAyzyr5o1pVFvbCzQoAFw9qzpiP7rm28knzpNGmDWLM7kU+hGjwZuuAHYvNmdPaV695bVtdKl5eaYKBTZsgGzZ8s5dPBg6QnnJmfOAM89JxMKnTrJJAhRKKpUkYIR8fFybJ08aTqi//rhB7vq4TvvANmzGw3HV3w/wFJKtVZKrVdKrT98+LDpcOgaRo2Sik3bt0tzR7c4dAho2FA2+b/yCnDPPaYjIj/ImhWYM0cq840YIaX/3eKjj2Q2MypKmgtzQoGc8OCDMnDXWm443dKAWGspPrNrF1CuHHtekXMGDgTuuAP47TfZ0+eWBsT//gs8+6wM/rp0kcEgOcf4AEsplV0pVeZaH6l9fa31FK11Ba11hTx58jgZOoVBpkzA++/LpuiYGJntNC0uTgZXf/8tJyBu8icn3XOPnS7VqJFchE3bvVt6dgGSCluhgtFwyGf69pUiAAcOyLk1Pt50RMDUqcCMGdLX8L33pMItkROio+W+JlMmmVCbNMl0RMClS7Lvav9+6Wvo5j1iXmV8gAWgHoDtyfiggLjtNmlOCUhPhvXrzcbTpw/w7bdA/vxy4Y2KMhsP+U/v3pIie/Qo8NRTZtNjz54F6taVVJZnnmEbAnJeunTA/PlA3rzAsmV2nyxT1q2TPYaAFJ+59Vaz8ZD/lC5t78fq1AlYudJsPAMHAp99BuTKBXzwAZA+vdl4/Mj4AEtrPU1rra71YTpOiqw2beQjNhZ48klJ0TNh6lRg+HDpa/H++zLIInJamjQys1mypOzHatFCZhgjLT5eVhS2bJFCLm+/Lb1ciJxWuDAwb55MWA0ZImmoJvz+O/D001JJ7aWXpKgLUTg0bCh9BOPiZBJr/34zcbz/vhTsUkquO9dfbyYOvzM+wCK6krFjJV//zz9ldv/Eici+/5dfAi++KI8nTQIeeiiy70/BkiMHsGiRlHCfN08uxJHM1ddaZlY//hjImRNYvFiKEhCFy0MPASNHyuPmzYGvvors+x89CtSoARw8CFSq5I2edORtQ4ZINdbDh6WISqRLA6xcaad/v/WWHP8UHp4bYCmlyiql6gJ4MuGpCkqpukqpyibjIudFRwMLFwKlSgGbNgGPPx651Kk1ayQ9Kj4e6NVLmk4ShVu5clJcIl06qTAYyVYAgwYBEyZIqsjHHwNlUr3zlSj5Ona0Z/WfflpKuEfC2bPAE09IQaVy5WRyIzo6Mu9NwZU2raTH3nKLFFSpWRM4dSoy771pk2QEXbggq7VM/w4vzw2wADwLYD4Aq/xB+4Q/u7SrBoUib16Z1SxUSEr6Pv10+AdZa9fKrM7p01LlauDA8L4fUWIPPyzFXZSSwf1bb4X/PQcMkH0wSkkLggcfDP97ElmGDZPWHKdPA9WqSenocDp1CnjsMWD1aklV/PxzWbUlioQcOaRibPHisse8Vq3wZ+j89BNQtSpw7JgMskaPZvp3uHlugKW17neFfVoPmY6NwuP662WQlTu3nJQeeURSO8Jh+XKgenW5ADdsKH0h0njut4S8rn59u9JU9+4y+AlHuqDVdqBvX7u/27PPOv8+RFeTJg0wc6YUeDl+XCYZwtUj69gxuYasXAkULCjXlsKFw/NeRFdSoIBsQyhYUI71qlXDly64dq1MXBw7Jqu2c+eyWFck8NaRPOGmm+QkVLSopO9VqiSlpJ0UE2MPrho0kJvNtGmdfQ+i5GrTRlayoqJkFbVZM2dXb8+ckYHcoEF2rytu8CdToqNl72HDhnIOfvhhYPp0Z9/jl1+kPPy6dUCxYnJNYSosmVKypKyiliwpK0wPPAD8/LOz7zFzpux1PH5cVq7mz2fFwEjhAIs8o0wZORmVLQts2waULy8X5FCdOSM3s23byp6rnj2lsg4HV2Ra48bAggXSm2fWLOC++5yZWNi5Uy7m8+dLs+NFi+TGlsikdOlkUuGll2SfyAsvyHn53LnQX/uDD6Tfz65dUoZ95UqgRInQX5coFMWKAatWAbffLuf2u++Wc32ozp2TokXNmsnvUrt28jvAfYaRwwEWeUrhwrKCVa+evdLUqFHqy7h/8YWc2KZMkRPPjBlS5YdpgeQWTz4pM+6lSkn59Ntuk31ZFy+m/LViY4GhQ+WY37xZZk7XrQNq13Y+bqLUiIoCxo2Tc3H69JJZcOutkr6dGn/+KWmv9evLZFrDhnINKVLE2biJUitf4becIQAACvFJREFUPpk8btpUBkZNm8p+8wMHUvd6K1bIBPTYsTJRPGmSFDBKl87RsOkaeBtJnpM9u6xcTZwIZMggPR1KlpS9Ksk5IV26JBfrRx+VYha//grcfDPw4492+VIiN7nlFjk+GzWSC3D37nbjyjNnrv3zZ84AkydLqm2vXsD583IR//FHeY7IbZo1k4FQuXJyjn74YTlff/118vYj7tsHvPyyXBvmzwcyZ5abzDlz5DGRm2TKJJMK06dLq46PPpJehJ07A3v3XvvntZZV2dq1gSpVpDpmmTLyO9S2bfjjp/+ldCQbrRhWoUIFvX79etNhkIP27ZNSo4sWyZ+VAipXlk3Mt98us5TR0ZJ/vHevbPZcvFiaSwKSHvXaa7KUzqVz8oLPPpNjfudO+XPWrFIRrUoVuRm97jr5PTh8WPL5v/0W+OQTeyBWrpz0+6le3dzfgSi5YmPleB0yxC5nXby4bNa/5x4ZQGXPLhMPf/4JbNwomQmrVtkDsWeekVXfYsWM/TWIku3PP6V1QeItEA8+KOfs22+XvegZMkjlwb17JQth8WJ7IJY5s0ykdesm6eWUOkqpDVrrCqn+eQ6wyA82bACGD5eBVmzstb+/cGHZd9WuHZArV/jjI3JSXJxUgpowAfj+++T9zL33ykRC3brcX0jec+SIZC3ExEhj4GuJjpbUwM6dJV2KyGs2b5bJhQULkrcPMX9+oHVroH17aXFDoeEAKwU4wPK/Y8ek7O7KlTLD/+efcjOaJYvM+lSoIDP9993HfVbkD7/+Kn18fvhBNvCfOCHFWnLnln1bd90lfVaKFzcdKVHo4uNlxv7LL6Vx6u+/AydPykx9njyS7l2xosz2Z8tmOlqi0J04IdsaVqwAduyQ+5rYWFm5LVRIJhAeekhWuVh+3TkcYKUAB1hERERERHQ1oQ6wOIdPRERERETkEA6wiIiIiIiIHMIBFhERERERkUM4wCIiIiIiInIIB1hEREREREQOCVQVQaXUKQA7TcdBgZEbwBHTQVCg8JijSOLxRpHE440iqbTWOmtqfzho7SZ3hlJykSgllFLrebxRJPGYo0ji8UaRxOONIkkpFVJfJ6YIEhEREREROYQDLCIiIiIiIocEbYA1xXQAFCg83ijSeMxRJPF4o0ji8UaRFNLxFqgiF0REREREROEUtBUsIiIiIiKisOEAi4iIiIiIyCGBGGAppcoqpZYrpc4qpQ4qpQYopaJMx0Xep5QqqZSKUUptVkrFK6VWXOZ7lFKqj1LqgFLqnFJqpVLqdgPhkocppeoppT5WSv2plDqtlNqglGp4me9rpZTarZQ6n/A91UzES96mlKqrlFqjlPo34VjaqZR6VSkVneh7eG6jsFBKFUo4z2mlVJZEz/OYI0copZolHF9JP9om+p5UH2++H2AppXICWAZAA6gDYACAbgD6m4yLfKMcgJoAdiV8XE4vAK8BGArgcQCnASxTSuWPSITkF10hx04XAE8A+AbAe0qpDtY3KKUaAJgMYBaAxwBsA7BUKXVz5MMlj7sOcoy9ADmW3gbwCoCRib6H5zYKl+GQ4ykpHnPktKoA7kv08WGir6X6ePN9kQulVG8APQBcr7U+mfBcDwD9AOS3niNKDaVUGq31pYTHCwDk1lo/lOjrGQD8DWCE1npAwnOZAewDEKO1fjXiQZMnKaVya62PJHnuPQD3aa2LJ/x5J4DVWusWCX9OA2AzgM1a68aRjpn8RSk1EEB7ADkBpAfPbRQGSqmKABYDGAQZaGXVWp/m9ZScpJRqBmAGEo6vy3w9pOPN9ytYkJm3L5IMpOYCyAigspmQyC+swdVV3A8gG4APEv3MGQBLIMcmUbIkHVwl2AggLwAopUoAuBH/PdYuAZgPHmvkjH8BWCmCPLeR4xK2b4yDZBslPefxmKNICul4C8IAqwyAHYmf0Fr/DuBswteIwqkMgHgAu5M8vx08/ih09wP4JeGxdTztSPI92wHkUkrliVhU5BtKqSilVCal1IMAOgKYpCX1hec2Coe2ADIAmHCZr/GYo3DYo5SKS9hn2ibR8yEdb2kdDNCtcgI4fpnnjyV8jSiccgI4rbWOT/L8MQCZlFLRWusLBuIij0soXlEHQIuEp6zzWdLz3bFEXz8cgdDIX85A0gEB2dvXPeExz23kKKXUdQDeANBYa31RKZX0W3jMkZMOQfZX/QAgCkBDAJOVUpm01qMQ4vEWhAEWIAUuklJXeJ7IaVc6/q70NaKrUkoVA/AegMVa63eSfDnpMcVjjUJxP4BMAO4G8DqA8QDaJXyN5zZy0kAA67TWn17le3jMkSO01l8A+CLRU58ppdIDeFUpNcb6tsv8aLKOtyAMsI4ByHGZ57Pj8itbRE46BiCrUioqySxIDgBntdYXDcVFHqWUygXgMwC/A0hcuMJaqcoB4ESi563zH893lGJa658SHq5SSh0BMFMpNQI8t5GDlFLlIKvxlZRS1jkrU8Ln7EqpePCYo/BbAOBZAMUQ4vEWhD1YO5AkV1IpVQRAZvzvXgUip+2ALD2XTPL8/+wNJLoWpVQmAEshhQZqJWy4tVjHU9Lc8DIAjmqtmR5IobIGW8XBcxs5qxSAdADWQm5sj8Heh/UHpPAFjzmKFI0Qj7cgDLA+A1BdKZU10XP1AZwD8K2ZkChA1gA4CaCe9UTCTfLjkGOTKFmUUmkhFQFLAXhMa/1P4q9rrfdCerElPtbSJPyZxxo54YGEz7+B5zZy1ioAVZJ8DE34Wk1IuXYecxRuz0CqV+5HiMdbEFIEJ0MqH32olBoKoASkB9ZI9sCiUCX8stVM+GMhANmUUnUT/vyp1vqsUmoIgNeUUscgsx5dIZMb4yIeMHnZRMix1glSFfDeRF/bqLWOhZzb3lVK7QOwGkBTyICsUWRDJa9TSn0OYBmkWXU8ZHDVDcA8rfWehO/huY0ckdCGYkXi5xL2mgLAd1afIh5z5BSl1EJIgYstkJWq+gkfHRNanJwP5Xjz/QBLa30sodrWeEjt+uMARkFuRIhClReyqpCY9efikIZ0QyC/kL0BXAdgPYBHtNZ/RyhG8odHEz6PuczXigPYp7V+XymVBUBPSHWkbQBqa61/jlCM5B8/AmgG2YsQB2Av5Bw2OdH38NxGkcZjjpyyE7LvrwikcMUvAJ7XWs9O9D2pPt6UtLMgIiIiIiKiUAVhDxYREREREVFEcIBFRERERETkEA6wiIiIiIiIHMIBFhERERERkUM4wCIiIiIiInIIB1hEREREREQO4QCLiIiIiIjIIRxgERFRICmlsiml+imlbjIdCxER+QcHWEREFFQVAPQFkM50IERE5B8cYBERUVDdASAWwC+mAyEiIv9QWmvTMRAREUWUUmo7gDJJnl6ota5rIh4iIvIPDrCIiChwlFJ3AZgLYBuAQQlPH9Ja7zcXFRER+UFa0wEQEREZsBlAYQDjtNbfmw6GiIj8g3uwiIgoiMoBiAbwk+lAiIjIXzjAIiKiILoTgAawyXQgRETkLxxgERFREN0BYI/W+qTpQIiIyF84wCIioiAqC5ZnJyKiMGCRCyIiCqLjAO5USlUHcALAbq31v4ZjIiIiH2CZdiIiChyl1M0ApgO4FUAGABW11qvMRkVERH7AARYREREREZFDuAeLiIiIiIjIIRxgEREREREROYQDLCIiIiIiIodwgEVEREREROQQDrCIiIiIiIgcwgEWERERERGRQzjAIiIiIiIicggHWERERERERA75P/0rGJ7s5KjuAAAAAElFTkSuQmCC",
+      "text/plain": [
+       "<Figure size 864x216 with 1 Axes>"
+      ]
+     },
+     "metadata": {
+      "needs_background": "light"
+     },
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "image/png": 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",
+      "text/plain": [
+       "<Figure size 864x216 with 1 Axes>"
+      ]
+     },
+     "metadata": {
+      "needs_background": "light"
+     },
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "import numpy as np\n",
+    "import pandas as pd\n",
+    "import matplotlib.pyplot as plt\n",
+    "\n",
+    "#xt=np.linspace(0,15*np.pi,num=500)\n",
+    "xt=np.linspace(0,50,num=1000)\n",
+    "##\n",
+    "fig, ax = plt.subplots(figsize=(12, 3))\n",
+    "\n",
+    "y1=0.62*np.cos(0.52*xt) + 1.09*np.sin(0.52*xt)\n",
+    "y2=3.75 + 4.33*np.cos(0.26*xt) + 2.5*np.sin(0.26*xt)\n",
+    "\n",
+    "ax.plot(xt, y1+y2, color='g', lw=2)\n",
+    "ax.tick_params(axis='both', labelsize= 15)\n",
+    "ax.set_xlabel('$t$', fontsize = 16)\n",
+    "ax.set_ylabel('$f\\,(t)$', fontsize = 16)\n",
+    "\n",
+    "ax.set_xlim(0,50)\n",
+    "fig.tight_layout()\n",
+    "fig.savefig('comp_signal.png')\n",
+    "\n",
+    "fig, ax = plt.subplots(figsize=(12, 3))\n",
+    "\n",
+    "ax.plot(xt, y1, color='b', lw=2)\n",
+    "ax.tick_params(axis='both', labelsize= 15)\n",
+    "ax.set_xlabel('$t$', fontsize = 16)\n",
+    "ax.set_ylabel('$y_1\\,(t)$', fontsize = 16)\n",
+    "\n",
+    "ax.set_xlim(0,50)\n",
+    "fig.tight_layout()\n",
+    "fig.savefig('y1_signal.png')\n",
+    "\n",
+    "fig, ax = plt.subplots(figsize=(12, 3))\n",
+    "\n",
+    "ax.plot(xt, y2, color='y', lw=2)\n",
+    "ax.tick_params(axis='both', labelsize= 15)\n",
+    "ax.set_xlabel('$t$', fontsize = 16)\n",
+    "ax.set_ylabel('$y_2\\,(t)$', fontsize = 16)\n",
+    "\n",
+    "ax.set_xlim(0,50)\n",
+    "fig.tight_layout()\n",
+    "fig.savefig('y2_signal.png')"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 3,
+   "metadata": {
+    "tags": [
+     "remove-cell"
+    ]
+   },
+   "outputs": [
+    {
+     "data": {
+      "application/vnd.jupyter.widget-view+json": {
+       "model_id": "a785b675781e44dcb130fe4aba0b9568",
+       "version_major": 2,
+       "version_minor": 0
+      },
+      "text/plain": [
+       "interactive(children=(FloatSlider(value=0.0, description='c', max=2.0, min=-2.0), Output()), _dom_classes=('wi…"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "application/vnd.jupyter.widget-view+json": {
+       "model_id": "b9af681dfe384b9fbd3b14793a2feede",
+       "version_major": 2,
+       "version_minor": 0
+      },
+      "text/plain": [
+       "interactive(children=(FloatSlider(value=0.0, description='c', max=2.0, min=-2.0), Output()), _dom_classes=('wi…"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "text/plain": [
+       "<function __main__.vtrans(c)>"
+      ]
+     },
+     "execution_count": 3,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "from ipywidgets import interact\n",
+    "import matplotlib.pyplot as plt\n",
+    "import numpy as np\n",
+    "\n",
+    "def func(x):\n",
+    "    return x**3\n",
+    "\n",
+    "def htrans(c):\n",
+    "    x = np.linspace(-3,3)\n",
+    "    \n",
+    "    y = func(x+c) #constant c in the argument of the function\n",
+    "    \n",
+    "    plt.plot(x,y, label='$f\\, (x+c)$')\n",
+    "    plt.title('Horizontal translation')\n",
+    "        \n",
+    "    plt.xlabel('x')\n",
+    "    plt.ylabel('y')\n",
+    "    plt.legend(fontsize=12)\n",
+    "    \n",
+    "    plt.ylim(-8, 8) #remember to change the limits for better visualisation if necessary\n",
+    "    \n",
+    "    plt.axhline(0, color='black', lw=1)\n",
+    "    plt.axvline(0, color='black', lw=1)\n",
+    "    \n",
+    "    \n",
+    "\n",
+    "    plt.show()\n",
+    "\n",
+    "def vtrans(c):\n",
+    "    x = np.linspace(-3,3)\n",
+    "    \n",
+    "    y = func(x)+c #constant c summing the whole function\n",
+    "    \n",
+    "    plt.plot(x,y,  label='$f\\, (x)+c$')\n",
+    "    plt.title('Vertical translation')\n",
+    "    \n",
+    "    plt.xlabel('x')\n",
+    "    plt.ylabel('y')\n",
+    "    plt.legend(fontsize=12)\n",
+    "    \n",
+    "    plt.ylim(-8, 8) #remember to change the limits for better visualisation if necessary\n",
+    "        \n",
+    "    plt.axhline(0, color='black', lw=1)\n",
+    "    plt.axvline(0, color='black', lw=1)\n",
+    "    \n",
+    "    plt.show()\n",
+    "    \n",
+    "# create a slider\n",
+    "\n",
+    "interact(htrans, c=(-2.0,2.0))\n",
+    "interact(vtrans, c=(-2.0,2.0))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 6,
+   "metadata": {
+    "tags": [
+     "remove-cell"
+    ]
+   },
+   "outputs": [
+    {
+     "data": {
+      "application/vnd.jupyter.widget-view+json": {
+       "model_id": "12d5a3f30eab4d5395e543cb666f5e70",
+       "version_major": 2,
+       "version_minor": 0
+      },
+      "text/plain": [
+       "interactive(children=(FloatSlider(value=1.05, description='c', max=2.0, min=0.1), Output()), _dom_classes=('wi…"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "application/vnd.jupyter.widget-view+json": {
+       "model_id": "bd6164c6fecf4c72b93b1a39f7947633",
+       "version_major": 2,
+       "version_minor": 0
+      },
+      "text/plain": [
+       "interactive(children=(FloatSlider(value=1.05, description='c', max=2.0, min=0.1), Output()), _dom_classes=('wi…"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "text/plain": [
+       "<function __main__.vstetcom(c)>"
+      ]
+     },
+     "execution_count": 6,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "from ipywidgets import interact\n",
+    "import matplotlib.pyplot as plt\n",
+    "import numpy as np\n",
+    "\n",
+    "def func(x):\n",
+    "    return x**3-2*x\n",
+    "\n",
+    "def hstetcom(c):\n",
+    "    x = np.linspace(-3,3)\n",
+    "    \n",
+    "    y = func(c*x) #constant c multiplying the argument of the function\n",
+    "    \n",
+    "    plt.plot(x,y, label='$f\\, (cx)$')\n",
+    "    plt.title('Horizontal stretching/compression')\n",
+    "        \n",
+    "    plt.xlabel('x')\n",
+    "    plt.ylabel('y')\n",
+    "    plt.legend(fontsize=12)\n",
+    "    \n",
+    "    plt.ylim(-4, 4) #remember to change the limits for better visualisation if necessary\n",
+    "    \n",
+    "    plt.axhline(0, color='black', lw=1)\n",
+    "    plt.axvline(0, color='black', lw=1)\n",
+    "    \n",
+    "    \n",
+    "\n",
+    "    plt.show()\n",
+    "\n",
+    "def vstetcom(c):\n",
+    "    x = np.linspace(-3,3)\n",
+    "    \n",
+    "    y = c*func(x) #constant c multiplying the whole function\n",
+    "    \n",
+    "    plt.plot(x,y,  label='$c\\, f\\, (x)$')\n",
+    "    plt.title('Vertical stretching/compression')\n",
+    "    \n",
+    "    plt.xlabel('x')\n",
+    "    plt.ylabel('y')\n",
+    "    plt.legend(fontsize=12)\n",
+    "    \n",
+    "    plt.ylim(-4, 4) #remember to change the limits for better visualisation if necessary\n",
+    "        \n",
+    "    plt.axhline(0, color='black', lw=1)\n",
+    "    plt.axvline(0, color='black', lw=1)\n",
+    "    \n",
+    "    plt.show()\n",
+    "    \n",
+    "# create a slider\n",
+    "\n",
+    "interact(hstetcom, c=(0.1,2.0))\n",
+    "interact(vstetcom, c=(0.1,2.0))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 48,
+   "metadata": {
+    "tags": [
+     "remove-cell"
+    ]
+   },
+   "outputs": [
+    {
+     "data": {
+      "image/png": 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",
+      "text/plain": [
+       "<Figure size 720x360 with 2 Axes>"
+      ]
+     },
+     "metadata": {
+      "needs_background": "light"
+     },
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "fig, ax = plt.subplots(1,2,figsize=(10, 5))\n",
+    "\n",
+    "xt = np.linspace(-3,5)\n",
+    "\n",
+    "# x-axis reflection\n",
+    "ax[0].plot(xt, (xt-1)**2, color ='r', lw=2, label = '$(x-1)^2$')\n",
+    "ax[0].plot(xt, -(xt-1)**2, color ='b', lw=2, label = '$-(x-1)^2$')\n",
+    "ax[0].tick_params(axis='both', labelsize= 12)\n",
+    "\n",
+    "ax[0].set_xlabel('$x$', fontsize = 14)\n",
+    "ax[0].set_ylabel(\"$y$\", fontsize = 14)\n",
+    "\n",
+    "ax[0].set_title('x-axis reflection')\n",
+    "\n",
+    "ax[0].set_ylim(-10,10)\n",
+    "\n",
+    "ax[0].legend()\n",
+    "\n",
+    "ax[0].axhline(0, color='black', lw=1)\n",
+    "ax[0].axvline(0, color='black', lw=1)\n",
+    "\n",
+    "# y-axis reflection\n",
+    "\n",
+    "xt = np.linspace(-4,4)\n",
+    "\n",
+    "ax[1].plot(xt, (xt-1)**2, color ='r', lw=2, label = '$(x-1)^2$')\n",
+    "ax[1].plot(xt, (-xt-1)**2, color ='b', lw=2,  label = '$[(-x)-1]^2$')\n",
+    "ax[1].tick_params(axis='both', labelsize= 12)\n",
+    "\n",
+    "ax[1].set_xlabel('$x$', fontsize = 14)\n",
+    "#ax[1].set_ylabel(\"$y$\", fontsize = 14)\n",
+    "\n",
+    "ax[1].set_title('y-axis reflection')\n",
+    "\n",
+    "ax[1].set_ylim(-1,10)\n",
+    "\n",
+    "ax[1].legend()\n",
+    "\n",
+    "ax[1].axhline(0, color='black', lw=1)\n",
+    "ax[1].axvline(0, color='black', lw=1)\n",
+    "\n",
+    "fig.tight_layout()\n",
+    "fig.savefig('reflection.png')"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Describing shapes and patterns  "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Measuring angles  \n",
+    "  \n",
+    "An **angle** is a primitive geometrical element, such as the *point*, the *curve* or the *plane*. It is defined as the figure formed by two line segments that extend from a common point.  \n",
+    "    \n",
+    "<br/>\n",
+    "    \n",
+    "```{image} angle.png  \n",
+    "\n",
+    "```  \n",
+    "  \n",
+    "<br/>\n",
+    "<br/>\n",
+    "      \n",
+    "The two most common units for measuring angles are the *degree* and the *radian*.  \n",
+    "  \n",
+    "  \n",
+    "`````{tabbed} Degrees \n",
+    "\n",
+    "````{panels}  \n",
+    "  \n",
+    "```{image} ./degree.png  \n",
+    "\n",
+    "```  \n",
+    "\n",
+    "---  \n",
+    "  \n",
+    "If we divide a full rotation by 360 equal parts, each of these parts is called a *degree* (noted as $1^{\\circ}$). Thus, the degree corresponds to $1/360$ of one full rotation. Each degree can also be divided in 60 equal parts called *minutes* ($1^{\\circ} = 60'$) and each minute further divided in 60 *seconds* ($1' = 60''$).  \n",
+    "  \n",
+    "+++  \n",
+    "In the figure, one degree (shown in red) and eighty nine degrees (shown in blue).\n",
+    "\n",
+    "  \n",
+    "````  \n",
+    "`````  \n",
+    "  \n",
+    "`````{tabbed} Radians  \n",
+    "\n",
+    "````{panels}  \n",
+    "  \n",
+    "```{image} ./Circle_radians.gif  \n",
+    "\n",
+    "```  \n",
+    "\n",
+    "---  \n",
+    "  \n",
+    "One *radian* is the angle comprised in a circunference when the size of the arc is equal to the radius of the circunference. If $C$ is the size of the arc and $R$ the radius of the circunference, the corresponding angle in radians $\\phi$ will be given by:  \n",
+    "  \n",
+    "$$ \\phi = \\frac{C}{R} $$\n",
+    "  \n",
+    "Thus, if we are considering the arc for the full circunference, $C = 2\\pi R$, and we will have:  \n",
+    "  \n",
+    "$$\\phi = \\frac{2\\pi R}{R} = 2\\pi, $$  \n",
+    "  \n",
+    "and the angle for a full rotation corresponds to $2\\pi$ radians.  \n",
+    "  \n",
+    "```{tip}  \n",
+    "When the angle is written in radians, the unit (radians or rad) is frequently omitted.\n",
+    "\n",
+    "```\n",
+    "  \n",
+    "````  \n",
+    "`````  \n",
+    "  \n",
+    "For a given angle, transforming between units of degrees and radians can be simply done with:  \n",
+    "  \n",
+    "$$\\frac{360^{\\circ}}{\\theta} = \\frac{2\\pi}{\\phi},$$  \n",
+    "  \n",
+    "where $\\theta$ is the angle measured in degrees and $\\phi$, measured in radians. In this lecture, we will consider the angles measured in radians (independent of the symbol used for the variable), unless stated otherwise. \n",
+    "  \n",
+    "```{admonition} Example  \n",
+    "If $\\theta = 90^{\\circ}$, then using the previous expression we have $\\phi = \\displaystyle\\frac{\\pi}{2}$. Since $\\theta$ and $\\phi are directly proportional, we also have:  \n",
+    "  \n",
+    "  $$\\theta = 180^{\\circ} \\longrightarrow \\phi = 2 \\cdot \\frac{\\pi}{2} = \\pi $$  \n",
+    "  $$\\theta = 270^{\\circ} \\longrightarrow \\phi = 3 \\cdot \\frac{\\pi}{2} = \\frac{3\\pi}{2} $$  \n",
+    "  $$\\theta = 360^{\\circ} \\longrightarrow \\phi = 4 \\cdot \\frac{\\pi}{2} = 2\\pi $$\n",
+    "  \n",
+    "```"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Trigonometric circle  \n",
+    "  \n",
+    "  \n",
+    "We obtain the trigonometric circle by superimposing a circle of radius $1$ to the cartesian system of coordinates:   \n",
+    "\n",
+    "```{image} trig_circ.png\n",
+    "```  \n",
+    "\n",
+    "  \n",
+    "The projections on the $x$ and $y$-axes have special meanings, if we consider the angle $\\theta$ between the line segment in the radial direction and the $x$-axis.  \n",
+    "  \n",
+    "From the geometric relations on the right triangle (triangle in which one internal angle is a right angle):  \n",
+    "  \n",
+    "````{panels}  \n",
+    "  \n",
+    "```{image} ./right_trig.png\n",
+    "\n",
+    "```  \n",
+    "\n",
+    "---  \n",
+    "  \n",
+    "The larger side of the triangle, opposing the right angle, is called *hypothenuse*, while the other two smaller side are called *catheti*. We define the *sine* and *cosine* of the angle $\\alpha$ as the ratios of the opposing and adjacent catheti, respectively, with the hypothenuse:  \n",
+    "  \n",
+    "$$\\mbox{sin}(\\alpha) = \\frac{a}{c}$$  \n",
+    "$$\\mbox{cos}(\\alpha) = \\frac{b}{c}$$\n",
+    "  \n",
+    "````  \n",
+    "  \n",
+    "Back to the trigonometric circle, it is possible to construct a right triangle where the hypothenuse corresponds to the radius of the superimposed circle. In this case, since this radius equals to $1$, we have:  \n",
+    "  \n",
+    "$$\\mbox{sin}(\\theta) = y$$  \n",
+    "$$\\mbox{cos}(\\theta) = x.$$  \n",
+    "  \n",
+    "In other words, the projections of the end of the radial line segment on the $y$ and $x$-axes correspond to the sine and cosine of the angle $\\theta$. Sine and cosine for this angle can be obtained by the projections even if $\\theta > \\displaystyle\\frac{\\pi}{2}$.  \n",
+    "  \n",
+    "#### Trigonometric identities  \n",
+    "  \n",
+    "  With the aid of the projections on the trigonometric circle, some trigonometric identities become imediately clear. Consider $\\theta > 0$ for counterclockwise angle measurements starting from the $x$-axis and $\\theta < 0$ for clockwise measurements. We have:  \n",
+    "    \n",
+    "$$\\mbox{sin}(0) = 0,\\ \\ \\ \\mbox{sin}\\left(\\frac{\\pi}{2}\\right) = 1,\\ \\ \\ \\mbox{sin}(\\pi) = 0,\\ \\ \\ \\mbox{sin}\\left(\\frac{3\\pi}{2}\\right) = -1,\\ \\ \\ \\mbox{sin}(2\\pi) = 0$$  \n",
+    "$$\\mbox{cos}(0) = 1,\\ \\ \\ \\mbox{cos}\\left(\\frac{\\pi}{2}\\right) = 0,\\ \\ \\ \\mbox{cos}(\\pi) = -1,\\ \\ \\ \\mbox{cos}\\left(\\frac{3\\pi}{2}\\right) = 0,\\ \\ \\ \\mbox{cos}(2\\pi) = 1$$\n",
+    "  \n",
+    "From the fact that a full rotation brings you back to the same point:  \n",
+    "  \n",
+    "$$\\mbox{sin}(\\theta) = \\mbox{sin}(\\theta + 2\\pi)$$  \n",
+    "$$\\mbox{cos}(\\theta) = \\mbox{cos}(\\theta + 2\\pi)$$  \n",
+    "  \n",
+    "or from Pythagoras' theorem on the right triangle in the trigonometric circle:  \n",
+    "  \n",
+    "$$\\begin{align}\n",
+    "[\\mbox{sin}(\\theta)]^2 + [\\mbox{cos}(\\theta)]^2 =& 1^2 \\\\\n",
+    "\\mbox{sin}^2(\\theta) + \\mbox{cos}^2(\\theta) =& 1\n",
+    "\\end{align}$$  \n",
+    "  \n",
+    "```{tip}\n",
+    "The notation $\\mbox{sin}^2(\\theta)$ is frequently used to express the square of $\\mbox{sin}(\\theta)$, or:  \n",
+    "  \n",
+    "$$ \\mbox{sin}^2(\\theta) = [\\mbox{sin}(\\theta)]^2 = [\\mbox{sin}(\\theta)]\\cdot[\\mbox{sin}(\\theta)],$$  \n",
+    "  \n",
+    "which is different from $\\mbox{sin}(\\theta^2)$.\n",
+    "```  \n",
+    "  \n",
+    "<!-- Now watch this [video](https://web.microsoftstream.com/video/b3023026-7d45-4378-a8ee-c166a43eb2e7) for a different application of the same trigonometric construct. -->\n",
+    "  \n",
+    "  "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Trigonometric functions  \n",
+    "  \n",
+    "Since sine and cosine are defined for any real value of a given angle $x$ ($x \\in {\\rm I\\!R}$), we can define the functions:  \n",
+    "  \n",
+    "$$f(x) = \\mbox{sin}(x)$$  \n",
+    "$$f(x) = \\mbox{cos}(x).$$  \n",
+    "  \n",
+    "Thus, the association of sine and cosine with pure geometric ratios gives place for a more generic definition of functions of real variables. The graphs of these functions are given by:  \n",
+    "  \n",
+    "```{image} ./sin_cos.png\n",
+    "\n",
+    "```  \n",
+    "  \n",
+    "All the other known trigonometric relations can also be defined as functions in a similar way. Take for example the tangent of an angle $\\theta$, which is:  \n",
+    "  \n",
+    "$$tan(\\theta) = \\displaystyle\\frac{\\mbox{sin}(\\theta)}{\\mbox{cos}(\\theta)}.$$\n",
+    "\n",
+    "If $x \\in {\\rm I\\!R}$, than we can define a function  \n",
+    "  \n",
+    "$$ f(x) = \\mbox{tan}(x) = \\displaystyle\\frac{\\mbox{sin}(x)}{cos(x)}, \\ \\ \\ (\\mbox{for }\\mbox{cos}(x) \\ne 0).$$  \n",
+    "  \n",
+    "The graph from the tangent function will be given by:  \n",
+    "  \n",
+    "```{image} ./tan.png\n",
+    "\n",
+    "```  \n",
+    "  \n",
+    "Note by the above graphs that the functions sine and cosine have patterns that repeat themselves every interval of $2\\pi$, while the tangent repeats itself every interval of $\\pi$. We say that these three functions are **periodic**.\n",
+    "\n",
+    "\n",
+    "A function $f(x)$ is periodic if there is a positive constant $a$, so that:  \n",
+    "  \n",
+    "$$f(x + a) = f(x),$$  \n",
+    "  \n",
+    "for all values of $x$ for which the function f(x) is defined.  \n",
+    "  \n",
+    "If $a$ is the smallest number with this property, we call it the *period* of f(x).\n",
+    "\n",
+    "```{admonition} Example  \n",
+    "  \n",
+    "Since $\\mbox{sin}(x + 2\\pi) = \\mbox{sin}(x)$ for any value of $x$ and since $2\\pi$ is the smallest number for which this property holds, then sine is a periodic function with period $2\\pi$. With the same reasoning, it can be shown that the tangent is periodic with period $\\pi$.\n",
+    "```  \n",
+    "```{admonition} Example  \n",
+    "  \n",
+    "Many natural processes can be modelled as periodic. Think for example of a pendulum, the tides, the seasons. Often periodicity can be a good first order approximation, even if it does not strictly hold. For example when describimg the shape of the cloud cover in the picture below.\n",
+    "```\n",
+    "```{image} ./cloud.png\n",
+    "```  \n",
+    "  \n",
+    "#### Amplitude and Period  \n",
+    "  \n",
+    "  \n",
+    "From the basic functions $\\mbox{sin}(x)$ or $\\mbox{cos}(x)$ we can write a more general function $f(x)$ given by  \n",
+    "  \n",
+    "$$f(x) = A\\mbox{sin}(\\omega x),\\ \\ \\ \\ \\ \\ A, \\omega \\in {\\rm I\\!R}\\ \\ (A, \\omega \\ne 0)$$  \n",
+    "  \n",
+    "Since the sine ranges from $-1$ to $1$ (see the graph for $sin(x)$ for reference), by its definition, $f(x)$ will range from $-A$ to $A$. We call $A$ the **amplitude** of $f(x)$ and it gives the extent of the variation of the oscillation of the function.  \n",
+    "  \n",
+    "Being sine a periodic function, $f(x)$ will also be, possibly with a different period. The period $T$ of $f(x)$, as discussed before, will be determined by the equation $f(x + T) = f(x)$, that should be valid for every $x$. Thus:  \n",
+    "  \n",
+    "$$\\begin{align}\n",
+    "    f(x + T) &= f(x) \\\\\n",
+    "    A\\mbox{sin}(\\omega(x+T)) &= A\\mbox{sin}(\\omega x) \\\\\n",
+    "    \\mbox{sin}(\\omega(x+T)) &= \\mbox{sin}(\\omega x) \\\\\n",
+    "    \\mbox{sin}(\\omega x+\\omega T)) &= \\mbox{sin}(\\omega x) \\\\\n",
+    "    \\mbox{sin}(z+\\omega T)) &= \\mbox{sin}(z)\\ \\ \\ \\ \\ \\ (\\mbox{where we change the variable to } z=\\omega x)\n",
+    "  \\end{align}$$\n",
+    "  \n",
+    "But since we know that sine is periodic with period $2\\pi$, we should have $|\\omega|T = 2\\pi$. Thus, the **period** of the function $f(x)$ will be:  \n",
+    "  \n",
+    "$$T = \\frac{2\\pi}{\\omega}.$$  \n",
+    "  \n",
+    "The constant $\\omega = \\displaystyle\\frac{2\\pi}{T}$ is called **angular frequency** and measures the number of full rotations the function $f(x)$ oscillates in one interval of $T$.  \n",
+    "  \n",
+    "```{admonition} Example  \n",
+    "  \n",
+    "The function $f(x) = 3.6\\, \\mbox{cos}\\left(\\displaystyle\\frac{\\pi}{3}x\\right)$ has an amplitude $A=3.6$, an angular frequency $\\omega = \\displaystyle\\frac{\\pi}{3}$, and a period $T = \\displaystyle\\frac{2\\pi}{\\omega} = \\displaystyle\\frac{2 \\pi}{\\frac{\\pi}{3}} = 6$.  \n",
+    "```"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Modelling complex periodic phenomena\n",
+    "  \n",
+    "<!-- Watch this [video](https://web.microsoftstream.com/video/02f373a2-621b-4313-a18a-265f5fb65656) about periodic functions decomposition. -->"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Transformation on graphs \n",
+    "  \n",
+    "Starting from the elementary functions we have discussed  \n",
+    "      \n",
+    "$$ax+b,\\ \\ ax^2+bx+c,\\ \\ \\sqrt{x},\\ \\ \\mbox{e}^x,\\ \\ \\mbox{log}(x),\\ \\ \\mbox{sin}(x),\\ \\ \\mbox{cos}(x)$$  \n",
+    "  \n",
+    "we can introduce several transformations to obtain new functions, by adding or multiplying constants to the argument (function variable) or to the whole function. These are the simplest cases of what is know as *function composition* with the resulting function called *composite function*.  \n",
+    "  \n",
+    "When working with mathematical models, this will help you to understand how the shape of a given function might change by changing one of the parameters, or how to construct a set of functions (that might be useful for your particular model) with small modifications of basic functions.  \n",
+    "  \n",
+    "  \n",
+    "#### Translations  \n",
+    "  \n",
+    "A given function can be dislocated, either vertically or horiontally, by *adding* a constant $c$ to the whole function or to its argument. In general, starting from a function $f(x)$, translations are built following the rules:  \n",
+    "  \n",
+    "- **Horizontal** (Constant $c$ summed to the argument of the function)  \n",
+    "\n",
+    "  $y = f(x + c)\\ \\ \\ \\  -  \\left\\{\n",
+    "\\begin{array}{ll}\n",
+    "      c > 0 & \\rightarrow \\ \\  \\mbox{function moves to the left} \\\\\n",
+    "      c < 0 & \\rightarrow \\ \\  \\mbox{function moves to the right} \\\\\n",
+    "\\end{array} \n",
+    "\\right. $  \n",
+    "  \n",
+    "- **Vertical** (Constant $c$ summed to the whole function)  \n",
+    "\n",
+    "  $y = f(x) + c\\ \\ \\ \\  -  \\left\\{\n",
+    "\\begin{array}{ll}\n",
+    "      c > 0 & \\rightarrow \\ \\  \\mbox{function moves up} \\\\\n",
+    "      c < 0 & \\rightarrow \\ \\  \\mbox{function moves down} \\\\\n",
+    "\\end{array} \n",
+    "\\right. $  \n",
+    "  \n",
+    "```{admonition} Example  \n",
+    "If we start with the function $f(x) = x^3$, the function $y = f(x + 2) = (x+2)^3$ will represent a horizontal translation, while the function $y = f(x) + 2 = x^3+2$ will represent a vertical translation.  \n",
+    "```  \n",
+    "  \n",
+    "To see how this works, try to implement the following code in your own jupyter notebook. Try to use different functions $f(x)$ (defined in 'func(x)') to see how different graphs behave to horizontal and vertical translations.\n",
+    "  \n",
+    "```{code}\n",
+    "from ipywidgets import interact\n",
+    "import matplotlib.pyplot as plt\n",
+    "import numpy as np\n",
+    "\n",
+    "def func(x):\n",
+    "    return x**3\n",
+    "\n",
+    "def htrans(c):\n",
+    "    x = np.linspace(-3,3)\n",
+    "    \n",
+    "    y = func(x+c) #constant c in the argument of the function\n",
+    "    \n",
+    "    plt.plot(x,y, label='$f(x+c)$')\n",
+    "    plt.title('Horizontal translation')\n",
+    "    plt.legend(fontsize=12)\n",
+    "    \n",
+    "    plt.ylim(-8, 8) #remember to change the limits for better visualisation if necessary\n",
+    "    \n",
+    "    plt.axhline(0, color='black', lw=1)\n",
+    "    plt.axvline(0, color='black', lw=1)\n",
+    "    \n",
+    "    \n",
+    "\n",
+    "    plt.show()\n",
+    "\n",
+    "def vtrans(c):\n",
+    "    x = np.linspace(-3,3)\n",
+    "    \n",
+    "    y = func(x)+c #constant c summing the whole function\n",
+    "    \n",
+    "    plt.plot(x,y,  label='$f(x)+c$')\n",
+    "    plt.title('Vertical translation')\n",
+    "    \n",
+    "    plt.xlabel('x')\n",
+    "    plt.ylabel('y')\n",
+    "    plt.legend(fontsize=12)\n",
+    "    \n",
+    "    plt.ylim(-8, 8) #remember to change the limits for better visualisation if necessary\n",
+    "        \n",
+    "    plt.axhline(0, color='black', lw=1)\n",
+    "    plt.axvline(0, color='black', lw=1)\n",
+    "    \n",
+    "    plt.show()\n",
+    "    \n",
+    "# create a slider\n",
+    "\n",
+    "interact(htrans, c=(-2.0,2.0))\n",
+    "interact(vtrans, c=(-2.0,2.0))\n",
+    "```  \n",
+    "  \n",
+    "  \n",
+    "#### Stretching / Compression  \n",
+    "  \n",
+    "A given function can also be stretched or compressed, either vertically or horiontally, by *multiplying* a constant $c$ to the whole function or to its argument. Starting from a function $f(x)$, stretchings/compressions are built following the rules:  \n",
+    "  \n",
+    "- **Horizontal** (Constant $c$ multiplied to the argument of the function)  \n",
+    "\n",
+    "  $y = f(cx)\\ \\ \\ \\  -  \\left\\{\n",
+    "\\begin{array}{ll}\n",
+    "      c > 1 & \\rightarrow \\ \\  \\mbox{function is horizontally compressed} \\\\\n",
+    "      0 < c < 1 & \\rightarrow \\ \\  \\mbox{function is horizontally stretched} \\\\\n",
+    "\\end{array} \n",
+    "\\right. $  \n",
+    "  \n",
+    "- **Vertical** (Constant $c$ multiplied to the whole function)  \n",
+    "\n",
+    "  $y = cf(x)\\ \\ \\ \\  -  \\left\\{\n",
+    "\\begin{array}{ll}\n",
+    "      c > 1 & \\rightarrow \\ \\  \\mbox{function is vertically stretched} \\\\\n",
+    "      0 < c < 1 & \\rightarrow \\ \\  \\mbox{function is vertically compressed} \\\\\n",
+    "\\end{array} \n",
+    "\\right. $  \n",
+    "  \n",
+    "```{admonition} Example  \n",
+    "For the function $f(x) = x^3-2x$, the function $y = f(2x) = (2x)^3-2(2x) = 8x^3-4x$ will represent a horizontal compression, while the function $y = 2f(x) = 2x^3 - 4x$ will represent a vertical stretching.  \n",
+    "```  \n",
+    "```{admonition} Example  \n",
+    "One example where we stretch or compress functions comes from chemical processes. Many chemical processes happen faster when the temperature is higher. If we describe the concentration of a certain chemical as a function of time $c(t)$, increasing the temperature will lead to a horizontal compression of the function.\n",
+    "```   \n",
+    "\n",
+    "Now play with the code below to see how different graphs behave to horizontal and vertical stretching/compression.\n",
+    "  \n",
+    "```{code}  \n",
+    "from ipywidgets import interact\n",
+    "import matplotlib.pyplot as plt\n",
+    "import numpy as np\n",
+    "\n",
+    "def func(x):\n",
+    "    return x**3-2*x\n",
+    "\n",
+    "def hstetcom(c):\n",
+    "    x = np.linspace(-3,3)\n",
+    "    \n",
+    "    y = func(c*x) #constant c multiplying the argument of the function\n",
+    "    \n",
+    "    plt.plot(x,y, label='$f\\, (cx)$')\n",
+    "    plt.title('Horizontal stretching/compression')\n",
+    "        \n",
+    "    plt.xlabel('x')\n",
+    "    plt.ylabel('y')\n",
+    "    plt.legend(fontsize=12)\n",
+    "    \n",
+    "    plt.ylim(-4, 4) #remember to change the limits for better visualisation if necessary\n",
+    "    \n",
+    "    plt.axhline(0, color='black', lw=1)\n",
+    "    plt.axvline(0, color='black', lw=1)\n",
+    "    \n",
+    "    \n",
+    "\n",
+    "    plt.show()\n",
+    "\n",
+    "def vstetcom(c):\n",
+    "    x = np.linspace(-3,3)\n",
+    "    \n",
+    "    y = c*func(x) #constant c multiplying the whole function\n",
+    "    \n",
+    "    plt.plot(x,y,  label='$c\\, f\\, (x)$')\n",
+    "    plt.title('Vertical stretching/compression')\n",
+    "    \n",
+    "    plt.xlabel('x')\n",
+    "    plt.ylabel('y')\n",
+    "    plt.legend(fontsize=12)\n",
+    "    \n",
+    "    plt.ylim(-4, 4) #remember to change the limits for better visualisation if necessary\n",
+    "        \n",
+    "    plt.axhline(0, color='black', lw=1)\n",
+    "    plt.axvline(0, color='black', lw=1)\n",
+    "    \n",
+    "    plt.show()\n",
+    "    \n",
+    "# create a slider\n",
+    "\n",
+    "interact(hstetcom, c=(0.1,3.0))\n",
+    "interact(vstetcom, c=(0.1,3.0))\n",
+    "```  \n",
+    "  \n",
+    "#### Reflection  \n",
+    "  \n",
+    "If the constant $c$ multiplying the whole function or its argument has the particular value $c = -1$, we will have a *reflection*. Reflections can be done in relation to the $x$ or $y$-axes and they are built, starting from a function $f(x)$, following the rules:  \n",
+    "  \n",
+    "  \n",
+    "- **About the x-axis** - $y = -f(x)$  \n",
+    "  \n",
+    "- **About the y-axis** - $y = f(-x)$  \n",
+    "\n",
+    "The figure below shows these two reflections using the function $f(x) = (x-1)^2$.  \n",
+    "  \n",
+    "```{image} ./reflection.png\n",
+    "\n",
+    "```  "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 2,
+   "metadata": {
+    "tags": [
+     "remove-cell"
+    ]
+   },
+   "outputs": [
+    {
+     "data": {
+      "image/png": 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",
+      "text/plain": [
+       "<Figure size 864x216 with 1 Axes>"
+      ]
+     },
+     "metadata": {
+      "needs_background": "light"
+     },
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "image/png": 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",
+      "text/plain": [
+       "<Figure size 288x216 with 1 Axes>"
+      ]
+     },
+     "metadata": {
+      "needs_background": "light"
+     },
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "#Codes for figures\n",
+    "\n",
+    "import numpy as np\n",
+    "import pandas as pd\n",
+    "import matplotlib.pyplot as plt\n",
+    "\n",
+    "def multiple_formatter(denominator=2, number=np.pi, latex='\\pi'):\n",
+    "    def gcd(a, b):\n",
+    "        while b:\n",
+    "            a, b = b, a%b\n",
+    "        return a\n",
+    "    def _multiple_formatter(x, pos):\n",
+    "        den = denominator\n",
+    "        num = np.int(np.rint(den*x/number))\n",
+    "        com = gcd(num,den)\n",
+    "        (num,den) = (int(num/com),int(den/com))\n",
+    "        if den==1:\n",
+    "            if num==0:\n",
+    "                return r'$0$'\n",
+    "            if num==1:\n",
+    "                return r'$%s$'%latex\n",
+    "            elif num==-1:\n",
+    "                return r'$-%s$'%latex\n",
+    "            else:\n",
+    "                return r'$%s%s$'%(num,latex)\n",
+    "        else:\n",
+    "            if num==1:\n",
+    "                return r'$\\frac{%s}{%s}$'%(latex,den)\n",
+    "            elif num==-1:\n",
+    "                return r'$\\frac{-%s}{%s}$'%(latex,den)\n",
+    "            else:\n",
+    "                return r'$\\frac{%s%s}{%s}$'%(num,latex,den)\n",
+    "    return _multiple_formatter\n",
+    "\n",
+    "class Multiple:\n",
+    "    def __init__(self, denominator=2, number=np.pi, latex='\\pi'):\n",
+    "        self.denominator = denominator\n",
+    "        self.number = number\n",
+    "        self.latex = latex\n",
+    "\n",
+    "    def locator(self):\n",
+    "        return plt.MultipleLocator(self.number / self.denominator)\n",
+    "\n",
+    "    def formatter(self):\n",
+    "        return plt.FuncFormatter(multiple_formatter(self.denominator, self.number, self.latex))\n",
+    "\n",
+    "xt=np.linspace(-4*np.pi,4*np.pi,num=500)\n",
+    "##\n",
+    "fig, ax = plt.subplots(figsize=(12, 3))\n",
+    "ax.plot(xt, np.sin(xt), color='b', lw=2, label='$sin(x)$')\n",
+    "ax.plot(xt, np.cos(xt), color='g', lw=2, label='$cos(x)$')\n",
+    "ax.tick_params(axis='both', labelsize= 15)\n",
+    "ax.set_xlabel('$x$', fontsize = 16)\n",
+    "ax.legend(fontsize=14)\n",
+    "\n",
+    "ax.axhline(0, color='black', lw=1)\n",
+    "ax.axvline(0, color='black', lw=1)\n",
+    "\n",
+    "ax.xaxis.set_major_locator(plt.MultipleLocator(np.pi / 2))\n",
+    "ax.xaxis.set_minor_locator(plt.MultipleLocator(np.pi / 12))\n",
+    "ax.xaxis.set_major_formatter(plt.FuncFormatter(multiple_formatter()))\n",
+    "\n",
+    "fig.tight_layout()\n",
+    "fig.savefig('sin_cos.png')\n",
+    "\n",
+    "##\n",
+    "xt=np.linspace(-2*np.pi,2*np.pi,num=500)\n",
+    "fig, ax = plt.subplots(figsize=(4, 3))\n",
+    "y = np.tan(xt)\n",
+    "y[:-1][np.diff(y) < 0] = np.nan\n",
+    "\n",
+    "ax.plot(xt,y, color ='r', lw=2)\n",
+    "ax.tick_params(axis='both', labelsize= 15)\n",
+    "\n",
+    "ax.set_xlabel('$x$', fontsize = 16)\n",
+    "ax.set_ylabel(\"$tan(x)$\", fontsize = 16)\n",
+    "\n",
+    "plt.ylim(-10,10)\n",
+    "ax.axhline(0, color='black', lw=1)\n",
+    "ax.axvline(0, color='black', lw=1)\n",
+    "\n",
+    "ax.xaxis.set_major_locator(plt.MultipleLocator(np.pi / 2))\n",
+    "ax.xaxis.set_minor_locator(plt.MultipleLocator(np.pi / 12))\n",
+    "ax.xaxis.set_major_formatter(plt.FuncFormatter(multiple_formatter()))\n",
+    "\n",
+    "fig.tight_layout()\n",
+    "fig.savefig('tan.png')"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 3,
+   "metadata": {
+    "tags": [
+     "remove-cell"
+    ]
+   },
+   "outputs": [
+    {
+     "data": {
+      "application/vnd.jupyter.widget-view+json": {
+       "model_id": "a785b675781e44dcb130fe4aba0b9568",
+       "version_major": 2,
+       "version_minor": 0
+      },
+      "text/plain": [
+       "interactive(children=(FloatSlider(value=0.0, description='c', max=2.0, min=-2.0), Output()), _dom_classes=('wi…"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "application/vnd.jupyter.widget-view+json": {
+       "model_id": "b9af681dfe384b9fbd3b14793a2feede",
+       "version_major": 2,
+       "version_minor": 0
+      },
+      "text/plain": [
+       "interactive(children=(FloatSlider(value=0.0, description='c', max=2.0, min=-2.0), Output()), _dom_classes=('wi…"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "text/plain": [
+       "<function __main__.vtrans(c)>"
+      ]
+     },
+     "execution_count": 3,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "from ipywidgets import interact\n",
+    "import matplotlib.pyplot as plt\n",
+    "import numpy as np\n",
+    "\n",
+    "def func(x):\n",
+    "    return x**3\n",
+    "\n",
+    "def htrans(c):\n",
+    "    x = np.linspace(-3,3)\n",
+    "    \n",
+    "    y = func(x+c) #constant c in the argument of the function\n",
+    "    \n",
+    "    plt.plot(x,y, label='$f\\, (x+c)$')\n",
+    "    plt.title('Horizontal translation')\n",
+    "        \n",
+    "    plt.xlabel('x')\n",
+    "    plt.ylabel('y')\n",
+    "    plt.legend(fontsize=12)\n",
+    "    \n",
+    "    plt.ylim(-8, 8) #remember to change the limits for better visualisation if necessary\n",
+    "    \n",
+    "    plt.axhline(0, color='black', lw=1)\n",
+    "    plt.axvline(0, color='black', lw=1)\n",
+    "    \n",
+    "    \n",
+    "\n",
+    "    plt.show()\n",
+    "\n",
+    "def vtrans(c):\n",
+    "    x = np.linspace(-3,3)\n",
+    "    \n",
+    "    y = func(x)+c #constant c summing the whole function\n",
+    "    \n",
+    "    plt.plot(x,y,  label='$f\\, (x)+c$')\n",
+    "    plt.title('Vertical translation')\n",
+    "    \n",
+    "    plt.xlabel('x')\n",
+    "    plt.ylabel('y')\n",
+    "    plt.legend(fontsize=12)\n",
+    "    \n",
+    "    plt.ylim(-8, 8) #remember to change the limits for better visualisation if necessary\n",
+    "        \n",
+    "    plt.axhline(0, color='black', lw=1)\n",
+    "    plt.axvline(0, color='black', lw=1)\n",
+    "    \n",
+    "    plt.show()\n",
+    "    \n",
+    "# create a slider\n",
+    "\n",
+    "interact(htrans, c=(-2.0,2.0))\n",
+    "interact(vtrans, c=(-2.0,2.0))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 6,
+   "metadata": {
+    "tags": [
+     "remove-cell"
+    ]
+   },
+   "outputs": [
+    {
+     "data": {
+      "application/vnd.jupyter.widget-view+json": {
+       "model_id": "12d5a3f30eab4d5395e543cb666f5e70",
+       "version_major": 2,
+       "version_minor": 0
+      },
+      "text/plain": [
+       "interactive(children=(FloatSlider(value=1.05, description='c', max=2.0, min=0.1), Output()), _dom_classes=('wi…"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "application/vnd.jupyter.widget-view+json": {
+       "model_id": "bd6164c6fecf4c72b93b1a39f7947633",
+       "version_major": 2,
+       "version_minor": 0
+      },
+      "text/plain": [
+       "interactive(children=(FloatSlider(value=1.05, description='c', max=2.0, min=0.1), Output()), _dom_classes=('wi…"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "text/plain": [
+       "<function __main__.vstetcom(c)>"
+      ]
+     },
+     "execution_count": 6,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "from ipywidgets import interact\n",
+    "import matplotlib.pyplot as plt\n",
+    "import numpy as np\n",
+    "\n",
+    "def func(x):\n",
+    "    return x**3-2*x\n",
+    "\n",
+    "def hstetcom(c):\n",
+    "    x = np.linspace(-3,3)\n",
+    "    \n",
+    "    y = func(c*x) #constant c multiplying the argument of the function\n",
+    "    \n",
+    "    plt.plot(x,y, label='$f\\, (cx)$')\n",
+    "    plt.title('Horizontal stretching/compression')\n",
+    "        \n",
+    "    plt.xlabel('x')\n",
+    "    plt.ylabel('y')\n",
+    "    plt.legend(fontsize=12)\n",
+    "    \n",
+    "    plt.ylim(-4, 4) #remember to change the limits for better visualisation if necessary\n",
+    "    \n",
+    "    plt.axhline(0, color='black', lw=1)\n",
+    "    plt.axvline(0, color='black', lw=1)\n",
+    "    \n",
+    "    \n",
+    "\n",
+    "    plt.show()\n",
+    "\n",
+    "def vstetcom(c):\n",
+    "    x = np.linspace(-3,3)\n",
+    "    \n",
+    "    y = c*func(x) #constant c multiplying the whole function\n",
+    "    \n",
+    "    plt.plot(x,y,  label='$c\\, f\\, (x)$')\n",
+    "    plt.title('Vertical stretching/compression')\n",
+    "    \n",
+    "    plt.xlabel('x')\n",
+    "    plt.ylabel('y')\n",
+    "    plt.legend(fontsize=12)\n",
+    "    \n",
+    "    plt.ylim(-4, 4) #remember to change the limits for better visualisation if necessary\n",
+    "        \n",
+    "    plt.axhline(0, color='black', lw=1)\n",
+    "    plt.axvline(0, color='black', lw=1)\n",
+    "    \n",
+    "    plt.show()\n",
+    "    \n",
+    "# create a slider\n",
+    "\n",
+    "interact(hstetcom, c=(0.1,2.0))\n",
+    "interact(vstetcom, c=(0.1,2.0))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 48,
+   "metadata": {
+    "scrolled": true,
+    "tags": [
+     "remove-cell"
+    ]
+   },
+   "outputs": [
+    {
+     "data": {
+      "image/png": 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",
+      "text/plain": [
+       "<Figure size 720x360 with 2 Axes>"
+      ]
+     },
+     "metadata": {
+      "needs_background": "light"
+     },
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "fig, ax = plt.subplots(1,2,figsize=(10, 5))\n",
+    "\n",
+    "xt = np.linspace(-3,5)\n",
+    "\n",
+    "# x-axis reflection\n",
+    "ax[0].plot(xt, (xt-1)**2, color ='r', lw=2, label = '$(x-1)^2$')\n",
+    "ax[0].plot(xt, -(xt-1)**2, color ='b', lw=2, label = '$-(x-1)^2$')\n",
+    "ax[0].tick_params(axis='both', labelsize= 12)\n",
+    "\n",
+    "ax[0].set_xlabel('$x$', fontsize = 14)\n",
+    "ax[0].set_ylabel(\"$y$\", fontsize = 14)\n",
+    "\n",
+    "ax[0].set_title('x-axis reflection')\n",
+    "\n",
+    "ax[0].set_ylim(-10,10)\n",
+    "\n",
+    "ax[0].legend()\n",
+    "\n",
+    "ax[0].axhline(0, color='black', lw=1)\n",
+    "ax[0].axvline(0, color='black', lw=1)\n",
+    "\n",
+    "# y-axis reflection\n",
+    "\n",
+    "xt = np.linspace(-4,4)\n",
+    "\n",
+    "ax[1].plot(xt, (xt-1)**2, color ='r', lw=2, label = '$(x-1)^2$')\n",
+    "ax[1].plot(xt, (-xt-1)**2, color ='b', lw=2,  label = '$[(-x)-1]^2$')\n",
+    "ax[1].tick_params(axis='both', labelsize= 12)\n",
+    "\n",
+    "ax[1].set_xlabel('$x$', fontsize = 14)\n",
+    "#ax[1].set_ylabel(\"$y$\", fontsize = 14)\n",
+    "\n",
+    "ax[1].set_title('y-axis reflection')\n",
+    "\n",
+    "ax[1].set_ylim(-1,10)\n",
+    "\n",
+    "ax[1].legend()\n",
+    "\n",
+    "ax[1].axhline(0, color='black', lw=1)\n",
+    "ax[1].axvline(0, color='black', lw=1)\n",
+    "\n",
+    "fig.tight_layout()\n",
+    "fig.savefig('reflection.png')"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 91,
+   "metadata": {
+    "tags": [
+     "remove-cell"
+    ]
+   },
+   "outputs": [
+    {
+     "data": {
+      "image/png": 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",
+      "text/plain": [
+       "<Figure size 432x288 with 1 Axes>"
+      ]
+     },
+     "metadata": {
+      "needs_background": "light"
+     },
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "import matplotlib.pyplot as plt\n",
+    "import numpy as np\n",
+    "    \n",
+    "time = np.linspace(0,15)\n",
+    "rat = 0.5 + ((1-0.5)/10)*time\n",
+    "\n",
+    "fig, ax = plt.subplots(figsize=(6, 4))\n",
+    "\n",
+    "\n",
+    "ax.plot(time,rat)\n",
+    "    \n",
+    "ax.set_xlabel('$N_t$',fontsize = 16)\n",
+    "ax.set_ylabel(r'$\\frac{N_t}{N_{t+1}}$',fontsize = 22)\n",
+    "\n",
+    "ax.set_yticks([0.5,1])\n",
+    "ax.set_yticklabels(['$1/R$','1'],fontsize=14)\n",
+    "\n",
+    "ax.set_xticks([0,10])\n",
+    "ax.set_xticklabels(['0','K'],fontsize=14)\n",
+    "\n",
+    "ax.axhline(1, ls='--', color='black', lw=0.5)\n",
+    "ax.axvline(10, ls='--', color='black', lw=0.5)\n",
+    "\n",
+    "ax.set_ylim(0, 1.5)\n",
+    "ax.set_xlim(0, 15)\n",
+    "\n",
+    "fig.tight_layout()\n",
+    "fig.savefig('parent-offspring.png')"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "metadata": {
+    "tags": [
+     "remove-cell"
+    ]
+   },
+   "outputs": [
+    {
+     "data": {
+      "image/png": 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",
+      "text/plain": [
+       "<Figure size 600x400 with 1 Axes>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "import matplotlib.pyplot as plt\n",
+    "import numpy as np\n",
+    "\n",
+    "lisg1=[]\n",
+    "lisg2=[]\n",
+    "lisg3=[]\n",
+    "\n",
+    "n0g1=1\n",
+    "n0g2=1\n",
+    "n0g3=1\n",
+    "\n",
+    "lisg1.append(n0g1)\n",
+    "lisg2.append(n0g2)\n",
+    "lisg3.append(n0g3)\n",
+    "\n",
+    "vect = range(1,25)\n",
+    "for t in vect:\n",
+    "    n0g1=1.1*n0g1\n",
+    "    lisg1.append(n0g1)\n",
+    "    \n",
+    "    n0g2=1*n0g2\n",
+    "    lisg2.append(n0g2)\n",
+    "    \n",
+    "    n0g3=0.9*n0g3\n",
+    "    lisg3.append(n0g3)\n",
+    "\n",
+    "fig, ax = plt.subplots(figsize=(6, 4))\n",
+    "time = range(25)\n",
+    "\n",
+    "ax.scatter(time,lisg1,  label='$R>1$')\n",
+    "ax.scatter(time,lisg2,  label='$R=1$')\n",
+    "ax.scatter(time,lisg3,  label='$0<R<1$')\n",
+    "    \n",
+    "ax.set_xlabel('t',fontsize = 16)\n",
+    "ax.set_ylabel('$N_t$',fontsize = 16)\n",
+    "\n",
+    "ax.tick_params(\n",
+    "    axis='both',          \n",
+    "    which='both',      \n",
+    "    bottom=False,      \n",
+    "    left=False,\n",
+    "    labelbottom=False,labelleft=False)\n",
+    "\n",
+    "\n",
+    "\n",
+    "#plt.yticks([0,1],['0','K'],fontsize = 13)\n",
+    "#plt.axhline(1, ls='--', color='black', lw=0.5)\n",
+    "\n",
+    "ax.set_ylim(0, 5)\n",
+    "ax.grid()\n",
+    "ax.legend(fontsize=12)\n",
+    "\n",
+    "fig.tight_layout()\n",
+    "fig.savefig('expgrowth.png')"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 3,
+   "metadata": {
+    "tags": [
+     "remove-cell"
+    ]
+   },
+   "outputs": [
+    {
+     "data": {
+      "image/png": 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",
+      "text/plain": [
+       "<Figure size 600x400 with 1 Axes>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "import matplotlib.pyplot as plt\n",
+    "import numpy as np\n",
+    "\n",
+    "lisg1=[]\n",
+    "lisg2=[]\n",
+    "lisg3=[]\n",
+    "\n",
+    "n0g1=0.1\n",
+    "n0g2=0.8\n",
+    "n0g3=1.4\n",
+    "\n",
+    "lisg1.append(n0g1)\n",
+    "lisg2.append(n0g2)\n",
+    "lisg3.append(n0g3)\n",
+    "\n",
+    "vect = range(1,16)\n",
+    "for t in vect:\n",
+    "    n0g1=1.5*n0g1/(1+((1.5-1)/1)*n0g1)\n",
+    "    n0g2=1.5*n0g2/(1+((1.5-1)/1)*n0g2)\n",
+    "    n0g3=1.5*n0g3/(1+((1.5-1)/1)*n0g3)\n",
+    "    \n",
+    "    lisg1.append(n0g1)\n",
+    "    lisg2.append(n0g2)\n",
+    "    lisg3.append(n0g3)\n",
+    "    \n",
+    "time = range(16)\n",
+    "\n",
+    "fig, ax = plt.subplots(figsize=(6, 4))\n",
+    "\n",
+    "ax.scatter(time,lisg1)\n",
+    "ax.scatter(time,lisg2)\n",
+    "ax.scatter(time,lisg3)\n",
+    "    \n",
+    "ax.set_xlabel('t',fontsize = 16)\n",
+    "ax.set_ylabel('$N_t$',fontsize = 16)\n",
+    "\n",
+    "ax.tick_params(\n",
+    "    axis='x',          \n",
+    "    which='both',      \n",
+    "    bottom=False,      \n",
+    "    top=False,\n",
+    "    labelbottom=False)\n",
+    "\n",
+    "\n",
+    "ax.set_yticks([0,1])\n",
+    "ax.set_yticklabels(['0','K'],fontsize=12)\n",
+    "ax.axhline(1, ls='--', color='black', lw=0.5)\n",
+    "\n",
+    "ax.set_ylim(0, 1.5)\n",
+    "\n",
+    "fig.tight_layout()\n",
+    "fig.savefig('logistic.png')"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Analysing change, one step at a time"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Imagine a colony of wasps is founded by a queen with the following pattern of eggs laid:  \n",
+    "\n",
+    "<br />  \n",
+    "    \n",
+    "| Days     |  Eggs  |\n",
+    "|----------|:------:|\n",
+    "| $0$      |  $100$ |\n",
+    "| $1$      |  $135$ |\n",
+    "| $2$      |  $170$ |\n",
+    "| $3$      |  $205$ |\n",
+    "| $4$      |  $240$ |  \n",
+    "    \n",
+    "\n",
+    "Since there is an additive pattern for the increase in the total number of eggs laid, with the number of eggs in any given day, we can determine the number of eggs the colony will have in the following day. If we index the days by a sequence of natural numbers *n*, we have for the above colony:  \n",
+    "  \n",
+    "$$a_{n+1} = a_{n} + 35,$$  \n",
+    "  \n",
+    "where $a_n$ represents the total amount of eggs in a given day $n$ and $a_{n+1}$ the same variable on the day following day $n$. The constant $35$ represents the daily increment (or decrement) on this total number. A sequence of number built with constant additive increments receives the name of *arithmetic progression*.  \n",
+    "  \n",
+    "The above expression defines an iterative process (also known as a *recursion*) with which we can construct the whole sequence starting from a single initial value. If we want to know how many eggs will have been laid on a given day $n$, this shall be given by the expression:  \n",
+    "  \n",
+    "$$a_n = 100 + 35n.$$  \n",
+    "  \n",
+    "Thus, the total amount of eggs laid on day $n=100$ is $a_{100} = 100 + 35 \\cdot 100 = 3600$.  \n",
+    "  \n",
+    "<br />\n",
+    "<br />\n",
+    "Now imagine a single *Escherichia coli* cell put on a Petri dish with a favorable growth medium. If given the right conditions, *E. coli* cells divide, generating two other cells, in a time of approximately $30$ minutes. For our experiment, we would have:  \n",
+    "\n",
+    "<br />  \n",
+    "    \n",
+    "| Minutes  |  # cells  |\n",
+    "|----------|:------:|\n",
+    "| $0$      |  $1$ |\n",
+    "| $30$      |  $2$ |\n",
+    "| $60$      |  $4$ |\n",
+    "| $90$      |  $8$ |\n",
+    "| $120$      |  $16$ |  \n",
+    "  \n",
+    "    \n",
+    "If $t$ represents the number of reproduction intervals since the start of the experiment (assuming values $t=0, 1, 2, \\ldots$), then we can construct the recursion:  \n",
+    "  \n",
+    "$$N_{t+1} = 2N_t,$$  \n",
+    "  \n",
+    "where $N_t$ represents the number of *E. coli* cells after $t$ intervals of reproduction (each interval corresponding to $30$ minutes) and $N_{t+1}$ represents the number of cells on the interval following $t$. See that this time the constant $2$ provides a multiplicative increment, so that the sequence formed by this multiplicative iteration is called *geometric progression*.  \n",
+    "  \n",
+    "If we want $N_t$ for a given interval $t$, we then have:  \n",
+    "  \n",
+    "$$N_t = 2^t,$$\n",
+    "    \n",
+    "leading to an exponential growth in discrete steps.  \n",
+    "  \n",
+    "In general, if we have a population growth described by a geometric progression, starting with a number of individuals $N_0$, the number of individuals on the subsequent generations can be described by the recursion:  \n",
+    "  \n",
+    "$$N_{t+1} = RN_t,$$  \n",
+    "  \n",
+    "where $t$ indicates the number of generations and $R$ is a positive constant ($R>0$).  \n",
+    "  \n",
+    "With this recursion, starting from $N_0$, we can obtain the full sequence by construction:  \n",
+    "  \n",
+    "$$\\begin{align}  \n",
+    "N_1 &= RN_0 \\\\\n",
+    "N_2 &= RN_1 = R(RN_0) = R^2N_0 \\\\\n",
+    "N_3 &= RN_2 = R(RN_1) = R[R(RN_0)] = R^3N_0 \\\\\n",
+    "    &\\vdots \\end{align}$$  \n",
+    "  \n",
+    "From this process of construction, we can obtain what we call the *solution* of the above recursion, given by:  \n",
+    "  \n",
+    "$$N_t = N_0R^t.$$  \n",
+    "  \n",
+    "In problems of population dynamics, $R$ is called **growth constant** and its value determines the long-term behaviour of the function:  \n",
+    "  \n",
+    "```{image} expgrowth.png\n",
+    "```"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Sequences  \n",
+    "  \n",
+    "The two last problems are examples of *sequences* for which a formal definition is given as:  \n",
+    "  \n",
+    "$$\\begin{align}\n",
+    "    f\\!:\\ &{\\rm I\\!N} \\rightarrow {\\rm I\\!R} \\\\\n",
+    "       & n \\rightarrow f(n),\\end{align}$$  \n",
+    "         \n",
+    "in other words, a sequence is a function $f$ from the set of natural numbers to the set of real numbers. For each value of the independent variable $n$ (a natural number), the sequence determines its image $f(n)$.  \n",
+    "  \n",
+    "```{admonition} Example  \n",
+    "The sequence defined by the rule $f(n) = \\displaystyle\\frac{1}{n+1}$ defines the sequence of numbers (starting from $n=0$):  \n",
+    "  \n",
+    "$$1,\\frac{1}{2},\\frac{1}{3},\\frac{1}{4},\\frac{1}{5},\\ldots,$$  \n",
+    "  \n",
+    "corresponding to the set of values $n=0,1,2,3,4,\\ldots$.\n",
+    "```  \n",
+    "  \n",
+    "  \n",
+    "\n",
+    "Sequences are usually written as ordered lists of numbers $a_0, a_1, a_2, a_3, \\ldots$ where $a_n=f(n)$ and referred to with the notation $\\{a_n\\}$ (which is short for $\\{a_n: n \\in {\\rm I\\!N}\\}$, making it explicit that $n$ is a natural number).  \n",
+    "  \n",
+    "As we have seen before, two forms of representation are possible:  \n",
+    "  \n",
+    "````{panels}\n",
+    "**Explicit description**\n",
+    "^^^\n",
+    "$$a_n=f(n)$$  \n",
+    "  \n",
+    "```{admonition} Examples  \n",
+    "$$a_n = 100 + 35n$$  \n",
+    "$$a_n = 100\\cdot2^n$$  \n",
+    "$$a_n = \\frac{4n^2-1}{n^2}$$  \n",
+    "$$a_n = \\mbox{cos}\\left(\\displaystyle\\frac{3\\pi n}{2}\\right)$$\n",
+    "```\n",
+    "+++\n",
+    "In this case, any term $a_n$ of arbitrary index $n$ can be obtained by direct calculation if we know the function $f(n)$. \n",
+    "---\n",
+    "\n",
+    "**Recursive description**\n",
+    "^^^\n",
+    "$$a_{n+1}=g(a_n)$$  \n",
+    "  \n",
+    "```{admonition} Examples  \n",
+    "$$a_{n+1} = a_n + 35,\\ \\ \\ \\mbox{with}\\ a_0 = 100$$  \n",
+    "$$a_{n+1} = 2a_n,\\ \\ \\ \\mbox{with}\\ a_0=100$$  \n",
+    "$$a_{n+1} = \\frac{3}{a_n},\\ \\ \\ \\mbox{with}\\ a_0=2$$ \n",
+    "```  \n",
+    "  \n",
+    "The recursive description is also called *difference equation*.\n",
+    "+++\n",
+    "Note that with the form $a_{n+1}=g(a_n)$ the following term $a_{n+1}$ can be determined only with the term one step back $a_n$, knowing the function $g$. In general, recursions can be dependent on several other backward terms. When it depends only on the immediate previous term it is called a **first-order** recursion. \n",
+    "````"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Limits  \n",
+    "  \n",
+    "  \n",
+    "In many applications, we are interested to know what is the *long-term behaviour* of a given sequence $\\{a_n\\}$. For example, is a given population tending towards a stable number of individuals as time passes? In other words, we want to know if there is a number $L$ for which the sequence tends to as $n$ increases. To represent this we use the following notation:  \n",
+    "  \n",
+    "$$\\lim_{n\\rightarrow \\infty} a_n = L\\ \\ (\\mbox{or simply}\\ \\lim a_n =L)$$  \n",
+    "  \n",
+    "If such a number exists, we say that the sequence $\\{a_n\\}$ is **convergent**.  \n",
+    "  \n",
+    "```{admonition} Examples  \n",
+    "1. $a_n = \\displaystyle\\frac{1}{n+1}: \\ \\ \\ \\left\\{1,\\displaystyle\\frac{1}{2},\\displaystyle\\frac{1}{3},\\displaystyle\\frac{1}{4},\\displaystyle\\frac{1}{5},\\ldots\\right\\}$  \n",
+    "    <br />\n",
+    "    Note that each term is smaller than the previous and they are all positive. We can conclude that:  \n",
+    "  \n",
+    "    $$\\lim a_n = 0.$$  \n",
+    "\n",
+    "2. $b_n = (-1)^n: \\ \\ \\ \\left\\{1,-1,1,-1,1,-1,\\ldots\\right\\}$  \n",
+    "    <br />\n",
+    "    The terms in this sequence are alternating between $1$ and $-1$, never convergeing for a specific value. Thus, the limit of sequence $\\{b_n\\}$ does not exist.  \n",
+    "    <br />\n",
+    "3. $c_n = 2^n: \\ \\ \\ \\left\\{1,2,4,8,16,32,\\ldots\\right\\}$  \n",
+    "    <br />\n",
+    "    For this sequence, the terms are ever increasing. We frequently say that the sequence tends to infinity (or tends to $\\infty$). However, $\\infty$ represents an abstraction (\"ever increasing\"), not a specific number. Therefore, the limit for this sequence also does not exist.  \n",
+    "    <br />\n",
+    "4. $d_n = \\displaystyle\\frac{n+2}{n+1}: \\ \\ \\ \\left\\{2,\\displaystyle\\frac{3}{2},\\displaystyle\\frac{4}{3},\\displaystyle\\frac{5}{4},\\displaystyle\\frac{6}{5},\\ldots\\right\\}$  \n",
+    "    <br />\n",
+    "    For this sequence, each term is smaller than the previous (look at the decimal forms of the fractions: $2, 1.5, 1.333\\cdots, 1.25, 1.2 \\ldots$) and they are greater than $1$. We can conclude that:  \n",
+    "  \n",
+    "    $$\\lim a_n = 1.$$  \n",
+    "  \n",
+    "```  \n",
+    "  \n",
+    "Although there is a formal way to *prove* that a sequence $\\{a_n\\}$ converges to a certain limit, in practice we use a set of rules to apply limits to sequences and composition of sequences.  \n",
+    "  \n",
+    "```{tip}  \n",
+    "Consider two sequences $\\{a_n\\}$ and $\\{b_n\\}$, and $C$ as a real constant. If these two sequences are convergent with  $\\lim a_n = L$ and $\\lim b_n = M$, we have:  \n",
+    "  \n",
+    "$$\\lim (a_n + b_n) = \\lim a_n + \\lim b_n = L + M$$  \n",
+    "$$\\lim (Ca_n) = C\\lim a_n = CL$$  \n",
+    "$$\\lim (a_nb_n) = (\\lim a_n)(\\lim b_n) = LM$$  \n",
+    "$$\\lim \\left(\\frac{a_n}{b_n}\\right) = \\frac{\\lim a_n}{\\lim b_n} = \\frac{L}{M},\\ \\ (\\mbox{if}\\ M \\ne 0)$$\n",
+    "  \n",
+    "In other other words, if you can decompose a given sequence into the sum, product or division of two other convergent sequences, the limit of this sequence will be the composition of the limits of the two others.\n",
+    "```  \n",
+    "  \n",
+    "```{admonition} Examples  \n",
+    "If $a_n = \\displaystyle\\frac{4n^2-1}{n^2}:$  \n",
+    "  \n",
+    "$$\\lim\\left(\\displaystyle\\frac{4n^2-1}{n^2}\\right) = \\lim\\left(4-\\displaystyle\\frac{1}{n^2}\\right) = \\lim 4 - \\left(\\lim\\displaystyle\\frac{1}{n}\\right)\\left(\\lim\\displaystyle\\frac{1}{n}\\right) = 4,$$  \n",
+    "  \n",
+    "since as we know: $\\ \\lim 4 = 4\\ $ and $\\ \\lim \\displaystyle\\frac{1}{n} = 0$.\n",
+    "\n",
+    "```  \n",
+    "  \n",
+    "For recursions, one way to find the limit is to find the solution (corresponding explicit description) and then calculate the limit of the sequence with the limit rules. However, finding the explicit form of a given sequence is not always possible.  \n",
+    "  \n",
+    "There is a procedure to find *candidates* for the long-term behaviour of a given sequence. We do this by calculating **fixed points**.  \n",
+    "  \n",
+    "Fixed points are specific values on a given sequence for which all subsequent values equal the first. That is, if $a$ is a fixed point of the sequence $\\{a_n\\}$, and if $a_0 = a$, then $a_1 = a$, $a_2 = a$, and all the other values will be $a$. Thus, given the recursion $a_{n+1} = g(a_n)$, if $a$ is a fixed point we will have:  \n",
+    "  \n",
+    "$$a = g(a)$$  \n",
+    "  \n",
+    "and the last equation provides a way of find possible values of $a$.  \n",
+    "  \n",
+    "```{admonition} Example  \n",
+    "  \n",
+    "Consider the following recursion  \n",
+    "  \n",
+    "$$a_{n+1} = \\sqrt{3a_n},\\ \\ \\ a_0 = 2.$$  \n",
+    "  \n",
+    "To obtain the fixed points we calculate:  \n",
+    "  \n",
+    "$$\\begin{align}\n",
+    "  a &= \\sqrt{3a} \\\\\n",
+    "  a^2 &= 3a \\ \\ \\longrightarrow a = 0\\ \\mbox{ or }\\ a = 3 \\end{align}$$  \n",
+    "  \n",
+    "Starting with $a_0 = 2$, we can use the recursion to construct the sequence as:  \n",
+    "  \n",
+    "  $$\\left\\{\\sqrt{6},\\sqrt{3\\sqrt{6}},\\sqrt{3\\sqrt{3\\sqrt{6}}},\\sqrt{3\\sqrt{3\\sqrt{3\\sqrt{6}}}},\\ldots \\right\\}$$  \n",
+    "    \n",
+    "or also: $\\{2.449\\cdots,2.711\\cdots,2.852\\cdots,2.925\\cdots, \\ldots\\}$. We see that starting with $a_0=2$, we have $a_n>2$ for all $n$, with increasing values. Since $a=3$ is a fixed point, we conclude that $\\lim a_n = 3$.  \n",
+    "```  \n",
+    "\n",
+    "```{admonition} Example\n",
+    "Whenever we are asking questions like 'where does this process stabilise' or 'how high/much will it be in the end,' we are interested in the limit of some series. This could be a variety of things, like population size, ground water levels, global mean temperature, number of trees, etc.\n",
+    "```  \n",
+    "  \n",
+    "```{warning}  \n",
+    "  \n",
+    "Remember that fixed points are only **candidates** for long-term behaviour. See for example the recursion:  \n",
+    "  \n",
+    "$$a_{n+1} = \\displaystyle\\frac{3}{a_n}$$  \n",
+    "  \n",
+    "For the fixed points, we calculate:  \n",
+    "  \n",
+    "$$\\begin{align}\n",
+    "  a &= \\displaystyle\\frac{3}{a} \\\\\n",
+    "  a^2 &= 3 \\\\\n",
+    "  &\\longrightarrow a = -\\sqrt{3}\\ \\mbox{ or }\\ a = \\sqrt{3}. \\end{align}$$  \n",
+    "    \n",
+    "Indeed, if we construct the sequence from the recursion starting with any of these two values, all the subsequent elements of the sequence will have the same value. However:  \n",
+    "  \n",
+    "$$\\mbox{if } a_0 = 2 \\ \\mbox{the sequence will be given by: }\\ \\left\\{2,\\displaystyle\\frac{3}{2},2,\\displaystyle\\frac{3}{2},\\ldots \\right\\}$$  \n",
+    "  \n",
+    "or  \n",
+    "  \n",
+    "$$\\mbox{If } a_0 = -3 \\ \\mbox{the sequence will be given by: }\\ \\left\\{-3,-1,-3,-1,\\ldots \\right\\}$$  \n",
+    "  \n",
+    "showing that for any initial value chosen (different from the fixed points), the sequence will oscillate between two values, without a clear tendency to any limit.\n",
+    "```\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Discrete-time population models  \n",
+    "  \n",
+    "Sequences are generally used in many applications in Biology. Important examples are models for **seasonally breeding** populations.  \n",
+    "  \n",
+    "  \n",
+    "#### Density-dependent population growth  \n",
+    "  \n",
+    "As we have seen before, the recursion relation for the exponential growth model is given by $N_{t+1} = RN_t$. Rearranging the terms of this equation we have:  \n",
+    "  \n",
+    "$$\\displaystyle\\frac{N_t}{N_{t+1}}=\\frac{1}{R}.$$  \n",
+    "  \n",
+    "The left-hand term of the last equation is known as *parent-offspring ratio* as it denotes the ratio between the population size at time $t$ (parents) and the population size at the following time $t+1$ (offspring). Here we are considering a population that breeds without overlapping generations. Each generation breeds, dies, and is replaced by their offspring on the following generation.  \n",
+    "  \n",
+    "On the equation we also see that the parent-offspring ratio is equal to $\\displaystyle\\frac{1}{R}$, which is a constant. Therefore, we say that this growth is *density-independent* as the parent-offspring ratio does not depend on the density of individuals at any time step. Relating the parent-offspring ratio to the value of $R$ allows us to predict the behaviour of this population in terms of growth or decline:  \n",
+    "  \n",
+    "$$\\begin{align}\n",
+    "R>1 \\ &- \\ \\mbox{there will be more offspring than parents (population grows)} \\\\\n",
+    "R<1 \\ &- \\ \\mbox{there will be more parents than offspring (population declines)} \\\\\n",
+    "R=1 \\ &- \\ \\mbox{there will be no change in population size} \\end{align}$$\n",
+    "  \n",
+    "We know, however, than in real scenarios, that the growth factor of a given population should depend on the current size of that population, due to space or resources limitation. These are called *density-dependent* effects. One of the simplest methods to include density-dependent effects in this population model is to consider the parent-offspring ratio as a linear function of $N_t$, instead of a constant. Let us assume that the parent-offspring ratio show the following linear behaviour:  \n",
+    "  \n",
+    "```{image} parent-offspring.png\n",
+    "```  \n",
+    "  \n",
+    "This means that the parent-offspring ratio is smaller than $1$ (population growth) for values of $N_t$ smaller than a given population threshold $K$, and greater than $1$ (population decline) if the population size is greater than this threshold. The graph above leads to the following expression:  \n",
+    "  \n",
+    "$$\\begin{align}\n",
+    "\\displaystyle\\frac{N_t}{N_{t+1}} = \\frac{1}{R} + \\frac{1-\\frac{1}{R}}{K}N_t \\\\\n",
+    "N_{t+1} = \\frac{N_t}{\\frac{1}{R} + \\frac{1-\\frac{1}{R}}{K}N_t}, \\end{align}$$  \n",
+    "  \n",
+    "which leads, after rearranging the terms, to:  \n",
+    "  \n",
+    "$$N_{t+1} = \\frac{RN_t}{1 + \\frac{(R-1)}{K}N_t}.$$  \n",
+    "  \n",
+    "The previous expression is known as **Beverton-Holt recruitment curve** (or Beverton-Holt model) and it describes a type of population growth called *logistic growth*.  \n",
+    "  \n",
+    "The fixed points $\\bar{N}$ for this recursion can be calculated as:  \n",
+    "  \n",
+    "$$\\bar{N} = \\frac{R\\bar{N}}{1 + \\frac{(R-1)}{K}\\bar{N}}\\ \\longrightarrow \\ \\bar{N}\\left(1 + \\frac{(R-1)}{K}\\bar{N}\\right) = R\\bar{N}$$  \n",
+    "  \n",
+    "From the last term we can show (try it as an exercise) that $\\bar{N}=0$ and $\\bar{N}=K$ are the two fixed points. Starting from different values for the initial population size $N_0$ it can be shown that the population tends to $K$ as time increases as:  \n",
+    "  \n",
+    "```{image} logistic.png\n",
+    "```  \n",
+    "  \n",
+    "The constant $K$ is know as *carrying capacity* and represents the maximum population size a given population can reach at a given place, due to intra-specific competition for resources or space.  \n",
+    "  \n",
+    "  \n",
+    "<!-- Watch this [video](https://www.youtube.com/watch?v=Kas0tIxDvrg) about application of growth models to epidemics.   -->\n",
+    "  \n",
+    "  \n",
+    "#### Annual plants\n",
+    "  \n",
+    "Suppose a given population of plants produces, on average $f$ seeds per individual in the reproductive period. Seeds survive winter with a probability $\\sigma$ and germinate on the next season with probability $\\alpha$ (these seeds reach reproductive maturity on the same season). If $p_n$ is the plant population size on season $n$ and $p_{n+1}$ the population size on the season following $n$, we shall have:  \n",
+    "  \n",
+    "$$p_{n+1} = (\\alpha\\sigma f)p_n$$  \n",
+    "  \n",
+    "which correspond to the density-independent model since $R=\\alpha\\sigma f$ is a constant.  \n",
+    "  \n",
+    "Now suppose that some of the seeds that were produced on the previous season and survived the previous winter, but did not germinate on the current season, might have a second chance of germinating. They might do so on the following season, provided they survive the winter and germinate on the following season. From the average number of seeds produced last season $fp_{n-1}$, a number $\\sigma fp_{n-1}$ of them survived last winter. From this number, a fraction $(1-\\alpha)$ did not germinate this season (1 minus the probability of germinating). These seeds will have a second chance provided they survive next winter (with probability $\\sigma$) and germinate on the next season (with probability $\\alpha$). Thus, the number plant population size on the season following $n$ will be:  \n",
+    "  \n",
+    "$$\\begin{align}\n",
+    "p_{n+1} = \\alpha\\sigma fp_n + \\alpha\\sigma(1-\\alpha)\\sigma fp_{n-1} \\\\  \n",
+    "p_{n+1} = (\\alpha\\sigma f)p_n + [\\alpha(1-\\alpha)\\sigma^2 f]p_{n-1} \\\\  \n",
+    "p_{n+1} = \\beta p_n + \\gamma p_{n-1}\\end{align}$$  \n",
+    "  \n",
+    "where $\\beta=\\alpha\\sigma f$ and $\\gamma = \\alpha(1-\\alpha)\\sigma^2 f$.  \n",
+    "  \n",
+    "The previous model is an example of a discrete population model with overlapping populations. Since the term $n+1$ depends not only on the term $n$ but also on $n-1$ this is called a *second-order difference equation* or *delay (or lagged) difference equation*.  \n",
+    "  \n",
+    "  \n",
+    "#### Fibonacci sequence   \n",
+    "  \n",
+    "The Fibonacci sequence is a sequence defined by the following rule: each number is equal to the sum of the previous two numbers (starting with $0$ and $1$). Thus the following sequence might be formed:  \n",
+    "  \n",
+    "$$\\left\\{ 0,1,1,2,3,5,8,13,21,\\ldots \\right\\}. $$  \n",
+    "  \n",
+    "It can also be defined by the second-order difference equation:  \n",
+    "  \n",
+    "$$F_{n+1} = F_{n} + F_{n-1}.$$  \n",
+    "  \n",
+    "Although this is clearly not a convergent sequence (with sustained growth for each element), if we define a new sequence given by the ratio between successive elements of the Fibonacci sequence as $b_n = \\displaystyle\\frac{F_{n+1}}{F_n}$, it can be show that this sequence is convergent.  \n",
+    "  \n",
+    "Assuming that $\\lim \\displaystyle\\frac{F_{n+1}}{F_n} = \\lambda$, we have:  \n",
+    "  \n",
+    "$$\\begin{align}\n",
+    "F_{n+1} &= F_{n} + F_{n-1}\\quad \\mbox{(Fibonacci seq.)} \\\\  \n",
+    "\\displaystyle\\frac{F_{n+1}}{F_{n}} &= 1 + \\frac{F_{n-1}}{F_{n}}\\quad \\mbox{(divide both sides by}\\ F_{n}\\mbox{)} \\\\\n",
+    "\\lim \\left(\\displaystyle\\frac{F_{n+1}}{F_{n}}\\right) &= \\lim\\left(1 + \\frac{F_{n-1}}{F_{n}}\\right) \\\\\n",
+    "\\lambda &= 1 + \\frac{1}{\\lambda} \\\\\n",
+    "\\lambda^2 &- \\lambda -1 = 0 \\end{align}$$\n",
+    "  \n",
+    "The previous equation is solved finding the roots of the second-degree polynomial, which gives $\\lambda = \\displaystyle\\frac{1+\\sqrt{5}}{2}$ or $\\lambda = \\displaystyle\\frac{1-\\sqrt{5}}{2}$. Since the solution $\\displaystyle\\frac{1-\\sqrt{5}}{2}$ is negative, it can be the limit of the ratio of Fibonacci terms (all of them positive). Hence we conclude:  \n",
+    "  \n",
+    "$$\\lim \\left(\\displaystyle\\frac{F_{n+1}}{F_{n}}\\right) = \\displaystyle\\frac{1+\\sqrt{5}}{2} \\approx 1.618.$$  \n",
+    "  \n",
+    "The previous number is known as the *Golden ratio* and it has fascinated mathematicians, phylosophers, and artists since ancient times. Fibonacci numbers and the Golden ratio can be found in several interesting natural patterns, as you can see in this [video](https://www.youtube.com/watch?v=ahXIMUkSXX0)."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## From table arrangements to powerful tools"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Systems of linear equations  \n",
+    "  \n",
+    "As we saw previously, the equation $ax + by = c$, with $a$, $b$, and $c$ constants, defines a straight line on the $xy$-plane (note that it can also be written as $y = mx + k$, $m$ and $k$ constants, by a suitable change of the coeffients). Since the coefficients do not depend on the variables $x$ and $y$, we call it a *linear equation*. A possible plot for this equation would be:  \n",
+    "  \n",
+    "```{image} lineq.png\n",
+    "```  \n",
+    "  \n",
+    "Note that all the points $(x,y)$ on the blue line *satisfy* or *solve* the lienar equation $ax + by = c$.  \n",
+    "  \n",
+    "Now consider that we have the following:  \n",
+    "  \n",
+    "$$\\begin{cases} ax + by = c \\\\ dx +ey = f \\end{cases}$$  \n",
+    "  \n",
+    "where $a, b, c, d, e,\\ \\mbox{and}\\ f$ are all constants.In this case we have a *system of linear equations*. Solving this system means to find all possible pairs $(x,y)$ that solve both equations **simultaneously**. For the previous system, we would have 3 options:\n",
+    "  \n",
+    "```{image} syslineq.png\n",
+    "```  \n",
+    "  \n",
+    "On the left panel, the two lines intersect at a *single point*. This point is the only one that solves both linear equations at the same time, and, therefore, constitutes the **only solution** for the system of two equations. On the central panel, the two lines are *parallel*, so there is no point of intersection. In this case there is **no solution** and the system of equations is said to be *inconsistent*. On the right panel, the two lines coincide. Thus, every point in those lines correspond to a possible solution for the system, so that it has **infinite solutions**.  \n",
+    "  \n",
+    "   \n",
+    "```{admonition} Example  \n",
+    "  \n",
+    "Given the system of linear equations  \n",
+    "  \n",
+    "$$\\begin{cases} 2x + 3y = 6 \\\\ 2x + y = 4 \\end{cases}$$\n",
+    "\n",
+    "we can obtain the solution by isolating one of the variables in one of the equations and substituting on the other equation. Isolating $x$ on the first equation gives:  \n",
+    "  \n",
+    "$$x = 3 - \\displaystyle\\frac{3}{2}y$$  \n",
+    "  \n",
+    "Then by substituting on the second we have:  \n",
+    "  \n",
+    "$$\\begin{align} \n",
+    "2\\left(3 - \\displaystyle\\frac{3}{2}y \\right) + y &= 4 \\\\\n",
+    "6 - 3y + y &= 4 \\\\\n",
+    "-2y = -2 \\\\\n",
+    "y = 1 \\end{align}$$  \n",
+    "  \n",
+    "The value of $x$ shall then be given by:  \n",
+    "  \n",
+    "$$x = 3 - \\displaystyle\\frac{3}{2}\\cdot 1 = 3 - \\frac{3}{2} = \\frac{3}{2}$$  \n",
+    "  \n",
+    "We see that the system has a single solution, given by $\\left(\\displaystyle\\frac{3}{2},1\\right)$.  \n",
+    "  \n",
+    "```  \n",
+    "  \n",
+    "````{admonition} Example  \n",
+    "  \n",
+    "Now consider the sytem  \n",
+    "  \n",
+    "$$\\begin{cases} 2x + 3y &= 6 \\\\ -4x + -6y &= -12 \\end{cases}$$\n",
+    "\n",
+    "Isolating $x$ on the first equation gives the same as before:  \n",
+    "  \n",
+    "$$x = 3 - \\displaystyle\\frac{3}{2}y$$  \n",
+    "  \n",
+    "Then by substituting on the second we have:  \n",
+    "  \n",
+    "$$\\begin{align} \n",
+    "-4\\left(3 - \\displaystyle\\frac{3}{2}y \\right) - 6y &= -12 \\\\\n",
+    "-12 + 6y - 6y &= -12 \\\\\n",
+    "0 = 0 \\end{align}$$  \n",
+    "  \n",
+    "which is just a trivial equation.  \n",
+    "  \n",
+    "When we look at the equations in our system we see that the second equation is simply a constant multiplied by the first $(\\text{Eqn.} 2) = -2\\cdot(\\text{Eqn.} 1)$, and thus they should represent the same straight line.  \n",
+    "  \n",
+    "This system has then infinite solutions which are represented by the set of points $\\left(3 - \\displaystyle\\frac{3}{2}y,y\\right)$\n",
+    "  \n",
+    "```{tip}\n",
+    "Note that writing the solution as $\\left(3 - \\displaystyle\\frac{3}{2}y,y\\right)$ represents an infinite set of points, since for every real value of $y$ we would have a different ordered pair. This set of points forms the straight line corresponding to $2x+3y = 6$.\n",
+    "\n",
+    "```\n",
+    "  \n",
+    "````  \n",
+    "  \n",
+    "Solving the system by this method of substitutions, however, can get much more complicated if we have more than two variables. For these cases, we need a simpler method.  \n",
+    "  \n",
+    "One strategy is to find equivalent systems, which have the same solution as the original one, but are much easier to solve. We can apply the following set of operations without changing the solutions of a given system:  \n",
+    "  \n",
+    "- Multiplication of an entire row by a non-zero constant  \n",
+    "- Adding/subtracting one row to another  \n",
+    "- Rearranging the order of the rows  \n",
+    "  \n",
+    "The examples below show this procedure applied to two systems.  \n",
+    "  \n",
+    "```{admonition} Examples  \n",
+    "  \n",
+    "(i) $\\begin{cases} 3x + 5y - z = 10 \\\\ 2x –  y + 3z = 9 \\\\ 4x + 2y - 3z = -1 \\end{cases}$  \n",
+    "    \n",
+    "(ii) $\\begin{cases} x - 3y + z = 4 \\\\ x – 2y + 3z = 6 \\\\ 2x - 6y + 2z = 8 \\end{cases}$  \n",
+    "  \n",
+    "\n",
+    "```  \n",
+    "\n",
+    "<!-- See the solution on this [video](https://web.microsoftstream.com/video/194b0152-903a-4614-b608-306ce71d9a39) -->\n",
+    "\n",
+    "\n",
+    "<!-- See the solution on this [video](https://web.microsoftstream.com/video/067c8938-5e45-4f08-bf90-783cb125c6f5) -->\n",
+    "\n",
+    "Note that by applying the rules of matrix multiplication, the system *(ii)* can also be written in the form:  \n",
+    "  \n",
+    "$$\\begin{pmatrix} 1 & -3 & 1\\\\ 1 & -2 & 3 \\\\ 2 & -6 & 2  \\end{pmatrix} \\begin{pmatrix} x \\\\ y \\\\ z   \\end{pmatrix} = \\begin{pmatrix} 4 \\\\ 6 \\\\ 8   \\end{pmatrix} \\quad \\longrightarrow \\quad A\\mathbf{x}=\\mathbf{b}$$  \n",
+    "  \n",
+    "where $A$ is the matrix of coefficients and $\\mathbf{x}$ is the vector of variables.  \n",
+    "  \n",
+    "```{tip}  \n",
+    "We will be using the common notation of uppercase letters ($A$) to represent matrices and lowercase bold letters ($\\mathbf{x}$) to represent vectors.\n",
+    "```\n",
+    "\n",
+    "In general, the system with $n$ variables $\\{x_1, x_2, \\ldots, x_n \\}$ and $m$ equations can be written as  \n",
+    "  \n",
+    "$$\\begin{cases} a_{11}x_1 + a_{12}x_2 + \\ldots + a_{1n}x_n = b_1 \\\\ a_{21}x_1 + a_{22}x_2 + \\ldots + a_{2n}x_n = b_2 \\\\ \\dots \\\\ a_{m1}x_1 + a_{m2}x_2 + \\ldots + a_{mn}x_n = b_m \\end{cases}$$  \n",
+    "  \n",
+    "can be written as $A\\mathbf{x}=\\mathbf{b}$, where  \n",
+    "  \n",
+    "$$A=\\begin{pmatrix} a_{11} & a_{12} & \\cdots & a_{1n} \\\\ a_{21} & a_{22} & \\cdots & a_{2n} \\\\ \\vdots & & \\ddots & \\vdots \\\\ a_{m1} & a_{m2} & \\cdots & a_{mn} \\end{pmatrix}$$  \n",
+    "  \n",
+    "and  \n",
+    "  \n",
+    "$$\\mathbf{x} = \\begin{pmatrix} x_1 \\\\ x_2 \\\\ \\vdots \\\\ x_n   \\end{pmatrix}\\quad \\mbox{and} \\quad \\mathbf{b} = \\begin{pmatrix} b_1 \\\\ b_2 \\\\ \\vdots \\\\ b_m   \\end{pmatrix}$$  \n",
+    "  \n",
+    "The matrix $A$ has size $m \\times n$ (since the number of equations can, in principle, be different from the number of variables), while the vectors $\\mathbf{x}$ and $\\mathbf{b}$ have sizes $n \\times 1$ and $m \\times 1$, respectively.  \n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Matrix operations  \n",
+    "  \n",
+    "Let us quickly remind the basic operations that can be performed with matrices. If $A$ and $B$ are both matrices of size $m \\times n$ ($m$ rows and $n$ columns), we have:  \n",
+    "  \n",
+    "- $A=B\\ $ if and only if $a_{ij} = b_{ij}$ for every $1 \\le i \\le m,  1 \\le j \\le n$ (elements at the same position have to be equal).  \n",
+    "  \n",
+    "- $C = A + B$ is the matrix for which $c_{ij} = a_{ij} + b_{ij}$ (sum is performed summing the elements at the same position)  \n",
+    "  \n",
+    "- If $k \\in \\rm I\\!R$ than $kA$ is the $m \\times n$ matrix with elements $ka_{ij}$  \n",
+    "  \n",
+    "- The following properties are valid:  \n",
+    "    \n",
+    "    $$\\begin{align}\n",
+    "    A+B &= B+A \\\\\n",
+    "    (A+B)+C &= A+(B+C) \\\\\n",
+    "    A + 0 &= A\\ (0\\ \\mbox{here represents the matrix with null elements}) \\end{align}$$  \n",
+    "  \n",
+    "    ```{admonition} Example  \n",
+    "    If $A=\\begin{pmatrix} 1 & 0  \\\\ -1 & 2  \\end{pmatrix}$ and $B=\\begin{pmatrix} 2 & 3  \\\\ -1 & 1  \\end{pmatrix}$, then:  \n",
+    "\n",
+    "    $$C = 2A+B = \\begin{pmatrix} 2 & 0  \\\\ -2 & 4  \\end{pmatrix} + \\begin{pmatrix} 2 & 3  \\\\ -1 & 1  \\end{pmatrix} = \\begin{pmatrix} 4 & 3  \\\\ -3 & 5  \\end{pmatrix}$$\n",
+    "    ```  \n",
+    "      \n",
+    "- If $A$ is the $m \\times n$ matrix with elements $a_{ij}$, the transpose of $A$ (denoted by $A'$) is the $n \\times m$ matrix with elements $a_{ij}' = a_{ji}$.  \n",
+    "  \n",
+    "    ```{admonition} Example  \n",
+    "    $$A=\\begin{pmatrix} 1 & 2 & 3 \\\\ 4 & 5 & 6  \\end{pmatrix} \\longrightarrow A'=\\begin{pmatrix} 1 & 4 \\\\ 2 & 5 \\\\ 3 & 6   \\end{pmatrix}$$\n",
+    "      \n",
+    "    $$B=\\begin{pmatrix} 1 & 2 \\end{pmatrix} \\longrightarrow B' = \\begin{pmatrix} 1 \\\\ 2 \\end{pmatrix}$$\n",
+    "    ```  \n",
+    "      \n",
+    "- **(Matrix multiplication)** Consider now that $A$ and $B$ are matrices with sizes $m \\times l$ and $l \\times n$ (number of columns of $A$ is equal to the number of rows of $B$). Then we can define $C = AB$  as the $m \\times n$ matrix with elements $c_{ij}$ formed by multiplying every element of the $i$-th row of $A$ with the corresponding element of the $j$-th column of $B$.  \n",
+    "    \n",
+    "    ```{admonition} Example  \n",
+    "      If $A=\\begin{pmatrix} 1 & 2  \\\\ -1 & 0  \\end{pmatrix}$ and $B=\\begin{pmatrix} 1 & 2 & 3  \\\\ 0 & -1 & 4  \\end{pmatrix}$, then:  \n",
+    "        \n",
+    "      $$C = AB = \\begin{pmatrix} 1 & 0 & 11  \\\\ -1 & -2 & -3  \\end{pmatrix}.$$  \n",
+    "        \n",
+    "      Note that the matrix $BA$ cannot be defined here since the number of columns of $B$ is different from the number of rows of $A$.\n",
+    "    ```  \n",
+    "      \n",
+    "    ```{warning}  \n",
+    "    If matrices $A$ and $B$ are *square matrices* (same number of rows and columns) then we can define both $AB$ and $BA$. But, in general, we **cannot assume** that $AB=BA$. For example, if  $A=\\begin{pmatrix} 2 & -1  \\\\ 4 & -2  \\end{pmatrix}$ and $B=\\begin{pmatrix} 1 & -1  \\\\ 2 & -2  \\end{pmatrix}$, then:  \n",
+    "      \n",
+    "    $$AB = \\begin{pmatrix} 0 & 0  \\\\ 0 & 0  \\end{pmatrix} \\quad and \\quad BA = \\begin{pmatrix} -2 & 1  \\\\ -4 & 2  \\end{pmatrix}$$\n",
+    "    ```  \n",
+    "      \n",
+    "- For matrix multiplication, the following properties are valid:  \n",
+    "    \n",
+    "    $$\\begin{align}\n",
+    "    (A+B)C&=AC+BC \\\\\n",
+    "    A(B+C)&=AB+AC \\\\\n",
+    "    (AB)C &= A(BC) \\\\\n",
+    "    A0 &= 0A = 0 \\end{align}$$  \n",
+    "  \n",
+    "- We can also define powers of $A$. If $k$ is a positive integer:  \n",
+    "    \n",
+    "    $$A^k = A^{k-1}A = AA^{k-1} = A\\cdot A\\cdot A \\cdots A\\ (\\mbox{product of}\\ k\\ \\mbox{matrices})$$\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Inverse matrix  \n",
+    "  \n",
+    "Suppose that $A$ is a $n \\times n$ matrix. If there exists a $n \\times n$ matrix $B$ so that  \n",
+    "  \n",
+    "$$ AB = BA = I_n,$$  \n",
+    "  \n",
+    "where $I=\\begin{pmatrix} 1 & 0 & \\cdots & 0 \\\\ 0 & 1 & \\cdots & 0 \\\\ \\vdots & & \\ddots & \\vdots \\\\ 0 & 0 & \\cdots & 1 \\end{pmatrix}$ is the $n \\times n$ *identity matrix* (with elements $1$ on the main diagonal and $0$ otherwise). \n",
+    "  \n",
+    "In this case we call the matrix $B$ the *inverse* of $A$ and denote $B=A^{-1}$. If $A$ has an inverse than it is called *invertible* or *non-singular*. If it does not have an inverse, $A$ is *singular*.\n",
+    "\n",
+    "Finding the inverse of $A$ automatically gives us the solution to the system of linear equations $A\\mathbf x = \\mathbf x$\n",
+    "\n",
+    "$$\\begin{align}  \n",
+    "A\\mathbf{x}&=\\mathbf{b} \\\\\n",
+    "A^{-1} A\\mathbf{x} &= A^{-1}\\mathbf{b} \\\\\n",
+    "\\mathbf{x} &= A^{-1}\\mathbf{b} \\end{align}$$  \n",
+    "\n",
+    "Finding ways of inverting matrices is therefore needed for a wide range of applications. A whole research field is dealing with optimising matrix inversions. \n",
+    "  \n",
+    "```{admonition} Example  \n",
+    "  \n",
+    "To find the inverse of matrix $A= \\begin{pmatrix} 2 & 5  \\\\ 1 & 3  \\end{pmatrix}$, we have to find the matrix $B = \\begin{pmatrix} b_{11} & b_{12}  \\\\ b_{21} & b_{22}  \\end{pmatrix}$ so that  \n",
+    "  \n",
+    "$$ AB = I \\quad (\\mbox{then check that }\\ BA = I.$$  \n",
+    "  \n",
+    "Thus  \n",
+    "  \n",
+    "$$\\begin{align}  \n",
+    "  \\begin{pmatrix} 2 & 5  \\\\ 1 & 3  \\end{pmatrix}\\begin{pmatrix} b_{11} & b_{12}  \\\\ b_{21} & b_{22}  \\end{pmatrix} = \\begin{pmatrix} 1 & 0  \\\\ 0 & 1  \\end{pmatrix} \\\\  \n",
+    "  \\begin{pmatrix} 2b_{11} + 5b_{21}  & 2b_{12} + 5b_{22} \\\\ b_{11} + 3b_{21} & b_{12} + 3b_{22}  \\end{pmatrix} = \\begin{pmatrix} 1 & 0  \\\\ 0 & 1  \\end{pmatrix}, \\\\\n",
+    "\\end{align}$$  \n",
+    "  \n",
+    "which leads to two systems of equations:  \n",
+    "\n",
+    "$$\\begin{cases} 2b_{11} + 5b_{21} &= 1 \\\\ b_{11} + 3b_{21} &= 0 \\end{cases}\\quad \\mbox{and}\\quad \\begin{cases} 2b_{12} + 5b_{22} &= 0 \\\\ b_{12} + 3b_{22} &= 1 \\end{cases}$$  \n",
+    "  \n",
+    "The solutions of these two systems, leads to $B= \\begin{pmatrix} 3 & -5  \\\\ -1 & 2  \\end{pmatrix}$\n",
+    "```  \n",
+    "  "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Determinant  \n",
+    "  \n",
+    "If $A = \\begin{pmatrix} a_{11} & a_{12}  \\\\ a_{21} & a_{22}  \\end{pmatrix}$ then the expression:  \n",
+    "  \n",
+    "$$\\mbox{det}A = a_{11}a_{22} - a_{21}a_{12}$$   \n",
+    "  \n",
+    "given by the product of the elements on the main diagonal minus the product of the elements on secondary diagonal. The term $\\mbox{det}A$ is called the *determinant* of $A$. The determinant can be calculated for matrices of any size, but the expression gets increasingly more complicated for larger matrices.  \n",
+    "  \n",
+    "The determinant establishes a clear criterium for the existence of $A^{-1}$. The matrix $A$ will be invertible if and only if $\\mbox{det}A \\ne 0$.  \n",
+    "  \n",
+    "```{note}  \n",
+    "The previous criterium has an important implication. Consider the system of linear equations defined by $A\\mathbf{x}=\\mathbf{b}$ with equal number of equations and variables. If $\\mbox{det}A \\ne 0$ then the inverse of $A$ exists. Thus, if we multiply $A^{-1}$ to the left of each side of the equation defining the linear system, we have:  \n",
+    "  \n",
+    "$$\\begin{align}  \n",
+    "A\\mathbf{x}&=\\mathbf{b} \\\\\n",
+    "A^{-1}(A\\mathbf{x}) &= A^{-1}\\mathbf{b} \\\\\n",
+    "(A^{-1}A)\\mathbf{x} &= A^{-1}\\mathbf{b} \\\\\n",
+    "\\mathbf{x} &= A^{-1}\\mathbf{b} \\end{align}$$  \n",
+    "  \n",
+    "which means that if $\\mbox{det}A \\ne 0$ the system will have a **single solution** with the variables assuming the values $\\mathbf{x} = A^{-1}\\mathbf{b}$.\n",
+    "```"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Structured models  \n",
+    "  \n",
+    "In the models we have seen until now, all the individuals in a given population are identical. However, real populations have importance differences between individuals according to their life stages. For example, some insects have stages of eggs, larvae, pupae, and adults, while many plants pass through stages of seeds, seedlings, saplings, and mature individuals. Thus, understanding the differences in the dynamics of each life stage is important to add realism to our models (especially in applied models, such as pest control or harvesting management).  \n",
+    "  \n",
+    "Suppose our population is composed by two classes: *immature* and *mature* individuals. Let us assume non-overlapping breeding seasons and that the timescales for reproduction and maturation are similar. Thus, time can be counted in discrete steps corresponding to this reproduction/maturation interval. At time $t$ we have $X_t$ immature individuals and $Y_t$ mature individuals. Immature individuals survive until maturity with a probability $p$, while mature individuals reproduce and generate, on average, $m$ immature individuals for the next generation. Mature individuals die after reproduction. With these assumptions, we can calculate the sizes of each population class on the step $t+1$ as:\n",
+    "  \n",
+    "$$\\begin{align}\n",
+    "X_{t+1} = m Y_t \\\\\n",
+    "Y_{t+1} = p X_t \\end{align}$$  \n",
+    "  \n",
+    "See that now we have a model with two equations describing the variable class sizes in our population. Moreover, the dynamics of each class depends on the other, so that we have a system of *coupled* difference equations.  \n",
+    "\n",
+    "Interestingly, we can also write this system in matrix form, as:  \n",
+    "  \n",
+    "$$\\begin{pmatrix} X_{t+1}  \\\\ Y_{t+1} \\end{pmatrix} = \\begin{pmatrix} 0 & m  \\\\ p & 0  \\end{pmatrix} \\begin{pmatrix} X_{t}  \\\\ Y_{t} \\end{pmatrix}, \\quad \\mbox{or}$$\n",
+    "\n",
+    "$$\\mathbf{n_{t+1}} = P\\mathbf{n_{t}}$$\n",
+    "  \n",
+    "where the sum of the components of the population vector $\\mathbf{n_{t}}$, gives the total population (immature plus mature individuals) at time $t$: $N_t = X_t + Y_t$. The matrix $P=\\begin{pmatrix} 0 & m  \\\\ p & 0  \\end{pmatrix}$ is called *projection matrix* or *Leslie matrix*. Multiplying matrix $P$ by the population vector at time $t$, $\\mathbf{n_{t}}$, we can project the next population size at time $t+1$. Thus, from an initial population state, say $\\mathbf{n_{0}}$, the projection matrix governs the dynamics of the model.  \n",
+    "  \n",
+    "<br />\n",
+    "\n",
+    "This model can be easily generalised for a larger number of age classes. Suppose we now have $q+1$ classes, $X_0, X_1, X_2, \\ldots, X_q$, from the youngest (meaningful) age $X_0$ until the oldest age $X_q$. At each time step $t$, individuals from class $X_i$ have a chance of surviving to the next age class $X_{i+1}$, with probability $p_{i+1\\ i}$ (imagine an arrow for direction of age class progession $p_{i+1\\leftarrow i}$. We are going to omit the arrow for simplicity). Additionally, suppose that each age class $X_i$ excluding the youngest, will, in principle, be able to reproduce, with average fecundity $m_i$ generating new individuals for class $X_0$. Thus, with the age class sizes at time $t$, and the parameters for survival probabilities and average fecundities, we have:  \n",
+    "  \n",
+    "$$\\begin{align}\n",
+    "X_0(t+1) &= m_1X_1(t) +  m_2X_2(t) + m_3X_3(t) + \\ldots + m_qX_q(t) \\\\\n",
+    "X_1(t+1) &= p_{10}X_0(t) \\\\\n",
+    "X_2(t+1) &= p_{21}X_1(t) \\\\\n",
+    "X_3(t+1) &= p_{32}X_2(t) \\\\\n",
+    "         &\\vdots \\\\\n",
+    "X_q(t+1) &= p_{q\\ q-1}X_{q-1}(t)\n",
+    "    \\end{align}$$  \n",
+    "    \n",
+    "This can also be written in matrix form as:  \n",
+    "  \n",
+    "$$\\begin{pmatrix} X_0(t+1)  \\\\ X_1(t+1) \\\\X_2(t+1)  \\\\X_3(t+1) \\\\ \\vdots \\\\X_q(t+1)  \\\\ \\end{pmatrix} = \\begin{pmatrix} 0 & m_1 & m_2 & m_3 & \\cdots & m_q \\\\ p_{10} & 0 & 0 & 0 & \\cdots &0 \\\\ 0 & p_{21} & 0 & 0 & \\cdots & 0 \\\\ 0 & 0 & p_{32} & 0 & \\cdots & 0 \\\\ \\vdots & \\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\ 0 & 0 & 0 & \\cdots & p_{q\\ q-1} & 0\\end{pmatrix} \\begin{pmatrix} X_0(t)  \\\\ X_1(t) \\\\X_2(t) \\\\X_3(t) \\\\ \\vdots \\\\X_q(t)  \\\\ \\end{pmatrix}$$  \n",
+    "  \n",
+    "$$\\mathbf{n_{t+1}} = P\\mathbf{n_{t}}$$\n",
+    "  \n",
+    "The generalised projection matrix $P$ has then all the surviving probabilities just below the main diagonal, and the fecundities for each class in its $0$-th row.  \n",
+    "  \n",
+    "```{note}  \n",
+    "Two important differences here. First, the rows and columns for the matrix are indexed starting from $0$ until $q$, not starting from $1$ as usual. Second, the time steps are written in parentheses for each age class, instead of a subscript as we have seen before for other population models, so that it does not conflict with the index for the age classes (going from $0$ to $q$).\n",
+    "```  \n",
+    "  \n",
+    "```{admonition} Example  \n",
+    "Consider a population with three age classes $X_0, X_1,\\mbox{ and}\\ X_2$, and the following projection matrix  \n",
+    "  \n",
+    "$$P=\\begin{pmatrix} 0 & 5 & 10\\\\ 0.5 & 0 & 0  \\\\ 0 & 0.2 & 0 \\end{pmatrix}$$\n",
+    "\n",
+    "  \n",
+    "If the population starts with $10$ individuals in the oldest category at time $t=0$, the population vector at time $t=1$ will be:  \n",
+    "  \n",
+    "$$\\mathbf{n_{1}} = P\\mathbf{n_{0}} \\longrightarrow \\begin{pmatrix} X_0(1)  \\\\ X_1(1) \\\\X_2(1) \\end{pmatrix}=\\begin{pmatrix} 0 & 5 & 10\\\\ 0.5 & 0 & 0  \\\\ 0 & 0.2 & 0 \\end{pmatrix}\\begin{pmatrix} 0  \\\\ 0 \\\\ 10 \\end{pmatrix} \\longrightarrow \\begin{pmatrix} 100  \\\\ 0 \\\\ 0 \\end{pmatrix}$$  \n",
+    "  \n",
+    "The population age distribution at each time step will then be given be the successive application of the matrix $P$ to the previous population distribution. For the first 10 time steps we have:  \n",
+    "  \n",
+    "$$\\begin{pmatrix}0 \\\\ 0 \\\\ 10 \\end{pmatrix} \\rightarrow \\begin{pmatrix} 100  \\\\ 0 \\\\ 0 \\end{pmatrix} \\rightarrow \\begin{pmatrix} 0  \\\\ 50 \\\\ 0 \\end{pmatrix} \\rightarrow \\begin{pmatrix} 250  \\\\ 0 \\\\ 10 \\end{pmatrix} \\rightarrow \\begin{pmatrix} 100  \\\\ 125 \\\\ 0 \\end{pmatrix} \\rightarrow \\begin{pmatrix} 625  \\\\ 50 \\\\ 25 \\end{pmatrix} \\rightarrow \\begin{pmatrix} 500  \\\\ 312 \\\\ 10 \\end{pmatrix} \\rightarrow \\begin{pmatrix} 1662  \\\\ 250 \\\\ 62 \\end{pmatrix} \\rightarrow \\begin{pmatrix} 1875  \\\\ 831 \\\\ 50 \\end{pmatrix} \\rightarrow \\begin{pmatrix} 4656  \\\\ 937 \\\\ 166 \\end{pmatrix}$$  \n",
+    "  \n",
+    "Two important things are worth noting here. First, the total population size (sum accros all age classes), follows the sequence $N_t = \\left\\{ 10,100,50,260,225,700,822,1975,2756,5759,\\ldots \\right \\}$. The reproduction process does not always lead to increasing population sizes in the beginning, but is subject to size fluctuations due to the demographic distribution of this population. After a few steps increases in population size become more consistent. Second, as the population size continues to increase, the population age distribution tends to a given proportion of the population classes given by: $X_0 \\approx 76\\%, X_1 \\approx 22\\%, \\mbox{and}\\ X_2 \\approx 2\\%$.  \n",
+    "  \n",
+    "Now consider that, instead of $10$ individuals in the oldest class, we initiate the population with the following distribution $\\mathbf{n_{0}} = \\begin{pmatrix} 7.6  \\\\ 2.2 \\\\ 0.2 \\end{pmatrix}$. The population progression by successive application of the projection matrix will be  \n",
+    "  \n",
+    "$$\\begin{pmatrix} 7.6  \\\\ 2.2 \\\\ 0.2 \\end{pmatrix} \\longrightarrow \\begin{pmatrix} 13  \\\\ 4 \\\\ 0.4 \\end{pmatrix} \\longrightarrow \\begin{pmatrix} 23  \\\\ 6 \\\\ 0.76 \\end{pmatrix} \\longrightarrow \\begin{pmatrix} 40  \\\\ 12 \\\\ 1 \\end{pmatrix} \\longrightarrow \\begin{pmatrix} 72  \\\\ 20 \\\\ 2 \\end{pmatrix} \\longrightarrow \\begin{pmatrix} 124  \\\\ 36 \\\\ 4 \\end{pmatrix} \\longrightarrow \\begin{pmatrix} 219  \\\\ 62 \\\\ 7 \\end{pmatrix} \\longrightarrow \\begin{pmatrix} 381  \\\\ 109 \\\\ 12 \\end{pmatrix} $$  \n",
+    "  \n",
+    "You can check that although each age cathegory is increasing, the percentage contribution of each class to the total population remains nearly constant. To this distribution of classes we give the name of **stable age distribution**.  \n",
+    "  \n",
+    "Moreover, at each time step, each age class increases with approximately the same factor in relation to the previous step, around $\\lambda = 1.75$. If for each age class $X_i(t+1) = \\lambda X_i(t)$ is valid,then we have:  \n",
+    "  \n",
+    "$$\\mathbf{n_{t+1}}=\\lambda\\mathbf{n_{t}},$$  \n",
+    "  \n",
+    "so that we recover the usual model for exponential growth for the total population size $N_{t+1} = \\lambda N_t$.  \n",
+    "\n",
+    "```  "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 40,
+   "metadata": {
+    "tags": [
+     "remove-cell"
+    ]
+   },
+   "outputs": [
+    {
+     "data": {
+      "image/png": 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",
+      "text/plain": [
+       "<Figure size 360x360 with 1 Axes>"
+      ]
+     },
+     "metadata": {
+      "needs_background": "light"
+     },
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "import matplotlib.pyplot as plt\n",
+    "import matplotlib.ticker as plticker\n",
+    "import numpy as np\n",
+    "\n",
+    "X = np.array((0,0,0,0))\n",
+    "Y= np.array((0,0,0,0))\n",
+    "U = np.array((3,2,1,8))\n",
+    "V = np.array((-1,3,7,12))\n",
+    "\n",
+    "fig, ax = plt.subplots(figsize=(5, 5))\n",
+    "q = ax.quiver(X, Y, U, V,units='xy', scale=1)\n",
+    "\n",
+    "ax.spines['left'].set_position('zero')\n",
+    "ax.spines['bottom'].set_position('zero')\n",
+    "\n",
+    "# Eliminate upper and right axes\n",
+    "ax.spines['right'].set_color('none')\n",
+    "ax.spines['top'].set_color('none')\n",
+    "\n",
+    "# Show ticks in the left and lower axes only\n",
+    "ax.xaxis.set_ticks_position('bottom')\n",
+    "ax.yaxis.set_ticks_position('left')\n",
+    "\n",
+    "ax.set_aspect('equal')\n",
+    "\n",
+    "ax.annotate('$\\mathbf{v_1}$', (3+0.2,-1-0.8),fontsize=14)\n",
+    "ax.annotate('$\\mathbf{v_2}$', (2+0.2,3-0.8),fontsize=14)\n",
+    "ax.annotate('$A\\mathbf{v_1}$', (1+0.2,7-0.8),fontsize=14)\n",
+    "ax.annotate('$A\\mathbf{v_2}$', (8+0.2,12-0.8),fontsize=14)\n",
+    "\n",
+    "ax.tick_params(axis='both', which='major', labelsize=13)\n",
+    "\n",
+    "loc = plticker.MultipleLocator(base=2) # this locator puts ticks at regular intervals\n",
+    "ax.xaxis.set_major_locator(loc)\n",
+    "\n",
+    "ax.set_ylim(-2,14)\n",
+    "ax.set_xlim(-2,10)\n",
+    "\n",
+    "plt.tight_layout()\n",
+    "plt.savefig('lin_trans.png')"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 46,
+   "metadata": {
+    "tags": [
+     "remove-cell"
+    ]
+   },
+   "outputs": [
+    {
+     "data": {
+      "image/png": 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",
+      "text/plain": [
+       "<Figure size 360x360 with 1 Axes>"
+      ]
+     },
+     "metadata": {
+      "needs_background": "light"
+     },
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "import matplotlib.pyplot as plt\n",
+    "import matplotlib.ticker as plticker\n",
+    "import numpy as np\n",
+    "\n",
+    "X = np.array((0,0))\n",
+    "Y= np.array((0,0))\n",
+    "U = np.array((1,2/3))\n",
+    "V = np.array((-1,1))\n",
+    "\n",
+    "fig, ax = plt.subplots(figsize=(5, 5))\n",
+    "q = ax.quiver(X, Y, U, V,units='xy', scale=1)\n",
+    "\n",
+    "ax.spines['left'].set_position('zero')\n",
+    "ax.spines['bottom'].set_position('zero')\n",
+    "\n",
+    "# Eliminate upper and right axes\n",
+    "ax.spines['right'].set_color('none')\n",
+    "ax.spines['top'].set_color('none')\n",
+    "\n",
+    "# Show ticks in the left and lower axes only\n",
+    "ax.xaxis.set_ticks_position('bottom')\n",
+    "ax.yaxis.set_ticks_position('left')\n",
+    "\n",
+    "ax.set_aspect('equal')\n",
+    "\n",
+    "ax.annotate('$\\mathbf{v_1}\\ (\\lambda=-1)$', (1+0.1,-1-0.1),fontsize=14)\n",
+    "ax.annotate('$\\mathbf{v_2}\\ (\\lambda=4)$', (2/3+0.1,1+0.1),fontsize=14)\n",
+    "\n",
+    "ax.tick_params(axis='both', which='major', labelsize=13)\n",
+    "\n",
+    "#loc = plticker.MultipleLocator(base=2) # this locator puts ticks at regular intervals\n",
+    "#ax.xaxis.set_major_locator(loc)\n",
+    "\n",
+    "ax.set_ylim(-2,2)\n",
+    "ax.set_xlim(-2,2)\n",
+    "\n",
+    "plt.tight_layout()\n",
+    "plt.savefig('eigenvec_01.png')"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "<!-- ## From table arrangements to powerful tools - part 2 -->"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Linear transformations\n",
+    "\n",
+    "In general terms, a given matrix $A$ is a representation of what we call a *linear transformation*, a function from a special kind of set called *vector space* into another vector space. Vector spaces are generalised sets with particular properties, but for the moment you can think of two-dimensional and three-dimensional cartesian spaces as usual vector spaces. Then vectors are gonna be represented as arrays of real numbers as before: $\\mathbf{n_{0}} = \\begin{pmatrix} 7.6  \\\\ 2.2  \\end{pmatrix}$ or $\\mathbf{v} = \\begin{pmatrix} x \\\\ y \\\\ z \\end{pmatrix}$.  \n",
+    "  \n",
+    "Back to linear transformations, they are usually denoted as:  \n",
+    "  \n",
+    "$$\\begin{align}\n",
+    "A:V &\\longrightarrow W \\\\\n",
+    "  \\mathbf{v} &\\rightarrow A(\\mathbf{v}) \\end{align}$$\n",
+    "  \n",
+    "where $V$ and $W$ are the domain and codomain vector spaces, and the transformation $A$ is a function that projects vectors $\\mathbf{v}$ into the images $A(\\mathbf{v})$. The property that defines linear transformations is the following:  \n",
+    "  \n",
+    "$$A(a\\mathbf{v}+b\\mathbf{w})=aA(\\mathbf{v})+bA(\\mathbf{w}),$$  \n",
+    "  \n",
+    "where $a$ and $b$ are constants. In other words,for a linear transformation, the image of the transformation $A$ of a linear combination of two vectors ($\\mathbf{v}$ and $\\mathbf{w}$) is equal to the linear combination of the images of this vectors by $A$.  \n",
+    "  \n",
+    "If we represent vectors with their components in a cartesian system, the images of the transformation $A$ will be given be the multiplication of the matrix representation of $A$ on the vectors.  \n",
+    "  \n",
+    "````{admonition} Example\n",
+    "\n",
+    "Given the matrix $A = \\begin{pmatrix} 1 & 2 \\\\ 3 & 2 \\end{pmatrix}$ and the vectors $\\mathbf{v_1} = \\begin{pmatrix} 3  \\\\ -1  \\end{pmatrix}$ and $\\mathbf{v_2} = \\begin{pmatrix} 2  \\\\ 3  \\end{pmatrix}$, then we have:  \n",
+    "  \n",
+    "$$\\begin{align} A\\mathbf{v_1} &= \\begin{pmatrix} 1 & 2 \\\\ 3 & 2 \\end{pmatrix}\\begin{pmatrix} 3  \\\\ -1  \\end{pmatrix} = \\begin{pmatrix} 1  \\\\ 7  \\end{pmatrix} \\\\  \n",
+    "  A\\mathbf{v_2} &= \\begin{pmatrix} 1 & 2 \\\\ 3 & 2 \\end{pmatrix}\\begin{pmatrix} 2  \\\\ 3  \\end{pmatrix} = \\begin{pmatrix} 8  \\\\ 12  \\end{pmatrix} = 4\\begin{pmatrix} 2  \\\\ 3  \\end{pmatrix} \\end{align}$$  \n",
+    "    \n",
+    "The vectors and their respective transformations are given by:  \n",
+    "  \n",
+    "```{image} lin_trans.png\n",
+    "```  \n",
+    "  \n",
+    "We see that under the transformation $A$, vector $\\mathbf{v_1}$ gets rotated (counterclockwise) and stretched, while vector $\\mathbf{v_2}$ only gets stretched without changing the direction.\n",
+    "\n",
+    "````  \n",
+    "  \n",
+    "### Eigenvalues and eigenvectors  \n",
+    "\n",
+    "  \n",
+    "As seen on the previous example, there might be some vectors that upon application of $A$ do not have their direction changed, only their magnitude (or *module* of the vector). For these vectors, we have, in general:  \n",
+    "  \n",
+    "$$A\\mathbf{v} = \\lambda\\mathbf{v}, \\quad \\mbox{for some constant }\\ \\lambda$$  \n",
+    "  \n",
+    "But this is equivalent to:  \n",
+    "  \n",
+    "$$\\begin{align}\n",
+    "A\\mathbf{v} - \\lambda\\mathbf{v} = 0 \\\\  \n",
+    "A\\mathbf{v} - \\lambda I\\mathbf{v} = 0 \\\\\n",
+    "\\left(A - \\lambda I \\right)\\mathbf{v} = 0 \\end{align},$$  \n",
+    "  \n",
+    "where $I$ is the identity matrix. The equation $\\left(A - \\lambda I \\right)\\mathbf{v} = 0$ defines a system of linear equations where $A - \\lambda I$ is the coefficient matrix, $\\mathbf{v}$ is the vector of variables, and $\\mathbf{b}=0$. See that since $\\mathbf{b}=0$ we already know one possible solution for this system, which is $\\mathbf{v}=0$ (all the variables equal to $0$). Therefore, either this system has an unique solution (the null solution) or it has infinite solutions, but definitely it is not inconsistent.  Thus, if we want to know all possible non-zero vectors $\\mathbf{v}$ that satisfy $A\\mathbf{v} = \\lambda\\mathbf{v}$, we have to impose that the system assumes infinite solutions. This criterium should be defined by the determinant of the matrix of coefficients, as:  \n",
+    "  \n",
+    "$$\\mbox{det}(A-\\lambda I) = 0.$$  \n",
+    "  \n",
+    "  \n",
+    "The determinant then defines a polynomial in the variable $\\lambda$, $\\mbox{det}(A-\\lambda I)=p(\\lambda)$, called *characteristic polynomial*. Since we are interested in the condition $p(\\lambda)=0$, what we want to know are the roots $\\lambda$ of this polynomial.  \n",
+    "  \n",
+    "The solutions $\\lambda$ for these equations are called **eigenvalues** and their corresponding vectors are **eigenvectors**. In summary, the eigenvectors are all the vectors that, upon application of $A$, remain on the same direction, only changing magnitude, with the eigenvalue being the stretching/compressing factor. \n",
+    "  \n",
+    "````{admonition} Example  \n",
+    "\n",
+    "Let us calculate the eigenvalues and eigenvectors of $A = \\begin{pmatrix} 1 & 2 \\\\ 3 & 2 \\end{pmatrix}$. First we need to obtain matrix $(A-\\lambda I)$ and to calculate its determinant. We have:  \n",
+    "  \n",
+    "$$A-\\lambda I = \\begin{pmatrix} 1 & 2 \\\\ 3 & 2 \\end{pmatrix} - \\lambda \\begin{pmatrix} 1 & 0 \\\\ 0 & 1 \\end{pmatrix} = \\begin{pmatrix} 1 -\\lambda & 2 \\\\ 3 & 2 -\\lambda  \\end{pmatrix}$$  \n",
+    "  \n",
+    "The characteristic polynomial will be given by $p(\\lambda)=\\mbox{det}(A-\\lambda I)$, and applying the rule for the calculus of determinants of matrices size 2, we have:  \n",
+    "  \n",
+    "$$p(\\lambda)=\\mbox{det}(A-\\lambda I) = \\mbox{det}\\begin{pmatrix} 1 -\\lambda & 2 \\\\ 3 & 2 -\\lambda  \\end{pmatrix} = (1 -\\lambda)(2 -\\lambda) - 3\\cdot 2 $$  \n",
+    "  \n",
+    "$$p(\\lambda) = 2 -\\lambda -2\\lambda +\\lambda^2 - 6 = \\lambda^2 - 3\\lambda -4.$$  \n",
+    "  \n",
+    "Thus, calculating the roots of this 2nd-degree polynomial, we find $\\lambda_1=-1$ and $\\lambda_2=4$. For each of the eigenvalues, to obtain the eigenvectors we have to solve the linear system  $A\\mathbf{v} = \\lambda\\mathbf{v}$ for some general vector $\\mathbf{v} = \\begin{pmatrix} x  \\\\ y  \\end{pmatrix}$ (with coordinates $x$ and $y$ to be specified). Thus:  \n",
+    "  \n",
+    "- $\\mathbf{\\lambda_1 = -1}$:  \n",
+    "    \n",
+    "    $\\begin{align}\n",
+    "A\\mathbf{v} = \\lambda\\mathbf{v} \\longrightarrow \\begin{pmatrix} 1 & 2 \\\\ 3 & 2 \\end{pmatrix}\\begin{pmatrix} x  \\\\ y \\end{pmatrix}= -1\\begin{pmatrix} x  \\\\ y \\end{pmatrix}\\longrightarrow \\cases{x + 2y = -x \\\\ 3x + 2y = -y} \\longrightarrow \\cases{2x + 2y = 0 \\\\ 3x + 3y = 0}\n",
+    "\\end{align}$  \n",
+    "    \n",
+    "    Note that both equations correspond to the same linear equation $x + y = 0$, which is exactly what we would expect since we built the system to be underdetermined when we did $\\mbox{det}(A-\\lambda I) = 0$. The eigenvector corresponding to $\\lambda_1$ will be such that $x + y = 0$ or $x = -y$. Thus, we can choose $\\mathbf{v_1} = \\begin{pmatrix} 1  \\\\ -1  \\end{pmatrix}$.  \n",
+    "    \n",
+    "    ```{note}  \n",
+    "    Remember that infinite solutions are possible for the same system. The important is that the condition $x = -y$ holds. In fact, multiplying a non-zero constant for the vector $\\begin{pmatrix} 1  \\\\ -1  \\end{pmatrix}$ would still give a vector in the same direction. So if $\\mathbf{v_1}$ is an eigenvector, $k\\mathbf{v_1}\\ (k \\in {\\rm I\\!R})$ also is.  \n",
+    "    \n",
+    "    ```  \n",
+    "    <br />  \n",
+    "    \n",
+    "    \n",
+    "- $\\mathbf{\\lambda_1 = 4}$:  \n",
+    "    \n",
+    "    $\\begin{align}\n",
+    "A\\mathbf{v} = \\lambda\\mathbf{v} \\longrightarrow \\begin{pmatrix} 1 & 2 \\\\ 3 & 2 \\end{pmatrix}\\begin{pmatrix} x  \\\\ y \\end{pmatrix}= 4\\begin{pmatrix} x  \\\\ y \\end{pmatrix}\\longrightarrow \\cases{x + 2y = 4x \\\\ 3x + 2y = 4y} \\longrightarrow \\cases{-3x + 2y = 0 \\\\ 3x - 2y = 0}\n",
+    "\\end{align}$  \n",
+    "    \n",
+    "    Since the eigenvector needs to satisfy $3x=2y$, we choose $\\mathbf{v_2} = \\begin{pmatrix} 2/3  \\\\ 1  \\end{pmatrix}$.  \n",
+    "    \n",
+    "    The representation of these vectors on the cartesian plane will be:  \n",
+    "      \n",
+    "```{image} eigenvec_01.png\n",
+    "```\n",
+    "\n",
+    "````  \n",
+    "  \n",
+    "  \n",
+    "  \n",
+    "````{admonition} Example  \n",
+    "  \n",
+    "Let us calculate the eigenvalues and eigenvectors of matrix $A = \\begin{pmatrix} -2 & 1 \\\\ 0 & -1 \\end{pmatrix}$. We have:  \n",
+    "  \n",
+    "$$p(\\lambda)=\\mbox{det}(A-\\lambda I) = \\mbox{det}\\begin{pmatrix} -2 -\\lambda & 1 \\\\ 0 & -1 -\\lambda  \\end{pmatrix} = (-2 -\\lambda)(-1 -\\lambda) - 1\\cdot 0 $$  \n",
+    "  \n",
+    "$$p(\\lambda)=(-2 -\\lambda)(-1 -\\lambda)$$  \n",
+    "  \n",
+    "Since we need $p(\\lambda)=0$ either one of the two factors has to be zero. So, $\\lambda_1=-2$ and $\\lambda_2=-1$.   \n",
+    "For the eigenvectors:  \n",
+    "  \n",
+    "- $\\mathbf{\\lambda_1 = -2}$:  \n",
+    "    \n",
+    "    $\\begin{align}\n",
+    "A\\mathbf{v} = \\lambda\\mathbf{v} \\longrightarrow \\begin{pmatrix} -2 & 1 \\\\ 0 & -1 \\end{pmatrix}\\begin{pmatrix} x  \\\\ y \\end{pmatrix}= -2\\begin{pmatrix} x  \\\\ y \\end{pmatrix}\\longrightarrow \\cases{-2x + y = -2x \\\\ -y = -2y} \\longrightarrow \\cases{y = 0 \\\\ y = 0}\n",
+    "\\end{align}$  \n",
+    "    \n",
+    "    Since $y=0$, we choose $\\mathbf{v_1} = \\begin{pmatrix} 1  \\\\ 0  \\end{pmatrix}$.  \n",
+    "      \n",
+    "- $\\mathbf{\\lambda_2 = -1}$:  \n",
+    "    \n",
+    "    $\\begin{align}\n",
+    "A\\mathbf{v} = \\lambda\\mathbf{v} \\longrightarrow \\begin{pmatrix} -2 & 1 \\\\ 0 & -1 \\end{pmatrix}\\begin{pmatrix} x  \\\\ y \\end{pmatrix}= -1\\begin{pmatrix} x  \\\\ y \\end{pmatrix}\\longrightarrow \\cases{-2x + y = -x \\\\ -y = -y} \\longrightarrow \\cases{-x + y = 0 \\\\ y = y}\n",
+    "\\end{align}$  \n",
+    "    \n",
+    "    The second equation is trivial and provides no information. From the first equation, since $x=y$, we choose $\\mathbf{v_2} = \\begin{pmatrix} 1  \\\\ 1  \\end{pmatrix}$.  \n",
+    "      \n",
+    "```{tip}  \n",
+    "  \n",
+    "Note that the two eigenvalues $\\lambda_1 = -2$ and $\\lambda_2 = -1$ correspond to the elements of the main diagonal of the matrix $A$. Whenever the matrix $A$ is *triangular*, meaning all the elements *below* the main diagonal are $0$, the eigenvalues will correspond to the elements of the main diagonal.  \n",
+    "  \n",
+    "```\n",
+    "\n",
+    "````\n",
+    "                             "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Vector decomposition  \n",
+    "  \n",
+    "Now suppose matrix $A$ has eigenvalues $\\lambda_1$ and $\\lambda_2$, with corresponding eigenvectors $\\mathbf{u_1}$ and $\\mathbf{u_2}$. If $\\lambda_1 \\ne \\lambda_2$, then the vectors $\\mathbf{u_1}$ and $\\mathbf{u_2}$ are *linearly independent* meaning one can not be written as a multiple (multiplied by a real constant) of the other. Geometrically, this means that the directions corresponding to this two eigenvectors are not on the same line.  \n",
+    "  \n",
+    "In two dimensions, this is enough to say that any given vector $\\mathbf{v}$ can be written as a linear combination of $\\mathbf{u_1}$ and $\\mathbf{u_2}$ (or that $\\mathbf{u_1}$ and $\\mathbf{u_2}$ consitute a *basis* of this space). We have:  \n",
+    "  \n",
+    "$$\\mathbf{v}= a_1\\mathbf{u_1} + a_2\\mathbf{u_2},$$  \n",
+    "  \n",
+    "for constants $a_1$ and $a_2$ to be determined.  \n",
+    "  \n",
+    "But given that $A$ is a linear transformation, it should hold that:  \n",
+    "  \n",
+    "$$A\\mathbf{v} = A(a_1\\mathbf{u_1} + a_2\\mathbf{u_2}) = a_1A(\\mathbf{u_1}) + a_2A(\\mathbf{u_2}) = a_1\\lambda_1\\mathbf{u_1} + a_2\\lambda_2\\mathbf{u_2},$$  \n",
+    "  \n",
+    "with the last step coming from the fact that $\\mathbf{u_1}$ and $\\mathbf{u_2}$ are eigenvectors of $A$. Since $A\\mathbf{v}$ is also a vector in this space, we can go ahead and apply the transformation $A$ again, and we will have:  \n",
+    "  \n",
+    "$$A(A\\mathbf{v}) = A(a_1\\lambda_1\\mathbf{u_1} + a_2\\lambda_2\\mathbf{u_2})=a_1\\lambda_1 A(\\mathbf{u_1}) + a_2\\lambda_2 A(\\mathbf{u_2}) = a_1\\lambda_1^2\\mathbf{u_1} + a_2\\lambda_2^2\\mathbf{u_2}.$$\n",
+    "  \n",
+    "  \n",
+    "So, in general:  \n",
+    "  \n",
+    "  \n",
+    "$$A(A(A \\cdots A\\mathbf{v}) \\cdots )) = A^n\\mathbf{v} = a_1\\lambda_1^n\\mathbf{u_1} + a_2\\lambda_2^n\\mathbf{u_2},$$  \n",
+    "or, in other words, if we know the eigenvectors and eigenvalues of $A$ (with $\\lambda_1 \\ne \\lambda_2$), and we know the coeffiencients $a_1$ and $a_2$ of the expansion of $\\mathbf{v}$ into the eigenvectors, then it is easy to calculate what is going to be the image ov vector $\\mathbf{v}$ by any number of successive applications of $A$. This is also equivalent of applying the $n$-th power of matrix $A$ to $\\mathbf{v}$.  \n",
+    "  \n",
+    "This fact will have many important applications, as we will se next.  \n",
+    "  \n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Structured models revisited  \n",
+    "  \n",
+    "Let us considered a structured population model defined by the projection matrix $P=\\begin{pmatrix} 1.5 & 2  \\\\ 0.08 & 0  \\end{pmatrix}$ and an initial population vector given by $\\mathbf{n_0} = \\begin{pmatrix} 105  \\\\ 1 \\end{pmatrix}$. If we calculate the eigenvalues and eigenvectors, we will have:  \n",
+    "  \n",
+    "$$p(\\lambda)=\\mbox{det}(P-\\lambda I) = \\mbox{det}\\begin{pmatrix} 1.5 -\\lambda & 2 \\\\ 0.08 & -\\lambda  \\end{pmatrix} = (1.5 -\\lambda)(-\\lambda) - 0.08\\cdot 2 $$  \n",
+    "  \n",
+    "$$p(\\lambda)=\\lambda^2-1.5\\lambda-0.16=(\\lambda-1.6)(\\lambda+0.1)$$  \n",
+    "  \n",
+    "Thus, we have $\\lambda_1=1.6$ and $\\lambda_2=-0.1$.  \n",
+    "  \n",
+    "- $\\mathbf{\\lambda_1 = 1.6}$:  \n",
+    "    \n",
+    "    $\\begin{align}\n",
+    "A\\mathbf{v} = \\lambda\\mathbf{v} \\longrightarrow \\begin{pmatrix} 1.5 & 2  \\\\ 0.08 & 0 \\end{pmatrix}\\begin{pmatrix} x  \\\\ y \\end{pmatrix}= 1.6\\begin{pmatrix} x  \\\\ y \\end{pmatrix}\\longrightarrow \\cases{1.5x + 2y = 1.6x \\\\ 0.08x = 1.6y} \\longrightarrow \\cases{x - 20y = 0 \\\\ x - 20y = 0}\n",
+    "\\end{align}$  \n",
+    "    \n",
+    "    Since $x=20y$, we choose $\\mathbf{u_1} = \\begin{pmatrix} 20  \\\\ 1  \\end{pmatrix}$.  \n",
+    "  \n",
+    "  <br />\n",
+    "  \n",
+    "- $\\mathbf{\\lambda_2 = -0.1}$:  \n",
+    "    \n",
+    "    $\\begin{align}\n",
+    "A\\mathbf{v} = \\lambda\\mathbf{v} \\longrightarrow \\begin{pmatrix} 1.5 & 2  \\\\ 0.08 & 0 \\end{pmatrix}\\begin{pmatrix} x  \\\\ y \\end{pmatrix}= -0.1\\begin{pmatrix} x  \\\\ y \\end{pmatrix}\\longrightarrow \\cases{1.5x + 2y = -0.1x \\\\ 0.08x = -0.1y} \\longrightarrow \\cases{0.8x + y = 0 \\\\ 0.08x + 0.1y = 0}\n",
+    "\\end{align}$  \n",
+    "    \n",
+    "    Since $-\\displaystyle\\frac{4}{5}x=y$, we choose $\\mathbf{u_2} = \\begin{pmatrix} 5  \\\\ -4  \\end{pmatrix}$.  \n",
+    "\n",
+    "<br />\n",
+    "      \n",
+    "We can also note that the initial population vector can be decomposed into the eigenvectors $\\mathbf{u_1}$ and $\\mathbf{u_2}$ of the matrix $P$ according to:  \n",
+    "  \n",
+    "$$\\mathbf{n_0}=\\begin{pmatrix} 105  \\\\ 1 \\end{pmatrix} = 5\\begin{pmatrix} 20  \\\\ 1 \\end{pmatrix} + \\begin{pmatrix} 5  \\\\ -4 \\end{pmatrix} = 5\\mathbf{u_1} + \\mathbf{u_2}$$  \n",
+    "  \n",
+    "We know that to find the population vector for the following step, we have to apply the projection matrix on the current population vector. Thus:  \n",
+    "  \n",
+    "$$\\mathbf{n_1} = P\\mathbf{n_0}= P(5\\mathbf{u_1} + \\mathbf{u_2})=5P(\\mathbf{u_1}) + P(\\mathbf{u_2})=5(1.6)\\begin{pmatrix} 20  \\\\ 1 \\end{pmatrix} + (-0.1)\\begin{pmatrix} 5  \\\\ -4 \\end{pmatrix} = \\begin{pmatrix} 159.5  \\\\ 8.4 \\end{pmatrix}$$  \n",
+    "  \n",
+    "If we perform successive applications of $P$, we will finally have:  \n",
+    "  \n",
+    "$$\\mathbf{n_t} = P^t\\mathbf{n_0} = P^t(5\\mathbf{u_1} + \\mathbf{u_2})=5(1.6)^t\\begin{pmatrix} 20  \\\\ 1 \\end{pmatrix} + (-0.1)^t\\begin{pmatrix} 5  \\\\ -4 \\end{pmatrix}.$$  \n",
+    "  \n",
+    "<br />  \n",
+    "  \n",
+    "By decomposing into eigenvalues and eigenvectors, we know the behaviour of our population for any time $t$, which is much easier than applying $P$ a number $t$ of times, or than calculating $P^t$. Furthermore, as $t\\rightarrow\\infty$ the first term will dominate (with the second term dying out) and the proportions for each age class in this population will be dictated by the vector $\\begin{pmatrix} 20  \\\\ 1 \\end{pmatrix}$ (or, if we divide by the sum of the components: $\\displaystyle\\frac{1}{21}\\begin{pmatrix} 20  \\\\ 1 \\end{pmatrix} = \\begin{pmatrix} 95\\%  \\\\ 5\\% \\end{pmatrix}$.  \n",
+    "  \n",
+    "In general the eigenvector corresponding to the **largest eigenvalue** (also called *dominant eigenvalue*) of the projection matrix, which is always *real* and *positive*, will correspond to the stable age distribution, for which our population vector converges after we wait enough time.  \n",
+    "  \n",
+    "As we have seen, after enough time, the dominant eigenvalue will also correspond to the growth factor for each of the age classes. Thus, if $\\lambda_1>1$ the population size will increase, whereas if $0<\\lambda_1<1$ the population size will decrease.\n",
+    "  \n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 13,
+   "metadata": {
+    "tags": [
+     "remove-cell"
+    ]
+   },
+   "outputs": [
+    {
+     "data": {
+      "image/png": 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OjGyJUwQJAPD++7IhcXMzMHEi8M03QP/+ekdFRCdr2TJg7lyZ5z9kiNzEJyXpHRUdzdq18n+2f7+si/vqKyA1Ve+oiOhk7dsno1Rbt0oH19tvyxYO5Lg4RZCO6uqrgcxMIDZWNrKbMEHeE5Fj0jTgpZeAs86S5Oq882RaMJMr+zd5skwXOu002SfrlFOAd97ROyoiOhm//y4d2Fu3AvHxcq/F5Mp5McEii8REuQE75RSgtBSYOVP2ZyEix2IwyHTfu+6Sve8eegj47jtZRE2OISQEWLJE1mIZjbJXWfuUIiJyHJoGvPYacMYZQFUVcPbZMkrNvUidGxMsshIaCvz6K3DbbXIhv+46eW006h0ZER2P4mLpHPngA8DbWyqGLlgAuLC1dzhubsCLL8r/paenLIo//XSgvFzvyIjoeLS0SMXm224D2tqABx4AfviBnV19Addg0RG9954svmxtlZ6XxYtlITYR2adNm2QqYHExMHCgjFqNG6d3VGQL69YBs2fL/21cHPDTT6wASWTPKivld3blStlv8N13OSXQGXENFp2wa64Bli8HwsOB//5XqpDl5ekdFRF15ccfpepccbE8Z2UxuXImkybJ/+mkSdIOp6QAv/2md1RE1JVdu2Tj8JUrpWBYRgaTq76GI1h0THl5wLnnAjk5UtHq++/lIk9E9uHVV2Wtjtks5X8XLpQpZeR8mppkL7Ovv5YphP/5j3SGEZF9yMgALrwQqK6WokI//KB/Veb6+nqUl5fDyPUeJ8Td3R3h4eEICAg44jlHGsFigkXHpbZWNr/87TdZ1/Hxx8BFF+kdFVHf1tYG3HOP7JkEAI8/Djz6KKCUrmFRDzObgQcfBJ57Tt7Pmwc89RTX2RHp7ZNPpMOjtVWma3/2GeDnp29M9fX1KCsrQ//+/eHt7Q3FC8Rx0TQNzc3NKCoqQkRExBGTLE4RpJPSrx/wyy/AtdfKXllz5gD//rdUxyGi3tfYKJ0c7RvTfvQR8NhjTK76AhcX4NlnZfTK1RX45z+Byy6TtpmIep+mAfPnywyC1lbg1luBb7/VP7kCgPLycvTv3x8+Pj5Mrk6AUgo+Pj7o378/yrtRWYgJFh03d3fZi+Wf/5TG5L77pBFpa9M7MqK+pawMSEuT6bpBQbJG8sor9Y6KetsNNwA//wz4+wNffCEVBqur9Y6KqG8xGmXUqn32wEsvybRtV1e9IxNGoxHe3t56h+GwvL29uzW1kgkWnRClZGrK55/LGo833gAuuUT23SGinrdnD5CaCmzYAAweLBuCp6XpHRXp5cwzgdWrgZgYeZ42DSgo0Dsqor6hsRH4059kKwUfH+Cbb4A77tA7qs44ctV93f23Y4JF3TJ3LrB0qZRt/+orYNYsWadFRD1n40apHrdnDzBhgiRXw4frHRXpbcwY+SyMGQPs2CHVy7Zv1zsqIudWWQmcdposnwgNBX7/XZItIoAJFp2EGTOkWk50NLBihbwvLtY7KiLn9NtvMlJVXi770v3vf7KFAhEgVcrS06VEf1GRjGStXKl3VETOKT9ffsfWrgViY4FVq1hdmawxwaKTkpAg01KGDwe2bpXe9Z079Y6KyLl88QVw9tlAQ4MUM/jxR1l3Q3SooCDg119lc9PaWknEv/tO76iInMu2bR33Ou33QPHxekdF9oYJFp209t6bKVOkVyc1VXp1iOjkvfoqcOmlspD6zjtliwQPD72jInvl7Q0sXgzceKOsjb3oIilOREQnb+XKjg3dZ8yQUePoaL2jcn7vvPMOBg8eDDc3N9x4442W4zU1NYiIiMCePXuO+v1z5szBCy+80NNhWmGCRTYREgIsWyYbEldVAaeeKvOSiah7NA14+GHg9tvl9b/+BbzwAvc6omNzdQXefFP2RTObpdrg/PncVoPoZHz/vYwK19bKKPGvv8oWNtSzcnNzcfPNN+P5559HQUEBnn/+ecvXnn76aZxzzjkYMmSI5di9996LWbNmWf2Mxx57DE899RTq6up6LW5eqslmfH2lgs7f/gY0NQEXXCBTm4joxLS1yU3xggVys/z++8ADD3CPKzp+Ssm+aP/5jyTljz4qm1IzySI6ce+9J0mVwSBt8+LFgJeX3lH1Dd9//z3GjBmD2bNnIyoqCn4HNxdramrCwoULce2111qdv379ekw6bEFcQkICBg8ejI8//rjX4maCRTbl7i4N0b33AiaTrBd59129oyJyHEYjcPnlwMKFMt3ru++k04KoO264QbbVcHcHXnwRuP567l1IdCJeegm49loZDX70UeCtt+xnjytnFx8fjwceeADZ2dlQSmH27NmWr/38889wcXFBamoqANnvy8PDA+np6Zg/fz6UUhg9erTl/AsuuACfffZZr8XOBItsTing2WeBp56SBum662RqExEdncEAXHyx3BAHBMhWCOeeq3dU5OjmzAF++EES9nfflY6v1la9oyKyb5om9zF33SXvX34ZeOIJ55hJoJQ+jxO1cuVKxMfH46mnnkJJSQk+/PBDy9cyMjKQnJxs2afK1dUVmZmZAIC1a9eipKQEKw8ppTpp0iSsW7cOzc3NJ/ePd5yYYFGPUAr4xz9kgT4gU1Mee4zTU4iOpKEBOO88uREODpYy7NOm6R0VOYuzzpKEPSBApjf96U8ylZuIOtM04MEHgUcekfuZd9+V9bDUuwICArB3716kpqYiMjISAQEBlq/l5+cjKirK8t7FxQUlJSXw9/fHxIkTERkZiaCgIMvXo6OjYTQaUdxL+wkxwaIedeutwIcfyhqAJ5+UKmhms95REdmX2lrgzDNlr6vISNlXLjlZ76jI2UybJpuhhoYCS5ZI0tWLa76JHILZLPcuzz4LuLkBn30GXHON3lHZlqbp8zhR27Ztg8lkwrhx4zp9rbm5GV6HLYTbtGkTEhMTLaNah/L29rZ8X29ggkU97qqrgC+/lNLSr7wic5lNJr2jIrIPFRVSdTMzExg4UMr+jhmjd1TkrMaPlw3iBwyQktOnnCKfQSKSe5NrrgHeeAPw9AS+/hq45BK9o+q7Nm/ejNjYWPTrolxjaGgoampqOp2flJTU5c+qrq4GAISFhdk+0C4wwaJeMXu2bI7q4wN88IHs69PSondURPoqLgbS0oBNm4Bhw+TGd9gwvaMiZzdihCRXQ4fKZ2/GDKCwUO+oiPTV2irrEz/8UO5VfvoJOP98vaPq2zZv3tzl6BUAJCUlIScnx+pYdnY2xo4d2+X527ZtQ3R0NCIiImweZ1eYYFGvOeMM4L//lX0jvvpKyrg3NuodFZE+8vJkw8odO2TEKj1dRrCIekNsrCT0CQlAbq5MH9y9W++oiPTR3AxceKHMtmkvMHTaaXpHRUdLsM466yzs2LEDVVVVlmMmkwm5ubkoLi5GbW2t1fkZGRmd9sfqSUywqFelpADLlwPh4dKAnXWWrD8h6kt27pQb2r17gQkT5HciMlLvqKiviYyUz97kyUB+viT8W7fqHRVR7zpwADjnHOCXX4CQEFmneLDyN+lI0zRs2bLliAlWQkICJk2ahEWLFlmOLViwAIsWLcKAAQMwb948y3GDwYBvvvkG119/fY/H3U5pdlDWTSml2UMc1Ht27QJOPx0oKACSkmRH9F6aFkukqy1bZDS3vFxuaH/8UXpMifTS0CBVBf/3P6lguWQJMHGi3lER9byaGuDss4G1a4GoKGDZMmDUKL2jsq0dO3Zg5MiReofRI5YsWYI77rgDOTk5cD3K5mSvv/46vvvuOyxdurRbf87R/g2VUhs0TZtw+HGbjWAppf6mlNK6eNxkqz+DnEd8vExPaV8DkJYm61GInNm6dcDMmZJcnXmm3MgyuSK9+fl1rDeprpapUenpekdF1LPKy6XIy9q1HVNmnS25cnazZs3CLbfcgsJjLCJ1d3fHq+37BvUSm41gKaX+BuB9AKcCOLQG4l5N08qP8b0cweqjSkrkRnPbNmDwYOk9GjRI76iIbG/FCtnnqqFB5vovWiRVqojshdEIXHmlbHTt7S0V1HpxyQJRrykslFk0O3dKh++yZUBMjN5R9QxnHsHqLbqOYB1ivaZpaw55HDW5or4tKkrWAEyYIOtRpk+XBddEzuSXX+RGtaEBuPxy4IsvmFyR/XF3Bz75RLbSaG6WQkRffaV3VES21X6vsXMnMHasjNY6a3JF+mGRC9JdSIhssDp9OlBUJCWDN2/WOyoi2/jqK1nfYjAAN9wAfPSR3MgS2SNXV+Cdd2RTeKMRmDtXPrNEzmDHDrnXyMsDJk2Sgha9VLWb+pieSLD2KKVMSqmdSqkbe+DnkxMKCJD1KLNmyaaXp5wiG68SObL/+z+5QTUagbvvBt56C3BhtxbZOaWAF14AHnkEMJuBv/5VNl4lcmTte74VF8vzsmVS1IWoJ9jyUl8C4BEAVwI4H8BaAG8ppe6y4Z9BTszHB/j2W+Dii6V0+xlnSFUrIkf05pvAVVfJDepjjwH//rfcuBI5AqWAJ58Enn1W3t9yS8drIkeTmSkdt5WV0pH7yy+Av7/eUZEzs1mCpWnar5qmPaVp2lJN037RNO0qAF8AeFgp1enPUUrdoJTKUkpl2SoGcnyenrL4/6qrZBPic86RMtZEjuS554C//73j9eOPM7kix3TffdJZoBTwwAPAww8DrElFjuR//5MO27o64KKLpCPXx0fvqMjZ9fRklS8BBAOIO/wLmqa9rWnahK4qb1Df5uYGvP++3KC2tACzZ0tVKyJ7p2kyWnX//XJD+uabwL336h0V0cm56SZZh+XqCixYIOuzzGa9oyI6tp9+ko7axsaOCpksMES9obdWA7C/i06Iiwvw2mvSY2oyAZddBrz7rt5RER2ZpgH33CPTqlxcgA8/lBtTImdwxRXA4sVSoOWVV4Drrwfa2vSOiujIFi+WLTFaWqQt/uAD6cAl6g09nWBdDKASQH4P/znkhJQC/vUv6THVNOC664CXXtI7KqLO2tqkQuCLL8oN6OLF0ltK5ExmzwZ++EH2yHrvPeAvfwFaW/WOiqiz994DLr1UOmjvvVeKtLDAEPUmm33clFJfKaUeUEqdrZQ6Tyn1fwAuAfCkpmmcTEDd9tBDwMsvy+u77gLmz+caALIf7ZuzLlwoN54//CDz/Imc0VlnAb/+KgUCvvhCPuvNzXpHRdThlVdkLzezuaNQC9fAUm+zZT6/E8A1AL4CsBjAKABXaZr2qg3/DOqjbr9dpgi6uACPPipTB5lkkd4MBmDOHOCzz+SGc8kSuQElcmbTp0vhgOBgWeNy7rmyiTaRnjRNZrzccYe8f/FF2WqAyZXje+eddzB48GC4ubnhxhuPvgNUTU0NIiIisGfPnqOeN2fOHLzwwgu2DNOK0uzgLlUppdlDHGT/vvgCuPxyGfa/6Sbg9dc57E/6aGiQ+f2//SY3mkuWABMn6h0VUe/Ztk2qs5WWAlOmAD//DAQF6R0V9UWaBsybBzzzjCRUb78tywoI2LFjB0aOHKl3GN2Wm5uLMWPGYPHixZgyZQr8/f3h5+eHxx9/HIGBgbjrLuvdoO677z5UVlbi/ffftzp+7733Ytu2bViyZAkAYOvWrUhLS8O+ffsQGBh41BiO9m+olNrQVcE+3pqSQ5k7V0qsenrKpq1//askW0S9qbYWOPNMSa4iIoAVK5hcUd8zZgyQkQEMHAisWSP7DJWX6x0V9TVmM3DrrZJcubkBn37K5MqZfP/99xgzZgxmz56NqKgo+Pn5AQC++eYbpKWlWZ3b1NSEhQsX4tprr+30c9avX49JkyZZ3ickJGDw4MH4+OOPeyRuJljkcM49VzYJ9PUFPv5Ykq6WFr2jor6iokJuJDMz5cYyI0NuNIn6oqFDgZUrgfh4IDsbmDEDKCzUOyrqK0wm4OqrpYiFpyfw9ddS3IKcQ3x8PB544AFkZ2dDKYXZs2cDAEpKSuDn54fx48dbnf/zzz/DxcUFqamplmNGoxEeHh5IT0/H/PnzoZTC6NGjAQAXXHABPvvssx6JnQkWOaRTTgGWLQP69QO++Qa44AKgqUnvqMjZFRXJDeTmzcCwYZJcDRumd1RE+oqJAdLTgYQEYOdOWaN1jOUPRCetpQW45BLZo83HR9YDnn++3lGRLa1cuRLx8fF46jSiUJoAACAASURBVKmnUFJSgg8//BAAsHTpUtx6662dzs/IyEBycjLUIQvvXF1dkZmZCQBYu3YtSkpKsHLlSgDApEmTsG7dOjT3QKUe7ghADmvKFGD5clkDsHSpFBf48UfgGFNpibpl717g9NOBffvkRnLpUiAyUu+oiOxDRIS0x2efDaxbJ0nWsmXAqFF6R0bOqKkJuPhiWfsaGCjr/1JS9I7Kcagn9Kn8oT12YvUWAgICsHfvXqSmpiLykAvuH3/8gUcffbTT+fn5+YiKirI65uLigpKSEvj7+2PixIlWyVd0dDSMRiOKi4sxZMiQE/zbHB1HsMihJSbKKMKAATJN5bTTgKoqvaMiZ7Njh9ww7tsna62WL2dyRXS44GBJqmbOBEpKZLR340a9oyJnU18vifySJUBYmLTHTK6c07Zt22AymTBu3Dir4/Pnz4e7u3un85ubm+Hl5dXp+KZNm5CYmGiVXAGAt7e35ftsjSNY5PCGD5ck6/TTgQ0bgLQ04L//BQ7rxCDqlrVrgXPOAaqr5Ybxhx+AgAC9oyKyT/7+MpowZ448n3KKPB+yJIKo28rKJLnatAno318S+hEj9I7K8ZzoSJJeNm/ejNjYWPTr1++4zg8NDUVNTU2XPycpKanT8erqagBAWFjYyQXaBY5gkVOIi5Mka/RoYPt2GW3Iz9c7KnJ0v/4KnHqqJFfnnSfFVZhcER2dt7esjf3zn2W04cwzpdOL6GTs3SuJ+qZNUlwlI4PJlbPbvHlzp9Gro0lKSkJOTk6n49nZ2Rg7dmyn49u2bUN0dDQiIiJOKs6uMMEipxEVJVMFkpNlgfW0acCuXXpHRY7qs88kqWpqku0Avv5aFlIT0bF5eMjv0NVXy+/QeecB332nd1TkqLZskeRqzx5g/Hhg1Spg0CC9o6KedqIJ1llnnYUdO3ag6rC1IiaTCbm5uSguLkZtba3leEZGBmbNmmWzeA/FBIucSmio7E00bZqUCp4+XUoHE52IV1/t2ND63nuB998HupjuTURH4eoKLFwI3HYb0NoqRQl6aMsZcmIZGTI9u7RUppz+/jsQHq53VNTTNE3Dli1bTijBSkhIwKRJk7Bo0SKr4wsWLMCiRYswYMAAzJs3DwBgMBjwzTff4Prrr7dp3O2Upuk/D1MppdlDHOQ8GhuBiy6SSm8BATJd5dRT9Y6K7J2mAY89BsyfL++ffRa47z59YyJydJoGPPww8PTT8v6554B77gGUPoXMyIF8/72UYjcYOhL0LmoY0FHs2LEDI0eO1DuMXrNkyRLccccdyMnJgaur6xHPe/311/Hdd99h6dKlx/yZR/s3VEpt0DRtwuHHOYJFTsnXVxrmuXNlDcCsWcBhHRpEVtragJtvluTKxQV47z0mV0S2oBSwYAHw73/L+/vuA+6+GzCb9Y2L7Nv770tHqcEA3Hgj8PnnTK7o2GbNmoVbbrkFhcfY8dzd3R2vvvpqj8XBESxyamaz9JS+9JK8f/55ubATHcpgAK68EvjyS7mAf/65bF5NRLb12WeyptFolA6wjz4CPD31jorsiaZJMn7//fL+kUeAJ57giGd39bURrJ7AESyiw7i4AC++2NFzes897Dkla1VVsln1l1/KhpW//srkiqinXHaZ7F/k7w988YXMLjhkzTn1cW1twO23dyRXr7wCPPkkkytyPEywqE+45x7gk0+kUMGLLwJ/+QvQ0qJ3VKS3vXtlg8qVK2VPlfR0WUxNRD3n1FOlcEF75dcZM4CiIr2jIr21r51+7bWOKpS33aZ3VETdwwSL+oy//EX2MfL3lylgs2YBdXV6R0V6WbcOmDJFSvmPHQusWSPPRNTzEhOBzEzZx2jrVmDqVKCL7WuojygrkwqB338PBAXJvmmXXqp3VETdxwSL+pTTTpNRishI6TlNSQH27dM7Kupt330HzJwJVFTIJqgZGcCAAXpHRdS3xMbKfkYpKUBBgTxzQ+K+JzdXOrvWr5e9rVav5kwCW2Odg+7r7r8dEyzqc8aNk57TUaOkx3TyZGnQqW949VVg9myguRm45hrgxx+llD8R9b7gYGDZMinBXVcHnH028NZbekdFvSU9XRLrvDxg4sSOUU2yHXd3dzQ3N+sdhsNqbm6Gezc2wmSCRX1SXJwkVWeeKaMYp5wia7TIeZlMsnj69tulStX8+bIJKjcQJtKXt7cUvHjooY7tEu66S16T8/roIykwVFMjhYV+/x2IiNA7KucTHh6OoqIiNDU1cSTrBGiahqamJhQVFSG8Gztbs0w79WkmE3DHHcAbb8j7Rx8FHn+cFYucTU2NlIRetkwWTy9cKGXZici+fPghcP31Usb93HOl0IG/v95RkS21tQHz5smG0wBw662ylcpR9oSlk1RfX4/y8nIYjUa9Q3Eo7u7uCA8PR8BRprkcqUw7EywiyLSxO++U8u2XXCIbHHp76x0V2cLOncD55wO7dwPh4cDXXwOpqXpHRURHkp4u03irq4GEBJnGO3Cg3lGRLdTXS8Gpn36ShOrVV2XEkshRMcEiOoaff5aqRQcOyFzwr74CYmL0jopOxtKlMnJVVydVy777ThbWE5F9++MPGcHatUs6RhYvZuEDR7d3r3R25eTI2rsvv5Tp+USOjBsNEx3DOedIRavYWKlmlJwMrFihd1TUHZomU07OPluSq9mzZa8rJldEjmHoUNk64fTTgfJyqQD76qvyu02O5/ffgUmTJLkaOVK2yWByRc6MCRbRIRISgKwsuahXVMhF/ZVXeFF3JA0NwGWXySJ5sxl45BHpKfXz0zsyIjoRQUGyd+F993UUqfnb36QCKDkGTQOefVauqVVV0pGZmQkMGaJ3ZEQ9i1MEibpgMgH/+IdcGADgiiuA//wH8PHRNy46utxcKfeckyMJ1XvvAX/+s95REdHJWrQIuPZaoKkJGD9e1lJyRNq+1dUBV18NfPONvJ83T6q3spgFOROuwSLqhi++kL2SGhtl/6zFi2XqCtmfr76S3u2GBpmC8vXX3E+FyJls2SLTfffuBUJDgY8/Bs46S++oqCvbtgEXXSTFhQIDpST7BRfoHRWR7XENFlE3zJ0r6wCGDgU2b5ae00WL9I6KDmU0yhSiOXMkuZo7V+b3M7kici5jx8r62FmzgMpKeZ43T9oAsh+ffgpMnizJ1dixMu2eyRX1NUywiI5hzBi5QMydKxUGL7tM9mlpatI7Mtq7F5g+Hfj3vwE3N+DFFyUB5norIucUHCwlvhcsAFxcgH/9C0hLA/Lz9Y6MGhpkxsfll8v18aqrZL0VZ31QX8QpgkTHSdOAt9+W/bIMBmD0aJlCOGqU3pH1TZ99Btx4oyS9MTHSazptmt5REVFvyciQDq+iIqBfP9m/8MIL9Y6qb9q4Uf4vdu0CvLyAl1+Wjkil9I6MqGdxiiDRSVJKbujbp59t3w5MmAC89RarDPamhgZZOP2Xv0hydfHFQHY2kyuivmb6dJm6fe65QG2trM+65RZZM0u9w2wGXngBmDJFkqv2Srw33MDkivo2JlhEJ6j9AtJeLvjmm6X0bFGR3pE5v4wMKTbywQeAt7dUdly8WMo5E1HfExoK/PAD8PzzMk34jTekjcjM1Dsy57d/vxQZueceWQd3yy3A2rUyu4Oor2OCRdQNvr4yHeXzz2VNwJIlslbr0085mtUTmppkX6u0NGDPno6F0+wlJSKlgLvv7ri5/+MPGdF+6CGgtVXv6JyPpgHvvCPXvGXLgJAQ4Ntvgddek44vIuIaLKKTVlIic81/+kneX3yxXGgiI/WNy1msXi2jhbt3y/4p8+bJ5sEeHvrFpGkajGYjDCYDWkwtaGlrsXo2mU0AAKUUFJTVs5uLG7zcvODt5g1vd294uXnBy80LLor9XUQnq6UFePRR4LnnJBFITATefRdITtY7MudQUCDXu19/lfezZwNvvglEROgbl8ls6rItbmlrQWubZNmHtsWAtM+uytXSBnu7e8PbTdpkVxdu1kXHh/tgEfUgTZNNbe+8U9YIBQYCzzwjFyIX3jd3S12dJFKvvSb/vqNHy9TACZ2asZNjbDOivLEc5Y3lKGssQ1lDGcoay1DeWI6KpgrUGepQ11KH+pZ6y+s6Qx2MZtvWhvZ09YSvhy+CvILQz6sfgrwPPnsFIcgrCCE+IYj0i0SkXySi/KIQ6ReJEJ8QJmZEXVi5EvjrX6XSqIsLcPvtssktK4x2T1ubTMmeNw+or5eZG6+9Blx6qW1nEbSZ21DVXCXt8SFtcVlDGSqaKlBrqLW0wfUt9ZbXzaZm2wUBwN3FHT7uPlZtcXt73P4c6ReJKP8oS5sc7hsOd1d3m8ZB9o8JFlEvyM8H/v534Oef5X1KilyUxozRNy5Homky9fKuu4DSUhm1uv9+4LHHAE/PE/1ZGsoby5Ffl4/9dfuRX5vf8bouHwV1BahqrupWnO09n55unvB09bR6dnNxs/z5GjS0t28aNJjMJhhMBjQbm9FsakazsRktbS3disHNxQ0RvhGI9ItETGAM4gLjENcvDoOCBiGun7wO8Azo1s8mcnQNDdJuvPSSFGOIiZGkgHsynZisLFlrnJUl7y+4QK5r3ZmlUWuo7bItzq+V12WNZTBr5hP+uS7KRdrjQ9piD1cPeLrJM9DRHh/6us3cBoPJIG3ywfbYYDJYzjtRoT6hiPKLQv+A/l22x2E+YVCc1+5UmGAR9RJNA778UnpMS0tl4fXddwP/+AcQwHvdo9q9WxLUZcvk/dSpMv0kMfHo31fdXI1dVbusHjurdmJ31e5j9my6KBeE+YQhwi8CEb4RiPCLQLhPOCL8IhDmE4Z+Xv0Q6BWIAM8ABHoefPYKhJebl43+1oBZM8NgMqChtQG1hlrLo6a5Rp4NNahorEBZYxlKGkpQ2lCKkgMlqDHUHPNnB3kFYVDQIAwLHoYRoSMwPGQ4RoSOQHxIPHw9fG32dyCyVxs3ynrNDRvk/YUXyt55Q4boG5e9q60FHn5YCodoGjBggJRfnz376KNWja2N+KP6j472uLqjXa5urj7mnxvsHYwI3wiE+4Zb2uVw33CE+4ZLe+wZiECvQKtnH3cfmyUumqahta0VjcZG1BnqUGOo6dQeVzVVobSxFKUNpZb2uLyx/JiJmY+7D+L6xWFo8FCMCBmB4aHDLe1yiE+ITeKn3sUEi6iX1dXJIus335SLU1iYTFG59lpJuqhDdTXw1FPA66/LovSgIJliee211lMs6wx12Fq+FVvLtspz+VbsqNhx1FGoIK8gxPaLRWygPAYGDrS8jwmMQbhvuMNOs2sxtcjFvaEEBXUF2Fe7D3m1eVaPoyWYMQExGB46HCNDRyIhPAGJkYkYHTaaiRc5nbY2aV/+8Q8Z2fLwkE6whx+WKd3UwWiUIhaPPw5UVMgsgjvvlPeHTrE0mAzYUbHDqk3eXrEdhfWFR/zZPu4+He1wYGxH29xPjkX5RTnsNDuT2YTKpkqUHChBYX0h8mrzOrXJR+sUC/EOwYjQERgROgIJ4QkYGzEWYyPGMvGyc3afYCW8kYCkqCQkRSZhXOQ4jIsch35e/fQOjeikrV8v091WrZL3Y8bIviFnnKFvXPbAYJApOwsWSG+pUlLQ4p//MqPWZTc2lGzAlrItlgt4QX1Blz/Hz8MP8SHx8giOt7weFjKsT7cj7VMk99bstYzq5VbmWkb3ulpHpqAwLGSYXNzDxyIxMhFjI8YiNjCWU1vI4RUVSZL14YfyPjQUePJJWS/b1zu+NA345hvgwQdlNgEg1RjfeAMIiStGVnEWtpRtsbTJu6t2o01r6/Rz3F3cMSR4SKf2OD4kHpF+kX26Hakz1GFf7T7srtptaYvbnxtaG7r8nmj/aCRGJFoSrrERYzE8ZLjDJqLOxu4TLDze+figfoMwLnIckiKTMCF6Aib1n8RMnhxS+7TB++8H8vLk2KmnAk880Tc3yDWZpKT9o48C+fvNQNAejD5zA8afm4X9pixsLNmIA60HOn2fl5sXRoWNwtiIsUgIT0BCeAJGh49GlF9Un75od4fJbEJebR5yK3ORU5GDLWVbkF2WjdzKXEsVxEMFeQVhQvQETIyeiIn9J2JC9AT09+/Pf3dySBs2SMdXRoa8HzZM2qPLLpMRm74mPV1mXKxaBcCvFFETsjBj7gY0BmYhqyQLpQ2lnb7HRblgWPAwJEQkWNrjMeFjMChokGUdKh0fTdNQfKAYO6t2IqciB1vLtiK7LBtby7eiydjU6XwvNy+MixyHidETLe3y8NDhDjsbw5HZfYKVWZCJTSWbsLl0MzaVbsLW8q0wmAydzh0aPBST+0/G5P6TMan/JIyLHAdPtxNc+U6kE4MBeOUV4OmnZQohICNZTzwh642cndEIvPpBCZ79NBNlbmuB6Cy4DNgAs0ddp3P7+/dHcnQyxkWMs1zAhwYPZfncHtZiakFuZS6yy7ItvdWbSzejoqmi07mRfpFWF/iJ/Sci1CdUh6iJTlz7iM0DD8jeWQAwYoQkWnPnOn+ipWnAD0vr8fCb67C1JrOjPfYr6nRuoGcgkqOTkRSZJMlURAJGho6Etzs3vupJZs2MvTV7pQOsNBtbyuV5X+2+Tuf6efghOSrZ0iZP6j8Jcf3i2AnWw+w+wTo8DpPZhNzKXGwu3YyNJRuxrmgdNpRs6JR0ebh6YFzkOEvSlRKTwg8U2b3aWuDFF6W6VX29HJs+XYphnH++81zYjW1GbCnbgt//yMSiVauxuSoTbf55nc6L9o9GclQyJkRPQHJUMpKjkxHpx43E7IWmaSisL8T64vXIKs6yPNcaajudGx8Sj9SYVKTEpCA1JpW9qmT3TCbg449lquC+g/etQ4bIuqOrr5aN5Z2BpmnYU7MHK/NX47OVq7EyPxNNflsBZX3/5e/hj+ToZEyImiDP0RMwOGgwf4/tSE1zDbKKsyzt8fri9V2ufYv0i0RKTApSBqQgdWAqkiKTOChhYw6XYHXF2GbE1vKtWFu4FuuK12Ft4VrsqNzR6bxo/2hMGzgN0wdOx7SB05AQnsBeb7JL1dWyHuu11zpGtIYOlQv7FVc43uLryqZKZBZkIrMwE6sLVmNd4Xo0t1lPb3Ax+mNUwGT8KXkqJg+YiOToZET7R+sUMXVX+83a+iLrpOvwohpBXkGYGjPVknRNjJ7IIhpkl4xGWZv19NMdiVZQEHDTTVKFMC5O1/BOWJOxCVnFWVhdsBqZhZlYlb8aVYZK65Pa3DDAbTzOS5yK6YMnYUL0BAwNHspkygGVNpRKW1y0HlklWVhbuLZTAShPV09MiJ5gaY+nxkxFuG+4ThE7B6dIsLpSZ6jD+uL1WFu4FmuK1mB1wepOZUD9PfyREpNiSbom9Z/EYW2yKwcOAO++KyNa+flyzNsb+POfgeuuk3Va9jgoW9pQihV5K7AiXx45FTmdT6oaBhRMxVCvFNx03lTcNnc0PNzZ4eGMjG1GbC7djNUFq7G6cDVW7V+FogPW041clSuSopKQGpOKtNg0zIidwbW1ZFfa2oBvvwWefx7IzJRjSsl07uuuA/70J6lCaG8aWhuwav8qS3u8rmhd5/WUDeFAQQpCmqfi8hkpePiaZIQF8X7IGWmahl1Vu6Q9LliNVQWruhyUGBY8DKkDUzFj4AykxaVhUL9BnAV2AnolwVJKjQLwKoCpAGoBLATwhKZ1UWbG+vtsVqbdrJmRW5mLlftXWh6Hz1V1d3FHcnQypsVMw8y4mZg2cBoCvRxsqICckskEfP018NZbwO+/dxwfOlSSrTlzgKQk/ZKtwvpCq4RqV9Uuq6+7mr2Bokloy5sKFKTAvXwK/vKnMNxxh8RNfYumaSioL5CL+/5VWF24Gtml2Z0qj40JH2O5uKfFpiHCL0KniImsZWbKDIOvvgJaDu4HHhIie0HNmSPFitx1KuZWZ6jDyv0rLe3xhuINVr9bSnOBW/VYGPekAIXSJp+WPAh33qFwzjnWW2BQ31DdXI01hWss7fHawrWdZh0MCBhg6fxKi01DfEg8E66j6PEESykVBGA7gBwAzwAYAuB5AC9qmvbwMb63R/fBKqwvxKr9qyThKliJ7NJsq83gXJQLxkeNR1psGmbGzcT0gdOZcJHu9uwB3nsPeP99oKSk4/jgwdKDesYZwIwZPbs+IK82zyqh2luz1+rrnsoXQQdSUb0pDa270oDiiUCbByZPBv76V+CSS4Dg4J6LjxxPQ2sD1hWtQ3p+Olbkr8CawjWd1tYODxnecYGPS8OAgAE6RUskamqATz6R/aG2bOk4HhQk62bPPBM47TQgsgeXjtY01yBjfwaW5y3HivwV2Fy6GWbNbPm6C1wRakxGU04aGrbNBPanAi2BiIuT9viqq+T6QdTO2GZEdlk2MvIzsCJ/BTL2Z3SaBRbpF4kZsTMsnWCjwkZxCukheiPBmgfgfgCxmqbVHzx2P4DHAUS2HzvC9/bqRsN1hjpkFmZaLvCHD6O7KBckRSZhZtxMpMWmYXrs9D69lw7py2SSUsJffim9qGVlHV9zd5fqg9OnAxMmABMnAtHR3Rvhal9Tc2hCtb9uv9U53i4BCGuahubcNFSsTwNKxgNm6b4dO1YSv8suA0aOPJm/MfUlLaYWrC9eb/ncrS5YjUZjo9U5g4MGW/WospAR6UXTgJwcYPFieeQcNis6IQGYOVPa4gkTgPj47hctqmyqlPuUg78bW8q2WHUOu8INkeZJ0PaloXRNGsz5KUCrPwCgf39pj+fOlesDR6voeJg1M7aXb7fcA6Tnp6O8sdzqnBDvEEm4DrbHYyPG9uk6B72RYKUDKNY07dJDjg0EkA/gAk3TfjjK9/ZqgnW4xtZGrC5YjRX5K7A8bznWFa2z2oBTQSEpKgkzY2fKCBcTLtJJW5vsU/Lrr8CyZUBWFmA2W58TEQGMGiVVsIYOlR7L8HCZ1hISIj2ubm6AUhr+qNmF3/etwLI/lmNVYTpKm6zXy3i09YNP5XQ0bE+DafdMoCwRMMv+Jj4+QGoqcPbZciFnzyjZgrHNiI0lGy0X+JX7V6K+xbp/LiYgxjKdcGbcTAwJGsKEi3SRkwP88gvw3//KXlLN1rOt4OcHjB4tbXH7Iyqqoz0ODpb1XK6uQHljGZbnrcBve1YgY/8K7KzZbvWzXDQP+NdNRsvONBh2pgEFUwGjTGFwdQWSk2Vmw4UXymv+StDJ0jQNO6t2WnW8Fh8otjqnn1c/S7I1M24mEiMS+1TC1RsJVjmANzRNe/yw440AHtc07bmjfK+uCdbhGlsbkVmYaRmGX1u4tlPCNS5yHGbGzbRMKQzyDtIxYuqramqA5cuBdesk2crKkhLwXdOAsBwgbgUQu0Ke/cqsT2kKAfJnAHlpQH4aUJYAaNJQxsUBiYnApEnAKadI76xeaw+o72gztyG7LNtygU/PT0eNocbqnGj/aKTFSsKVFpeG4SHDmXBRrzMYZM1WZmZHe1xQcJRv8C/uaItjVwBhudZfN3oBhVOkLc5Lk9cmKUgRGirt8YQJMmKWmgr4+/fYX40IgCRce2v2Wtri5XnLkV+Xb3VOoGcgZsTOsMwCGxc5zqkTrt5IsIwA7tM07aXDjhcC+EjTtIcOO34DgBsOvk22SRBERERERES9o8sEy83Gf0hX2Zrq6rimaW8DeBuwvxGsY2kyNmFN4Rosz1uO5XnLsbZoLVrbWq3OGR022jJkOj12Ovf5oR53eE9/V4tVo/yiLFOr0mLTMDxkBADF+fnksDRNQ05FjmX6yoq8FShrtB6ZDfUJtbTHabFpSIhI4CJt6lHtPf3ta71X5K9AXm2e1Tl+Hn6W7QrS4tKQHDUB7i4ebI/JoeXX5luW3HRVHCvAMwDTB063zAIbFzkObi62Tkd6z5FmS9h6iuDrmqY9cdjxBkipdoeZIniimo3NloQrfX96l1WxhgYPtVRgmRE7A7GBsZzCQifFZDbJWpW8jrUqdS11VucculYlLTYNQ4OH8nNHTq1975dDE67D9+IK8grC9Njplt8LZ5/CQj3veD537TeW7QlVUmQS3F05z5qc2/66/ViR15Fw7anZY/V1fw9/TI+dbqlzkBSV5FAJV28VuSjSNO2yQ47FANgPOy9yYWstphZkFWdZeq5WFaxCQ2uD1TkxATFWVVi4zwAdS7OxGeuL12PV/lVI35+OlftXdvpcDeo3yCqhYrU16uvaRxLaL+5dVccM9AzEtIHTLDe+46PGO9QFnnpfm7kN2yu2W/ahSs9PR2lDqdU5wd7BViOnfb3aGhEAFNQVWEa4luct75Rw5d6Si+Ghw3WK7sT1Vpn2+yBl2g8cPHYvgCdhZ2Xae5vJbMLm0s1YkbcC6fvTkZGf0WmRdrhvOGbEzkBqTCqmDpiKpKgkeLja4Vbx1GtKG0qxav8qrCqQx8aSjVbbCQAyMnro4v6BgQN1ipbIcRxrfzc/Dz+kxKQgZUAKpsZMxeT+k7k3Yh93oOUA1hatlU2zC1ZhTeGaTtUt26/j7W3y6PDRnIpKdAyF9YWWEa6cyhysvHqlQ3UM99ZGwzkAtkE2Gh4M4AUAL+m90bC9OXSfgfT8dKTnp3daM+Dp6onk6GRMHTAVUwdMxZQBU9A/oL9OEVNPM2tm5FTkWCVUh9/0uSgXJIQnIDUmVXrb49K4to/IBtp7VNuTrt3Vu62+rqAwOny0pT1OiUnhrAMnpmka9tftx6qCVZaEakvZFqtNfQEgNjAWqQNTLdP/Wb2SqO/p8QTr4B8yCsBrAKYCqAWwEFKive0Y39enEqzDaZqG3dW7kZ6fjsyCTGQWZmJH5Y5O58UExGBqzFTLRT4xMhFebl46REwnQ9M0FNYXYl3ROqwvXo91ReuQVZyFA60HrM7z8/DDlAFTkDIgBakDUzFlwBQEeAboFDVR31F8oBirC1ZjdcFqZBZmYmPJxk6FjIK9gy2/n5MHTMb4qPEI9g7WKWI6GTXNNcgqzsL64vWWNvnwvX7cXNyQFJmEGaj7XQAADuVJREFU1JhUGd2MSWGnJxH1ToLVXX09wepKTXMN1hattSRca4vWdpqO4ObihtFho5EclYzk6GQkRyVjbMRYeLt76xQ1daWqqQpZxVlWCdXhI5aAJNCpA1ORGiOPhIgErgMhsgMGkwEbSzYisyATqwtXI7MgEyUNJZ3OG9RvECZET7Bqk7lHon1pbG3ExpKNWF+83pJU/VH9R6fzgryCkBKTYkmoJvafCB93Hx0iJiJ7xgTLwbWZ27CjcodVwpVbmdtpyoKrcsWosFGWi3tiRCLGhI/hRb4XtJnb8Ef1H8guy8bm0s3ILstGdml2p0pSgFy8J/WfhInRE+W5/0RE+kXqEDURnaj2KWTtI1xZxVnYVLqpU/VYABgcNBjJUckYHzUeCeEJSIhIQExADKeS9TBN01B0oAjZpdnYUrYF2WXyvLNqZ6frppebF8ZFjsPE6Iny6D8R8SHxXD9FRMfEBMsJNbY2YnPpZmwo2YCNJRuxoWQDcipyOl08ACDaPxpjwsdgTNgYeQ4fg1Fho+Dr4atD5I5N0zQU1BcgtzIXOyp2YHvFdmSXZWNb+TY0GZs6ne/r7otxkeOsEqrBQYN5g0XkRExmE3ZU7MCGkg3IKs7ChpIN2Fy6ucukK8AzAGPCx0jCdTDpGhM+hlMMu6mqqQo7q3ZiR8UObC3fakmmDt8HEJBOyISIBKtkanTYaJZLJ6JuYYLVRzQZm5Bdmm1JuLaVb8P2iu1d3vgrKMT1i8OwkGEYFnzwcfB1XL+4Pn/BqTXUYm/NXuyt2YvcylxJqCp3YGflTjQaG7v8npiAGCRGJiIxQh7jIsdhSPAQ9oQS9UEmswk5FTnYUCzJ1raKbdhathUVTRVdnh/mE4b4kHjEh8RjWPAwy+uhwUP7/NRvg8mA/XX7sbtqt6U9zq2S58qmyi6/J8gryKo9HhsxFqPCRvX5f0sish0mWH2YWTNjX80+bCvfJo8Kec6tzO1U9rudq3JFXL84DA4ajIGBAxETECPPgTGW9458kTJrZlQ2VaL4QDFKDpQgvy4fe2v2Yl/tPktSVWuoPeL3h/uGY0ToCIwMHYkRoSPkAh6ZyB5oIjqmsoYybC3fim3lknBtLd96xI6wdgMCBiA2MBax/WIxMGAgBgYOlNeB8tqRC+Bomob6lnqUNpSi6EAR9tXsQ15tHvbV7sO+Wnl9eNGJQ/m6+2JE6AgMDx2OMWFjkBgpyVR///6cKUBEPYoJFnXS2taKPdV78Ef1H9hdvRu7q3bLc/VuFNQVQMPR/09CfUIR4RuBcN9whPmGIcwnTF77hCHMNwyhPqEI9AyEv6c/AjwD4O/hDy83L5tf8MyaGQaTAQdaDqDGUIPq5mpUN1ejprnjdUVThSRTDSUoPlCM0obSIyaX7XzcfTA4aDAG9RuE4SHDJaEKG4nhIcMR4hNi078DEfVtZs2M4gPF2FW1C7urdmNX1S7sqt6FXVW7sLdm7zHbK38Pf4T7hiPCT9rkcJ9DXvuGI9g7GAGeAQj0DESAZwACPAPg4+7TIwmIsc2IWkMtagw1qGmu6fRc2lCKkoYSy3PJgRI0m5qP+jNdlSsGBg7E4KDBGBk6EsNDpU0eETqCiRQR6YYJFp0Qg8mAPdV7kF+Xj/11+1FQV4D99fJcUF+AgroCGM3GE/65bi5u8Pfwh5+HHzxcPSwPd1d3y2s3FzeYNTPazG0wa2Z5rclrk9mEJmOT5dHY2njMC/ORhHiHIMo/CtH+0YgJiLEkU4OCBmFw0GCE+YTxok1EujOZTcivlba4/dHeNrc/utMOuipXBHgGwNvdGx6uHvB09ZRnN0/LewDQoEHTNJg1s9Vrg8mAZlMzmo3NaDY1o8nYhGZjM9qOvjNLl3zcfRDlJ+1xXL84DOo3SJ6D5HlAwABWVSUiu8MEi2zKrJlR3lhueVQ0VqCiqcLyurypHJVNlTjQcgD1LfU40CrPh+8lYytebl7w8/BDsHcwgr2DEeQVZHkd7B2MEO8QRPtHI9o/GlH+UYj0i+QeYkTkFDRNQ62h1qpNLm8sR1ljmeV1raEWdS11qG+pR31LPeoMdd3unDoWF+WCfl79EOQVJM/eQQjyOvjwDkKEbwSi/KMQ5RdlaY/9PfzZoUVEDocJFtmFFlMLDrQeQENrA4xtRrS2tcJolufWtlYY24wwmU1wUS5wUS5wdXHteK1c4eriCl93X/i4+1ge3u7eLCJBRHSCjG1G1LfUw2AyoLWtFS1tLWgxtVi9BgClFFyUCxSU1WtPN094u3lb2mFvN294u3vD3cWdyRIR9QlMsIiIiIiIiGzkSAkWu/2JiIiIiIhshAkWERERERGRjTDBIiIiIiIishEmWERERERERDbCBIuIiIiIiMhG7GXXvgal1E69gyCnEgqgUu8gyGnw80S2xs8U2RI/T2RL/Dwdv9iuDtpLgrWzqxKHRN2llMriZ4pshZ8nsjV+psiW+HkiW+Ln6eRxiiAREREREZGNMMEiIiIiIiKyEXtJsN7WOwByOvxMkS3x80S2xs8U2RI/T2RL/DydJKVpmt4xEBEREREROQV7GcEiIiIiIiJyeEywiIiIiIiIbES3BEspNUop9ZtSqkkpVayUelIp5apXPORYlFJDlVL/UUplK6XalFLLuzhHKaUeUkoVKKWalVLpSqlxOoRLdk4p9Wel1PdKqSKlVINSaoNS6rIuzrteKbVbKWU4eM5pesRL9k0pNUcptVopVXXws7JTKfWwUsrjkHPYPlG3KKX6H2ynNKWU3yHH+Zmi46KU+tvBz8/hj5sOOYefp5OgS4KllAoCsAyABuBPAJ4EcA+AJ/SIhxzSaADnANh18NGVBwE8AuAZAOcDaACwTCkV2SsRkiO5G/L5uAvABQB+B/CpUuq29hOUUpcCeAvAR/j/9u421LKyCuD4f82MOl2bccYxCUpyJGNKCZSKnLI3KBlL+pAlhZRIUBQYFBmSlRSKEioyVtOH3ijMSgVJnAb6UDT2jjbRvIiMjRKIpd1x0CnTcfXheY7t9tyZzrlnd/e95/x/cLhnP8++3PVhzZqzzt7Ps2ETsBO4KyLOXPhwtcito+TQhym58k3gs8ANjXOsT5qvL1Pypc2c0qjeBpzTeN3RmDOfxtDLJhcRcQVwOfCyzDxQxy4HrgJePBiTjiQilmXmc/X9bcBJmfmWxvxK4FHg+sz8Yh07HtgHfD0zr1zwoLVoRcRJmflYa+wW4JzMXF+P7wfuycxL6/EyYAewIzMvXuiYtbRExNXAx4G1wHFYnzQPEXEucCdwDaXRWpWZT/p/nkYREZcA36Lmzxzz5tOY+rpFcBOwrdVI3Qq8AHhzPyFpKRk0V0exEVgN/LDxO08BP6bkn/S8dnNV3QecDBARpwGv4L/z6TngR5hPGs7jwOAWQeuTRlaXUWym3PXTrlnmlLpkPo2prwZrA7CnOZCZDwMH65w0rg3AIeCB1vhuzDENZyOwq74f5Mye1jm7gRMj4kULFpWWjIhYHhEzEfFG4DLga1luG7E+aT4+CqwEvjLHnDml+dgbEc/WdaIfaYybT2Na0dPfXQvsn2N8ts5J41oLPJmZh1rjs8BMRBybmf/qIS4tAXXzincDl9ahQV1q163ZxvzfFiA0LS1PUW4HhLJ279P1vfVJI4mIdcCXgIsz85mIaJ9iTmkUj1DWV/0WWA68H9gSETOZeSPm09j6arCgbHDRFkcYl+bjSDl2pDmJiDgVuAW4MzO/3Zpu5435pKPZCMwArwM+D9wMfKzOWZ80iquB32Tm3Uc5x5zSUDJzG7CtMbQ1Io4DroyImwanzfGr5tOQ+mqwZoE1c4yfwNxXtqRRzQKrImJ56xuYNcDBzHymp7i0iEXEicBW4GGguXHF4ErVGuCJxvigjlm3dJjMvLe+3R4RjwHfiYjrsT5pBBFxBuVq+psiYlBzZurPEyLiEOaUxncb8D7gVMynsfW1BmsPrXs4I+IU4HgOX+MgzcceymXvl7fGD1v/JwFExAxwF2UjgnfWBb0Dg5xp33u+Afh7Znp7oP6XQbO1HuuTRnM6cAzwK8oH31n+sw7rL5SNL8wpdSUxn8bWV4O1FTgvIlY1xi4C/gH8vJ+QNGF+CRwA3jsYqB+gL6Dkn/S8iFhB2RHwdGBTZv61OZ+ZD1Ket9bMp2X12HzSMN5Qf/4Z65NGsx14a+t1XZ07n7Jduzmlcb2HsjvlQ5hPY+vrFsEtlB2V7oiI64DTKM/AusFnYGkY9R/6+fXwJcDqiLiwHt+dmQcj4lrgcxExS/nG5ZOULxU2L3jAWuy+SsmnT1B2BXx9Y+6+zHyaUqO+FxH7gHuAD1Easg8sbKha7CLiJ8BPKQ+jPkRprj4F/CAz99ZzrE8aSn2MxM+aY3WtKMAvBs8xMqc0rIi4nbLBxR8pV6ouqq/L6iNI/mk+jaeXBiszZ+suXTdT9tTfD9xI+QAjDeNkyhWHpsHxesrD8K6lFIMrgHXA74G3Z+ajCxSjlo531J83zTG3HtiXmd+PiBcCn6HsvrQTeFdm/mmBYtTS8TvgEspahmeBByl1aEvjHOuTumZOaVj3U9b1nULZuGIX8MHM/G7jHPNpDFEeySFJkiRJGldfa7AkSZIkaeLYYEmSJElSR2ywJEmSJKkjNliSJEmS1BEbLEmSJEnqiA2WJEmSJHXEBkuSJEmSOmKDJUmaSBGxOiKuiohX9h2LJGl62GBJkibVa4AvAMf0HYgkaXrYYEmSJtVZwNPArr4DkSRNj8jMvmOQJKlTEbEb2NAavj0zL+wjHknS9LDBkiRNnIh4LXArsBO4pg4/kpkP9ReVJGkarOg7AEmS/g92AC8FNmfmr/sORpI0PVyDJUmaRGcAxwL39h2IJGm62GBJkibR2UACf+g7EEnSdLHBkiRNorOAvZl5oO9AJEnTxQZLkjSJXoXbs0uSeuAmF5KkSbQfODsizgOeAB7IzMd7jkmSNAXcpl2SNHEi4kzgG8CrgZXAuZm5vd+oJEnTwAZLkiRJkjriGixJkiRJ6ogNliRJkiR1xAZLkiRJkjpigyVJkiRJHbHBkiRJkqSO2GBJkiRJUkdssCRJkiSpIzZYkiRJktSRfwPkhD7LPwV0AQAAAABJRU5ErkJggg==",
+      "text/plain": [
+       "<Figure size 864x216 with 1 Axes>"
+      ]
+     },
+     "metadata": {
+      "needs_background": "light"
+     },
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "import numpy as np\n",
+    "import pandas as pd\n",
+    "import matplotlib.pyplot as plt\n",
+    "\n",
+    "def multiple_formatter(denominator=2, number=np.pi, latex='\\pi'):\n",
+    "    def gcd(a, b):\n",
+    "        while b:\n",
+    "            a, b = b, a%b\n",
+    "        return a\n",
+    "    def _multiple_formatter(x, pos):\n",
+    "        den = denominator\n",
+    "        num = np.int(np.rint(den*x/number))\n",
+    "        com = gcd(num,den)\n",
+    "        (num,den) = (int(num/com),int(den/com))\n",
+    "        if den==1:\n",
+    "            if num==0:\n",
+    "                return r'$0$'\n",
+    "            if num==1:\n",
+    "                return r'$%s$'%latex\n",
+    "            elif num==-1:\n",
+    "                return r'$-%s$'%latex\n",
+    "            else:\n",
+    "                return r'$%s%s$'%(num,latex)\n",
+    "        else:\n",
+    "            if num==1:\n",
+    "                return r'$\\frac{%s}{%s}$'%(latex,den)\n",
+    "            elif num==-1:\n",
+    "                return r'$\\frac{-%s}{%s}$'%(latex,den)\n",
+    "            else:\n",
+    "                return r'$\\frac{%s%s}{%s}$'%(num,latex,den)\n",
+    "    return _multiple_formatter\n",
+    "\n",
+    "class Multiple:\n",
+    "    def __init__(self, denominator=2, number=np.pi, latex='\\pi'):\n",
+    "        self.denominator = denominator\n",
+    "        self.number = number\n",
+    "        self.latex = latex\n",
+    "\n",
+    "    def locator(self):\n",
+    "        return plt.MultipleLocator(self.number / self.denominator)\n",
+    "\n",
+    "    def formatter(self):\n",
+    "        return plt.FuncFormatter(multiple_formatter(self.denominator, self.number, self.latex))\n",
+    "\n",
+    "#xt=np.linspace(-4*np.pi,4*np.pi,num=500)\n",
+    "xt=np.linspace(0,52,num=1000)\n",
+    "##\n",
+    "fig, ax = plt.subplots(figsize=(12, 3))\n",
+    "\n",
+    "fun = 5 + 5*np.cos((np.pi/12)*(xt-3))\n",
+    "dfun = -(5*np.pi/12)*np.sin((np.pi/12)*(xt-3))\n",
+    "\n",
+    "ax.plot(xt, fun, color='b', lw=2, label='$f\\,(t)$')\n",
+    "ax.plot(xt, dfun, color='g', lw=2, label=\"$f\\,'\\,(t)$\")\n",
+    "ax.tick_params(axis='both', labelsize= 15)\n",
+    "ax.set_xlabel('$t$', fontsize = 16)\n",
+    "ax.legend(fontsize=14)  \n",
+    "\n",
+    "ax.axhline(0, color='black', lw=1)\n",
+    "ax.axvline(0, color='black', lw=1)\n",
+    "\n",
+    "ax.set_xlim(0,55)\n",
+    "ax.set_ylim(-2,12)\n",
+    "\n",
+    "#ax.xaxis.set_major_locator(plt.MultipleLocator(np.pi / 2))\n",
+    "#ax.xaxis.set_minor_locator(plt.MultipleLocator(np.pi / 12))\n",
+    "#ax.xaxis.set_major_formatter(plt.FuncFormatter(multiple_formatter()))\n",
+    "\n",
+    "fig.tight_layout()\n",
+    "fig.savefig('func_diff.png')"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Analysing change, continuously"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Average rate of change\n",
+    "\n",
+    "Watch this [video](https://web.microsoftstream.com/video/2953e661-e1c1-43ff-ba78-ca52099ed1ec)\n",
+    "for a short introduction to this lecture."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Instantaneous rate of change  \n",
+    "  \n",
+    "If we have a continuous function $f(x)$, for each two points $x_{0}$ and $x_{1}$, we can define the average rate of change between these points by the ratio between how much $f$ has changed and how much $x$ has changed:  \n",
+    "  \n",
+    "$$\\frac{\\Delta f(x)}{\\Delta x} = \\frac{f(x_{1}) - f(x_{0})}{x_{1} - x_{0}}$$  \n",
+    "  \n",
+    "For points $x_{0}$ and $x_{1}$ this can be graphically represented by:  \n",
+    "  \n",
+    "```{image} sec_tan.png\n",
+    "```  \n",
+    "  \n",
+    "The line that passes through the points $(x_0, f(x_0))$ and $(x_1, f(x_1))$ (yellow line) is named *secant line*. As we have seen before in our treatment of linear functions, the slope of the secant line will be given by the average rate of change between this points (remember that the slope for linear functions is defined as $m=\\displaystyle\\frac{\\Delta y}{\\Delta x})$.\n",
+    "  \n",
+    "Now consider that we change the value of $x_1$ so that it slowly approaches $x_0$ from the right. In this case, the secant lines also change (from yellow to red), as we change the image of $x_1$ the function, $f(x_1)$. In the limit where $x_1$ is very close to $x_0$, the secant line approaches the *tangent* (sark red line) of the function $f(x)$ at the point $x_0$ (the tangent is the straight line that \"just touches\" the curve at that point).  \n",
+    "  \n",
+    "If we represent the slope of the tangent line at the point $x_0$ by the term $f'(x_0)$, we will have:  \n",
+    "  \n",
+    "$$\\lim_{x_{1} \\to x_{0}} \\frac{\\Delta f(x)}{\\Delta x} = \\frac{f(x_{1}) - f(x_{0})}{x_{1} - x_{0}} = f'(x_{0})$$  \n",
+    "  \n",
+    "Since we can do the same process as we change the value $x_0$, we define a new function $f'(x_{0})$ which we call the **derivative** of the function $f$ at the point $x_0$, and its value tells us about the *instantaneous* rate of change of $f$ at the point $x_0$.  \n",
+    "  \n",
+    "If we look instead at the interval between $x_{0}$ and $x_{1}$ and define $h = x_1 - x_0$, if $x_1$ approaches $x_0$, then $h$ approaches $0$. and we would have for the derivative of $f(x)$:  \n",
+    "  \n",
+    "$$\\lim_{h \\to 0} \\frac{f(x + h) - f(x)}{h} = f'(x)$$  \n",
+    "\n",
+    "Visit this [link](https://www.demonstrations.wolfram.com/SecantAndTangentLines/) for an interactive demonstration of the limit of the tangent curve.\n",
+    "  \n",
+    "``````{note}  \n",
+    "Note that in the previous expression the index $0$ was omitted from $x_0$. We will assume that the derivative is defined for all the points in the domain of the function $f(x)$ for which the limit in the expression exists. \n",
+    "\n",
+    "It is good to have a graphical intuition for when the derivative is not well defined. Some cases are shown below.  \n",
+    "  \n",
+    "`````{tabbed} Case 1   \n",
+    "\n",
+    "````{panels}  \n",
+    "  \n",
+    "```{image} ./cases01a.png  \n",
+    "\n",
+    "```  \n",
+    "  \n",
+    "```{image} ./cases01b.png  \n",
+    "\n",
+    "``` \n",
+    "\n",
+    "---  \n",
+    "  \n",
+    "For the points where the function $f(x)$ presents discontinuities, or for the points where it is simply not defined, the derivative will also not be defined.\n",
+    "\n",
+    "  \n",
+    "````  \n",
+    "`````  \n",
+    "  \n",
+    "`````{tabbed} Case 2   \n",
+    "\n",
+    "````{panels}  \n",
+    "  \n",
+    "```{image} ./cases02.png  \n",
+    "\n",
+    "```  \n",
+    "\n",
+    "---  \n",
+    "  \n",
+    "For the points where the curve of the function $f(x)$ presents kinks, note that there are two tangent lines, one coming from the right (red line) and another coming from the left (yellow line). Since the function $f'(x)$ must have only one value for the image of $x$ (in other words, only one value for the slope of the tangent at point $f(x)$), then the derivative is not defined in this case.\n",
+    "  \n",
+    "````  \n",
+    "`````  \n",
+    "  \n",
+    "`````{tabbed} Case 3\n",
+    "\n",
+    "````{panels}  \n",
+    "  \n",
+    "```{image} ./cases03.png  \n",
+    "\n",
+    "```  \n",
+    "\n",
+    "---  \n",
+    "\n",
+    "For the points where the function $f(x)$ is locally vertical (in this case, for $x=0$), the slope of the tangent curve is infinite. Since infinite is not a real number, the derivative is not defined for those points.\n",
+    "  \n",
+    "````  \n",
+    "`````  \n",
+    "\n",
+    "``````  \n",
+    "  \n",
+    "  \n",
+    "````{admonition} Example  \n",
+    "  \n",
+    "By the definition of the derivative of $f(x)$:  \n",
+    "  \n",
+    "$$f'(x)=\\lim_{h \\to 0} \\frac{f(x + h) - f(x)}{h}$$  \n",
+    "  \n",
+    "Thus, if $f(x)=x^2$, we will have:  \n",
+    "  \n",
+    "$$f'(x)=\\lim_{h \\to 0} \\frac{(x + h)^{2} - x^{2}}{h} = \\lim_{h \\to 0} \\frac{x^{2} + 2xh + h^{2} - x^{2}}{h} = \\lim_{h \\to 0} (h + 2x) = 2x$$  \n",
+    "  \n",
+    "So that the derivative of $f(x)=x^2$ is the function $f'(x)=2x$.  \n",
+    "  \n",
+    "  \n",
+    "```{note}  \n",
+    "Although limits of functions can be defined in a more rigorous way, in this course we will treat them more intuitively, and use, if required, the known rules for limits. Thus, if $C$ is a constant, $\\lim_{x \\rightarrow a} f(x) = L$, and $\\lim_{x \\rightarrow a} g(x) = M$, it is generally valid that:  \n",
+    "  \n",
+    "$$\\lim_{x \\rightarrow a} [f(x) + g(x)] = \\lim_{x \\rightarrow a} f(x) + \\lim_{x \\rightarrow a} g(x) = L + M$$  \n",
+    "$$\\lim_{x \\rightarrow a} [Cf(x)] = C\\lim_{x \\rightarrow a} f(x) = CL$$  \n",
+    "$$\\lim_{x \\rightarrow a} [f(x)g(x)] = [\\lim_{x \\rightarrow a} f(x)][\\lim_{x \\rightarrow a} g(x)] = LM$$  \n",
+    "$$\\lim_{x \\rightarrow a} \\left[\\frac{f(x)}{g(x)}\\right] = \\frac{\\lim_{x \\rightarrow a} f(x)}{\\lim_{x \\rightarrow a} g(x)} = \\frac{L}{M},\\ \\ (\\mbox{if}\\ M \\ne 0)$$  \n",
+    "$$\\lim_{x \\rightarrow a} [C^{f(x)}] = C^{\\lim_{x \\rightarrow a} f(x)} = C^L$$  \n",
+    "\n",
+    "```\n",
+    "  \n",
+    "````\n",
+    "  \n",
+    "Other common notations for the derivative are given by:  \n",
+    "  \n",
+    "$$f'(x) = \\frac{df}{dx} = \\frac{d}{dx}f(x) = \\dot{f}(x)$$  \n",
+    "\n",
+    "```{Example}\n",
+    "Derivatives (or rate of change) are needed everywhere in science. Some common examples are the velocity as the derivative of position and acceleration as the derivative of velocity. When asking yourself 'how fast is the population growing?' or 'how steep is this mountain?' you can find the answer by calculating the derivative of the population number or the height profile.\n",
+    "```\n",
+    "\n",
+    "Visit this [link](https://the-learning-machine.com/article/calculus/derivatives) for examples that provide demonstrations of the intuitive interpretation of the derivative."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Calculating derivatives  \n",
+    "  \n",
+    "  \n",
+    "It is possible to generalise the previous example for $f(x)=x^2$ and obtain the derivative for any power function $f(x)=x^n$ ($n \\ne 0$):  \n",
+    "  \n",
+    "$$f(x) = x^{n} \\implies f'(x) = nx^{n-1}$$  \n",
+    "  \n",
+    "  \n",
+    "```{admonition} Example  \n",
+    "  \n",
+    "$$x^{2} \\implies 2x^{2-1} = 2x$$\n",
+    "$$\\ x^{3} \\implies 3x^{3-1} = 3x^{2}$$\n",
+    "$$\\ x^{4.2} \\implies 4.2x^{4.2-1} = 4.2x^{3.2}$$  \n",
+    "\n",
+    "```  \n",
+    "  \n",
+    "It is possible to show, from the definition of derivative and the rules for limits, that there are simple rules for derivatives when calculating them for a sum of functions or a product by a constant. If $C$ is a constant, we generally have:  \n",
+    "  \n",
+    "$$\\begin{align}\\frac{d}{dx}[f(x)+g(x)] &= \\frac{df}{dx} + \\frac{dg}{dx} \\\\\n",
+    "   \\frac{d}{dx}[Cf(x)] &= C\\frac{df}{dx}\\end{align}$$  \n",
+    "  \n",
+    "  \n",
+    "Thus, differentiating a given polynomial p(x) stands for differentiating each term separately and summing the result.  \n",
+    "  \n",
+    "```{admonition} Example  \n",
+    "  $$f(x) = 3x^{2} - 4x + 6 \\implies f'(x) = 3(2x) - 4(x^{0}) + 0 = 6x - 4$$  \n",
+    "  $$f(x) = x^{60} - 3x^{20} - 4x \\implies f'(x) = 60x^{59} - 60x^{19} - 4$$\n",
+    "```  \n",
+    "  \n",
+    "For products and quotients of functions, however, we have different rules:  \n",
+    "  \n",
+    "$$\\frac{d}{dx}[f(x)g(x)] = \\left(\\frac{df}{dx}\\right) g(x) + f(x)\\left(\\frac{dg}{dx}\\right)$$  \n",
+    "$$\\frac{d}{dx}\\left[\\frac{f(x)}{g(x)}\\right] = \\frac{\\left(\\displaystyle\\frac{df}{dx}\\right)g(x) - f(x)\\left(\\displaystyle\\frac{dg}{dx}\\right)}{[g(x)]^{2}}$$  \n",
+    "\n",
+    "```{tip}\n",
+    "If you do not want to remember the quotients rule, you can use the product rule on $f(x)j(x)$, with $j(x)=g(x)^{-1}$ and use the chain rule to evaluate $\\frac{d}{dx} j(x)$.\n",
+    "```\n",
+    "  \n",
+    "```{admonition} Examples  \n",
+    "  $$f(x) = (x + 2)(x - 4) \\implies f'(x) = (1)(x-4) + (x+2)(1) = x - 4 + x + 2 = 2x - 2$$  \n",
+    "  $$f(x) = \\frac{\\alpha x}{1 + \\beta x} \\implies f'(x) =  \\frac{(\\alpha)(1 + \\beta x) - (\\alpha x)(\\beta)}{(1 + \\beta x)^{2}} = \\frac{\\alpha + \\alpha\\beta x - \\alpha \\beta x}{(1 + \\beta x)^{2}} =\\frac{\\alpha}{(1 + \\beta x)^{2}}$$  \n",
+    "  \n",
+    "```  \n",
+    "  \n",
+    "Moreover, expressions for the derivatives of the other elementary functions can also be found:\n",
+    "\n",
+    "- $f(x) = a^{x} \\implies f'(x) = a^{x} \\cdot (\\ln a) ,\\ (a > 0, a \\neq 1)$  \n",
+    "    $\\left(\\mbox{in particular: }\\ (e^{x})' = e^{x}\\right)$  \n",
+    "    \n",
+    "- $f(x) = \\log_{a}x \\implies f'(x) = \\displaystyle\\frac{1}{(\\ln a)x}$  \n",
+    "    $\\left(\\mbox{in particular: }\\ (\\ln x)' = \\displaystyle\\frac{1}{x}\\right)$  \n",
+    "    \n",
+    "- $f(x) = \\mbox{sin}(x) \\implies f'(x) = \\mbox{cos}(x)$  \n",
+    "\n",
+    "- $f(x) = \\mbox{cos}(x) \\implies f'(x) = -\\mbox{sin}(x)$\n",
+    "  \n",
+    "  \n",
+    "```{admonition} Example  \n",
+    "If $f(x) = \\mbox{tan}(x) = \\displaystyle\\frac{\\mbox{sin}(x)}{\\mbox{cos}(x)}$, then:  \n",
+    "  \n",
+    "$$f'(x) = \\frac{(\\mbox{cos}(x))\\mbox{cos}(x) - \\mbox{sin}(x)(-\\mbox{sin}(x))}{(\\mbox{cos}(x))^{2}} = \\frac{\\mbox{sin}^{2}(x) + \\mbox{cos}^{2}(x)}{\\mbox{cos}^{2}(x)} = \\frac{1}{\\mbox{cos}^{2}(x)} = \\mbox{sec}^{2}(x)$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Chain rule  \n",
+    "  \n",
+    "The calculation of derivatives for more complicated functions can generally be made easier by breaking the function into components, for example:  \n",
+    "  \n",
+    "- $f(x) = (3x^{2} - 2x + 1)^{3} = (g(x))^{3}$, where $g(x) = 3x^{2} - 2x + 1$\n",
+    "- $f(x) = \\displaystyle\\frac{\\sqrt{2x - 1}}{1 + \\sqrt{2x - 1}} = \\displaystyle\\frac{g(x)}{1 + g(x)}$, where $g(x) = \\sqrt{h(x)}$, and $h(x) = 2x - 1$  \n",
+    "  \n",
+    "We have seen this before when discussing transformation of graphs.  When one function is nested within the other, we have a composite function, or as for the last example above:\n",
+    "\n",
+    "$$f(x) = f(g(h(x)))$$  \n",
+    "  \n",
+    "The derivative of composite functions in relation to the dependent variable is then given by the *chain rule*:  \n",
+    "  \n",
+    "- $\\displaystyle\\frac{d}{dx}f(x) = \\displaystyle\\frac{d}{dx}f(g(x)) = \\left(\\displaystyle\\frac{df}{dg}\\right)\\left(\\displaystyle\\frac{dg}{dx}\\right)$\n",
+    "- $\\displaystyle\\frac{d}{dx}f(x) = \\displaystyle\\frac{d}{dx}f(g(h(x))) = \\left(\\displaystyle\\frac{df}{dg}\\right) \\left(\\displaystyle\\frac{dg}{dh}\\right) \\left(\\displaystyle\\frac{dh}{dx}\\right)$  \n",
+    "  \n",
+    "The strategy then consists in identifying the nested functions, calculating their derivatives and multiplying the results.  \n",
+    "  \n",
+    "```{admonition} Examples  \n",
+    "  \n",
+    "- If $f(x) = (3x^{2} - 2x + 1)^{3}$, we have:  \n",
+    "      \n",
+    "    $$f(g) = g^{3} \\implies \\displaystyle\\frac{df}{dg} = 3g^2$$\n",
+    "    $$g(x) = 3x^{2} - 2x + 1 \\implies \\displaystyle\\frac{dg}{dx} = 6x - 2$$  \n",
+    "    \n",
+    "    Thus:  \n",
+    "      \n",
+    "    $$\\displaystyle\\frac{df}{dx} = \\left(\\displaystyle\\frac{df}{dg}\\right)\\left(\\displaystyle\\frac{dg}{dx}\\right) = 3g^{2} \\cdot g' = 3(3x^{2} - 2x + 1)^2(6x - 2)$$  \n",
+    "      \n",
+    "    <br />  \n",
+    "      \n",
+    "- If $f(x) = \\displaystyle\\frac{\\sqrt{2x - 1}}{1 + \\sqrt{2x - 1}}$, then:  \n",
+    "      \n",
+    "    $$f(g) = \\displaystyle\\frac{g}{1+g} \\implies \\displaystyle\\frac{df}{dg} = \\frac{(1+g) - g}{(1+g)^2} = \\frac{1}{(1+g)^2}$$  \n",
+    "    $$g(h) = \\sqrt{h} \\implies \\displaystyle\\frac{dg}{dh} = \\frac{1}{2}h^{-1/2}$$  \n",
+    "    $$h(x) = 2x-1 \\implies \\displaystyle\\frac{dh}{dx} = 2$$  \n",
+    "      \n",
+    "    Thus:  \n",
+    "      \n",
+    "    $$\\displaystyle\\frac{df}{dx} =  \\left(\\displaystyle\\frac{df}{dg}\\right) \\left(\\displaystyle\\frac{dg}{dh}\\right) \\left(\\displaystyle\\frac{dh}{dx}\\right) = \\frac{1}{(1+\\sqrt{2x - 1})^2}\\cdot \\frac{1}{2(\\sqrt{2x - 1})}\\cdot 2 = \\frac{1}{(\\sqrt{2x - 1})(1+\\sqrt{2x - 1})^2}$$  \n",
+    "      \n",
+    "    <br />\n",
+    "      \n",
+    "- If $f(t) = 5 + 5\\mbox{cos}\\left[\\displaystyle\\frac{\\pi}{12}(t - 3)\\right]$, we have:  \n",
+    "      \n",
+    "    $$f(g) = 5 + 5g \\implies \\displaystyle\\frac{df}{dg} = 5$$  \n",
+    "    $$g(h) = \\mbox{cos}(h) \\implies \\displaystyle\\frac{dg}{dh} = -\\mbox{sin}(h)$$  \n",
+    "    $$h(t) = \\frac{\\pi}{12}(t - 3) \\implies \\displaystyle\\frac{dh}{dt} = \\frac{\\pi}{12}$$  \n",
+    "      \n",
+    "    Thus:  \n",
+    "      \n",
+    "    $$\\displaystyle\\frac{df}{dx} =  \\left(\\displaystyle\\frac{df}{dg}\\right) \\left(\\displaystyle\\frac{dg}{dh}\\right) \\left(\\displaystyle\\frac{dh}{dx}\\right) = 5 \\cdot \\left(-\\mbox{sin}(h)\\right) \\cdot \\frac{\\pi}{12} = -\\frac{5\\pi}{12}\\mbox{sin}\\left[\\displaystyle\\frac{\\pi}{12}(t - 3)\\right]$$  \n",
+    "      \n",
+    "    For this last example if we plot the function $f(t)=5 + 5\\mbox{cos}\\left[\\displaystyle\\frac{\\pi}{12}(t - 3)\\right]$ together with its derivative $f'(t)=-\\displaystyle\\frac{5\\pi}{12}\\mbox{sin}\\left[\\displaystyle\\frac{\\pi}{12}(t - 3)\\right]$ we will have:  \n",
+    "      \n",
+    "    ```{image} func_diff.png  \n",
+    "    ```  \n",
+    "      \n",
+    "    Try to compare the two graphs. What happens with the function $f(t)$ on the intervals where the derivative $f'(t)$ is positive or negative? What happens at the points where the derivative is zero?"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Higher order derivatives  \n",
+    "\n",
+    "\n",
+    "If the derivative of a given function is also a \"well-behaved\" function (without kinks or discontinuities), then we can also calculate the derivative of the derivative. Thus:  \n",
+    "  \n",
+    "$$\\frac{d}{dx}\\left(\\frac{df}{dx}\\right) = \\frac{d^{2}f}{dx^{2}} = f''(x)$$  \n",
+    "  \n",
+    "The function $f''(x)$ is called the *second derivative* with respect to $x$.  \n",
+    "  \n",
+    "We can continue this process and get higher-order derivatives:  \n",
+    "  \n",
+    "$$\\frac{d^{n}f}{dx^{n}} = f^{(n)}(x),$$  \n",
+    "  \n",
+    "assuming again that these functions are also well-behaved. Function $f^{(n)}(x)$ is called the $n$-th derivative of $f$ with respect to $x$ (note the parenthesis on $(n)$ to distinguish from powers of $f$).  \n",
+    "\n",
+    "```{admonition} Examples\n",
+    "The velocity is the derivative of the position, and the acceleration is the derivative of the velocity. Therefore, acceleration is the second-order derivative of the position\n",
+    "```\n",
+    "\n",
+    "```{admonition} Examples  \n",
+    "  \n",
+    "- $f(x) = 3x^{2} - 6x + 2$, then:  \n",
+    "      \n",
+    "    $$f'(x) = 6x - 6$$  \n",
+    "    $$f''(x) = 6$$  \n",
+    "      \n",
+    "    Polynomials are always well-behaved and they are infinitely differentiable. However, if $n$ is the degree of the polynomial (*i.e.* the largest exponent in the polynomial), we will have: $p^{(k)}(x) = 0,\\ \\mbox{for}\\ k \\ge n+1$.  \n",
+    "    \n",
+    "    <br />\n",
+    "      \n",
+    "- $f(x) = e^{-x^{2}}$, then:  \n",
+    "      \n",
+    "    $$f'(x) = e^{-x^{2}} \\cdot (-2x) = -2xe^{-x^{2}}$$  \n",
+    "      \n",
+    "    $$\\begin{align}f''(x)\n",
+    "&= -2[e^{-x^{2}} + x(-2x)(e^{-x^{2}})] \\\\\n",
+    "&= -2e^{-x^{2}} + 4x^{2}e^{-x^{2}} \\\\\n",
+    "&= -2e^{-x^{2}}(1 - 2x^{2})\\end{align}$$  \n",
+    "  \n",
+    "```  \n",
+    "  "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Functions of several variables  \n",
+    "  \n",
+    "A function $z = f(x,y)$ of two dependent variables $x$ and $y$ defines a surface on the three dimensional $xyz$-space. If this function is well-behave it will be smooth, like if we applied different deformations on a rubber sheet. For smooth functions, we can calculate derivatives with respect to each of the dependent variables. This is written as:  \n",
+    "  \n",
+    "$$ \\frac{\\partial f}{\\partial x} \\text{ or } \\frac{\\partial f}{\\partial x}(x, y) \\text{ or } \\frac{\\partial}{\\partial x}f(x,y)$$  \n",
+    "  \n",
+    "$$\\frac{\\partial f}{\\partial y}, \\frac{\\partial f}{\\partial{y}}(x,y), \\frac{\\partial}{\\partial y}f(x,y)$$  \n",
+    "  \n",
+    "The partial derivative informs us how the function $f$ varies along the $x$ (or the $y$) direction at a specific point.  \n",
+    "  \n",
+    "```{admonition} Example  \n",
+    "Consider $f(x,y) = 4x^2y-2xy^3$. To compute the partial derivative with respect to $x$ we consider all terms in $y$ constants, and calculate the derivative as usual. For the partial derivative with respect to $y$, we consider the $x$ terms constant. We have:  \n",
+    "  \n",
+    "$$\\frac{\\partial f}{\\partial x} = (4y) \\cdot (2x) - 2y^{3} \\cdot (1) = 8xy - 2y^{3}$$  \n",
+    "$$\\frac{\\partial f}{\\partial y} = 4x^{2} - 2x(3y^{2}) = 4x^{2} - 6xy^{2}$$\n",
+    "```  \n",
+    "  \n",
+    "Higher order partial derivatives can also be defined, and represented with the common notations:  \n",
+    "  \n",
+    "$$\\frac{\\partial}{\\partial x}\\left(\\frac{\\partial f}{\\partial x}\\right) = \\frac{\\partial^{2}f}{\\partial x^{2}} = \\partial_{xx} f = f_{xx}(x,y)$$\n",
+    "$$\\frac{\\partial}{\\partial y}\\left(\\frac{\\partial f}{\\partial y}\\right) = \\frac{\\partial^{2}f}{\\partial y^{2}} = \n",
+    "\\partial_{yy} f = f_{yy}(x,y)$$  \n",
+    "  \n",
+    "  \n",
+    "with subscripts representing the variable and the order of differentiation.  \n",
+    "  \n",
+    "\n",
+    "For well-behaved functions, the order of the variables chosen to differentiate does not matter (a special case of what is called Schwarz's theorem):  \n",
+    "  \n",
+    "$$f_{yx}(x,y) = \\frac{\\partial}{\\partial y}\\left(\\frac{\\partial f}{\\partial x}\\right) = \\frac{\\partial^{2}f}{\\partial y \\partial x} = \\frac{\\partial^{2} f}{\\partial x \\partial y}= \\frac{\\partial}{\\partial x}\\left(\\frac{\\partial f}{\\partial y}\\right) = f_{xy}(x,y)$$\n",
+    "\n",
+    "```{admonition} Example \n",
+    "Often, a quantity of interest does not depend only on one variable but on many. The population size of a species depends on various aspects, including the availability of food, the number of predators, and other environmental factors.\n",
+    "```"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": []
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": []
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 24,
+   "metadata": {
+    "tags": [
+     "remove-cell"
+    ]
+   },
+   "outputs": [
+    {
+     "data": {
+      "image/png": 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",
+      "text/plain": [
+       "<Figure size 360x360 with 1 Axes>"
+      ]
+     },
+     "metadata": {
+      "needs_background": "light"
+     },
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "import matplotlib.pyplot as plt\n",
+    "import numpy as np\n",
+    "import matplotlib.ticker as plticker\n",
+    "\n",
+    "# Create figure and add axes\n",
+    "#fig = plt.figure(figsize=(5, 5))\n",
+    "#ax = fig.add_subplot(111)\n",
+    "\n",
+    "fig, ax = plt.subplots(figsize=(5, 5)) \n",
+    "\n",
+    "# Move left y-axis and bottim x-axis to centre, passing through (0,0)\n",
+    "ax.spines['left'].set_position('zero')\n",
+    "ax.spines['bottom'].set_position('zero')\n",
+    "\n",
+    "# Eliminate upper and right axes\n",
+    "ax.spines['right'].set_color('none')\n",
+    "ax.spines['top'].set_color('none')\n",
+    "\n",
+    "# Show ticks in the left and lower axes only\n",
+    "ax.xaxis.set_ticks_position('bottom')\n",
+    "ax.yaxis.set_ticks_position('left')\n",
+    "\n",
+    "\n",
+    "#loc = plticker.MultipleLocator(base=3.0) # this locator puts ticks at regular intervals\n",
+    "#ax.xaxis.set_major_locator(loc)\n",
+    "#loc = plticker.MultipleLocator(base=2.0) # this locator puts ticks at regular intervals\n",
+    "#ax.yaxis.set_major_locator(loc)\n",
+    "\n",
+    "x = np.linspace(-3, 3, 100)\n",
+    "y = x**3\n",
+    "ax.plot(x, y, color='b', linewidth=2.5, label=('$f(x) = x^3$'))\n",
+    "\n",
+    "ylim = ax.get_ylim()\n",
+    "xlim = ax.get_xlim()\n",
+    "\n",
+    "ax.tick_params(axis='both', which='major', labelsize=13)\n",
+    "\n",
+    "\n",
+    "ax.legend(fontsize=13)\n",
+    "# Add legend\n",
+    "\n",
+    "#ax.legend(bbox_to_anchor=(1.05, 0.3),loc='lower left', \n",
+    "#          frameon=False, labelspacing=0.2, fontsize=13)\n",
+    "\n",
+    "plt.tight_layout()\n",
+    "plt.savefig('xcube.png')"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 25,
+   "metadata": {
+    "tags": [
+     "remove-cell"
+    ]
+   },
+   "outputs": [
+    {
+     "data": {
+      "image/png": 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",
+      "text/plain": [
+       "<Figure size 360x360 with 1 Axes>"
+      ]
+     },
+     "metadata": {
+      "needs_background": "light"
+     },
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "import matplotlib.pyplot as plt\n",
+    "import numpy as np\n",
+    "import matplotlib.ticker as plticker\n",
+    "\n",
+    "# Create figure and add axes\n",
+    "#fig = plt.figure(figsize=(5, 5))\n",
+    "#ax = fig.add_subplot(111)\n",
+    "\n",
+    "fig, ax = plt.subplots(figsize=(5, 5)) \n",
+    "\n",
+    "# Move left y-axis and bottim x-axis to centre, passing through (0,0)\n",
+    "ax.spines['left'].set_position('zero')\n",
+    "ax.spines['bottom'].set_position('zero')\n",
+    "\n",
+    "# Eliminate upper and right axes\n",
+    "ax.spines['right'].set_color('none')\n",
+    "ax.spines['top'].set_color('none')\n",
+    "\n",
+    "# Show ticks in the left and lower axes only\n",
+    "ax.xaxis.set_ticks_position('bottom')\n",
+    "ax.yaxis.set_ticks_position('left')\n",
+    "\n",
+    "\n",
+    "#loc = plticker.MultipleLocator(base=3.0) # this locator puts ticks at regular intervals\n",
+    "#ax.xaxis.set_major_locator(loc)\n",
+    "#loc = plticker.MultipleLocator(base=2.0) # this locator puts ticks at regular intervals\n",
+    "#ax.yaxis.set_major_locator(loc)\n",
+    "\n",
+    "x = np.linspace(-3, 3, 100)\n",
+    "y = x**3-(3/2)*x**2-6*x+3\n",
+    "ax.plot(x, y, color='b', linewidth=2.5, label=('$f\\,(x)$'))\n",
+    "\n",
+    "x = np.linspace(-3, 3, 100)\n",
+    "y = 3*x**2-3*x-6\n",
+    "ax.plot(x, y, color='g', linewidth=2.5, label=('$f\\,\\'(x)$'))\n",
+    "\n",
+    "ylim = ax.get_ylim()\n",
+    "xlim = ax.get_xlim()\n",
+    "\n",
+    "ax.tick_params(axis='both', which='major', labelsize=13)\n",
+    "\n",
+    "\n",
+    "ax.legend(fontsize=15)\n",
+    "# Add legend\n",
+    "\n",
+    "#ax.legend(bbox_to_anchor=(1.05, 0.3),loc='lower left', \n",
+    "#          frameon=False, labelspacing=0.2, fontsize=13)\n",
+    "\n",
+    "plt.tight_layout()\n",
+    "plt.savefig('func_diff.png')"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 31,
+   "metadata": {
+    "tags": [
+     "remove-cell"
+    ]
+   },
+   "outputs": [
+    {
+     "data": {
+      "image/png": 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okbyFhsozXx8f4OJFecvMBI4eBSwW+6/18QG6dAHuvhsYPRqIinLJR6kzo9EIk8mkugz94QKa7maweycDWH9++w2YPRtYsgQoKCj/WIsWQI8eQPfuQFyc3A8Orvp4xcXAiRPAr78Cqany9uOPFc+i/fyA224DJk4EBgyQ4awaA7gOGL4qMID17uhR4JlngKVLy9/fujUwciRwzz3A9dc7570sFmDrVuDbb4GVK4HDh8s/ft11wGOPAWPGAIGBznnPumAA1xLDVxUGsF7l5ACzZgFvvQVcumS7f/Bg4IkngF69AIPdX69zlJTIMF6yBPj0U9l0YdWwIfDvfwOJiUC9eq6roTIM4Fpg+KrEANajdeuABx8E/vrLdt9ddwEvvAC0bev+erKygIULgTlzytfUpAkwfTowfjzg6+u+ehjANcTwVc1uAGugFY/sycmRba39+9uCrksX2cPhs8/UhC8AhIcDTz4JHDoEfPyx7LoGAMePAxMmyHbnTZvU1EaVYPhqFgNYg375RQZZcrLcDwkB3n9fXhjr0kVtbVZ+fsB998m+xosWAU2byvvT0mSTyN13AydPKi2RAE6so3EMYI1Zvhzo1EkOngCAPn3koIqJE13bzltXfn7AAw/InhkvvGC7IPf55/KC4IIFQNWtXOQyHGSheQxgjSgpAZKSgFGjgPx8GbYzZ8o24GbNVFdXvcBA4NlnZRDffbe8LztbXpzr06diLwpyMYavLjCANaCwUH6df/11uR8ZCaxZI7ucaaGvbW00bSp7SnzzjbwwB8jRebGxctBIZWfD48aNg7+/P0JCQkpv7733nrvK9iwMX93Q2T9vz5ObKwc3LFsm96+7DjCZ5EAHPbv9dmD/fuAf/5D7ubmyh8SIEcD58/ZfM3bsWOTl5ZXeJk2a5L6CPQXDV1cYwAplZQG33AL88IPc79JF9re99lq1dTlLWBgwbx7w3XeyvzAgB3XcdBOwY4fa2jwSw1d3GMCK5ObKiW527ZL7gwfLIPbEmcgGDZIXEocMkfvHjsmh0m++Wb5JYsWKFYiMjETr1q2RlJSEvLw8u8dLTk6G0WiE0WhEZmamGz6BDjB8dYkDMRS4eFGG79atcn/ECNkE4efhc9MJAbzzjuxHbJ38Z8QIYPFi4LffdqNJkyZo0KABDhw4gPHjx6NFixZYZm2bqQQHYoDhqw8cCacFly4Bt94KbNgg9++4Q04j6e+vti532rVL9pQ4ckTu33AD8NVXcuIgqx9//BG9evVCXl4e6lUxxtnrA5jhqxccCaeaELLPrDV8Bw6Uo9q8KXwB4Oab5fSZ/fvL/V9+kfdZ28IBwOfv7h/VnCB4N4av7jGA3eg//wH++1+53a2bvCClYgIbLYiMlBfnpkyR+xcuAAMHCiQnA3/88QcmT56MO+64A1dccYXaQrWK4esRGMBu8sEHcmAFIOdP+OortdM4aoGvL/DKK7L922AoRHGxARMnAkbjJsTHd8HixYtVl6hNXMnCY3j4ZR9t2LJFDiUG5PI+332nvZUlVBo5EmjWLAB33inXrsvJeQhHj3pf00yN8MzXo/AM2MVOnZIXnIqKgIAAeebbqpXqqrSnc2e5Eke7dnL/q6+Avn0rH7ThlRi+HocB7EIWi1yl4vRpuT93LpCQoLYmLWvWTM741qeP3N+2TbaVHz2qtCxtYPh6JAawCz31lGx+AIBx44CHHlJaji7Ury+baEaNkvsHD8oQ/vVXtXUpxfD1WAxgF/n6aznSCwA6dADee0+b00lqUb16wCefAJMny/0TJ+RCo9ZRg16F4evRGMAucOqUXEYIkGd0K1awx0Nt+fjI2eFmzZL7587JpglrH2qvwN4OHo8B7GTWwRbnzsn9998HmjdXW5OePfWU/DM0GIC8PDmK8LvvVFflBlxGyCswgJ3svfeAlBS5fe+9sosVOWbiRDmAxc9PDuUeMkQ28Xgshq/XYAA70cGDcqIZAIiJkb0eyDlGjrTNmWGxyEl8PvtMdVUuwPD1KgxgJykpke2+ZrP8uvzRR3IFYXKeO++UZ75XXCH7VY8a5WH9hBm+XocB7CTvvy/7rQLAY48BPXuqrcdTDRoErF4tL2qWlMi15pYvV12VEzB8vRKno3SCv/6SI7hyc+Vggl9+AYKDVVfl2TZulBfkCgqM8PExYelSHbe3M3y9AaejdAUhgEmTZPgC8kyY4et6vXrJ3hAGgzwTvvde4PPPVVdVBwxfr8YAdtAXX8ivxAAwZoz+F9PUk1695Lwa1uaI0aN11juC4ev1GMAOyMsDHn9cbkdF2Ua+kfuEhgLffCNHzxUVAXfdBaxZo7qqGmD4EhjADpk1Sw6TBYBXX+UUk6r07Qt8+aWcbc5iAYYOBdavV11VFSoML2b4eisGcB39+accKgvI5XTGjlVbj7cbNEi2AVsHa9x5J7B9u+qq7ODcDlQGA7iOHn8cKCyU2++8I+cuILXuuANYulT+Li5elKG8Z4/qqspg+NJlGBt1sGYNsGqV3B43DujUSWk5VMbddwMLF8rt7Gy58OeBA2prAsDwJbsYwLVUVGQbbhwaaputi7Rj/Hhgzhy5ffasDGGlk7ozfKkSDOBa+ugj2+TgzzwDNGyoth6y71//si2Cevw40K8fcOaMgkIYvlQFBnAt5OfLpeUBoEkT4NFH1dZDVfv3v22Tuv/xBzBwIJCV5cYCGL5UDQZwLbzzjq3b2QsvcJJ1rTMYgNdek/MzA8DevfJCXUGBG96c4Us1wACuofPnbe297doB99+vth6qGYMBmD8fGDZM7m/ZIueMKCpy4ZsyfKmGGMA1NGuWvKoOAC+/DPj6qq2Has7PT3ZP691b7n/zDZCYKOfxcDqGL9UCA7gGTp+2Ta7evTtw221q66Hau+IK4KuvgI4d5f7ixXK5I6di+FItMYBr4NVX5UTrgLyyztWN9SksTPbhbtVK7r/6KjB7tpMOzvClOmAAV+P0aWDePLndp49cHp30Kzoa+P57W/fBxx8Hli1z8KAMX6ojBnA1yp79Tp+uthZyjmuvlWfCYWFyf+xYYN26Oh6M4UsOYABXgWe/nqtDBzl3sHUGtWHDgJ9+quVBGL7kIAZwFXj269l69ZK9IwwGObfzoEFylrsaYfiSEzCAK5GZKZcXAnj268lGjLDNG5GRIUfLZWRU8yKGLzkJA7gSc+faRkxNm6a2FnKtRx4Bnn5abqeny26GeXmVPJnhS07EALbj4kVbv9+bb5ZfVcmzvfiinFoUAEwmOa2lxXLZkyosI8TwJccwgO1YuFAOPQaAqVPZ79ddiouLkZSUhAYNGiA0NBTDhw/H2bNn3fLeBgOQnCybIADZS+If/ygzWo7LCJELMIAvY7HYFtds1QoYMkRtPd7k5Zdfxtdff43U1FQcP34cADBmzBi3vb+/v1zWKC5O7n/wwd8XX9nsQC7CAL7M8uXAsWNy+8knOeeDOyUnJ2Pq1Klo3rw56tevj1dffRUpKSk4cuSI22oICQG+/RZo3lzu/7R6NYo3MnzJNQyiihlJBg4cKBz5CpiZmYkGDRrU+fUq/PqrvPjm5we0b29rftDjZ6mMFj9LcXEx9u7di+uuuw5BQUGl9+/ZswfXXnstwsPDS+/LzMwsbZq4dOkSOnTo4PR6Ll0CTh7JRrOoP2EwCAAGILQF4F/f6e9VlhZ/N3XFz2Kze/fu74UQAys8IISo6uaQuLg4Rw/hVuvWCSFb/YR46aXyj+nts1RFi5/l2LFjAoA4dOhQuftjYmLExx9/XOnrgoKCXFPQ8VWieKm/EEshzB8GiGGdVokdO1zzVmVp8XdTV/ws5djNWDZBlPH22/JnYCAwcaLaWrxNaGgoACDbOufn37KyshBmHTPsLn+3+frAgmIE4K45K7AydTAGDwZ+/929pZBnYwD/LT1dtv0BcrL1yEi19Xib8PBwxMTE4Kcy44EPHTqEnJwctG/f3n2FXHbBzbfnCtz+D9nme/as7CWhZG058kguDeDExERXHt6p3nnH1uXI3lpvevos1dHqZ0lMTMQrr7yCw4cPIycnB1OnTsWAAQPQrFmzSl8TFRXlvAIq6e0wYYJtLcDDh4FbbwVyc533tmVp9XdTF/ws1avyIhwAV6wZoDk5OXKRzdxcuXru2rWqK/JOxcXFmDp1Kj788ENcunQJ/fr1Q3JycpUhazQaYTKZHH/zarqaCQE89JDsmgYAAwYAq1bJrmtENWB3NAEDGLLt97HH5Pbq1VzxQk+cEsA17Odrsch+4d99J/fvvx/48EMO1KEasfu3xOvbgEtKZPMDALRsKWfEIi9Si0EW/v7Ap5/K4ekA8NFHnCeEHOP1Abx2rW0Kwn/9C/Dx+j8RL1KHEW4hIfJbUsuWcv+ll4D33nNDreSRXB43DzzwAJo2bYqwsDBcffXVeOCBB3DhwgVXv22NWaecDA62TcZSmUuXLmHixIlo1aoVQkNDERMTg6SkJJitkwbrzJw5c9CpUycEBQWhpTVRdMI6b0RaWlrd5o1wYHhxdDSQkiJ/AnI2tZUr6/Ah/rZ8+XJ0794dYWFh8PPzq/uBNGDq1Klo164dwsLC0KhRI0yYMAHnrROr6NAzzzyDa6+9FmFhYYiOjsaIESNwzDpU1glcHsBPPPEEDh48iJycHBw4cAD5+fl4+OGHXf22NXL8uDybAYBRo2xL1FSmqKgIUVFRWLVqFbKysrBlyxZs2LABU6dOdX2xLtCoUSNMmTIFzzzzjOpSas06b0Tbtm1rP2+EE+Z2aNFCdlsMDpYX6EaPBjZvru2nkCIiIjBp0iTMdtoKoer4+vrik08+wblz55CWlobjx49j/PjxqsuqszFjxmDv3r3IycnBkSNHEBMTg5EjRzrvDSoboSGcMBLuchcuXBCjRo0SHTt2dPah6+S552wj30ymuh1j7ty5on379s4tzM0WL14sWrRoobqMWomJiRELFy4sHaGUnp4uAIjDhw9X/cLjq4RYJke4iWUBct8BKSlC+PnJv0P16wvx8891P9b//vc/4evr61A9WrN69WoRFhamugyHvPSSEDNnCnH6dJ6YPHmyiIyMrMth1I2Ee/nllxEaGoqIiAh89dVXmjjjKiqS004CcvYr6wxYtbV+/Xr3DhQgZGdn49ixY4gr80tr0aIFwsLC8PPPP1f+QhfMajZggK1rWna2HKhx9KhDh/Qoev/3cfo08PzzRZg2DWjYcD3efvttPPfcc047fp0DeNy4cTAYDJXeppW5PPzUU08hNzcXhw4dwuTJkzXR3vjdd7IJApDzvtbm81jNnj0bW7duxYsvvujm6qtWl8+iJzk5OQCA+vXLT4wTHh5e+lgFLpxScswYuX4gAJw8KUPZTdMYa9qKFSuwYMECvG0d469Ds2YBly7JdvkvvkjAc889hxtvvNF5b1DZqbGopgkiNzdXZGZmVnq7ePGi3delpqaKxo0bi+Li4rqcxjvNoEHya2NoqBC5ubX/PG+++aa46qqrxL59+xR9gsrV9rPorQniwoULAoDYs2dPuUlSwsLCxNdff13xBU5udrCnpESIxx+3NWnFx8u/V7XhSU0Qn332mYiIiBAbNmxQXUqdHT0qRECA/H3ecou878yZMyI4OFicO3eutoezm7F1vuQaEhKCkJCQWr+uqKgIJ06cwMWLF0snYHG3o0flVWxAnr3Ij1HzzzNjxgzMnz8fmzZtQps2bVxWZ13V9XejF7WaN8JNk6kbDMDrr8vFXD/5BNi5Uy74+c03QECA099O0xYvXozJkydj1apV6Natm+py6mzGDKCwUG5bv+QWFRXh4sWLOHnyJCKdMWFMZcksnHAR7syZM2LJkiXiwoULQgghfvvtN9GtWzeRkJDg6KEdMn267Uxl797avfbJJ58UMTExIj093SW1uZPFYhEFBQUiOTlZNG/eXBQUFIiCggLVZdXIzJkzRevWrcUNN9wgsrOzxYgRI8SAAQPKP8kNZ76XKywUYuBA29+vUaOEqO7LXlFRkSgoKBDff/+98PX1Lf09lJSUuLxeZ3v77bdFZGSk2Llzp+pSHPL770L4+pYIQIj+/c1CCCH++usvMWTIENGsWTNhsVhqe0i7GevSAM7IyBC9e/cWERERIigoSMTExIiJE+GgFhUAABjCSURBVCeK06dPO3roOisuFuKaa+Qnr+0Un0eOHBEAREBAgAgODi69XX/99S6p1dWmT58uIIebl7vpQVFRkZg8ebLw9fUVISEhYujQoSIzM9P2BAXha5WXJ0SnTrYQfvhh2URRmcWLF9v9PVTbo0ODAAg/P79y/z6Cg4NVl1Vro0fbfn8REb1EUFCQaNSokRg9enRdT77sZqzXzQWxYQNwyy1ye+5cQCNdkqmO7M4FoYE13M6dA3r0kCusAHI2teefd2sJVEd79wIdO8r4veceuUyZE3AuCABYvFj+DAiQgy/Iw2ggfAHgyivlMPdrrpH7L7xgm/CftEsIYPJk+dPPT/7eXMmrAjg7G1ixQm4PGcJJ1z2ORsLXqnFjYN06wLqU2GOPydnTSLtWr5bfkgFg0iSgdWvXvp9XBfBnn8kFNwHggQfU1kJOVi58/ZWHr1WrVvJM2Npl+cEHHZs3glzHYpEroQNAeLhtEn5X8qoAtjY/NGkC9O2rthZyogpnvis1Eb5WHTrIeSOCguT0pyNHAt9/r7oquty8ebY1//7zH9mM5GpeE8AHDwLbt8vt++8HfH3V1kNOYsnWVLNDZbp1A778Us4pbLEAQ4cCmzaproqszp+3XSRt2dJ9F+e9JoA//ti2Xd20k6QTJ1YDuX9qPnyt+veXV9R9fWVT2ODBwI4dqqsiAHj6aRnCAPDKK+4bPOMVAVxSAvz3v3K7c2fZLkc6Z212gNBF+FoNGwYsWSJHzuXlycl7ygzoIwV27ACSk+V2v37y24m7eEUAb9sGHDkit++9V2kp5Axl23xh0E34Wt17L7BggdzOzpb/6NPS1NbkrYqKgH/+U27Xqwe8+6571/jzigBeulT+9PUF7r5bbS3koMsvuIW20FX4Wj34IDBnjtw+f15eFP7lF7U1eaN335UDLwDgqafc/+3Y4wO4sFB2PwPkNIHWZWRIh+z18/WvX/3rNOpf/wLefFNunz0L9OljGzlHrnfiBPDss3K7RQsZwO7m8QGckmJrXGfzg45pbJCFszz+OPDaa3I7MxPo3RvYv19tTd5ACCAxEcjNlfvvvgtccYX76/D4ALY2PwQHA3feqbYWqiMPDV+rJ5+UE38DQEYG0KsXUNXCHuS4JUvkogyAPDEbMEBNHR4dwDk5cj5WQF7ZDA5WWw/VgYeHr9VTT8nuT4CtOcLaNknOdfw48H//J7evvtrWFq+CRwfwypWAdcV4Nj/okJeEr9WUKcAbb8jtc+dkc0RqqtqaPI0QwIQJ8uQMkN3PVM4J49EBbL341qABhx7rjpeFr9UTT9hmTcvKkn9vOWLOeZKTbavhjB0rB8Oo5LEBfP68nIkKAIYPl1PLkU54afhaPfqoDIqygzXWrFFdlf7t2ydnpAPkTHVvvaW2HsCDA/irr2Qna4B9f3XFy8PXasIEubacr69sRrvzTmDZMtVV6dfFi3JydbMZ8PGRF+cjIlRX5cEBbG1+iI6WKxOQDjB8yxk9GvjiCzkvgcUi91VeMNKzRx8FDhyQ2//5D9Czp9p6rDwygM+eBX74QW6PGMGZz3SB4WvXkCGy+cG6gPj//R8wbZq8mEQ18/HHwAcfyO1eveSfn1Z4ZAB/+SVQXCy32fygAwzfKvXpA2zcaBvF+eKLckY/65LpVLmdO2VzDgBERcmmBy2dkHlkAFubH66+GkhIUFsLVaNM+F6yACPeNiCkzUiEhIRg9erVqqvTjI4dgR9/BJo3l/sffQQMGiR7SpB9J0/K/v+XLsnQ/fRToFEj1VWV53EBnJlpW9OJzQ8aVyZ8C4sN+PLcffhiuxl5eXnIy8vDYNV9hDSmZUu5qECnTnJ/wwZ5gmGd6Y9szGYZvidPyv3Zs+U3Ca3xuABeuVLO/wuw+UHTLmt2mL6uHdLz26iuSvOio2XwDhki9/fvB26+Gdi8WW1dWlJSIptodu6U+xMmuG+Fi9ryuAC2rnrcqBHQtavaWqgSdtp8tx+7ErNnz0ZkZCTatWuHWbNmwWKxqK5Uk4KCZO+IJ56Q+2fPygEbixaprUsLhJAXKj/9VO4nJABz57p3jt9aEUJUddOV8+eF8PMTAhBi0iTV1ZAQQowdO1YAKL3ddhPEpSUQYimE+UOIj2bdLYQQYtu2beL8+fOiqKhIbN++XTRv3lw89dRTdo85f/58ERcXJ+Li4kRMTIw7P47mfPCBEP7+8u88IMQ//ymE2ay6KnVeeMH2Z3H99UKcO6e6olJ2M9ajAvijj2x/+D/8oLoaEkKI3NxckZmZKTIzM0XWr0tFyX/9hVgKUfLfAJH161Jx8eJFu6/75JNPRJMmTao9flxcnLNL1p2tW4WIjrb93Y+PF+LYMdVVud/cubY/g5gYIf76S3VF5djNWI9qgvjyS/kzIoKDL7QiJCQEUVFRiLq0A/XTxsEgZLODoccK1L9uNIKCguy+zsfHR54hULW6dQN277ZdnNu5U/aasM554A3eegt45BG5HRUFrF0LNGmitqaa8JgAzs+3/YW7/Xa5/DdpRDX9fLOysrB69Wrk5eVBCIE9e/bgueeewz333KOwaH1p0kRO2jNpktw/e1Z2U5s8WXbD8mSzZtnawyMiZA600cn1XI8J4LVr5VLfgFx5ljSiBoMsLBYLZs6cicaNGyMsLAz33HMPRo8ejVnWWcqpRqyLSn7yCRASIu978015MfrgQbW1uUJJCfDMM8C//y33GzQA/vc/IC5ObV21Yajma55uvgOOHSs7pwcFyf/9AwNVV0TuGOFmNBphMpmcekxPkJ4OjBoFWP9o6tUDZs6USyB5Qt/4ggLZ1azsoKsffgCuv15pWVWx2w/DI86ALRbbyhcDBzJ8NYHDi5Vq2VKOnJsyRXbBunQJSEoCunfX/9nw6dNysnpr+LZsKZtfNBy+lfKIAN60yTYkc+hQtbUQGL4aERAglznaskWGFCBH0rVvL7+65+erra8uNm4EjEbbSiE9ewI7drh/OXln8YgAtvZ+8PNTP8O912P4ak63bkBampyM3GCQ3xhfekmeMa5cqY+Z1SwWOYtZnz5yOXkAeOABee3nyivV1uYI3QewEMCqVXK7d28gPFxtPV6tXPj6M3w1JChIdtXatUueQQLA0aNytZiEBGDbNrX1VWXvXlnjiy/Kf++BgcD8+cDChfIsX890H8BpacBff8nt229XW4tXq3Dmu5Lhq0FxcfIr+7x5tsUot22TZ8l33KGtRUBzcuRZe1ycbV6HG2+UFxYTEzU8vLgWdB/A1rNfgAGsDJsddMXXF/jHP2RPiSlTZA8JQP5b6txZziuxbp1tUit3KyiQs5e1bi0XKC0pkc2LSUnyPwg9XmyrjMcE8A03AM2aKS3FOzF8dSsiQl6k+/134KGHbIOX1q8H+vcH2raV/YjPnXNPPefPA2+8Iec8fvxx4MwZeX/37sCePcCrr3peDyddB/CpU7JNC+DZrxIMX48QEwMsWAAcOiRnErOG3B9/yJF0DRsCt94KLF4sQ9KZLBY5vea998oZDJ98UnYzA+QZ8NKlspfTDTc49321QtcDMRYtkv9zA7Idq0sXtfV4FY2ELwdiON/583JQ0/vvA7/9Vv4xHx85z0SfPnK+lfbt5TDomrbHWiyyH/KuXXLI8Nq1QHZ2+ee0bg08+6wcSOIJg0b+ZvdPSNcBPGQI8PXXcgjiqVMe9cvSNo2EL8AAdiUh5ETvy5fLebYzM+0/LzxcNldcfTVw1VWyW5j132JREZCRIZsTjh8Hfv3V/twUAQFyCoEJE+TCmT66/m5ul2cFsNksf9H5+XJI4uLFqivyEhoKX4AB7C5FRTKM166VTQa7dzt+kS4iAhgwQDZv3Hqrvvvz1oDdAPZzdxXOsmGDbSQPB1+4icbCl9zHz082O1jXVcvKkhfG9u0DfvlFthdbz3TPn7cN7vDxkd9Qr7pKtiVfdx1w001Ahw5Au3byuN5Mtx/f2vshIEBesSUXY/hSGeHhcuBT796qK9E3Xba0CAF8+63c7tULCA1VWo7nY/gSuYQuA/jXX22j3269VW0tHo/hS+QyugzgNWts24MGqavD4zF8iVxK1wHcvLl+p6HTPIYvkcvpLoBzc+X8poA8+/WECTk0h+FL5Ba6C+ANG+RoGkCufkFOxvAlchvdBbC1+SEggF1gnI7hS+RWugpgIWwB3LMnEBysth6PwvAlcjtdBfDBg8CxY3KbvR+ciOFLpISuApjdz1yA4UukjC4DuFkzoE0bpaV4BoYvkVK6CeD8fDkbEyB7P7D7mYNOrAa2DGf4EimkmwDesgUoLJTbAwaorUX3SsO3kOFLpJBuAnjtWvnT15fdzxzC8CXSDN0E8Lp18md8PFC/vtpadIvhS6Qpugjg06flxM8A0K+f2lp0i+FLpDm6COAffrBtc/L1OmD4EmmSLlbEsDY/hIbKJgiqBXY1I9IszZ8BC2EL4N69AX9/tfXoCsOXSNM0H8D798sl5wG2/9YKw5dI8zQfwNazX4DtvzWmsfCdM2cOOnXqhKCgILRs2dLucz766CO0aNECQUFB6NSpE3bv3u3mKoncTzcBHBPD1S9qRGPhCwCNGjXClClT8Mwzz9h9fOvWrfjnP/+JefPm4cKFCxg+fDhuvfVW5OTkuLlSIvfSdAAXFgKbNsntfv04/LhaGgxfABgxYgSGDx+Oxo0b2318wYIFGDZsGPr374969eohKSkJ9erVw5dffunmSoncS9MBnJoq54AAgL591daieRoN35pIS0tDXFxc6b7BYMBNN92EtLQ0u89PTk6G0WiE0WhEZmamu8okcjpNB/CGDbZtDj+ugqKJdcaNGweDwVDpbdq0aTU6Tm5uLupfNrwxPDy80iaIxMREmEwmmEwmNGjQwOHPQaSKpvsB/+9/8me7dsBVV6mtRbMUDrKYO3cuXn/99UofDwoKqtFxQkNDkZ2dXe6+rKwstGjRwqH6iLROswGcnw9s3y63+/RRW4tmKR7hFhISgpCQEIePExsbi59++ql0XwiBvXv3YtiwYQ4fm0jLNNsEsW2bbfpJBrAdOhpeXFRUBLPZDIvFAiEEzGYzzGZz6eMTJkzAypUrsX79ehQWFuKNN96A2WzG0KFDFVZN5HqaPQO2tv8aDHIBTiqjXPj6azp8AWDmzJl4/vnnS/cDAwMByDNdAEhISMB7772HCRMm4NSpU7jxxhvx3XffISwsTEm9RO5isP4jqESVD7pS586yF0THjgD75JdRIXxXajp8Xc1oNMJkMqkug6g6djvRarIJIjsb2LVLbrP5oQyGL5FH0WQTxJYtQEmJ3GYA/61cP1+GL5En0OQZsLX7mZ8fkJCgthZNYPgSeSRNBrD1AtzNN8s5gL0aw5fIY2kugM+dA/buldte3/xQYXgxw5fIk2gugLdssW336qWsDPV0PLcDEdWM5gLYOvuZnx/QpYvaWpRh+BJ5Bc0G8M03A8HBamtRguFL5DU0FcDZ2bb2X68c/cbwJfIqmgrgrVvlIpyAFwYww5fI62gqgK3NDz4+QNeuamtxK4YvkVfSZAB37Ah4zTwsDF8ir6WZAM7Ls0264zXNDwxfIq+mmQDetg0oLpbbXhHADF8ir6eZALY2PxgMXjD/A8OXiKChAN68Wf5s3x6IiFBbi0sxfInob5oI4IICYOdOue3RzQ8MXyIqQxMBnJpqW/+tRw+1tbiMoqXjiUi7NBHAW7fatrt3V1eHy+hoAU0ich9NrIhhDeBWrYDoaLW1OB2bHYioEsrPgIuLge3b5bbH9X5g+BJRFZQH8C+/ADk5ctujApjhS0TVUB7AZdt/PSaAGb5EVAOaCeAGDWQbsO4xfImohpQGsBC2JYgSEuQoOF1j+BJRLSgN4GPHgBMn5Lbumx8YvkRUS0oDuGz7b7du6upwGMOXiOpAaQD/+KP8GRgI3HSTykocwPAlojrSxBlwp05AQIDKSuqI4UtEDlAWwBcuyD7AgE7bfxm+ROQgZQG8fbttAU7dBTDDl4icQFkAb9smfxoMQOfOqqqoA4YvETmJ0jNgALj+eqB+fVVV1BLDl4icSEkAFxfbJmDv0kVFBXXA8K2zOXPmoFOnTggKCkLLli0rPP7hhx/Cx8cHISEhpbdRo0YpqJTIvZRMR/nLL3IVZADo2lVFBbXE8HVIo0aNMGXKFBw8eBCLFy+2+5zmzZsjPT3dzZURqaUkgK3ND4AOzoAZvg4bMWIEAHmmS0Q2SpogrAEcEQG0bq2ighpi+LrNX3/9hYYNG6Jp06YYOXIkDh8+XOlzk5OTYTQaYTQakZmZ6cYqiZxLSQBbe0B07gz4KJ+PrRIM32qNGzcOBoOh0tu0adNqdJwePXpg3759OHnyJHbt2oUrrrgC/fr1w8WLF+0+PzExESaTCSaTCQ0aNHDmRyJyK7c3QWRmAtamPs02PzB8a2Tu3Ll4/fXXK308KCioRsdp3rx56XbDhg2xYMEC1K9fHzt27MAtt9zicJ1EWuX2AN6xw7atyQBm+NaYtceCs1nPoIV1pA6Rh3J7A4C1/ddgAOLj3f3u1WD4ukRRURHMZjMsFguEEDCbzTCbzaWPf/vttzh+/DiEEDh//jwefvhhREVFobOuRugQ1Z6yAL7hBiAszN3vXgWGr8vMnDkTgYGBSExMxKFDhxAYGIjAwMDSxzdu3Ij4+HiEhISgXbt2OHfuHNatW+eSs2siLTFU8zXPqd8Bi4rkqLf8fCAxEZg/35lHdwDDV7eMRiNMJpPqMoiqY3e9H7eeAe/bJ8MX0FD7L8OXiBRxawBrbgAGw5eIFHJrAKemyp+aGIDB8CUixZQEcHy84hWQGb5EpAFuC+CsLOC33+R2p07uelc7yoWvP8OXiJRxWwDv2mXbVtb/t8KZ70qGLxEp47YAts7/CygKYDY7EJHGuC2Are2/zZsDbp8/heFLRBrklgAWovwFOLdi+BKRRrklgI8dAzIy5LZbL8AxfIlIw9wSwNazX8CNZ8AMXyLSOLcGsJ8fcNNNbnhDhi8R6YBbAtjaAyI2FigzCZZrMHyJSCdcHsAWC7B7t9x2efPDidXAluEMXyLSBZcH8P79QEGB3HbpBbjS8C1k+BKRLrg8gMtegHNZADN8iUiHXL4mnLX9NyzMRTOgsc2XiHTK5WfA1sUKjEYXLEHP8CUiHXNpAOfnyzZgQAawUzF8iUjnXBrAe/cCxcVy++abnXhghi8ReQCXBnDZtRKddgbM8CUiD+GWAL7ySuCaa5xwQIYvEXkQlwawdRL2m292whJEXMmCiDyMywI4J8e2BJHDzQ8VwpcrWRCR/rksgPfskfMAAw5egGP4EpGHclkAl10Drs5nwAxfIvJgLgtg6wW4q68GGjWqwwEYvkTk4VwewHVqfmD4EpEXcEkAX7gA/Pmn3K518wPDl4i8hEsCuM4DMBi+RORFtBPADF8i8jIuDeBrrgEaNKjBCxi+ROSFXDIfcNOmQIsWNVyAk+FLRF7KJWfAs2cD6enAsmXVPJHh6/EuXbqEiRMnolWrVggNDUVMTAySkpJgNpvLPe+1115D48aNERwcjL59++LQoUOKKiZyH5fOBeFX1fk1w9crFBUVISoqCqtWrUJWVha2bNmCDRs2YOrUqaXPWbp0KV577TWsWrUKmZmZuP7663HHHXeg2DqXKZGHMgjreGH7qnywzhi+Xu3dd99FcnIy0tLSAAA9e/ZEjx49MGPGDABAXl4eoqOjsWbNGvTs2bPKYxmNRpjKXvUl0ia705G5fEmiCipMKcnw9Tbr169H+/btS/fT0tIQFxdXuh8SEoJWrVqVBvTlkpOTYTQaYTQakZmZ6fJ6iVzFvQHM+Xw9yrhx42AwGCq9TZs2rcJrZs+eja1bt+LFF18svS83Nxf169cv97zw8HDk5OTYfd/ExESYTCaYTCY0qFE3GyJtcvmqyKUYvh5n7ty5eP311yt9PCgoqNz+W2+9hVdeeQUbNmxATExM6f2hoaHIzs4u99ysrCyEhYU5t2AijXFPADN8PVJISAhCQkJq9NwZM2Zg/vz52LRpE9q0aVPusdjYWPz0008YMmQIANkG/McffyA2NtbpNRNpieubIBi+Xi8pKQkLFy60G76AbFKYP38+9uzZg4KCAkybNg3XXnstEhISFFRL5D6uPQNm+Hq9o0eP4vXXX0dAQEC5M9prrrkG+/fvBwDce++9OHHiBG677TZkZWWhS5cu+Oabb+Dr66uqbCK3cF03NIYvuQG7oZFOuLEbGsOXiKhazg/gogJgZyLDl4ioGs4PYL9AoNcaIPBqhi8RURVccxEuIha4/U8ZxkREZJfruqExfImIquT+uSCIiAgAA5iISBkGMBGRIgxgIiJFGMBERIowgImIFGEAExEpwgAmIlKEAUxEpAgDmIhIEQYwEZEiDGAiIkWqWxGDSNMMBkOKEGKg6jqI6oIBTESkCJsgiIgUYQATESnCACYiUoQBTESkCAOYiEiR/wfRuMltw634cQAAAABJRU5ErkJggg==",
+      "text/plain": [
+       "<Figure size 360x360 with 1 Axes>"
+      ]
+     },
+     "metadata": {
+      "needs_background": "light"
+     },
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "import matplotlib.pyplot as plt\n",
+    "import numpy as np\n",
+    "import matplotlib.ticker as plticker\n",
+    "\n",
+    "# Create figure and add axes\n",
+    "#fig = plt.figure(figsize=(5, 5))\n",
+    "#ax = fig.add_subplot(111)\n",
+    "\n",
+    "fig, ax = plt.subplots(figsize=(5, 5)) \n",
+    "\n",
+    "# Move left y-axis and bottim x-axis to centre, passing through (0,0)\n",
+    "ax.spines['left'].set_position('zero')\n",
+    "ax.spines['bottom'].set_position('zero')\n",
+    "\n",
+    "# Eliminate upper and right axes\n",
+    "ax.spines['right'].set_color('none')\n",
+    "ax.spines['top'].set_color('none')\n",
+    "\n",
+    "# Show ticks in the left and lower axes only\n",
+    "ax.xaxis.set_ticks_position('bottom')\n",
+    "ax.yaxis.set_ticks_position('left')\n",
+    "\n",
+    "\n",
+    "#loc = plticker.MultipleLocator(base=3.0) # this locator puts ticks at regular intervals\n",
+    "#ax.xaxis.set_major_locator(loc)\n",
+    "#loc = plticker.MultipleLocator(base=2.0) # this locator puts ticks at regular intervals\n",
+    "#ax.yaxis.set_major_locator(loc)\n",
+    "\n",
+    "x = np.linspace(-3, 3, 100)\n",
+    "y = x**3-(3/2)*x**2-6*x+3\n",
+    "ax.plot(x, y, color='b', linewidth=2.5, label=('$f\\,(x)$'))\n",
+    "\n",
+    "x = np.linspace(-3, 3, 100)\n",
+    "y = 6*x-3\n",
+    "ax.plot(x, y, color='orange', linewidth=2.5, label=('$f\\,\\'\\'(x)$'))\n",
+    "\n",
+    "ylim = ax.get_ylim()\n",
+    "xlim = ax.get_xlim()\n",
+    "\n",
+    "ax.tick_params(axis='both', which='major', labelsize=13)\n",
+    "\n",
+    "\n",
+    "ax.legend(fontsize=15)\n",
+    "# Add legend\n",
+    "\n",
+    "#ax.legend(bbox_to_anchor=(1.05, 0.3),loc='lower left', \n",
+    "#          frameon=False, labelspacing=0.2, fontsize=13)\n",
+    "\n",
+    "plt.tight_layout()\n",
+    "plt.savefig('plot_concav.png')"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 37,
+   "metadata": {
+    "tags": [
+     "remove-cell"
+    ]
+   },
+   "outputs": [
+    {
+     "data": {
+      "image/png": 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",
+      "text/plain": [
+       "<Figure size 360x360 with 1 Axes>"
+      ]
+     },
+     "metadata": {
+      "needs_background": "light"
+     },
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "import matplotlib.pyplot as plt\n",
+    "import numpy as np\n",
+    "import matplotlib.ticker as plticker\n",
+    "\n",
+    "# Create figure and add axes\n",
+    "#fig = plt.figure(figsize=(5, 5))\n",
+    "#ax = fig.add_subplot(111)\n",
+    "\n",
+    "fig, ax = plt.subplots(figsize=(5, 5)) \n",
+    "\n",
+    "# Move left y-axis and bottim x-axis to centre, passing through (0,0)\n",
+    "ax.spines['left'].set_position('zero')\n",
+    "ax.spines['bottom'].set_position('zero')\n",
+    "\n",
+    "# Eliminate upper and right axes\n",
+    "ax.spines['right'].set_color('none')\n",
+    "ax.spines['top'].set_color('none')\n",
+    "\n",
+    "# Show ticks in the left and lower axes only\n",
+    "ax.xaxis.set_ticks_position('bottom')\n",
+    "ax.yaxis.set_ticks_position('left')\n",
+    "\n",
+    "\n",
+    "#loc = plticker.MultipleLocator(base=3.0) # this locator puts ticks at regular intervals\n",
+    "#ax.xaxis.set_major_locator(loc)\n",
+    "#loc = plticker.MultipleLocator(base=2.0) # this locator puts ticks at regular intervals\n",
+    "#ax.yaxis.set_major_locator(loc)\n",
+    "\n",
+    "x = np.linspace(-3, 4, 100)\n",
+    "\n",
+    "y = (3/2)*x**4-2*x**3-6*x**2+2\n",
+    "ax.plot(x, y, color='b', linewidth=2.5, label=('$f\\,(x)$'))\n",
+    "\n",
+    "y = 6*x**3-6*x**2-12*x\n",
+    "ax.plot(x, y, color='g', linewidth=2.5, label=('$f\\,\\'(x)$'))\n",
+    "\n",
+    "y = 18*x**2-12*x-12\n",
+    "ax.plot(x, y, color='orange', linewidth=2.5, label=('$f\\,\\'\\'(x)$'))\n",
+    "\n",
+    "#ylim = ax.get_ylim()\n",
+    "#xlim = ax.get_xlim()\n",
+    "ax.set_ylim(-50,50)\n",
+    "\n",
+    "ax.tick_params(axis='both', which='major', labelsize=13)\n",
+    "\n",
+    "\n",
+    "ax.legend(fontsize=15)\n",
+    "# Add legend\n",
+    "\n",
+    "#ax.legend(bbox_to_anchor=(1.05, 0.3),loc='lower left', \n",
+    "#          frameon=False, labelspacing=0.2, fontsize=13)\n",
+    "\n",
+    "plt.tight_layout()\n",
+    "plt.savefig('plot_extrema.png')"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 51,
+   "metadata": {
+    "tags": [
+     "remove-cell"
+    ]
+   },
+   "outputs": [
+    {
+     "data": {
+      "image/png": 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",
+      "text/plain": [
+       "<Figure size 432x288 with 1 Axes>"
+      ]
+     },
+     "metadata": {
+      "needs_background": "light"
+     },
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "import matplotlib.pyplot as plt\n",
+    "import numpy as np\n",
+    "    \n",
+    "dens = np.linspace(0,4)\n",
+    "\n",
+    "fig, ax = plt.subplots(figsize=(6, 4))\n",
+    "\n",
+    "\n",
+    "ax.plot(dens,2*dens**2/(1+dens**2))\n",
+    "    \n",
+    "ax.set_xlabel('$N$',fontsize = 16)\n",
+    "ax.set_ylabel('$f\\,(N)$',fontsize = 16)\n",
+    "\n",
+    "ax.set_yticks([])\n",
+    "ax.set_yticklabels([])\n",
+    "\n",
+    "ax.set_xticks([np.sqrt(1/3)])\n",
+    "ax.set_xticklabels([r'$\\sqrt{\\frac{1}{3ah}}$'],fontsize=16)\n",
+    "\n",
+    "ax.axvline(np.sqrt(1/3), ls='--', color='black', lw=0.5)\n",
+    "\n",
+    "ax.set_ylim(0, 2)\n",
+    "ax.set_xlim(0, 4)\n",
+    "\n",
+    "fig.tight_layout()\n",
+    "fig.savefig('hollingIII.png')"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 57,
+   "metadata": {
+    "tags": [
+     "remove-cell"
+    ]
+   },
+   "outputs": [
+    {
+     "data": {
+      "image/png": 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",
+      "text/plain": [
+       "<Figure size 360x360 with 1 Axes>"
+      ]
+     },
+     "metadata": {
+      "needs_background": "light"
+     },
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "import matplotlib.pyplot as plt\n",
+    "import numpy as np\n",
+    "import matplotlib.ticker as plticker\n",
+    "\n",
+    "# Create figure and add axes\n",
+    "#fig = plt.figure(figsize=(5, 5))\n",
+    "#ax = fig.add_subplot(111)\n",
+    "\n",
+    "fig, ax = plt.subplots(figsize=(5, 5)) \n",
+    "\n",
+    "# Move left y-axis and bottim x-axis to centre, passing through (0,0)\n",
+    "ax.spines['left'].set_position('zero')\n",
+    "ax.spines['bottom'].set_position('zero')\n",
+    "\n",
+    "# Eliminate upper and right axes\n",
+    "ax.spines['right'].set_color('none')\n",
+    "ax.spines['top'].set_color('none')\n",
+    "\n",
+    "# Show ticks in the left and lower axes only\n",
+    "ax.xaxis.set_ticks_position('bottom')\n",
+    "ax.yaxis.set_ticks_position('left')\n",
+    "\n",
+    "\n",
+    "#loc = plticker.MultipleLocator(base=3.0) # this locator puts ticks at regular intervals\n",
+    "#ax.xaxis.set_major_locator(loc)\n",
+    "#loc = plticker.MultipleLocator(base=2.0) # this locator puts ticks at regular intervals\n",
+    "#ax.yaxis.set_major_locator(loc)\n",
+    "\n",
+    "x = np.linspace(-3, 3, 100)\n",
+    "y = np.exp(x)\n",
+    "ax.plot(x, y, color='b', linewidth=2.5, label=('$e^x$'))\n",
+    "\n",
+    "y = 1+x\n",
+    "ax.plot(x, y, color='orange', linewidth=2.5, label=('$1+x$'))\n",
+    "\n",
+    "\n",
+    "ax.tick_params(axis='both', which='major', labelsize=13)\n",
+    "\n",
+    "\n",
+    "ax.legend(fontsize=15)\n",
+    "ax.set_ylim(-2,10)\n",
+    "# Add legend\n",
+    "\n",
+    "#ax.legend(bbox_to_anchor=(1.05, 0.3),loc='lower left', \n",
+    "#          frameon=False, labelspacing=0.2, fontsize=13)\n",
+    "\n",
+    "plt.tight_layout()\n",
+    "plt.savefig('exp_linapp.png')"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Analysing change, continuously (Part 2)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "In many different contexts we may be interested in finding the maximum or minimum values a given function may assume. This process is referred to as *optimization*.  \n",
+    "  \n",
+    "In Ecology, some of the most common uses for optimization methods are:\n",
+    "\n",
+    "1 - *Statistical estimation and inference*: optimizing the likelihood of a model and its parameters being true, for a given set of observations.\n",
+    "\n",
+    "2 - *Resources management*: finding a balance between different wildlife and economic priorities (market, legal, natural constraints)\n",
+    "\n",
+    "3 - *Optimality theory*: how can the behavioural or life-history trade-offs result in optimal strategies for different organisms"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Extreme values of functions  \n",
+    "  \n",
+    "For a given function $f(x)$ that is **continuous** in an interval $a \\le x \\le b$, there will always be a global maximum and a global minimum, points at which the function assumes its maximum and minimum values in the interval.  \n",
+    "```{image} extrema.png  \n",
+    "  \n",
+    "```\n",
+    "  \n",
+    "A function $f(x)$ presents a *local maximum* value if for a given vicinity of the point $x_0$, $f(x_0)$ is greater than all the other $f(x)$ for $x$ in this vicinity. The analogous definition can be made for a *local minimum* value when $f(x_0)$ is smaller than all the other $f(x)$ in the vicinity. Local minima and local maxima constitue the *local extrema* for $f(x)$, and with the set of all extrema we can find the global maximum and the global minimum.  \n",
+    "  \n",
+    "An important result is that, if the function $f$ has a local extremum at a given point $x=c$ in that interval ($a<c<b$) and the derivative of the function exists at that point (in other words, if $f'(c)$ is well defined), then we will have:  \n",
+    "  \n",
+    "$f'(c)=0$  \n",
+    "    \n",
+    "  \n",
+    "`````{warning}  \n",
+    "\n",
+    "````{panels}  \n",
+    "  \n",
+    "```{image} ./xcube.png  \n",
+    "\n",
+    "```  \n",
+    "\n",
+    "---  \n",
+    "  \n",
+    "Note, however, that the contrary is not always true. If $f'(c)=0$, not necessarily $f(c)$ will be an extremum point. On the left plot, $f(x)=x^3 \\implies f'(x)=2x^2$, thus $f'(0)=0$ but there is no extremum at $x=0$ .\n",
+    "      \n",
+    "```` \n",
+    "`````  \n",
+    "  \n",
+    "Thus the points $x=c$, for which $f'(c)$, will constitute **candidates** for local extrema, and will have to be evaluated under other sets of conditions. Also important to have in mind that point for which the derivative is not definided (kinks or discontinuities) may also constitute local extrema (see on the first figure).  \n",
+    "  \n",
+    "  \n",
+    "In general, when searching for local extrema, we have to adopt the following rules:  \n",
+    "\n",
+    "- Do not assume points $x$ for which $f'(x) = 0$ are extrema. They are candidates.\n",
+    "- Check points where derivative is not defined.\n",
+    "- Check endpoints of domain.\n",
+    "  \n",
+    "```{tip}\n",
+    "If possible, create a plot. Visuals are powerful tools and can help verify whether calculations have been done correctly.\n",
+    "```"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Monotonicity and concavity  \n",
+    "  \n",
+    "Imagine a function $f(x)$ defined in a given interval $a \\le x \\le b$. For every $x_1$ and $x_2$ in this interval, if $x_{2} > x_{1}$ implies that $f(x_{2}) > f(x_{1})$ then the function is **increasing** in this interval. If on the other hand, $x_{2} > x_{1}$ implies that $f(x_{2}) < f(x_{1})$ then the function is **decreasing** in this interval.  This can be graphically shown as:  \n",
+    "  \n",
+    "```{image} incr_decr.png  \n",
+    "```  \n",
+    "  \n",
+    "\n",
+    "  \n",
+    "```{note}  \n",
+    "Since we are using the stric inequalities (\"$>$\" or \"$<$\") it can be added that the function is **monotonic**, or monotonically increasing or decreasing. If we had $x_{2} > x_{1}$ implying that $f(x_{2}) \\ge f(x_{1})$ (or $f(x_{2}) \\le f(x_{1})$), the function would be *non-decreasing* (or *non-increasing*).  \n",
+    "  \n",
+    "```\n",
+    "  \n",
+    "Now suppose that $f$ is well-behaved in a given interval $a \\le x \\le b$ (in mathematical terms, we say: $f$ is continuous and differentiable in $(a,b)$), then an important results is that\n",
+    "\n",
+    "$$\\mbox{If }\\ f'(x) > 0\\ \\mbox{ for}\\ a \\le x \\le b \\implies f(x)\\ \\mbox{is increasing for}\\ a \\le x \\le b$$\n",
+    "\n",
+    "$$\\mbox{If }\\ f'(x) < 0\\ \\mbox{ for}\\ a \\le x \\le b \\implies f(x)\\ \\mbox{is decreasing for}\\ a \\le x \\le b$$  \n",
+    "  \n",
+    "  \n",
+    "In other words, if the derivative of function $f$ is positive (negative) for every point $x$ in a given interval then the function is increasing (decreasing) in this interval.  \n",
+    "  \n",
+    "````{admonition} Example  \n",
+    "Let us calculate the intervals where the function $f(x) = x^{3} - \\displaystyle\\frac{3}{2}x^{2} - 6x + 3$ is increasing and decreasing. Since $f(x)$ is continuos and differentiable for all real values of $x$ (remember that all polynomial functions are well-behved), we can use the first derivative test:  \n",
+    "  \n",
+    "$$f'(x) = 3x^{2}-3x-6 = 3(x^{2}-x-2) = 3(x+1)(x-2)$$  \n",
+    "  \n",
+    "and we see that $f'(x)$ has roots in $x=-1$ and $x=2$. Since $f'(x)$ is a polynomial of degree 2 with the coefficient of the leading term positive, it is *concave up*. From this, we can conclude that:  \n",
+    "  \n",
+    "$$x<-1\\ \\mbox{or}\\ x>2 \\implies f'(x) > 0 \\implies f(x)\\ \\mbox{is}\\ increasing$$  \n",
+    "$$-1<x<2 \\implies f'(x) < 0 \\implies f(x)\\ \\mbox{is}\\ decreasing$$  \n",
+    "  \n",
+    "The following plot shows the function $f(x)$ and its derivative. Compare the intervals where the derivative is positive (negative) with the intervals where the function $f(x)$ is increasing (decreasing).  \n",
+    "  \n",
+    "```{image} func_diff 2.png  \n",
+    "```  \n",
+    "\n",
+    "````  \n",
+    "  \n",
+    "When we discussed polynomials of degree 2, depending on the sign of the coefficient of the leading term, we had the two cases:  \n",
+    "  \n",
+    "```{image}  concav.png  \n",
+    "```\n",
+    "  \n",
+    "Let us now generalise the notion of concavity for any arbitrary function. Suppose $f(x)$ is differentiable in a given interval $a \\le x \\le b$ (in other words, the derivative $f'(x)$ is well-defined for every point in this interval). Then we have the following results:  \n",
+    "  \n",
+    "$$\\mbox{If }\\ f'(x)\\ \\mbox{is increasing for}\\ a \\le x \\le b \\implies f(x)\\ \\mbox{is}\\ concave\\ up$$\n",
+    "\n",
+    "$$\\mbox{If }\\ f'(x)\\ \\mbox{is decreasing for}\\ a \\le x \\le b \\implies f(x)\\ \\mbox{is}\\ concave\\ down$$  \n",
+    "  \n",
+    "This means that if for a given function $f(x)$ the slopes of the tangent lines are increasing in a given interval of $x$, in this interval the function should resemble an \"upwards U-shape\". If on the other hand, the slopes of the tangent lines are decreasing in this interval, the function should resemble a \"downwards U-shape\". This can be visualised on the plots below.  \n",
+    "  \n",
+    "```{image}  slopes_inc_dec.png  \n",
+    "```\n",
+    "  \n",
+    "If the function $f$ is twice differentiable in $a \\le x \\le b$ ($f''(x)$ is well-define for $a \\le x \\le b$), we can use the last two results to find a general criterium for concavity. This will be given by:  \n",
+    "  \n",
+    "$$\\mbox{If }\\ f''(x) > 0\\ \\mbox{ for}\\ a \\le x \\le b \\implies f'(x)\\ \\mbox{is increasing for}\\ a \\le x \\le b \\implies f(x)\\ \\mbox{is}\\ concave\\ up$$\n",
+    "\n",
+    "$$\\mbox{If }\\ f''(x) < 0\\ \\mbox{ for}\\ a \\le x \\le b \\implies f'(x)\\ \\mbox{is decreasing for}\\ a \\le x \\le b \\implies f(x)\\ \\mbox{is}\\ concave\\ down$$  \n",
+    " \n",
+    "```{admonition} Example\n",
+    "If you have a population curve, these tools will help you analyze questions such as 'How fast is the population growing?' 'Is the growth accelerating?' or 'Are any of these factors changing over different periods?\n",
+    "```\n",
+    "  \n",
+    "````{admonition} Example  \n",
+    "Let us analyse the function $f(x) = x^{3}-\\displaystyle\\frac{3}{2}x^{2}-6x+3$ again. We have:  \n",
+    "  \n",
+    "$$\\begin{align} \n",
+    "f'(x) = 3x^{2}-3x-6 \\\\\n",
+    "f''(x) = 6x -3 \\end{align}$$  \n",
+    "  \n",
+    "The function f''(x) has a root in $x=\\displaystyle\\frac{1}{2}$. Since its slope is positive, we have:  \n",
+    "  \n",
+    "$$x<\\displaystyle\\frac{1}{2} \\implies f''(x) < 0 \\implies f(x)\\ \\mbox{is}\\ concave\\ down$$  \n",
+    "$$x>\\displaystyle\\frac{1}{2} \\implies f''(x) > 0 \\implies f(x)\\ \\mbox{is}\\ concave\\ up$$  \n",
+    "  \n",
+    "On the plots below, compare the intervals where the second derivative is positive (negative) with the intervals where the function $f(x)$ is concave up (concave down).  \n",
+    "  \n",
+    "```{image}  plot_concav.png  \n",
+    "```  \n",
+    "\n",
+    "````"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Finding extrema  \n",
+    "  \n",
+    "  \n",
+    "Back to our strategy for finding maxima or minima of functions, we first determine the set:  \n",
+    "  \n",
+    "- find all points $c$ for which $f'(c) = 0$ or all point $c$ where $f'(c)$ does not exist. These are called **critical points**.\n",
+    "- find endpoints of domain of $f$.  \n",
+    "  \n",
+    "Additionally, if $f$ is twice differentiable (in some interval containing $c$):  \n",
+    "  \n",
+    "- If $f'(c) = 0$ and $f''(c) < 0 \\implies c$ is a local maximum (since the function is concave down)\n",
+    "- If $f'(c) = 0$ and $f''(c) > 0 \\implies c$ is a local minimum (since the function is concave up)  \n",
+    "  \n",
+    "````{admonition} Example  \n",
+    "Let us analyse the function $f(x) = \\displaystyle\\frac{3}{2}x^{4}-2x^{3}-6x^{2}+2$, and find its local extrema. We have:  \n",
+    "  \n",
+    "$$\\begin{align}\n",
+    "f'(x)&= 6x^{3}-6x^{2}-12x=6x(x-2)(x+1) \\\\\n",
+    "f''(x)&=18x^{2}-12x-12=6(3x^{2}-2x-2)\\end{align}$$  \n",
+    "  \n",
+    "Since this is a polynomial function, the derivative is well-defined for all points. Also, for the critical points (roots of $f'(x)$) we have $x=0$, $x=-1$, and $x=2$. Calculating the value of the second derivative at the critical points:  \n",
+    "  \n",
+    "$$\\begin{align}\n",
+    "&x=0 \\implies f''(0)=-12 \\text{ (local maximum)} \\\\\n",
+    "&x=-1 \\implies f''(-1)=18 \\text{ (local minimum)} \\\\\n",
+    "&x=2 \\implies f''(2)=36 \\text{ (local minimum)}\\end{align}$$  \n",
+    "  \n",
+    "Now we can check on the plot of $f(x)$ that we have successfully determined the local extrema.  \n",
+    "  \n",
+    "```{image} plot_extrema.png  \n",
+    "```\n",
+    "\n",
+    "````"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Inflection points  \n",
+    "  \n",
+    "  \n",
+    "As we saw before, given a function $f(x)$, for some values of $x$, there might be a change in the concavity of the function:  \n",
+    "  \n",
+    "```{image} inflection.png  \n",
+    "```\n",
+    "    \n",
+    "The points $x$ for where the function changes its concavity are called **inflection points**. If $f$ is twice differentiable and $c$ is an inflection point, then $f''(c)=0$.  \n",
+    "\n",
+    "Inflection points are the points where the function is (locally) steepest. It therefore helps us answer questions like when the growth/decline was strongest.\n",
+    "  \n",
+    "```{note}  \n",
+    "Note again that the points $x=c$ where $f''(c)=0$ are only candidates for inflection points. See that if $f(x)=x^{4}$, $f''(x)=12x^{2}$. Then, $f''(0)=0$, but $x=0$ is not an inflection point of $f$.  \n",
+    "  \n",
+    "After calculating the candidate points $x=c$, if the concavities of the function (given by the sign of the second derivative) to the left and to the right of $c$ change, then $c$ is an inflection point.  \n",
+    "\n",
+    "```  \n",
+    "\n",
+    "````{admonition} Example  \n",
+    "\n",
+    "Consider the Holling Type III functional response   \n",
+    "  \n",
+    "$$f(N)=\\frac{aN^{2}}{1+ahN^{2}},$$  \n",
+    "  \n",
+    "where $f(N)$ is prey consumption rate per predator, $N$ is the prey population density, $a$ is the attack rate, and $h$ is handling time. Let us check if this function presents any inflection point. We have:  \n",
+    "  \n",
+    "$\\begin{align}f'(N) \n",
+    "&= \\frac{2aN(1+ahN^{2})-aN^{2}(2ahN)}{(1+ahN^{2})^{2}} \\\\\n",
+    "&= \\frac{2aN}{(1+ahN^{2})^{2}}\\end{align}$  \n",
+    "  \n",
+    "<br />\n",
+    "  \n",
+    "$\\begin{align}f''(N)\n",
+    "&=\\frac{2a(1+ahN^{2})^{2}-2aN[2(1+ahN^{2})\\cdot2ahN]}{(1+ahN^{2})^{4}} \\\\\n",
+    "&=\\frac{2a(1+2ahN^{2}+a^{2}h^{2}N^{4})-2a[4ahN^{2}+4a^{2}h^{2}N^{4}]}{(1+ahN^{2})^{4}} \\\\\n",
+    "&=\\frac{2a(1-3ahN^{2})}{(1+ahN^{2})^{3}}\\end{align}$  \n",
+    "  \n",
+    "See that the function $f''(N)$ has two roots: $N=-\\sqrt{\\displaystyle\\frac{1}{3ah}}$ and $N=\\sqrt{\\displaystyle\\frac{1}{3ah}}$. Since $N$ is a population density, it only makes sense to analyse $N \\ge 0$. It is also possible to show that:\n",
+    "  \n",
+    "$$N<\\sqrt{\\displaystyle\\frac{1}{3ah}} \\implies f''(N) > 0$$  \n",
+    "$$N>\\sqrt{\\displaystyle\\frac{1}{3ah}} \\implies f''(N) < 0$$  \n",
+    "  \n",
+    "This means that we have a change of concavity at the point $N=\\sqrt{\\displaystyle\\frac{1}{3ah}}$ and thus this is and inflection point of this curve.  \n",
+    "  \n",
+    "Analyse how the function $f(N)$ changes with $N$.  \n",
+    "  \n",
+    "```{image} hollingIII.png  \n",
+    "```\n",
+    "  \n",
+    "How does this function compares with the Holling type II functional response?\n",
+    "\n",
+    "````  \n",
+    "  \n",
+    "```{tip}  \n",
+    "  \n",
+    "For the intervals where the function is concave up you can also think of it as representing \"accelerated change\" of that function (since the first derivative, the \"velocity\" is increasing), while the intervals where it is concave down represent \"deccelerated change\" of the function. Understanding this can be particularly important if you function has a specific biological meaning, as in the previous example.  \n",
+    "  \n",
+    "```"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Series expansions \n",
+    "  \n",
+    "From the formal definition of a derivative  \n",
+    "  \n",
+    "$$f'(a)= \\lim_{x \\to a} \\frac{f(x)-f(a)}{(x-a)} \\xrightarrow{\\text{$x$ very close to $a$}} f'(a) \\approx \\frac{f(x)-f(a)}{(x-a)},$$  \n",
+    "  \n",
+    "which means that if $x$ is sufficiently close to $a$, the average rate of change is sufficiently close to the value of the derivative at that point. From the previous approximation, we have:  \n",
+    "  \n",
+    "$$f(x) = f(a)+f'(a)(x-a),$$  \n",
+    "  \n",
+    "which provides a linear approximation of $f(x)$ around $a$ (see that if $\\alpha= f(a)$ and $\\beta=f'(a)$, both constants, then $f(x) = \\alpha+\\beta(x-a)$, which can easily be arranged on the form $y=mx+b$).  \n",
+    "  \n",
+    "`````{admonition} Example  \n",
+    "  \n",
+    "````{panels}  \n",
+    "  \n",
+    "```{image} ./exp_linapp.png\n",
+    "\n",
+    "```  \n",
+    "\n",
+    "---  \n",
+    "  \n",
+    "Say we have $f(x)=e^{x}$ and we want to use this linear expression to approximate the function close to $a=0$: \n",
+    "\n",
+    "$$f(x)=f(0)+f'(0)(x-0)=1+x,$$  \n",
+    "  \n",
+    "since $f'(x) = e^{x}$ and thus $f(0)=f'(0)=1$.  \n",
+    "  \n",
+    "The approximation is reasonable if $x$ very close to $0$, but fails when $x$ increasingly departs from $0$. \n",
+    "  \n",
+    "````  \n",
+    "  \n",
+    "`````"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "One way of trying to make the approximation better is to add terms of higher order of $x$. Suppose we want to approximate the function $f(x)$ at point $a$ by a given polynomial of degree $n$  \n",
+    "  \n",
+    "$$P(x)=a_{0}+a_{1}(x-a)+a_{2}(x-a)^{2}+\\cdots+a_n(x-a)^{n}$$  \n",
+    "  \n",
+    "so that the derivatives of $f(x)$ and $P(x)$ are the same at $x=a$:\n",
+    "\n",
+    "$$\\begin{align}f(a)&=P(a) \\\\\n",
+    "f'(a)&=P'(a)\\\\\n",
+    "&\\vdots \\\\\n",
+    "f^{(n)}(a)&=P^{(n)}(a)\\end{align}.$$  \n",
+    "  \n",
+    "But for the polynomial, we have:  \n",
+    "  \n",
+    "$$\\begin{align}  \n",
+    "P(a)&=a_{0} \\\\\n",
+    "P'(a)&=1 \\cdot a_{1} \\\\\n",
+    "P''(a)&=2 \\cdot 1 \\cdot a_{2} \\\\\n",
+    "P'''(a)&=3 \\cdot 2 \\cdot 1 \\cdot a_{3} \\\\  \n",
+    "P''''(a)&=4 \\cdot 3 \\cdot 2 \\cdot 1 \\cdot a_{4} \\\\\n",
+    "&\\vdots \\\\\n",
+    "P^{(n)}(a)&= \\underbrace{n(n-1)(n-2) \\cdots 3 \\cdot 2 \\cdot 1}_{\\displaystyle\\text{$$n!$$}}\\cdot a_{n} \\end{align}$$  \n",
+    "  \n",
+    "where $n! = n\\cdot(n-1)\\cdot(n-2) \\cdots 3 \\cdot 2 \\cdot 1$ is the factorial of $n$.  \n",
+    "  \n",
+    "Thus, by making derivatives of $f(x)$ and $P(x)$ equal at $x=a$, we obtain: \n",
+    "  \n",
+    "$$f^{(n)}(a) = P^{(n)}(a) \\implies f^{(n)}(a) = n!\\, a_n \\implies a_{n} = \\frac{f^{(n)}(a)}{n!}$$  \n",
+    "  \n",
+    "The polynomial with these coefficients is called **Taylor polynomial** of degree $n$ at $x = a$:  \n",
+    "  \n",
+    "$$P(x)=f(a)+f'(a)(x-a)+\\frac{f''(a)}{2!}(x-a)^{2}+\\cdots+\\frac{f^{(n)}(a)}{n!}(x-a)^{n}.$$  \n",
+    "  \n",
+    "````{admonition} Example  \n",
+    "  \n",
+    "To find approximation of degree 3 for $f(x)=e^{x}$, at $x=0$, we have:  \n",
+    "  \n",
+    "$$f(0)=1 \\\\\n",
+    "f'(0)=1 \\\\\n",
+    "f''(0)=1 \\\\\n",
+    "f'''(0)=1$$  \n",
+    "  \n",
+    "Thus, \n",
+    "  \n",
+    "$$P_{3}(x)=1+(x-0)+\\frac{1}{2!}(x-0)^2+\\frac{1}{3!}(x-0)^3=1+x+\\frac{x^2}{2}+\\frac{x^3}{6}$$  \n",
+    "  \n",
+    "See how the approximation to the function $f(x)=e^{x}$  at $x=0$ gets increasingly better when more terms are added to the polynomial.  \n",
+    "  \n",
+    "```{image} exp_series.gif  \n",
+    "```\n",
+    "  \n",
+    "````  \n",
+    "  \n",
+    "  \n",
+    "```{admonition} Example  \n",
+    "For the function $f(x)=\\displaystyle\\frac{1}{1-x}\\ (x\\neq1)$ at $x=0$, we have:  \n",
+    "  \n",
+    "$$f(x)=\\frac{1}{1-x} \\implies f(0)=1 \\\\\n",
+    "f'(x)=\\frac{1}{(1-x)^{2}} \\implies f'(0)=1 \\\\\n",
+    "f''(x)=\\frac{2}{(1-x)^{3}} \\implies f''(0)=2 \\\\\n",
+    "f'''(x)=\\frac{3\\cdot2}{(1-x)^{4}} \\implies f'''(0)=3! \\\\\n",
+    "f^{(n)}(x)=\\frac{n!}{(1-x)^{n+1}} \\implies f^{(n)}(0)= n!$$  \n",
+    "  \n",
+    "Thus the Taylor polynomial of degree $n$ will be:  \n",
+    "  \n",
+    "$$\\begin{align}P(x)\n",
+    "&=1+1(x-0)+\\frac{2}{2}(x-0)^{2}+\\frac{3!}{3!}(x-0)^{3}+\\cdots+\\frac{n!}{n!}(x-0)^{n} \\\\\n",
+    "&=1+x+x^{2}+x^{3}+\\cdots+x^{n}\\end{align}$$\n",
+    "\n",
+    "  \n",
+    "```  \n",
+    "  \n",
+    "For some functions, the approximation $f(x)=P_n(x)$ at a given reference point $x=a$ can be made increasingly better by adding more terms to $P(x)$, so that in the limit $n\\rightarrow \\infty$ the Taylor polynomials (called **Taylor series** in this limit) for the functions $e^{x}$, $\\mbox{sin}(x)$, $\\mbox{cos}(x)$ converge exactly for all values of $x$. For other functions, such as $\\mbox{ln}(1+x)$ and $\\displaystyle\\frac{1}{1-x}$, there is a restricted range of values of $x$ for which the approximation can be made better by adding terms.  \n",
+    "  \n",
+    "We have the following approximations by Taylor series, with ranges in $x$, at $x = 0$:\n",
+    "\n",
+    "$$e^x=\\sum_{n=0}^\\infty \\frac{x^n}{n!} =1+x+\\frac{x^2}{2}+\\frac{x^3}{3!}+\\cdots,\\ -\\infty<x<\\infty$$\n",
+    "\n",
+    "$$\\mbox{sin}(x)=\\sum_{n=0}^\\infty \\frac{(-1)^nx^{1+2n}}{(1+2n)!} = x-\\frac{x^3}{3}+\\frac{x^5}{5!}-\\frac{x^7}{7!}+\\cdots,\\ -\\infty<x<\\infty$$\n",
+    "\n",
+    "$$\\mbox{cos}(x)=\\sum_{n=0}^\\infty \\frac{(-1)^nx^{2n}}{(2n)!} = 1-\\frac{x^2}{2}+\\frac{x^4}{4!}-\\frac{x^6}{6!}+\\cdots,\\ -\\infty<x<\\infty$$\n",
+    "\n",
+    "$$\\mbox{ln}(1+x)=-\\sum_{n=1}^\\infty \\frac{(-1)^n x^n}{n} = \\frac{x^2}{2!}+\\frac{x^3}{3!}-\\frac{x^4}{4!}+\\frac{x^5}{5!}+\\cdots,\\ -1 < x \\leq 1$$\n",
+    "\n",
+    "$$\\frac{1}{1-x} = \\sum_{n=0}^\\infty x^n = 1+x+x^2+x^3+\\cdots,\\ -1 \\leq x < 1$$  \n",
+    "  \n",
+    "Knowing the Taylor series often gives insight into the properties of a function."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Calculating accumulated change"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Series  \n",
+    "  \n",
+    "For every sequence $\\{a_n\\}$ of real numbers, it is possible to define another sequence $\\{S_n\\}$ for which the elements are given by the sum of the first $n$ terms of the sequence $\\{a_n\\}$:\n",
+    "\n",
+    "$$\\begin{align}\n",
+    "S_0 &= a_0 \\\\\n",
+    "S_1 &= a_0 + a_1 \\\\\n",
+    "S_2 &= a_0 + a_1 + a_2 \\\\\n",
+    "&\\vdots \\\\\n",
+    "S_n &= a_0 + a_1 + \\cdots + a_{n-1} + a_n \\\\\n",
+    "S_n &= \\sum_{i = 0}^n a_i \\end{align}$$  \n",
+    "  \n",
+    "The sequence $\\{S_n\\}$ is called a *series* (or an infinite series, to emphasize the sum over increasing number of terms $\\{a_n\\}$).  \n",
+    "  \n",
+    "The same way as we asked about $\\lim a_n$, we can ask if $\\lim S_n$ exists. If it exists, or, in other words, if the sum of infinite terms $\\{a_n\\}$ is equal to a finite number, we say that the series $S_n$ is convergent.  \n",
+    "  \n",
+    "  \n",
+    "```{admonition} Examples  \n",
+    "  \n",
+    "The following expressions are example of convergent series:  \n",
+    "  \n",
+    "$$\\begin{align}&\\sum_{n=0}^\\infty \\frac{x^n}{n!} \\hspace{0.1cm}, \\hspace{0.5cm} x \\in \\mathbb{R} \\\\\n",
+    "&\\sum_{n=0}^\\infty x^n \\hspace{0.1cm}, \\hspace{0.5cm} -1<x<1 \\\\\n",
+    "&\\sum_{n=0}^\\infty (-1)^n \\frac{x^{2n+1}}{(2n+1)!} \\hspace{0.1cm}, \\hspace{0.5cm} x \\in \\mathbb{R}\\end{align}$$  \n",
+    "  \n",
+    "```  \n",
+    "  "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Calculating areas  \n",
+    "  \n",
+    "  \n",
+    "The idea of summing infinite quantities can be used to solve the problem of determining the area below a given curve. Let $f(x)$ be a function defined on the interval $a \\le x \\le b$ as:\n",
+    "\n",
+    "```{image}  sum_areas.png\n",
+    "```\n",
+    "\n",
+    "The area below the curve of $f(x)$ can be found dividing the interval $a \\le x \\le b$ into $n$ subintervals:\n",
+    "\n",
+    "$$a=x_0<x_1<x_2<\\cdots<x_{n-1}<x_n=b$$  \n",
+    "  \n",
+    "where each interval defines a rectangle of height $f(x_i)$ and width $(x_{i+1}-x_i) = \\Delta x_i = \\displaystyle\\frac{(b-a)}{n}$. The area below the curve will then be approximately the sum of the areas of the rectangles\n",
+    "\n",
+    "$$A \\approx \\sum_{i=0}^n f(x_i)\\Delta x_i.$$  \n",
+    "  \n",
+    "As we increasethe number of intervals $n$, while decreasing the widths of each rectangle, the sum of the areas of the rectangles gives a better approximation of the area below the curve. The exact result is given by the limit $n \\rightarrow \\infty$. In this limit, the sum is represented by the following symbol:\n",
+    "\n",
+    "$$\\lim_{n \\to \\infty} \\sum_{i=0}^n f(x_i) \\Delta x_i = \\int_a^b f(x)dx$$  \n",
+    "  \n",
+    "If this limit exists, we say that the function $f$ is **integrable** on the interval $a \\lt x \\lt b$ and denote the limit by the **definite integral** $\\int_a^b f(x)dx$.  \n",
+    "  \n",
+    "The interpretation of the integral as area can be generalized for when $f(x)$ assumes negative values  \n",
+    "  \n",
+    "```{image}  defin_int.png\n",
+    "``` \n",
+    "  \n",
+    "In the case of the previous figure, we will have:  \n",
+    "  \n",
+    "$$\\begin{align} \\int_a^b f(x)dx \n",
+    "&= A_1 - A_2 + A_3 \\hspace{0.1cm}, \\hspace{0.5cm} (A_i \\geq 0) \\\\\n",
+    "&= [\\text{area above x-axis}] - [\\text{area below x-axis}] \\end{align}$$  \n",
+    "  \n",
+    "Other intuitive results related to the calculation of the area are:\n",
+    "\n",
+    "$$\\int_a^a f(x)dx = 0\\quad \\mbox{(size of the interval equal to zero)}$$\n",
+    "\n",
+    "$$\\int_a^b f(x)dx = -\\int_b^a f(x)dx\\quad  \\mbox{(orientation of the integral)}$$   \n",
+    "  \n",
+    "Furthermore, some of the properties of the integral follows directly from the properties of limits:\n",
+    "\n",
+    "$$\\begin{align}\n",
+    "\\int_a^b kf(x)dx &= k \\int_a^b f(x)dx \\\\\n",
+    "\\int_a^b [f(x) + g(x)]dx &= \\int_a^b f(x)dx + \\int_a^b g(x)dx \\\\\n",
+    "\\int_a^b f(x)dx &= \\int_a^c f(x)dx + \\int_c^b f(x)dx ,\\ \\  (a \\le c \\le b) \\end{align}$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Fundamental theorem of Calculus  \n",
+    "  \n",
+    "  \n",
+    "The problem of calculating integrals as limits of sums requires specific methods for each function and becomes quite impractical. Fortunately, this problem can be solved with the very important result that the area problem is intrinsically related to the tangent problem.\n",
+    "\n",
+    "Suppose we define a function $F(x)$ that gives the area below the curve $f(u)$ from $a$ until a variable value $x$. By definition:  \n",
+    "  \n",
+    "$$F(x) = \\int_a^x f(u)du$$  \n",
+    "  \n",
+    "```{image} fund_calc.png  \n",
+    "```\n",
+    "  \n",
+    "If we consider a small increment $h$ to $x$, the new area below the curve shall be given by:  \n",
+    "  \n",
+    "$$F(x+h) = \\int_a^{x+h} f(u)du$$  \n",
+    "  \n",
+    "  \n",
+    "The difference between these two areas shall be approximately given by the area of the retangle with width $h$ and height $f(x)$:  \n",
+    "  \n",
+    "$$F(x+h) - F(x) \\approx hf(x) \\implies \\displaystyle\\frac{F(x+h) - F(x)}{h} \\approx f(x)$$  \n",
+    "  \n",
+    "But see that in the limit where the increment $h$ gets very small, the approximation gets exactly equal to the increment in area, and we obtain the definition of the derivative of $F(x)$:  \n",
+    "  \n",
+    "$$\\lim_{h\\rightarrow 0} \\displaystyle\\frac{F(x+h) - F(x)}{h} = \\displaystyle\\frac{dF}{dx} = f(x).$$  \n",
+    "  \n",
+    "This results is known as the *Fundamental theorem of Calculus*, and it say that the instantaneous rate of change of the area below a given curve $f$ at a point $x$ is given by the value of the function at that point. In other words, the area $F(x)$ is a function that, if differentiated, gives the function $f(x)$.  \n",
+    "  \n",
+    "The function $F(x)$ is then called an **antiderivative** of $f(x)$. In general, there is a *family* of antiderivates of $f(x)$, since $\\displaystyle\\frac{d(F(x) + C)}{dx}=\\displaystyle\\frac{dF}{dx}=f(x)$, independent of the value of the constant $C$. We therefore call **indefinite integral**:\n",
+    "\n",
+    "$$\\int f(x)dx = F(x) + C$$  \n",
+    "   \n",
+    "It also directly follows that\n",
+    "\n",
+    "$$\\int_b^c f(x)dx = F(c) - F(b)$$  \n",
+    "  \n",
+    "Thus the problem of calculating the area below the curve $f(x)$ from $b$ to $c$, which, by definition, is given by the infinite sum of terms $f(x_i)\\Delta x_i$ (with $\\Delta x_i \\rightarrow 0$), can be reduced to obtaining the antiderivative of $f$, $F(x)$, and subtracting the values of this function at points $c$ and $b$. The main problem then becomes how to calculate the antiderivates of a given function $f(x)$, for which different methods can be used."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Calculating integrals  \n",
+    "  \n",
+    "The simplest integrals can be calculated by inverting the differentiation of fundamental functions, as\n",
+    "\n",
+    "$$\\int x^n \\,dx = \\frac{x^{n+1}}{n+1} + C$$\n",
+    "\n",
+    "$$\\int e^x \\,dx = e^x + C$$\n",
+    "\n",
+    "$$\\int \\frac{1}{x}\\,dx = \\ln |x| + C$$\n",
+    "\n",
+    "$$\\int \\mbox{cos}(x)\\, dx = \\mbox{sin}(x) + C$$\n",
+    "\n",
+    "$$\\int \\mbox{sin}(x)\\, dx = -\\mbox{cos}(x) + C$$  \n",
+    "  \n",
+    "This are obtained simply because we know that the derivative of the functions on the righ-hand sides of the equations give exactly the function that is being integrated. When the integrals are not so simple, we must apply other methods to turn them into integrals we know. \n",
+    "  \n",
+    "  \n",
+    "#### Integration by substitution  \n",
+    "  \n",
+    "  \n",
+    "This method consists in finding an appropriate substitution of the variable of integration so that we end up with a simpler integral to calculate.  \n",
+    "  \n",
+    "  \n",
+    "````{admonition} Examples  \n",
+    "  \n",
+    "- $\\int\\frac{1}{\\sqrt{x+4}}dx$  \n",
+    "    \n",
+    "    Note that if $u=x+4 \\implies \\displaystyle\\frac{du}{dx}=1 \\implies du=dx$. Thus, the integral can be given by:  \n",
+    "    \n",
+    "    $$\\int u^{-\\frac{1}{2}}du = \\frac{1}{\\left(-\\frac{1}{2}+1\\right)} \\cdot u^{\\frac{1}{2}} + C = 2\\sqrt{x+4} + C.$$  \n",
+    "      \n",
+    "      \n",
+    "    \n",
+    "    ```{note}  \n",
+    "    Note that three mathematicians die everytime we operate with $du$ and $dx$ as if they were numbers on a fraction. Formally, the correct substitution on the integral would be given by a function $u=u(x)$, so that $ \\int f(u(x)) u'(x)dx = \\int f(u)du$. But we can procede with the heresy above as long as we have this in mind.  \n",
+    "    ```  \n",
+    "    <br />\n",
+    "      \n",
+    "- $\\int xe^{(x^2-1)}dx$  \n",
+    "      \n",
+    "    See that $u=x^2-1 \\implies \\displaystyle\\frac{du}{dx}=2x \\implies  dx=\\displaystyle\\frac{du}{2x}$. Then:  \n",
+    "    \n",
+    "    $$\\int \\frac{1}{2}e^{u}du = \\frac{1}{2}e^u + C = \\frac{e^{(x^2-1)}}{2} + C $$  \n",
+    "      \n",
+    "````\n",
+    "\n",
+    "#### Integration by parts  \n",
+    "  \n",
+    "When integrating by parts, we have to find an appropriate association $(u=u(x), v=v(x))$, so that:  \n",
+    "  \n",
+    "$$\\int uv'dx=uv-\\int u'vdx$$  \n",
+    "  \n",
+    "```{admonition} Example  \n",
+    "  \n",
+    "- $\\int x\\ln x\\, dx $  \n",
+    "      \n",
+    "    Note that  \n",
+    "      \n",
+    "    $$\\begin{align}\n",
+    "    u=\\ln x \\implies u' = \\displaystyle\\frac{1}{x} \\\\\n",
+    "    v'=x \\implies v=\\displaystyle\\frac{x^2}{2} \\end{align}$$  \n",
+    "      \n",
+    "    Thus:  \n",
+    "      \n",
+    "    $$\\begin{align}\\int x\\ln x\\, dx \n",
+    "&= \\frac{x^2}{2}\\ln x - \\int \\frac{x^2}{2} \\cdot \\frac{1}{x}dx \\\\\n",
+    "&= \\frac{x^2 \\ln x}{2} - \\frac{1}{2}\\int xdx \\\\\n",
+    "&= \\frac{x^2 \\ln x}{2} - \\frac{1}{4}x^2 + C \\\\\n",
+    "&= \\frac{1}{4} x^2 (2\\ln x - 1) + C \\end{align}$$  \n",
+    "  \n",
+    "- $\\int xe^x \\,dx$  \n",
+    "      \n",
+    "    Note that  \n",
+    "      \n",
+    "    $$\\begin{align}\n",
+    "    u=x \\implies u'=1 \\\\\n",
+    "    v'=e^x \\implies v=e^x \\end{align}$$  \n",
+    "      \n",
+    "    Thus:  \n",
+    "      \n",
+    "    $$\\begin{align}\\int xe^xdx  \n",
+    "    &= xe^x - \\int e^x dx \\hspace{0.1cm} \\\\\n",
+    "    &= xe^x-e^x + C \\\\\n",
+    "    &= e^x (x-1) + C \\end{align}$$  \n",
+    "      \n",
+    "```\n",
+    "\n",
+    "```{tip}\n",
+    "For complicated integrals, computer algebra programs like Mathematica or Sympy can be helpful. An especially user-friendly option is the website WolframAlpha.\n",
+    "```\n",
+    "  \n",
+    "#### Integration by partial fractions  \n",
+    "  \n",
+    "If the integral has the form \n",
+    "\n",
+    "$$\\int \\frac{P(x)}{Q(x)}dx$$ \n",
+    "\n",
+    "where the degree of $P(x)$ is smaller than the degree of $Q(x)$, and $Q(x)=a(x-x_1)(x-x_2)\\cdots(x-x_n)$, we can write\n",
+    "\n",
+    "$$\\frac{P(x)}{Q(x)} = \\frac{1}{a}\\left[\\frac{A_1}{x-x_1}+\\frac{A_2}{x-x_2}+\\cdots+\\frac{A_n}{x-x_n}\\right]$$ \n",
+    "\n",
+    "with $\\{A_n\\}$ to be determined.  \n",
+    "  \n",
+    "```{admonition} Example  \n",
+    "  \n",
+    "- $\\int \\frac{1}{x(x-1)}\\, dx$  \n",
+    "      \n",
+    "    First, note that $\\displaystyle\\frac{1}{x(x-1)} =  \\displaystyle\\frac{A}{x} + \\displaystyle\\frac{B}{(x-1)} \\implies 1 = A(x-1) + Bx \\implies 1 = x(A+B) - A$. By comparing the coeffients of the terms in $x$ and the terms independent of $x$, we obtain:  \n",
+    "      \n",
+    "    $$\\begin{cases}\n",
+    "    A + B &= 0\\\\\n",
+    "       -A &= 1\n",
+    "    \\end{cases} \\implies  \n",
+    "    \\begin{cases}\n",
+    "    A &= -1 \\\\\n",
+    "    B &= 1\\end{cases}$$  \n",
+    "      \n",
+    "    Thus:  \n",
+    "      \n",
+    "    $$\\frac{1}{x(x-1)} = -\\frac{1}{x} + \\frac{1}{x-1}$$\n",
+    "      \n",
+    "    Then we will have for the integral:  \n",
+    "      \n",
+    "    $$\\begin{align} \\int\\frac{1}{x(x-1)}dx  \n",
+    "    &= \\int\\left(-\\frac{1}{x}+\\frac{1}{x-1}\\right)dx \\\\\n",
+    "    &= -\\int \\frac{1}{x}dx + \\int\\frac{1}{x-1}dx \\\\\n",
+    "    &= - \\ln|x| + \\ln|x-1| + C \\\\\n",
+    "    &= \\ln \\left|\\frac{x-1}{x}\\right| + C\\end{align}$$  \n",
+    "      \n",
+    "```\n",
+    "    "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "tags": [
+     "remove-cell"
+    ]
+   },
+   "source": [
+    "### Fourier coefficients  \n",
+    "  \n",
+    "For a given periodic function $f(t)$ (period $P$), we have shown\n",
+    "\n",
+    "$$f(t)=a+\\sum_{n=1}^N[\\beta_ncos(\\omega nt) + \\gamma_n sin(\\omega nt)]$$  \n",
+    "  \n",
+    "For well-behaved functions $N \\rightarrow \\infty$ gives the exact correspondence between the function $f(t)$ and the series.  \n",
+    "  \n",
+    "If $$\\omega_n=\\frac{2\\pi}{P} \\cdot n$$\n",
+    "\n",
+    "then:\n",
+    "\n",
+    "$$f(t) = a + \\sum_{n=1}^\\infty \\beta_n cos\\left[\\frac{2\\pi nt}{P}\\right] + \\gamma_n sin\\left[\\frac{2\\pi nt}{P}\\right]$$  \n",
+    "  \n",
+    "Multiplying both sides by $cos\\left[\\frac{2\\pi nt}{P}\\right]$ and integrating over one period $P$:\n",
+    "\n",
+    "$$\\beta_n = \\frac{2}{P} \\int_{P} f(t) cos\\left[\\frac{2\\pi nt}{P}\\right]dt$$\n",
+    "\n",
+    "since \n",
+    "\n",
+    "$$\\begin{align} \\int_{-\\pi}^\\pi cos(nx)cos(mx)\n",
+    "&=\\pi\\gamma_{mn} \\\\\n",
+    "\\int_{-\\pi}^\\pi sin(mx)cos(nx)&=0\\end{align}$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Exercise Set 1"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "1.  An autocatalytic reaction uses its resulting product for the formation of a new product, as in the reaction  \n",
+    "  \n",
+    "    $$A + X \\rightarrow X$$  \n",
+    "    \n",
+    "    If we assume that this reaction occurs in a closed vessel, then the reaction rate is given by\n",
+    "    \n",
+    "    $$\n",
+    "       R(x) = kx(a - x)\n",
+    "    $$\n",
+    "    \n",
+    "    for $0 \\le x \\le a$, where $a$ is the initial concentration of $A$ and $x$ is the concentration of $X$.\n",
+    "    \n",
+    "    <br/>\n",
+    "    \n",
+    "    (a) Show that $R(x)$ is a polynomial and determine its degree.\n",
+    "    <br/>\n",
+    "    (b) Without using any graphic visualiser, graph $R(x)$ for $k = 2$ and $a = 6$. Find the value of $x$ at which the reaction rate is maximal.  \n",
+    "    \n",
+    "    <br/>\n",
+    "      \n",
+    "2. Find the maximum/minimum value of the following polynomials  \n",
+    "  \n",
+    "    <br/>\n",
+    "    \n",
+    "    (a) $p(x) = x^2 - 2x - 3$\n",
+    "    \n",
+    "    (b) $p(x) = -x^2 - 6x + 8$\n",
+    "    \n",
+    "    (*Hint:* Calculate the roots and consider the symmetry of the parabola)  \n",
+    "    \n",
+    "    <br/>\n",
+    "      \n",
+    "3. There is a function that is frequently used to describe the per capita growth rate of organisms when the rate depends on the concentration of some nutrient and becomes saturated for large enough nutrient concentrations. If we denote the concentration of the nutrient by $N$, then the per capita growth rate $r(N)$ is given by the *Monod growth function*\n",
+    "\n",
+    "    $$\n",
+    "        r(N)=\\frac{aN}{k+N},\\ \\ \\ N \\ge 0\n",
+    "    $$\n",
+    "    \n",
+    "    where $a$ and $k$ are positive constants.\n",
+    "    \n",
+    "    <br/>\n",
+    "    \n",
+    "    (a) Use Python or R to investigate the Monod growth function. Graph $r(N)$ for (i) $a = 5$ and $k = 1$, (ii) $a = 5$ and $k = 3$, and (iii) $a = 8$ and k = 1. Place all three graphs in the same coordinate system.\n",
+    "    \n",
+    "    (b) On the basis of the graphs in (a), describe in words what happens when you change $a$.\n",
+    "    \n",
+    "    (c) On the basis of the graphs in (a), describe in words what happens when you change $k$.  \n",
+    "    \n",
+    "    <br/>\n",
+    "      \n",
+    "      \n",
+    "4. The function \n",
+    "\n",
+    "    $$\n",
+    "        f(x) = \\frac{x^n}{b^n + x^n},\\ \\ \\ x \\ge 0\n",
+    "    $$ \n",
+    "    \n",
+    "    where $n$ is a positive integer and $b$ is a positive real number, is used in biochemistry to model reaction rates as a function of the concentration of some reactants    \n",
+    "    \n",
+    "    <br/>\n",
+    "    \n",
+    "    (a) Use Python or R to graph $f(x)$ for $n = 1, 2,\\ \\mbox{and}\\ 3$ in the same coordinate system when $b = 2$.\n",
+    "    \n",
+    "    (b) Where do the three graphs in (a) intersect?\n",
+    "    \n",
+    "    (c) What happens to $f(x)$ as $x$ gets larger?\n",
+    "\n",
+    "    (d) For an arbitrary positive value of $b$, show that $f (b) = 1/2$. On the basis of this demonstration and your answer in C. ,  explain why $b$ is called the half-saturation constant.  \n",
+    "    \n",
+    "    <br/>\n",
+    "  \n",
+    "  \n",
+    "5. The file [words_moby_dick.csv](https://imperialcollegelondon.box.com/s/940y521dq1owarx8pt4d4mu2d0f2cz85) contains data on the cumulative distribution of the number of times specific words occur in the text of the novel *Moby Dick*, by Herman Melville. Assume that this distribution is a power law of the form \n",
+    "\n",
+    "    $$\n",
+    "        y = x^D\n",
+    "    $$\n",
+    "    \n",
+    "    (a) Using the provided data, and R/Python, estimate the exponent of this power law. \n",
+    "    (*Hint:* If you plot this data in log-log scale, what should correspond to the exponent $D$?)\n",
+    "    \n",
+    "    (b) Can you think of a better way to estimate D?  \n",
+    "    \n",
+    "    <br/>\n",
+    "  \n",
+    "  \n",
+    "6. Assume that a population size at time $t$ is $N(t)$ and that\n",
+    "\n",
+    "    $$\n",
+    "        N(t) = 40 \\cdot 2^t,\\ \\ \\ t \\ge 0\n",
+    "    $$    \n",
+    "\n",
+    "    (a) Find the population size at time t = 0.\n",
+    "\n",
+    "    (b) Show that\n",
+    "\n",
+    "    $$\n",
+    "        N(t) = 40\\, \\mbox{e}^{t\\, ln 2},\\ \\ \\ t \\ge 0\n",
+    "    $$    \n",
+    "\n",
+    "    (c) How long will it take until the population size reaches 1000?\n",
+    "    \n",
+    "   (*Hint*: Find $t$ so that $N(t) = 1000$.)  \n",
+    "   \n",
+    "   <br/>\n",
+    "     \n",
+    "7. (*Adapted from Moss, 1980*) Hall (1964) investigated the change in population size of the zooplankton species *Daphnia galeata mendota* in Base Line Lake, Michigan. The population size $N(t)$ at time $t$ was modeled by the equation\n",
+    "\n",
+    "    $$ \n",
+    "        N(t) = N_0\\mbox{e}^{rt}\n",
+    "    $$\n",
+    "    \n",
+    "    where $N_0$ denotes the population size at time $0$. The constant $r$ is called the **intrinsic rate of growth**.\n",
+    "    \n",
+    "    <br/>\n",
+    "    \n",
+    "    (a) Plot $N(t)$ as a function of $t$ if $N_0 = 100$ and $r = 2$. Compare your graph against the graph of $N(t)$ when $N_0 = 100$ and $r = 3$. Which population grows faster?\n",
+    "    \n",
+    "    (b) The constant $r$ is an important quantity because it describes how quickly the population changes. Suppose that you determine the size of the population at the beginning and at the end of a period of length $1$, and you find that at the beginning there were $200$ individuals and after one unit of time there were $250$ individuals. Determine $r$. (*Hint*: Consider the ratio $N(t+1)/N(t)$.)  \n",
+    "    \n",
+    "    <br/>\n",
+    "      \n",
+    "8. Fish are indeterminate growers; that is, they grow throughout their lifetime. The growth of fish can be described by the von Bertalanffy function\n",
+    "\n",
+    "    $$\n",
+    "        L(x) = L_{\\infty}(1-\\textrm{e}^{-kx} )\n",
+    "    $$\n",
+    "    \n",
+    "    for $x \\ge 0$, where $L(x)$ is the length of the fish at age $x$ and $k$ and $L_{\\infty}$ are positive\n",
+    "    \n",
+    "    <br/>\n",
+    "    \n",
+    "    (a) Using Python or R, graph $L(x)$ for $L_{\\infty} = 20$, for (i) $k = 1$ and (ii) $k = 0.1$.\n",
+    "    \n",
+    "    (b) For $k = 1$, find $x$ so that the length is 90\\% of $L_{\\infty}$. Repeat for 99\\% of $L_{\\infty}$. Can the fish ever attain length $L_{\\infty}$? Interpret the meaning of $L_{\\infty}$.\n",
+    "    \n",
+    "    (c) Compare the graphs obtained in A. Which growth curve reaches 90\\% of $L_{\\infty}$ faster? Can you explain what happens to the curve of $L(x)$ when you vary $k$ (for fixed $L_{\\infty}$)?  \n",
+    "    \n",
+    "    <br/>\n",
+    "    \n",
+    "9. For the following pairs of functions, plot the ratio (quotient) between the two using R or Python. Based on the behaviour of the ratio when $x \\rightarrow \\infty$ (very large values of $x$), how does each of these functions compare to the other in the velocity that they grow?\n",
+    "\n",
+    "    <br/>  \n",
+    "    \n",
+    "    (a) $ f(x) = e^x \\quad \\text{ and } \\quad g(x) = x^2 $\n",
+    "\n",
+    "    (b) $ f(x) = e^x \\quad \\text{ and } \\quad g(x) = x^{20} $\n",
+    "    \n",
+    "    (c) $ f(x) = x \\quad \\text{ and } \\quad g(x) = \\log(x) $  \n",
+    "    \n",
+    "    <br/>\n",
+    "      \n",
+    "10. A community measure that takes both species abundance and species richness into account is the Shannon diversity index $H$. To calculate $H$, the proportion $p_i$ of species $i$ in the community is used. Assume that the community consists of $S$ species. Then\n",
+    "\n",
+    "    $$\n",
+    "        H = -\\sum_{i=1}^S p_i\\, \\textrm{ln}\\, p_i = -( p_1 \\textrm{ln}\\, p_1 + p_2 \\textrm{ln}\\, p_2 + \\cdots + p_S \\textrm{ln}\\, p_S )\n",
+    "    $$\n",
+    "    \n",
+    "    <br/>\n",
+    "    \n",
+    "    (a) Assume that $S = 5$ and that all species are equally abundant; that is, $p_1 = p_2 = \\cdots = p_5$. Compute $H$.\n",
+    "    \n",
+    "    (b) Assume that $S = 10$ and that all species are equally abundant; that is, $p_1 = p_2 = \\cdots = p_{10}$. Compute $H$.\n",
+    "    \n",
+    "    (c) A measure of equitability (or evenness) of the species distribution can be measured by dividing the diversity index $H$ by $\\textrm{ln}\\, S$. Compute $H/\\textrm{ln}\\, S$ for $S = 5$ and $S = 10$."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Exercise Set 2\n",
+    "\n",
+    "1. Solve the following systems of linear equations\n",
+    "    <br/>\n",
+    "    \n",
+    "    $$\n",
+    "        \\mbox{(a)}  \\ \n",
+    "        \\begin{cases}\n",
+    "            5x - y + 2z = 6 \\\\\n",
+    "            x + 2y - z = -1 \\\\\n",
+    "            3x + 2y - 2z = 1\\\\\n",
+    "        \\end{cases}\n",
+    "    $$\n",
+    "    \n",
+    "    <br/>\n",
+    "    \n",
+    "    $$\n",
+    "        \\mbox{(b)}  \\ \n",
+    "        \\begin{cases}\n",
+    "            2x - y + 3z = 3 \\\\\n",
+    "            2x + y + 4z = 4 \\\\\n",
+    "            2x - 3y + 2z = 2\\\\\n",
+    "        \\end{cases}\n",
+    "    $$\n",
+    "    \n",
+    "    <br/>\n",
+    "    \n",
+    "2. Three different species of insects are reared together in a laboratory cage. They are supplied with two different types of food each day. Each individual of species 1 consumes 3 units of food $A$ and 5 units of food $B$, each individual of species 2 consumes 2 units of food $A$ and 3 units of food $B$, and individual of species 3 consumes 1 unit of food $A$ and 2 units of food $B$. Each day, 500 units of food $A$ and 900 units of food $B$ are supplied. How many individuals of each species can be reared together? Is there more than one solution? What happens if we add 550 units of a third type of food, called $C$, and each individual of species 1 consumes 2 units of food $C$, each individual of species 2 consumes 4 units of food $C$, and each individual of species 3 consumes 1 unit of food $C$?\n",
+    "\n",
+    "    <br/>\n",
+    "\n",
+    "3. Networks are abstractions of real systems used to identify and study patterns of connectivity between the individual units that compose those systems. Networks (often referred to as *graphs*) are mathematically defined as a set of *nodes*, or vertices, (representing the individual units) and a set of *links*, or connections, between these nodes, which account for the interaction patterns within the studied systems. In Ecology, graphs have been widely used in studies investigating food-webs, plant-pollinator interactions, and disease transmission in populations.  \n",
+    "    \n",
+    "    <br />\n",
+    "  \n",
+    "    If the nodes of a network of size $N$ are indexed with positive integers such as $i = 1, 2, \\ldots, N$, an useful method for representing the structure of a network is to construct a matrix $A$ for which the elements $a_{ij}$ are given by:  \n",
+    "      \n",
+    "    $$a_{ij} = \\begin{cases}1,\\ \\mbox{if nodes}\\ i \\ \\mbox{and}\\ j \\ \\mbox{are linked} \\\\ 0,\\ \\mbox{otherwise}  \\end{cases}$$  \n",
+    "    \n",
+    "    Matrix $A$ is called the *adjacency matrix* of the network. One example of a network with $6$ nodes is shown below.\n",
+    "\n",
+    "    <br/>  \n",
+    "    \n",
+    "    ```{image} graph_example.png  \n",
+    "    ```  \n",
+    "          \n",
+    "    (a) Based on the graphical representation of this network, write down its adjacency matrix.  \n",
+    "        \n",
+    "    <br/>\n",
+    "    \n",
+    "    (b) By the definition of the adjacency matrix, the sum of the matrix elements on row $i$, $k_i = \\sum_{j=1}^N a_{ij}$, corresponds to the number of links of node $i$. The quantity $k_i$ is called the *degree* of node $i$. Calculate the corresponding degrees for each node on the previous network.\n",
+    "        \n",
+    "    <br/>\n",
+    "    \n",
+    "    (c) The Laplacian matrix is a mathematical object that can be used to find many useful properties of a network. By definition, $L$ is constructed as  \n",
+    "    \n",
+    "    <br/>  \n",
+    "    \n",
+    "    $$ L = D - A $$  \n",
+    "    \n",
+    "    <br/>  \n",
+    "    \n",
+    "    Where $A$ is the adjacency matrix, and $D$ is the degree matrix. The degree matrix $D$ is a diagonal matrix which contains information about the degree of each node. With your answer for itens (a) and (b), calculate $L$.\n",
+    "        \n",
+    "    <br/>\n",
+    "    \n",
+    "    (d) Use R/Python to numerically calculate the eigenvalues of the Laplacian matrix.\n",
+    "    \n",
+    "    <br/>\n",
+    "   \n",
+    "4. Write each system in matrix form\n",
+    "   <br/>\n",
+    "    \n",
+    "    $$\n",
+    "        \\mbox{(a)}  \\ \n",
+    "        \\begin{cases}\n",
+    "            2x_2-x_1 = x_3 \\\\\n",
+    "            4x_1 + x_3 = 7x_2 \\\\\n",
+    "            x_2 - x_1 = x_3\\\\\n",
+    "        \\end{cases}\n",
+    "    $$\n",
+    "    \n",
+    "    <br/>\n",
+    "    \n",
+    "    $$\n",
+    "        \\mbox{(b)}  \\ \n",
+    "        \\begin{cases}\n",
+    "            2x_1 - 3x_2 = 4 \\\\\n",
+    "            -x_1 + x_2 = 3 \\\\\n",
+    "            3x_1 = 4\\\\\n",
+    "        \\end{cases}\n",
+    "    $$\n",
+    "    \n",
+    "    <br/>\n",
+    "    \n",
+    "5. Suppose that \n",
+    "\n",
+    "    <br/>\n",
+    "    \n",
+    "    $$A = \\begin{pmatrix}\n",
+    "            2 & 4 \\\\\n",
+    "            3 & 6 \n",
+    "            \\end{pmatrix}\n",
+    "        \\ \\ \\ \\text{,}\\ \\ \\\n",
+    "        X = \\begin{pmatrix}\n",
+    "            x \\\\\n",
+    "            y \n",
+    "            \\end{pmatrix} \\ \\ \\ \\text{and}\\ \\ \\\n",
+    "            B = \\begin{pmatrix}\n",
+    "            b_1 \\\\\n",
+    "            b_2 \\\\\n",
+    "            \\end{pmatrix}$$\n",
+    "\n",
+    "    (a) Write $AX = B$ as a system of linear equation.\n",
+    "        \n",
+    "    <br/>\n",
+    "    \n",
+    "    (b) Show that if \n",
+    "   \n",
+    "    $$\n",
+    "        B = \\begin{pmatrix}\n",
+    "            3 \\\\\n",
+    "            \\frac92 \\\\\n",
+    "            \\end{pmatrix}\n",
+    "    $$\n",
+    "    \n",
+    "    then $AX = B$ has infinitely many solutions. Graph the two straight lines associated with the corresponding system of linear equations, and explain why the system has infinitely many solutions.\n",
+    "        \n",
+    "    <br/>\n",
+    "    \n",
+    "    (c) Find a column vector\n",
+    "    \n",
+    "    <br/>\n",
+    "    \n",
+    "    $$B = \\begin{pmatrix}\n",
+    "            b_1 \\\\\n",
+    "            b_2 \\\\\n",
+    "            \\end{pmatrix}$$\n",
+    "\n",
+    "    so that $AX = B$ has no solutions.\n",
+    "    \n",
+    "    <br/>\n",
+    "    \n",
+    "6. Assume that a population is divided into three age classes and that 80\\% of the individuals age 0 and 10\\% of the individuals age 1 survive until the end of the next breeding season. Assume further that individuals age 1 have an average of 1.6 offspring and individuals age 2 have an average of 3.9 offspring. If, at time 0, the population consists of 1000 individuals age 0, 100 individuals age 1, and 20 individuals age 2, find the projection matrix and the age distribution at time 3.\n",
+    "\n",
+    "    <br/>\n",
+    "\n",
+    "7. Consider the following projection matrix representing a population with five age classes:\n",
+    "\n",
+    "    <br/>  \n",
+    "    \n",
+    "    $$\n",
+    "        P = \\begin{pmatrix}\n",
+    "        0 & 5 & 3 & 2 & 1 \\\\\n",
+    "        0.9 & 0 & 0 & 0 & 0 \\\\\n",
+    "        0 & 0.3 & 0 & 0 & 0 \\\\\n",
+    "        0 & 0 & 0.1 & 0 & 0 \\\\\n",
+    "        0 & 0 & 0 & 0.05 & 0 \\\\\n",
+    "    \\end{pmatrix}\n",
+    "    $$\n",
+    "    \n",
+    "    <br/>\n",
+    "    \n",
+    "    (a) Making use of Python or R code, begin with a population distributed as $(0,0,0,0,10)$ individuals at time $t=0$, project the population at time $t=20$, and plot the total number over time. Plot the logarithm of the total population over time.\n",
+    "        \n",
+    "    <br/>\n",
+    "    \n",
+    "    (b) Repeat the above but begin with $(80,16,5,1,1)$ individuals. How do these results compare to the results in (a) ?\n",
+    "    \n",
+    "    <br/>\n",
+    "    \n",
+    "8. In the problems below, find the eigenvalues $\\lambda_1$ and $\\lambda_2$ and corresponding eigenvectors $\\mathbf{v_1}$ and $\\mathbf{v_2}$ for each matrix $B$. Determine the equations of the lines through the origin in the direction of the eigenvectors $\\mathbf{v_1}$ and $\\mathbf{v_2}$, and graph the lines together with the eigenvectors $\\mathbf{v_1}$ and $\\mathbf{v_2}$ and the vectors $A\\mathbf{v_1}$ and $A\\mathbf{v_2}$.\n",
+    "\n",
+    "    <br/>\n",
+    "    \n",
+    "    $$\n",
+    "    \\text{(a)} \\ \\ B = \\begin{pmatrix}\n",
+    "    2 & 3\\\\\n",
+    "    0 & -1\n",
+    "    \\end{pmatrix}\n",
+    "    $$\n",
+    "\n",
+    "    <br/>\n",
+    "\n",
+    "    $$\n",
+    "    \\text{(b)} \\ \\ B = \\begin{pmatrix}\n",
+    "    3 & 6\\\\\n",
+    "    -1 & -4\n",
+    "    \\end{pmatrix}  \n",
+    "    $$\n",
+    "    \n",
+    "    <br/>\n",
+    "    \n",
+    "9. Suppse that \n",
+    "\n",
+    "    <br/>\n",
+    "    \n",
+    "    $$\n",
+    "    P = \\begin{pmatrix}\n",
+    "    0.5 & 2.3\\\\\n",
+    "    a & 0\n",
+    "    \\end{pmatrix}\n",
+    "    $$\n",
+    "    \n",
+    "    <br/>\n",
+    "    \n",
+    "    is the projection matrix of a population with two age classes. For which values of $a$ does this population grow?\n",
+    "    \n",
+    "    <br/>\n",
+    "    \n",
+    "10. Suppose that \n",
+    "\n",
+    "    <br/>\n",
+    "    \n",
+    "    $$\n",
+    "    P = \\begin{pmatrix}\n",
+    "    0.5 & 2.0\\\\\n",
+    "    0.1 & 0\n",
+    "    \\end{pmatrix}\n",
+    "    $$\n",
+    "    \n",
+    "    <br/>\n",
+    "    \n",
+    "    is the projection matrix of a population with two age classes.\n",
+    "        \n",
+    "    <br/>\n",
+    "    \n",
+    "    (a) If you were to manage this population, would you need to be concerned about its long-term survival?\n",
+    "    \n",
+    "    <br/>\n",
+    "    \n",
+    "    (b) Suppose that you can improve either the fecundity or the survival of the zero-year-olds, but due to physiological and environmental constraints, the fecundity of zero-year-olds will not exceed 1.5 and the survival of zero-year-olds will not exceed 0.4. Investigate how the growth rate of the population is affected by changing either the survival or the fecundity of zero-year-olds, or both. What would be the maximum achievable growth rate?\n",
+    "    \n",
+    "    <br/>\n",
+    "    \n",
+    "    (c) In real situations, what other factors might you need to consider when you decide on management strategies?"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "editable": true,
+    "slideshow": {
+     "slide_type": ""
+    },
+    "tags": []
+   },
+   "source": [
+    "## Exercise Set 3"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "editable": true,
+    "slideshow": {
+     "slide_type": ""
+    },
+    "tags": []
+   },
+   "source": [
+    "1. **Island Model** Assume that a species lives in a habitat that consists of many islands close to a mainland. The species occupies both the mainland and the islands, but, although it is present on the mainland at all times, it frequently goes extinct on the islands. Islands can be recolonized by migrants from the mainland. The following model keeps track of the fraction of islands occupied: Denote the fraction of islands occupied at time $t$ by $p(t)$. Assume that each island experiences a constant risk of extinction and that vacant islands (the fraction $1-p$) are colonized from the mainland at a constant rate. Then\n",
+    "\n",
+    "    $$\n",
+    "        \\frac{dp}{dt}=c(1-p)-ep\n",
+    "    $$\n",
+    "\n",
+    "     where $c$ and $e$ are positive constants.\n",
+    "\n",
+    "    <br/>\n",
+    "    \n",
+    "    (a) The gain from colonization is $f (p) = c(1 - p)$ and the loss from extinction is $g(p) = ep$. Graph (without graph visualisers) $f(p)$ and $g(p)$ for $0 \\le p \\le 1$ in the same coordinate system. Explain why the two graphs intersect whenever $e$ and $c$ are both positive. Compute the point of intersection and interpret its biological meaning.\n",
+    "    \n",
+    "     <br/>\n",
+    "     \n",
+    "    (b) The parameter $c$ measures how quickly a vacant island becomes colonized from the mainland. The closer the islands, the larger is the value of $c$. Use your graph in (a) to explain what happens to the point of intersection of the two lines as c increases. Interpret your result in biological terms.  \n",
+    "      \n",
+    "    <br />  \n",
+    "      \n",
+    "2. **Biotic Diversity** (Adapted from Valentine, 1985.) Walker and Valentine (1984) suggested a model for species diversity which assumes that species extinction rates are independent of diversity but speciation rates are regulated by competition. Denoting the number of species at time $t$ by $S(t)$, the speciation rate by $b$, and the extinction rate by $a$, they used the model\n",
+    "\n",
+    "    $$\n",
+    "        \\frac{dS}{dt} = S\\left[b\\left(1-\\frac{S}{K}\\right)-a \\right]\n",
+    "    $$\n",
+    "      \n",
+    "    where $K$ denotes the number of \"niches\", or potential places for species in the ecosystem.\n",
+    "    \n",
+    "    <br/>\n",
+    "    \n",
+    "    (a) Find possible equilibria ($dS/dt = 0$) under the condition $a < b$.\n",
+    "    \n",
+    "    <br/>\n",
+    "    \n",
+    "    (b) Use your result in (a) to explain the following statement by Valentine (1985):\n",
+    "    \n",
+    "     ```{admonition} Quotation \n",
+    "     \"In this situation, ecosystems are never 'full', with all potential niches occupied by species so long as the extinction rate is above zero.\"\n",
+    "     ```\n",
+    "    \n",
+    "    <br/>\n",
+    "    \n",
+    "    (c) What happens when $a \\ge b$?  \n",
+    "      \n",
+    "    <br />  \n",
+    "      \n",
+    "3. Find a point on the curve\n",
+    "    \n",
+    "    $$y = 1 - 3x^3$$\n",
+    "    \n",
+    "     whose tangent line is parallel to the line $y = -x$. Is there more than one such point? If so, find all other points with this property.  \n",
+    "       \n",
+    "    <br />  \n",
+    "      \n",
+    "4. Find a point on the curve\n",
+    "\n",
+    "    $$y = 2x^3 - 4x + 1$$\n",
+    "    \n",
+    "    whose tangent line is parallel to the line $y-2x = 1$. Is there more than one such point? If so, find all other points with this property.  \n",
+    "      \n",
+    "    <br />  \n",
+    "      \n",
+    "5. In the following, use the product and quotient rules to find the derivative with respect to the independent variable.\n",
+    "\n",
+    "    <br/>\n",
+    "    \n",
+    "    (a) $f(x) = (2x^3 - 1)(3 + 2x^2)$\n",
+    "    \n",
+    "    <br/>\n",
+    "    \n",
+    "    (b) $h(s) = (4 - 3s^2 + 4s^3)^2$\n",
+    "    \n",
+    "    <br/>\n",
+    "    \n",
+    "    (c) $f(x) =\\sqrt{3x}(x^2 - 1)$\n",
+    "    \n",
+    "    <br/>\n",
+    "    \n",
+    "    (d) $g(t) = 4(3t^2 - 1)(2t + 1)$\n",
+    "    \n",
+    "    <br/>\n",
+    "    \n",
+    "    (e) $f(x) = 3(x^2 + 2)(4x^2 - 5x^4) - 3$\n",
+    "    \n",
+    "    <br/>\n",
+    "    \n",
+    "    (f) $g(x) = \\displaystyle\\frac{1 - 4x^3}{1 - x}$\n",
+    "    \n",
+    "    <br/>\n",
+    "    \n",
+    "    (g) $h(x) = \\displaystyle\\frac{1 + 2x^2 - 4x^4}{3x^3 - 5x^5}$\n",
+    "    \n",
+    "    <br/>\n",
+    "    \n",
+    "    (h) $f(t) =\\displaystyle\\frac{3 - t^2}{(t + 1)^2}$\n",
+    "    \n",
+    "    <br/>\n",
+    "    \n",
+    "    (i) $h(s) = \\displaystyle\\frac{2s^3 - 4s^2 + 5s - 7}{(s^2 - 3)^2}$\n",
+    "    \n",
+    "    <br/>\n",
+    "    \n",
+    "    (j) $g(s) = \\displaystyle\\frac{s^{1/7} - s^{2/7}}{s^{3/7} + s^{4/7}}$\n",
+    "      \n",
+    "    <br />  \n",
+    "      \n",
+    "6. **Logistic Growth**\n",
+    "\n",
+    "    <br/>\n",
+    "    \n",
+    "    (a) Find the derivative of the logistic growth curve  \n",
+    "      \n",
+    "    $$N(t)=\\frac{K}{1+\\left(\\frac{K}{N(0)}-1\\right)e^{-rt}}$$  \n",
+    "    \n",
+    "    where $r$ and $K$ are positive constants and $N(0)$ is the population size at time $0$.\n",
+    "        \n",
+    "    <br/>\n",
+    "            \n",
+    "    (b) Show that $N(t)$ satisfies the equation  \n",
+    "      \n",
+    "    $$\\frac{dN}{dt}=rN\\left(1-\\frac{N}{K}\\right)$$  \n",
+    "      \n",
+    "    [*Hint:* Use the function $N(t)$ given in (a) for the right-hand side, and simplify until you obtain the derivative of N(t) that you computed in A.]\n",
+    "        \n",
+    "    <br/>\n",
+    "        \n",
+    "    (c) What happens to the *per-capita* rate of growth $\\frac{1}{N}\\frac{dN}{dt}$ as you increase N? What biological principle and what type of interaction is your model describing?  \n",
+    "      \n",
+    "    <br />  \n",
+    "      \n",
+    "7. In the following, find the derivative with respect to the independent variable (*Hint*: use the chain rule).\n",
+    "    \n",
+    "    <br/>\n",
+    "    \n",
+    "    (a) $f(x) = \\mbox{sin}(e^{2x} + x)$\n",
+    "    \n",
+    "    <br/>\n",
+    "    \n",
+    "    (b) $h(x) = \\mbox{exp}[x^2 - 2 \\mbox{cos}\\, x]$\n",
+    "    \n",
+    "    <br/>\n",
+    "    \n",
+    "    (c) $f(x) =\\displaystyle\\frac{x-2^{−x}}\n",
+    "{1+x2^{−x}}$\n",
+    "    \n",
+    "    <br/>\n",
+    "    \n",
+    "    (d) $g(t) = \\mbox{log}\\left(\\displaystyle\\frac{1 - t}{1 + 2t}\\right)$\n",
+    "    \n",
+    "    <br/>\n",
+    "    \n",
+    "    (e) $f(t) = \\mbox{sin}(\\mbox{ln}(3t))$\n",
+    "    \n",
+    "    <br/>\n",
+    "    \n",
+    "    (f) $g(x) = \\mbox{ln}(\\mbox{cos}(1 - x))$\n",
+    "    \n",
+    "    <br/>\n",
+    "    \n",
+    "    (g) $h(x) = \\displaystyle\\frac{\\sqrt{x^2 - 1}}{2+\\sqrt{x^2+1}}$ \n",
+    "    \n",
+    "    <br/>\n",
+    "    \n",
+    "    (h) $f(t) =\\sqrt[3]{t^2+\\sqrt{1-t}}$\n",
+    "    \n",
+    "    <br/>\n",
+    "      \n",
+    "8. The functional responses of some predators are sigmoidal; that is, the number of prey attacked per predator as a function of prey density has a sigmoidal shape. If we denote the prey density by $N$, the predator density by $P$, the time available for searching for prey by $T$, and the handling time of each prey item per predator by $T_h$, then the number of prey encounters per predator as a function of $N$, $T$, and $T_h$ can be expressed as\n",
+    "\n",
+    "    $$\n",
+    "        f (N, T, T_h) = \\frac{b^2N^2T}{1 + cN + bT_hN^2}\n",
+    "    $$\n",
+    "    \n",
+    "    where $b$ and $c$ are positive constants.\n",
+    "    \n",
+    "    <br/>\n",
+    "    \n",
+    "    (a) Calculate the first-order partial derivative of $f(N, T, T_h)$ with respect to the prey density $N$. If $N$ increases, what happens to $f(N, T, T_h)$?\n",
+    "     \n",
+    "    <br/>\n",
+    "    \n",
+    "    (b) Calculate the first-order partial derivative of $f(N, T, T_h)$ with respect to the the time $T$ available for search. If $T$ increases, what happens to $f(N, T, T_h)$?\n",
+    "     \n",
+    "    <br/>\n",
+    "    \n",
+    "    (c) Calculate the first-order partial derivative of $f(N, T, T_h)$ with respect to the handling time $T_h$. If $T_h$ increases, what happens to $f(N, T, T_h)$?\n",
+    "    \n",
+    "    [*Hint:* The function $f(N, T, T_h)$ will be an increasing (decreasing) function of the independent variable if the first-order partial derivative of $f$ with respect to this variable is positive (negative).]"
+   ]
+  }
+ ],
+ "metadata": {
+  "celltoolbar": "Tags",
+  "kernelspec": {
+   "display_name": "Python 3 (ipykernel)",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.10.12"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 4
+}
diff --git a/_sources/notebooks/Ecosystems.ipynb b/_sources/notebooks/Ecosystems.ipynb
new file mode 100644
index 00000000..cdf73aa0
--- /dev/null
+++ b/_sources/notebooks/Ecosystems.ipynb
@@ -0,0 +1,178 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Ecosystems"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "* Communities, Food-webs and Ecosystems\n",
+    "  * Community vs Ecosystem\n",
+    "    * Microbial community vs microbiome\n",
+    "  * Food-webs\n",
+    "  * Random matrix theory\n",
+    "  * Evolutionary dynamics (!!)\n",
+    "    * Microbial systems as an example\n",
+    "  * Applications/Examples"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "\n",
+    "## Introduction\n",
+    "Ecological and mutualistic networks are fundamental to understanding how species interact and coexist within ecosystems. These networks, encompassing trophic (consumer-resource) and mutualistic (e.g., plant-pollinator) interactions, provide insights into biodiversity, ecosystem functioning, and resilience to disturbances.\n",
+    "\n",
+    "> \"The presence of a feline animal in large numbers in a district might determine, through the intervention first of mice and then of bees, the frequency of certain flowers in that district.\"  \n",
+    "> -- Charles Darwin, 1859\n",
+    "\n",
+    "This quote illustrates the intricate interdependence within ecological systems.\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "\n",
+    "## Ecological Networks\n",
+    "Ecological networks represent interactions where nodes correspond to individuals or species populations and links denote interactions such as predation, competition, or mutualism. A classic example is the Silwood Park food web, which illustrates trophic relationships among species.\n",
+    "\n",
+    "### Trophic Networks\n",
+    "Trophic networks describe feeding relationships among species. These are often represented as directed graphs, with arrows indicating the flow of energy from prey to predator. For example, a primary producer (e.g., a plant) might feed a herbivore, which in turn feeds a carnivore. Trophic networks often exhibit hierarchical structures, with top predators occupying the apex of the food web.\n",
+    "\n",
+    "### Mutualistic Networks\n",
+    "Mutualistic networks describe interactions where species mutually benefit, such as:\n",
+    "- **Plant-fungal networks:** Mycorrhizal fungi exchange nutrients with plants.\n",
+    "\n",
+    "- **Bacterial networks:** Exchange of metabolites, such as in yogurt and cheese production.\n",
+    "\n",
+    "- **Plant-pollinator networks:** Pollinators gain nectar, while plants achieve reproduction.\n",
+    "\n",
+    "- **Seed dispersal networks:** Animals disperse seeds in exchange for fruit.\n",
+    "\n",
+    "These networks contribute to critical ecosystem services, including pollination and nutrient cycling. They are often characterized by nested structures, where specialists interact with subsets of generalist species.\n",
+    "\n",
+    "### Competitive Networks\n",
+    "Competitive networks involve interactions where species compete for the same limited resources, such as nutrients, light, or space. For example, plants in dense forests compete for sunlight, while microbes in soil compete for nutrients. These networks are often depicted as undirected graphs where the links indicate negative interactions.\n",
+    "\n",
+    "### Behavioral Networks\n",
+    "Behavioral networks describe interactions within social or cooperative groups, such as ant colonies or primate troops. These interactions can include cooperation, altruism, or dominance hierarchies. Behavioral networks are critical for understanding group dynamics and evolutionary strategies.\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "\n",
+    "## Structure and Stability\n",
+    "The structure of ecological networks is shaped by modules or motifs, small groups of interconnected nodes that enhance stability. Stability in ecological networks can be formally defined as:\n",
+    "\n",
+    "- **Local Stability:** The ability of a system to return to equilibrium after a small perturbation.\n",
+    "\n",
+    "- **Global Stability:** The capacity of a system to remain functional across a wide range of disturbances.\n",
+    "\n",
+    "### Trophic Networks\n",
+    "In trophic networks, stability is often influenced by modularity (the division of the network into relatively independent clusters) and connectivity (the number of interactions between species). Highly connected but modular networks tend to be more resilient to species loss.\n",
+    "\n",
+    "### Mutualistic Networks\n",
+    "Mutualistic networks often exhibit nestedness, a structural pattern where specialist species interact with subsets of generalist species. This nestedness enhances stability by reducing competition and fostering redundancy. Modularity in mutualistic networks can also increase resilience by containing disturbances within specific clusters.\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "\n",
+    "## Mathematical Modeling\n",
+    "\n",
+    "### Trophic Networks\n",
+    "Trophic interactions are often modeled using Lotka-Volterra equations:\n",
+    "\n",
+    "$$\n",
+    "\\frac{dN}{dt} = r_m N \\left(1 - \\frac{N}{K}\\right) - aNC,\n",
+    "$$\n",
+    "$$\n",
+    "\\frac{dC}{dt} = eaNC - zC,\n",
+    "$$\n",
+    "\n",
+    "where $N$ and $C$ represent resource and consumer populations, respectively, and parameters include growth rates ($r_m$), carrying capacity ($K$), attack rate ($a$), conversion efficiency ($e$), and mortality rate ($z$). These models help predict population dynamics and the effects of perturbations.\n",
+    "\n",
+    "### Mutualistic Networks\n",
+    "The dynamics of mutualistic networks can be described using interaction matrices that quantify the strength and symmetry of benefits between species. For example, the dynamics of plant-pollinator networks can be modeled as:\n",
+    "\n",
+    "$$\n",
+    "\\frac{dP}{dt} = r_P P \\left(1 - \\frac{P}{K_P}\\right) + \\beta_{PM} M,\n",
+    "$$\n",
+    "$$\n",
+    "\\frac{dM}{dt} = r_M M \\left(1 - \\frac{M}{K_M}\\right) + \\beta_{MP} P,\n",
+    "$$\n",
+    "\n",
+    "where $P$ and $M$ represent plant and pollinator populations, and $\\beta_{PM}$ and $\\beta_{MP}$ are mutualistic interaction coefficients.\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "\n",
+    "## Conclusion\n",
+    "Ecological and mutualistic networks are integral to understanding biodiversity and ecosystem functioning. By studying their structure and stability, ecologists can predict responses to environmental changes and design conservation strategies. A comprehensive understanding of these networks is crucial for maintaining ecosystem resilience in the face of anthropogenic pressures.\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "\n",
+    "## Discussion Questions\n",
+    "1. What sorts of interactions might ancient microbes have had? What type(s) of Ecological Network(s)?\n",
+    "\n",
+    "2. What are the differences and similarities between microbial networks (who-eats-what type) and food webs (who-eats-whom type)?\n",
+    "\n",
+    "3. What role do indirect interactions play in trophic network / food web dynamics?\n",
+    "\n",
+    "4. How are trophic/biomass/ecological pyramids related to body size structure in food webs?\n",
+    "\n",
+    "5. How do you think climatic temperature (and warming) affects trophic networks / food webs?\n",
+    "\n",
+    "6. What are the differences between trophic (e.g., food web) and mutualistic networks?\n",
+    "\n",
+    "7. Can we truly study Mutualistic and Trophic networks separately?\n",
+    "\n",
+    "8. What role might body size play in Mutualistic (e.g., plant-pollinator) networks?\n",
+    "\n",
+    "9. How do you think global climate change might affect plant-pollinator networks and pollination services?\n",
+    "\n",
+    "10. How do you think increasing mutualism increases the rate of microbial community functioning (e.g., decomposition of leaf litter)? What role can temperature play in this? How does this relate to the global carbon cycle?\n",
+    "\n"
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3 (ipykernel)",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.10.12"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 2
+}
diff --git a/_sources/notebooks/Interactions.ipynb b/_sources/notebooks/Interactions.ipynb
new file mode 100644
index 00000000..4acb462c
--- /dev/null
+++ b/_sources/notebooks/Interactions.ipynb
@@ -0,0 +1,201 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Species Interactions"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "* Species Interactions\n",
+    "  * Consumer-resource interactions\n",
+    "    * Predator-prey (VJ)\n",
+    "    * Bacterial-substrate\n",
+    "    * MacArthur  \n",
+    "  * The Lotka-Volterra equations\n",
+    "    * Every LV system is \"effective\"\n",
+    "      * Normal form?\n",
+    "    * Bifurcations (VJ) \n",
+    "  * Evolutionary dynamics\n",
+    "  * Stochasticity\n",
+    "  * Applications/Examples"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "\n",
+    "## Introduction\n",
+    "Consumers \"live to eat\" and \"eat to live.\" They are heterotrophs that harvest energy from other organisms, such as tigers, or from organic sources, such as bacteria. As Ludwig Boltzmann noted in 1886:\n",
+    "\n",
+    "> \"The struggle for existence of living beings is not for the fundamental constituents of food, but for the possession of the free energy obtained, chiefly by means of the green plant, from the transfer of radiant energy from the hot sun to the cold earth.\"\n",
+    "\n",
+    "Consumer-resource interactions play a crucial role in the flow of energy through ecosystems. The metabolic theory of ecology provides a foundation for understanding these interactions, linking individual metabolic processes to population and ecosystem dynamics.\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "\n",
+    "## Energy, Metabolism, and Consumption\n",
+    "All living organisms must maintain an energy balance:\n",
+    "\n",
+    "$$\n",
+    "\\text{Energy Consumed} = \\text{Energy Used (Maintenance + Growth + Movement)}.\n",
+    "$$\n",
+    "\n",
+    "Different organisms allocate energy to maintenance, growth, and movement in varying proportions. To do more than just maintain themselves, organisms must consume energy at a rate faster than their metabolic rate. This requires investment in activities such as foraging, which links energy acquisition to body size. For instance, autotrophs utilize photosynthesis, while heterotrophs depend on consuming organic material.\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "\n",
+    "## Size and Metabolic Needs\n",
+    "Resting metabolic rate ($B$) increases with body size ($M$) following a power-law relationship:\n",
+    "\n",
+    "$$\n",
+    "B = B_0 M^b,\n",
+    "$$\n",
+    "\n",
+    "where $B_0$ is a normalization constant and $b$ is the scaling exponent, typically around $3/4$. Larger organisms have higher absolute metabolic rates but lower mass-specific rates:\n",
+    "\n",
+    "$$\n",
+    "\\frac{B}{M} = B_0 M^{b-1}.\n",
+    "$$\n",
+    "\n",
+    "This relationship implies that larger organisms need to consume more energy overall but are more efficient per unit of body mass. This efficiency has significant implications for energy flow in ecosystems, as larger consumers typically occupy higher trophic levels and have lower population densities.\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "\n",
+    "### Biomechanical Implications of Size\n",
+    "Size also affects biomechanics and interactions with the physical environment. As Haldane (1926) observed:\n",
+    "\n",
+    "> \"You can drop a mouse down a thousand-yard mine shaft; and, on arriving at the bottom, it gets a slight shock and walks away, provided that the ground is fairly soft. A rat is killed, a man is broken, a horse splashes.\"\n",
+    "\n",
+    "Size influences how organisms move, forage, and interact with resources, ultimately affecting their consumption rates. Smaller organisms, for example, often forage faster but consume smaller amounts per event, while larger organisms are slower but consume larger quantities.\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "\n",
+    "## Dissecting Consumer-Resource Interactions\n",
+    "Components of consumer-resource interactions determine consumption rate ($c$):\n",
+    "\n",
+    "$$\n",
+    " c = \\frac{aR}{1 + ahR},\n",
+    "$$\n",
+    "\n",
+    "where $a$ is the search rate, $R$ is resource density, and $h$ is handling time. This is known as the Type II functional response, where resource handling dominates at high resource densities, while searching dominates at low densities.\n",
+    "\n",
+    "These interactions are further influenced by environmental factors, such as temperature and resource availability, which alter the rates of searching and handling.\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "\n",
+    "## Predicting Consumer-Resource Interactions\n",
+    "Key components of consumer-resource interactions scale with size:\n",
+    "- Velocity and detection distance increase with consumer size.\n",
+    "- Handling time decreases with size.\n",
+    "- Consumption rate scales positively with size and interaction dimensionality (e.g., 2D vs. 3D).\n",
+    "\n",
+    "For example, in 3D environments like oceans, consumption rates scale more steeply with size compared to 2D environments. Studies have shown that aquatic predators exhibit 10x higher consumption rates at a given size compared to terrestrial predators due to the higher dimensionality of their interaction space.\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "\n",
+    "## Temperature and Consumer-Resource Interactions\n",
+    "Temperature affects interaction components via the Boltzmann-Arrhenius equation:\n",
+    "\n",
+    "$$\n",
+    "k = A e^{-\\frac{E_a}{k_B T}},\n",
+    "$$\n",
+    "\n",
+    "where $k$ is the reaction rate, $A$ is a constant, $E_a$ is activation energy, $k_B$ is Boltzmann's constant, and $T$ is temperature. Higher temperatures generally increase foraging velocity and reduce handling time, leading to higher consumption rates. For example, ectothermic organisms show a strong temperature dependence in their foraging activities, while endotherms have a more stable consumption rate due to thermal regulation.\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "\n",
+    "## Consumer-Resource Dynamics\n",
+    "The metabolic basis of consumer-resource interactions can predict population dynamics using models like the Lotka-Volterra equations:\n",
+    "\n",
+    "$$\n",
+    "\\frac{dN}{dt} = r_m N \\left(1 - \\frac{N}{K}\\right) - aNC,\n",
+    "$$\n",
+    "$$\n",
+    "\\frac{dC}{dt} = eaNC - zC,\n",
+    "$$\n",
+    "\n",
+    "where $r_m$ is the resource's maximum growth rate, $K$ is carrying capacity, $a$ is search rate, $e$ is conversion efficiency, and $z$ is mortality rate. These parameters depend on size and temperature, linking individual metabolism to population-level outcomes. Additionally, such models help explain phenomena like population cycles and the effects of environmental changes on consumer-resource interactions.\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "\n",
+    "## Summary\n",
+    "- Metabolic rate and biomechanics shape species' consumer-resource interactions and consumption rates.\n",
+    "- Size, temperature, and dimensionality influence these interactions.\n",
+    "- Models incorporating metabolic constraints can predict population dynamics and ecological patterns.\n",
+    "- Understanding these principles provides insights into energy flow and stability in ecosystems.\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "\n",
+    "## Discussion Questions\n",
+    "\n",
+    "1. What are the main advantages of a consumer (heterotrophic) lifestyle compared to an producer (autotrophic) one? What are the disadvantages?\n",
+    "2. Which is the most common and diverse organisms on planet earth ? Why? Think in terms of what type of consumer they are, and what their size is.\n",
+    "3. What is the largest consumer on Earth? What is the smallest? In what respects is their “struggle for existence” similar? In what respects is it different?\n",
+    "4. Which consumer(s) on earth operate(s) at the hottest temperatures? Which operate at the coldest? How is their “struggle for existence” similar or different?"
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3 (ipykernel)",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.10.12"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 2
+}
diff --git a/_sources/notebooks/MathsForBiologists/MathsForBiologists.ipynb b/_sources/notebooks/MathsForBiologists/MathsForBiologists.ipynb
new file mode 100644
index 00000000..9276ba27
--- /dev/null
+++ b/_sources/notebooks/MathsForBiologists/MathsForBiologists.ipynb
@@ -0,0 +1,5931 @@
+{
+ "cells": [
+  {
+   "cell_type": "code",
+   "execution_count": 3,
+   "metadata": {
+    "tags": [
+     "remove-cell"
+    ]
+   },
+   "outputs": [],
+   "source": [
+    "from ipywidgets import interact\n",
+    "import matplotlib.pyplot as plt\n",
+    "import numpy as np\n",
+    "\n",
+    "def fun(a,b):\n",
+    "    x = np.linspace(-2,2)\n",
+    "    plt.plot(x,a*x+b)\n",
+    "    plt.ylim(-8, 8)\n",
+    "    plt.show()\n",
+    "    \n",
+    "# create a slider\n",
+    "\n",
+    "interact(fun, a=(-6.0,4.0),b=(-1.0,1.0))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 62,
+   "metadata": {
+    "tags": [
+     "remove-cell"
+    ]
+   },
+   "outputs": [
+    {
+     "data": {
+      "image/png": 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",
+      "text/plain": [
+       "<Figure size 432x432 with 1 Axes>"
+      ]
+     },
+     "metadata": {
+      "needs_background": "light"
+     },
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "import matplotlib as mpl\n",
+    "import matplotlib.pyplot as plt\n",
+    "import numpy as np\n",
+    "\n",
+    "\n",
+    "# Create figure and add axes\n",
+    "fig = plt.figure(figsize=(6, 6))\n",
+    "ax = fig.add_subplot(111)\n",
+    "\n",
+    "\n",
+    "m = np.linspace(-2, 2, 11)\n",
+    "\n",
+    "# Get colors from coolwarm colormap\n",
+    "colors = plt.get_cmap('coolwarm', 11)\n",
+    "# Create variable reference to plot\n",
+    "f_d, = ax.plot([], [], linewidth=2.5)\n",
+    "# Add text annotation and create variable reference\n",
+    "temp = ax.text(1, 1, '', ha='right', va='top', fontsize=24)\n",
+    "\n",
+    "# Move left y-axis and bottim x-axis to centre, passing through (0,0)\n",
+    "ax.spines['left'].set_position('center')\n",
+    "ax.spines['bottom'].set_position('center')\n",
+    "\n",
+    "# Eliminate upper and right axes\n",
+    "ax.spines['right'].set_color('none')\n",
+    "ax.spines['top'].set_color('none')\n",
+    "\n",
+    "# Show ticks in the left and lower axes only\n",
+    "ax.xaxis.set_ticks_position('bottom')\n",
+    "ax.yaxis.set_ticks_position('left')\n",
+    "\n",
+    "\n",
+    "\n",
+    "labels = ['-2','-1.6','-1.2','-0.8','-0.4','0.0','0.4','0.8','1.2','1.6','2']\n",
+    "\n",
+    "for i in range(len(m)):\n",
+    "    x = np.linspace(-2, 2, 100)\n",
+    "    y = m[i]*x\n",
+    "    ax.plot(x, y, color=colors(i), linewidth=2.5, label=labels[i])\n",
+    "    \n",
+    "ax.tick_params(axis='both', which='major', labelsize=13)\n",
+    "\n",
+    "    \n",
+    "# Add legend\n",
+    "\n",
+    "leg = ax.legend(bbox_to_anchor=(1.05, 0.3),loc='lower left', \n",
+    "          frameon=True, labelspacing=0.2, fontsize=13)\n",
+    "leg.set_title(r\"$m$\", prop = {'size':'x-large'})\n",
+    "\n",
+    "plt.tight_layout()\n",
+    "plt.savefig('slope_01.png')"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 59,
+   "metadata": {
+    "tags": [
+     "remove-cell"
+    ]
+   },
+   "outputs": [
+    {
+     "data": {
+      "image/png": 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85jeIjIzUuoyA9NOfyq+HD8sTxxINRgwo8osTJ05gy5Yt2LBhg9alBCQloAC2omjwYkCRzzmdTixduhSvvPIK4uLiPF5rNpthMplgMplgtVpVqlD/pkyRu/oA4I03OJqPBicGFPncpk2bkJKSgnvvvbfPa3NycmCxWGCxWJCUlKRCddpTAtkTSQJycuT9774DDhxQoTAinWFAkU+VlZXh3/7t37B582atS9Gt0NBQhIaG9nnd448DYWHy/u9+5+eiiHSIAUU+dfDgQVitVmRmZsJoNCI7OxuAvIrsli1bNK5OHyorK1FZWdnndUlJwM9+Ju9/+CHgZoUOogFNEp47t9nzTf1y6dIlXHSZhvvs2bO4+eab8c0332DChAluV5JVmEwmWAbBDKl5eXkAgGnTpvV57eHD8v0oAJg/H/jjH/1ZGZFmJHcH2YIin4qMjMTw4cM7tpSUFABASkqKx3Ai966/Hnj4YXn//feBgwe1rYdITQwo8qv09HQIITB8+HCtSwlYv/oVEB4u7//850BLi7b1EKmFAUWkcyNGAP/4j/J+cTGwcqW29RCphQFFFADWrAFuvlne37wZeOcdbeshUgMHSZBuDJZBEg6HAwAQHBzcr687fVoeMFFfDwwZAuzZA9x2mz8qJFIdB0kQ6UFwcHC/wwkARo2Sh5sPGQK0twP33AP8+c9+KJBIJxhQRCrra7kNT265BdixAwgKAi5dkpfm2LqVUyHRwMSAIlJZXwsW9uWBB+TnoUJD5ZZUTg6weDFQV+e7Gon0gAFFFIDmzQM+/RS4/JgZ3noLyMwE3nuPrSkaOBhQRAFq+nTg0CHg9tvl9+fOAfffLx/ft49BRYGPAUUUwK65Rh4o8cYbQGKifCwvTw6tG26QW1Z8sJcCFQOKKMBJEvDYY0BZmfwQr7KI8bffAosWySH2+ONyq6q9XdtaifqDz0GRbgyW56D87cIF+WHe118Hqqu7nouLA+64Q95uuw1IS9OmRqJu3D4HxYAi3WBA+VZbG/DRR8D27cDeve5bT+npwLRpwI03yl2CkycDUVGql0rEgCJ9GywBdfLywk4ZGRmq/Zu1tcDu3XJg/eUvvQ9JlyRg7FjguuuAa68FJkwAxo8HxowBYmNVK5cGHwYU6dtgCaj+rAflDw6HfH/q88/l5Tvy8uRuwb4kJsqzWaSnAyNHyt2Dw4fL2zXXyEPehwzxe/k0MLkNqBC1qyAibQUHAz/6kbytWCEPR6+oACwWeYHE774DjhyR5/5z/fu1pkbePP0NYTQCycnylpTUuSUmyltCAhAf37nFxgIh/L8Q9YI/GkSDnCTJraL0dGDBgs7jLS1AaSlw/Li83PzJk3JoVVQAZ87I97i6u3BB3oqLvf/3DQY5qJQtOrrnZjDIW1RU52tkpLwp+xER8hYZKbfkJLd/k1MgYUARkVsREfKgicmTe55zOgGrVX44+Nw5oKpK3qqrO7fz5+Wwqq/3/O80NcnbuXO+qz0oqDOwwsN7bmFhna/dt9DQzld325AhPV9720JCer533YKDGaSeMKDIp1atWoXdu3ejsrISBoMBP/nJT7B+/XokJCRoXZpuXMlM5noTFNTZlZed7fna9nbg4kW5e/DiRXnAhvJaXy8P2Kiv79waG3tuTmf/6nM6geZmedM717ByF2Cu+8rm7pg3W1BQ5/7KlcC4cVp/es8YUORTwcHBePvtt5GZmYm6ujo8+uijWLJkCXbt2qV1abpx4403al2CqoYM6QyzKyEE0NoqB5USOsrW0tLz1XWz2Tpfla21VT7W2up+a2uTX/sbilfKbpc3tS1axIDqoIxccpWamor09HQ4HA7k5+f3OD9ixAiMGDECbW1tbkd3paenIzU1FS0tLSgoKOhxPiMjA8nJyWhqakJRUVGP82PHjkVSUhIaGhpw5MiRHucnTJiAhIQEXLx4ESUlJT3OZ2ZmIiYmBlarFaWlpT3OT548GQaDAdXV1R1Di11NmTIFERERqKqqcrv8gslkQmhoKCorK1FZWdnj/I033ojg4GCUl5e7nR1bGSV28uRJVHd7YjM4OLjjf5QnTpzAhW7DuEJDQ2EymQAAx44dQ21tbZfz4eHhyL78p3NxcTHqL/fj3H333WhpacHRo0eRlZWFp59+Gg888ECP//6xsbGYNGkSAODQoUOw2WxoamrquC4+Ph4TJ04EAFgsFrR1u+FhNBox7vJvV35+fscigIrk5OSOYdz82Rt4P3s/+lHPnz1FVFQUsrKyAACFhYVo7taMcvez50r52XM4gC+//BbNzXbY7RLs9iC0t0uIjk7AsGGj0NYGFBQcQVub6Dhvt0uIjo5DYuI1aG8Hjh0rvXxOgsMhvxoMsYiOTkBbmxMVFec6zjmdEhwOIDIyGmFh0Whrc6C6uubyOcDplL8+PDwSISERaGtzoK6usct5h0PCkCHhCAoagvZ2B5qbW7ucczqB4OAhECIYdnsT8vJ6/mx6+7OnhoBoQRUVSfjjH1MQHCwgSUBQkEBQEJCUFIHERMBuD0ZlpRGSJB+XJIHgYKCsLAyJiUBrazAqKmIv9/eKjte6uhAYjcClS0EoL4+6/H3F5aawQHh40OUuBgkXLgy53DQWHdfZbHIft1p/aQWiTz/9FGPHju31vNlsxsaNG+F0OlE3SNaLOHPmDIKCgpCUlKR1KeRBcDAQHi4QFNT1jx+jsb2j5dHe3uzmj6NQKI+45eVZe3zf1FQJ6ekJcDgE8vN7hr/8x5EcUBZLWY/z8h9HEWhpaUNBwdEe5zv/OGrp448jJ9z8baQrAfEc1MaNwLPPal2FZ5LkfV+ycuPU3Wv3Y66buxu03fddt+43fZXN9Saxsvnjtsj777+PxYsX4/PPP+9obXnC56CIBq3AfQ4qEFooQsg3gwN1Ms6QkM6wch39pAzfVYb0KsN6laG9yrBfg6HrcOAvv9yLf/qnl/DOOx/j+uv7Diciou4CIqCefBJ45BE5qByOzlchOt93P6Zsru+7f333471tdrt8jd3e9Vj3axwOOaCUY8p+91e7Xd5XXpVzyqYc97S5ewblatjtncN9fWMOgDm45x75XXS0/IxLXJy8KQ9qJiTIW2KiPKrrs8/kBzuHDpWP8SFOosErIH79lb/oqZMSvG1tcmC5jj7q/r77vjKSqfuoJtcRT8ooqEuXOvddR0spx72lDBc+e9bzdbfd1rkvSfLMBCkpQGqqvA0bJk+tk5Ymb+npnNyUaKAKiICinlzveWnF4egc7tvUJL9OmTIDQUFxGDIkAUIYAMRAiFjk5q7ueN6ltlZ+9kV5Fqa3VpsQ8sOgVqs8/U5vhg4FMjLkCU3HjQMmTpQnOh0zRp9zw4WGhmpdAlFACIhBEjSwtbXJD3HOmmXC5s0WWK3yLATKjAQ//NA5U8EPP3i/lHloqBxUU6YAU6fKS0pkZekztIgGOc5mTvrmzSi+9nbg++/lueAqKoDycuDUKXmeuNJSOcQ8iYgAbr4ZmDkTmD0bMJn8M4KRiPqFAUX65oth5o2NQEmJPFnpd9/Js3MfOtT7+kdGI/CTnwD33Qfcfrvc6vK3Y8eOAUDHg8hExIAinfPXc1BCAGVlwFdfAV98Ia+D5OYBeSQkAA8/DOTkAJcnGvALPgdF1IPbgApSuwoitSmrxD7yCPD668CxY0BlJfDv/w787d/Kz3sB8qCN3/4WyMwE5swB9u/XtGyiQY8BRYPS8OHA0qXAhx/KAzLefhuYNavz/J//DNxyi9zt52aqPSJSAQOKBr3oaOChh4C//EXuCnzmGXmWDADYt08eSPHss758iJmIvMGAInKRkSF38506JQdVSIg8i8jGjXJQeXoey1vh4eEIV/oViahXHCRBuqHHyWKLi4EnnpAHVwDy/aodO7oujU5EV42DJIj6a9IkedTfP/+zvBqpzQbcfz/w2mtaV0Y08DGgiPoQHAy88AKwZ488758QQG6uPArwShQXF6O4uNi3RRINQAwoIi/Nni0PPU9IkN8/8QSwd2//v099fX2PVWCJqCcGFFE/mEzARx/J96IcDrm7z82K6UTkAwwoon6aNk1+bgqQp1ZavDgwFtUkCjQMKKIrMH++3MUHyIMoNm7Uth6igYgBRXSFNmwARo+W9198Ebhwwbuvi4qKQhRXWSTqEwOK6AoZDMDvfy/vNzUB69d793VZWVnIysryX2FEAwQDiugq3HEHMH26vL95s7xWFRH5BgOKfM7hcGDFihVISkpCdHQ05s+fjwve9n8FGEkCfvlLed9mA15+ue+vKSwsRGFhoX8LIxoAGFDkc7/+9a+xa9cu5Ofn4+zZswCARx55ROOq/Odv/kZeoReQp0Fqa/N8fXNzM5qbm/1eF1GgY0CRz5nNZqxatQqjR49GbGwsXn75ZezduxflA/iBoWXL5NeaGuDjj7WthWig8DhZ7Jw5c4SeumasViuSkpK0LkM1gfh5HQ4HDh8+jIkTJyJSWbMCQEFBAUaNGoW4uLgu11ut1o7uv9bWVlx//fWq1usrTidQWCi/xsYCY8b0fm1TUxPq6uowfPhw9QrUWCD+LF8Nft7++fbbb/8shJjT44QQwtOmK1OnTtW6BFUF4uc9c+aMACBOnTrV5XhaWprYsWOHx6+NjIz0Z2l+99hjQgBChIQIcf5879d98cUXYvz48eoVpgOB+LN8Nfh5+81tBrGLj3wqOjoaAHrMNVdXV4eYmBgtSlLNokXyq90OvPde79fFxsYiODhYnaKIAhgDinwqLi4OaWlpOHToUMexU6dOoaGhAZMnT9awMv+bPh1ITpb3P/us9+smTZqEiIgIdYoiCmABFVA5OTlal6CqQP28OTk5WL9+PU6fPo2GhgasWrUKd9xxB9LT0z1+ndFoVKdAP5GkztF8n3/ueX6+QP1ve6X4eQc2f31erqhLPudwOLBq1Sq8+eabaG1txe233w6z2dxnAOlxRd3+ev114Mkn5f3vvgMyM3teo7Qus7OzVayMSNe4oi6pIzg4GBs2bMCFCxfQ2NiIDz74IOBbR95SWlCAvHaUOzabDTabTY1yiAIaA4rIh8aPB1JS5P3eAoqIvMOAIvKh/tyHIiLPAjagdu7cicmTJyMqKgopKSn4pTIh2gCzdOlSjBgxAjExMbjmmmuwdOlS1NbWal2Wz7jO21dQUDAg5u1TAurCBeDo0a7n1q5di/nz5+P222/H0KFDsWDBApw5c0b1GtW0b98+3HTTTTAYDDAajcjNzdW6JL+pqanBokWLkJKSgtjYWDz44IMD5vd1586dmDFjBmJiYhASEtLl3CeffIJbb70VRqMR8fHxmDFjBg4cOHDV/2ZABtSOHTvw7LPP4tVXX0V9fT1KS0txzz33aF2WXzz33HMoKSlBQ0MDjh07hkuXLuGpp57SuiyfcZ23TxmGHujz9t10U+f+d991PffII49g165dsFgsKC8vR1paGn72s5+pW6CK9u/fjwULFuD5559HTU0Nzp49i2XKvFAD0KOPPoqmpiaUlpbi9OnTqKmpCfifZ0V8fDxyc3Ox0c3qnLW1tXjmmWdQVlYGq9WKBx98EHfeeScqKyuv7h/t7QleocOZJIQQwuFwiNTUVPHaa69pXYrqamtrxcKFC0V2drbWpfhMWlqaeOONN4QQ8tPoZWVlAoA4ffq0toVdheZmeUYJQIgXX+z9uqamJrF8+XKRkJCgWm1qu+mmm8SqVau0LkMVTU1NQpIkcfjw4Y5j+/fvFwBEeXm5hpX51l//+lcRHBzc53VGo1F88MEH3n7bgTGTxIkTJ1BVVYWmpiZMmDABQ4cOxd13342ysjKtS/ObX//614iOjkZ8fDz+9Kc/Ye3atVqX5BP19fU4c+YMpk6d2nEsIyMDMTExKCoq0rCyqxMZCaSlyfvHj/c8/8477yA2NhYGgwGbNm3CSy+9pGp9amlubsbXX3+N8PBwZGdnw2g0YubMmQH/KEFvXP/HqnBevgk52JZXKSoqQk1NDTLdPWfRD7oKqMWLF0OSpF63devWddyf2L59O/bs2dPRTTJ37lzY7XaNP4H3vPmsitWrV6OxsRGnTp3C8uXLMcbTTKQBpKGhAYA89Y+ruLi4jnOBavx4+dVdQI0bNw6ffvopvv/+ezDRyZ8AAA9/SURBVLz00ku47rrr1C1OJbW1tXA6ndi6dSvefPNNVFVVYfbs2bjrrrtQV1endXk+ZzAYMHPmTLz00kuoq6uD1WrFr371KwAI+J/n/jh//jwWLFiAlStXYuzYsVf3zXprWgkNuvgaGxuF1WrtdWtubhaHDx8WAMTWrVs7vq62tlYAEMXFxWqXfMW8+azu5Ofni2HDhgmHw6Fyxb6n/HcrKCgQQnROOBkTEyN27dqlZWlX7emn5S6+qCghnM6u57744gvxxRdfCCGEqK6uFlFRUaKmpkaDKv2rrq5OABBr167tOOZ0OkVcXJz4+OOPNazMf86ePSvuv/9+kZKSItLT08Xvfvc7AUDs2bNH69J8xlMX37lz58S1114rnnrqKeHs/oPvmdsMCvGYXiozGAwwGAwerxk/fjwiIiIgST0fPHZ3TK+8+azu2O12nDt3Ds3NzR0TswYq13n7lGU2Bsq8fUoLqrkZOHcO6G1lDbvdjubmZlRVVSEhIUG9AlUQGxuL9PT0gP9d7Y9hw4bhv/7rvzref/zxxwgPD8dNriNnBqjy8nLcdtttmDdvHjZs2OCT76mrLj5vhIeHY8mSJdi0aRMqKyvR2tqKF154AZMmTcK4ceO0Ls+nzp8/j7feequjO+TEiRNYuXIlpk+fHvDhpHCdt0+ZIsmbefv0TgkooLObz+l0YvPmzR3Djs+ePYunnnoK6enpmDBhggZV+l9ubi62bduGo0ePwm6345VXXkF4eDimTZumdWl+cfz4cVy8eBFOpxPffPMN/uEf/gGrV6/usQ5aIHI4HLDZbGi7vGS0MiOKEAIlJSWYPn06Fi5c6LNwAqCvLj5v2Ww28eSTT4r4+HiRmJgo5s6d22P9oYHg/Pnz4pZbbhHx8fEiMjJSpKWliSeeeEL88MMPWpfmM3a7XSxfvlwkJiaKoKAgMW/ePGG1WrUu66pVVHSO5Pv97+VjDodD3HnnnSIuLk6EhYWJ1NRU8eCDD4qysjJti/Ujp9MpXnjhBZGcnCxiY2PFzJkzO7p0ByKz2SxSUlJERESEGDNmjNi4caPWJfnMtm3bBOT5Wbtsp0+fFosXLxYARFRUVJft7bff9vbbu80gThZLujEQJotVOJ2AwQC0tAB/93fApk2d506cOAEAA67FT3QV3Pb56uoeFNFAERQEjBsnLwPffSQfg4nIOwF3D4ooUHgaak5EfWNAEfmJElAVFUBra+fx/Px85Ofna1MUUQBhQBH5ybBh8qsQ8sSxCofDAYfDoU1RRAGEAUXkJ65rNAb4BO1EmmBAkc+0trbiiSeewNixYxEdHY20tDSsWLFi0K4ey4AiujoMKPIZu90Oo9GIjz76CHV1dThw4AA+++wzrFq1SuvSNMGAIro6HGZOPhMVFdVl4ciRI0di6dKlMJvNGlalnd4CKjk5Wf1iiAIQA4r86tNPP/U4r57ZbO4IMKvVqlZZqkhM7Nx3DaiMjAz1iyEKQOziI6/0Z3kQxcaNG3Hw4MEurarucnJyYLFYYLFYkJSU5M+PoLqQECA+Xt5nFx9R/7EFRV7ZvHmzx0kgIyMju7z/zW9+g/Xr1+Ozzz5DmrJ63yBkNAK1tV0DKi8vDwAG7ISpRL7CgCKv9Gd5kF/84hf4wx/+gM8//xzjXaf1HoSMRqC0lC0ooivBgCKfWrFiBf77v/8bn3/+Oe+1oHOgxAC7vUakCgYU+UxFRQU2bNiA0NBQZGVldRwfOXIkiouLNaxMO0pAsQVF1H8MKPKZkSNHoo/lWwYd14ASAhigC8kS+QUDisiPlIBqbZWXfzcYgNTUVG2LIgoQDCgiP+r+sK7BgIBfzp5ILXwOisiP3M0mwdnMibzDgCLyI3cBxfWgiLzDgCLyI04YS3TlGFBEfsSAIrpyDCgiP4qLA4Iu/5YxoIj6hwFF5EdBQZ2zmjOgiPqHw8yJ/MxolKc6UgJqxIgR2hZEFCAYUER+1n26IwYUkXfYxUfkZ90Dqq2tDW1tbdoVRBQgGFBEfqYsldXSIr8qCzQSkWcMKCI/CwuTX1tbta2DKNAwoIj8jAFFdGUYUER+xoAiujIMKCI/Y0ARXRkOMyfys/Bw+bWtTV60kMttEHmHLSjyi+bmZmRkZCAkhH8DKS0oQG5FpaamctFCIi8woMgvVq9ejVGjRmldhi50D6iWlha0KGPOiahXDCjyuf/7v//DgQMHsGrVKq1L0YXuAVVQUICCggLtCiIKEOx/IZ+6dOkSHn/8cbz99ttobm7Wuhxd6B5QROQdtqDIK4sXL4YkSb1u69atAwCsWbMGc+fOxQ033ODV9zWbzTCZTDCZTLBarf78CJphQBFdGbagyCubN2/Ghg0bej0fGRmJgwcP4pNPPkFhYaHX3zcnJwc5OTkAAJPJdNV16hEDiujKMKDIKwaDAQaDweM1+/btw9mzZ5GWlgYAaG9vh8PhgNFoxLZt2zB37lw1StUdZZg5ANhs2tVBFGgYUOQzzz33HJYtW9bx/ssvv8TChQtx+PBhJCqr9g1C3VtQGRkZ2hVDFEAYUOQzMTExiImJ6XiflJQEABg+fLhWJelC94BKTk7WrhiiAMJBEuQ3M2fOhN1u17oMzXUPqKamJjQ1NWlXEFGAYEAR+Vn3gCoqKkJRUZF2BREFCAYUkZ9xFB/RlWFAEfmZ6yg+BhSR9xhQRH7m2oLiMHMi7zGgiPyMXXxEV4bDzIn8rHtAjR07VrtiiAIIA4rIz7oHlPJ8GBF5xi4+Ij8LDe3cb20FGhoa0NDQoF1BRAGCAUXkZ5LUGVKtrcCRI0dw5MgRbYsiCgAMKCIVKEPNOYqPyHsMKCIVKPehOIqPyHsMKCIVMKCI+o8BRaQCBhRR/3GYOZEKXANqwoQJ2hZDFCAYUEQqcA2ohIQEbYshChDs4iNSgRJQNhtw8eJFXLx4UduCiAIAA4pIBcow89ZWoKSkBCUlJdoWRBQAGFDkczt37sTkyZMRFRWFlJQU/PKXv9S6JM1xkARR//EeFPnUjh07sHLlSuzYsQMzZ85ES0sLysvLtS5Lcwwoov5jQJHPOJ1OrF69Gi+++CJmzZoFAIiOjsZ1112ncWXaY0AR9R+7+MhnTpw4gaqqKjQ1NWHChAkYOnQo7r77bpSVlWldmuYYUET9x4AiryxevBiSJPW6rVu3DhcuXAAAbN++HXv27EF5eTnS0tIwd+5c2O12t9/XbDbDZDLBZDLBarWq+ZFU5TqKLzMzE5mZmdoWRBQAJCGEp/MeT9Lg0dTUBJuHmU4jIyNRWlqK66+/Hlu3bsWyZcsAAHV1dYiPj0dxcTGuvfZaj/+GyWSCxWLxad168fTTwO9/DyQkADU1WldDpDuSu4O8B0VeMRgMMBgMHq8ZP348IiIiIEk9f9bcHRtMXLv4lJYiFy4k8oxdfOQz4eHhWLJkCTZt2oTKykq0trbihRdewKRJkzBu3Dity9OUa0CVlpaitLRU24KIAgADinzq1VdfxfTp05GVlYVhw4ahoqICH330EYKDg7UuTVNKQNntgNOpbS1EgYJdfORTYWFh2LJlC7Zs2aJ1KbqiBBQAtLcHISyMKUXUF7agiFTgGlBtbYP7fhyRtxhQRCroGlD8tSPyBrv4iFSgTBYLABkZ1yItjU9wEPWFAUWkAtcWVEhIFPoYsU9EYBcfkSpcA6qqqgbV1dXaFUMUINiCIlKBa0CdPHkWoaHNSE5O1q4gogDAFhSRCroPMyeivvE3hUgFHMVH1H/8TSFSAZ+DIuo/BhSRClyHmbOLj8g7HCRBpALXFtSwYaMxZcpI7YohChAMKCIVuAaUEKGIiNCuFqJAwb4GIhW4BlR1dR2qqqq0K4YoQDCgiFTgGlDff38R5eXlmtVCFCgYUEQq4DBzov7jbwqRCjiKj6j/+JtCpIKQEEC6/PgTn4Mi8g4Dinzq+PHjuOuuu5CYmAij0Yif/vSnvN8COZyUbj62oIi8w98U8qmFCxfCaDSisrISFRUViI6OxkMPPaR1WbqgBFR8fApMJpO2xRAFAAYU+VRZWRkefvhhREZGIioqCo888ggKCwu1LksXlICy24MRGhqqbTFEAYABRT61evVqvPXWW2hsbERDQwPefPNNzJs3T+uydEEJqJqaZlRWVmpbDFEAYECRVxYvXgxJknrd1q1bBwCYM2cOSkpKEBcXh7i4OBw7dgwbNmzo9fuazWaYTCaYTCZYrVa1Po4mwsPlkGppaWJAEXlBEkJ4Ou/xJA0eTU1NsNlsvZ6PjIxEa2srRo0aheeffx7Lly+HEAIvv/wy3nnnHRQVFSHcday1GyaTCRaLxdel64YQ8mCJvLw8AMC0adM0rohIN9wObWVAkc9YLBbccMMNuHTpEiIuTzbX2NiImJgYHD58GFlZWR6/fqAHlIIBRdSD24BiFx/5zIQJE5CQkIBNmzahra0Nra2tePXVVxETE4OMjAytyyOiAMOAIp8xGAzYvXs39u7di5SUFKSkpGDfvn3YvXs3DAaD1uURUYBhFx/pxmDp4nM4HACA4OBgjSsh0g23XXxcD4pIZQwmIu+wi49IZeXl5Zz+icgLDCgilVVVVXHBQiIvMKCIiEiXGFBERKRLDCgiItIlBhQREekSh5kTqYxTHBF5hy0oIiLSJQYUERHpEgOKiIh0iQFFRES6xIAiIiJdYkAREZEuMaCIiEiXGFBERKRLDCgiItIlBhR57be//S1uvPFGREZGYsyYMW6veeutt5CRkYHIyEjceOON+Pbbb1WukogGCgYUeS01NRUrV67E2rVr3Z4/ePAgnnzySbz22muora3F/Pnzcdddd6GhoUHlSoloIGBAkdcWLFiA+fPnY9iwYW7Pb926Fffeey9mz56NsLAwrFixAmFhYfjwww9VrpSIBgIGFPlMYWEhpk6d2vFekiRMmTIFhYWFGlZFRIGKAUVYvHgxJEnqdVu3bp1X36exsRGxsbFdjsXFxXns4jObzTCZTDCZTLBarVf1OYhoYJGEEFrXQBqTJMkAINzDJZeEEJdcrl8MYJ0QostICUmSDgN4Uwix0eXYLgAnhRDPeVHHXiHEnP7WT0QDE9eDIgghmgA0+eBbFQLIVt5IkiQBuB7AB17WwXAiog7s4iOvSZIUIklSOIAh8lsp/PJ7xVYA90qSdJskSaEAlkNumXGUBBH1G1tQ1B/rALzo8r7l8qsEAEKIg5Ik5UIOqmsAfAfgLiEEx5kTUb/xHhQREekSu/iIiEiXGFBERKRLDCgiItIlBhQREekSA4qIiHSJAUVERLrEgCIiIl1iQBERkS4xoIiISJf+P2H1X8/qcdN9AAAAAElFTkSuQmCC",
+      "text/plain": [
+       "<Figure size 432x360 with 1 Axes>"
+      ]
+     },
+     "metadata": {
+      "needs_background": "light"
+     },
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "import matplotlib.ticker as plticker\n",
+    "\n",
+    "# Create figure and add axes\n",
+    "fig = plt.figure(figsize=(6, 5))\n",
+    "ax = fig.add_subplot(111)\n",
+    "\n",
+    "\n",
+    "# Move left y-axis and bottim x-axis to centre, passing through (0,0)\n",
+    "ax.spines['left'].set_position('zero')\n",
+    "ax.spines['bottom'].set_position('zero')\n",
+    "\n",
+    "# Eliminate upper and right axes\n",
+    "ax.spines['right'].set_color('none')\n",
+    "ax.spines['top'].set_color('none')\n",
+    "\n",
+    "# Show ticks in the left and lower axes only\n",
+    "ax.xaxis.set_ticks_position('bottom')\n",
+    "ax.yaxis.set_ticks_position('left')\n",
+    "\n",
+    "\n",
+    "loc = plticker.MultipleLocator(base=3.0) # this locator puts ticks at regular intervals\n",
+    "ax.xaxis.set_major_locator(loc)\n",
+    "loc = plticker.MultipleLocator(base=2.0) # this locator puts ticks at regular intervals\n",
+    "ax.yaxis.set_major_locator(loc)\n",
+    "\n",
+    "x = np.linspace(3.1, 11, 100)\n",
+    "y = 2 + 1/(x-3)\n",
+    "ax.plot(x, y, color='b', linewidth=2.5)\n",
+    "\n",
+    "x = np.linspace(-6, 2.9, 100)\n",
+    "y = 2 + 1/(x-3)\n",
+    "ax.plot(x, y, color='b', linewidth=2.5)\n",
+    "\n",
+    "ylim = ax.get_ylim()\n",
+    "plt.vlines(3, ylim[0], ylim[1], ls='dashed',alpha=0.3)\n",
+    "\n",
+    "xlim = ax.get_xlim()\n",
+    "plt.hlines(2, xlim[0], xlim[1], ls='dashed',alpha=0.3)\n",
+    "\n",
+    "ax.tick_params(axis='both', which='major', labelsize=13)\n",
+    "    \n",
+    "# Add legend\n",
+    "\n",
+    "#ax.legend(bbox_to_anchor=(1.05, 0.3),loc='lower left', \n",
+    "#          frameon=False, labelspacing=0.2, fontsize=13)\n",
+    "\n",
+    "plt.tight_layout()\n",
+    "plt.savefig('rational_01.png')"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 66,
+   "metadata": {
+    "tags": [
+     "remove-cell"
+    ]
+   },
+   "outputs": [
+    {
+     "data": {
+      "image/png": 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",
+      "text/plain": [
+       "<Figure size 432x360 with 1 Axes>"
+      ]
+     },
+     "metadata": {
+      "needs_background": "light"
+     },
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "import matplotlib as mpl\n",
+    "import matplotlib.pyplot as plt\n",
+    "import numpy as np\n",
+    "\n",
+    "\n",
+    "# Create figure and add axes\n",
+    "fig = plt.figure(figsize=(6, 5))\n",
+    "ax = fig.add_subplot(111)\n",
+    "\n",
+    "alphaset = [-2.5,-2,-1,-0.5, 0.0, 0.5, 1.0, 2.0, 2.5]\n",
+    "colors = plt.get_cmap('coolwarm', 9)\n",
+    "\n",
+    "labels = ['-5/2','-2','-1','-1/2','0','1/2','1','2','5/2']\n",
+    "\n",
+    "plt.ylim(top=10)\n",
+    "\n",
+    "ax.spines['right'].set_color('none')\n",
+    "ax.spines['top'].set_color('none')\n",
+    "\n",
+    "for i in range(len(alphaset)):\n",
+    "    \n",
+    "    x = np.linspace(0.1, 2, 100)\n",
+    "    y = x**alphaset[i]\n",
+    "    ax.plot(x, y, color=colors(i), linewidth=2.5, label=labels[i])\n",
+    "    \n",
+    "ax.tick_params(axis='both', which='major', labelsize=13)\n",
+    "\n",
+    "    \n",
+    "# Add legend\n",
+    "\n",
+    "leg = ax.legend(bbox_to_anchor=(1.05, 0.3),loc='lower left', \n",
+    "          frameon=True, labelspacing=0.2, fontsize=13,)\n",
+    "leg.set_title(r\"${\\alpha}$\", prop = {'size':'x-large'})\n",
+    "\n",
+    "plt.tight_layout()\n",
+    "plt.savefig('powerlaw_01.png')"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 65,
+   "metadata": {
+    "tags": [
+     "remove-cell"
+    ]
+   },
+   "outputs": [
+    {
+     "data": {
+      "image/png": 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",
+      "text/plain": [
+       "<Figure size 432x360 with 1 Axes>"
+      ]
+     },
+     "metadata": {
+      "needs_background": "light"
+     },
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "import matplotlib as mpl\n",
+    "import matplotlib.pyplot as plt\n",
+    "import numpy as np\n",
+    "import matplotlib.ticker as plticker\n",
+    "\n",
+    "\n",
+    "# Create figure and add axes\n",
+    "fig = plt.figure(figsize=(6, 5))\n",
+    "ax = fig.add_subplot(111)\n",
+    "\n",
+    "alphaset = [-5,-2,-1,1, 2, 5]\n",
+    "colors = plt.get_cmap('coolwarm', 6)\n",
+    "\n",
+    "temp1 = colors(1)\n",
+    "temp0 = colors(0)\n",
+    "\n",
+    "labels = ['-5','-2','-1','1','2','5']\n",
+    "\n",
+    "plt.ylim(top=4)\n",
+    "\n",
+    "# Move left y-axis and bottim x-axis to centre, passing through (0,0)\n",
+    "ax.spines['left'].set_position('zero')\n",
+    "ax.spines['bottom'].set_position('zero')\n",
+    "\n",
+    "# Eliminate upper and right axes\n",
+    "ax.spines['right'].set_color('none')\n",
+    "ax.spines['top'].set_color('none')\n",
+    "\n",
+    "# Show ticks in the left and lower axes only\n",
+    "ax.xaxis.set_ticks_position('bottom')\n",
+    "ax.yaxis.set_ticks_position('left')\n",
+    "\n",
+    "loc = plticker.MultipleLocator(base=1.0) # this locator puts ticks at regular intervals\n",
+    "ax.yaxis.set_major_locator(loc)\n",
+    "\n",
+    "for i in range(len(alphaset)):\n",
+    "    \n",
+    "    x = np.linspace(-2, 2, 100)\n",
+    "    y = np.exp(alphaset[i]*x)\n",
+    "    #if i==0: ax.plot(x, y, color=temp1, linewidth=2.5, label=labels[i])\n",
+    "    #if i==1: ax.plot(x, y, color=temp0, linewidth=2.5, label=labels[i])\n",
+    "    #if i==2 or i==3: ax.plot(x, y, color=colors(i), linewidth=2.5, label=labels[i])\n",
+    "    ax.plot(x, y, color=colors(i), linewidth=2.5, label=labels[i])\n",
+    "    \n",
+    "ax.tick_params(axis='both', which='major', labelsize=13)\n",
+    "\n",
+    "    \n",
+    "# Add legend\n",
+    "\n",
+    "leg = ax.legend(bbox_to_anchor=(1.05, 0.3),loc='lower left', \n",
+    "          frameon=True, labelspacing=0.2, fontsize=13,)\n",
+    "leg.set_title(r\"${\\lambda}$\", prop = {'size':'x-large'})\n",
+    "\n",
+    "plt.tight_layout()\n",
+    "plt.savefig('exp_lamb_01.png')"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 55,
+   "metadata": {
+    "tags": [
+     "remove-cell"
+    ]
+   },
+   "outputs": [
+    {
+     "data": {
+      "image/png": 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",
+      "text/plain": [
+       "<Figure size 432x324 with 1 Axes>"
+      ]
+     },
+     "metadata": {
+      "needs_background": "light"
+     },
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "import matplotlib as mpl\n",
+    "import matplotlib.pyplot as plt\n",
+    "import numpy as np\n",
+    "from math import log\n",
+    "import matplotlib.ticker as plticker\n",
+    "\n",
+    "\n",
+    "# Create figure and add axes\n",
+    "fig = plt.figure(figsize=(6, 4.5))\n",
+    "ax = fig.add_subplot(111)\n",
+    "\n",
+    "alphaset = [0.1,0.2,0.5,2,5,10]\n",
+    "colors = plt.get_cmap('coolwarm', 6)\n",
+    "\n",
+    "\n",
+    "labels = ['1/10','1/5','1/2','2','5','10']\n",
+    "\n",
+    "#plt.ylim(top=4)\n",
+    "\n",
+    "# Move left y-axis and bottim x-axis to centre, passing through (0,0)\n",
+    "ax.spines['left'].set_position('zero')\n",
+    "ax.spines['bottom'].set_position('zero')\n",
+    "\n",
+    "# Eliminate upper and right axes\n",
+    "ax.spines['right'].set_color('none')\n",
+    "ax.spines['top'].set_color('none')\n",
+    "\n",
+    "# Show ticks in the left and lower axes only\n",
+    "ax.xaxis.set_ticks_position('bottom')\n",
+    "ax.yaxis.set_ticks_position('left')\n",
+    "\n",
+    "#loc = plticker.MultipleLocator(base=1.0) # this locator puts ticks at regular intervals\n",
+    "#ax.yaxis.set_major_locator(loc)\n",
+    "\n",
+    "\n",
+    "for i in range(len(alphaset)):\n",
+    "    \n",
+    "    x = np.linspace(0.1, 10, 100)\n",
+    "    y = np.log(x) / np.log(alphaset[i])\n",
+    "    #if i==0: ax.plot(x, y, color=temp1, linewidth=2.5, label=labels[i])\n",
+    "    #if i==1: ax.plot(x, y, color=temp0, linewidth=2.5, label=labels[i])\n",
+    "    #if i==2 or i==3: ax.plot(x, y, color=colors(i), linewidth=2.5, label=labels[i])\n",
+    "    ax.plot(x, y, color=colors(i), linewidth=2.5, label=labels[i])\n",
+    "    \n",
+    "ax.tick_params(axis='both', which='major', labelsize=13)\n",
+    "\n",
+    "    \n",
+    "# Add legend\n",
+    "\n",
+    "leg = ax.legend(bbox_to_anchor=(1.05, 0.3),loc='lower left', \n",
+    "          frameon=True, labelspacing=0.2, fontsize=13,)\n",
+    "leg.set_title(r\"$a$\", prop = {'size':'x-large'})\n",
+    "\n",
+    "plt.tight_layout()\n",
+    "plt.savefig('log_base_01.png')"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 53,
+   "metadata": {
+    "tags": [
+     "remove-cell"
+    ]
+   },
+   "outputs": [
+    {
+     "data": {
+      "image/png": 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sRzqi1TrP32YLfzjOv78+wrWJ4bx2R3e8tM7VVSlcj4STcHpms8K8V49QVFIHwMNT25IQ66tyVdbz6ndHefW7Y4zqEcXLt3ZF6+48oStcl4STcHoffpbDtp2WozAGXR7KiKHhKldkXX0SQpg4IJ7Z13d26q2XhGuRcBJObf/hMt5+Nx2AiFbezLjfOU55NZrMbE0r4vJ2ofRv05L+bVqqXZIQViXtf+G0SkrrmP3iQYxGBa1Ww5wZnfDzdfy/x+qMZv7xwW7Gv7ODg7nlapcjRLNw/N9UIc7DaFJ49t+HKCiyjDM9eE8bp5g2rjeYmL5qJ5uPFDDr+k50jnT8z0mI85FwEk5pycp0du0tBWDoVWGMGh6pckVNV1lrZPKK39iRUcyLo7pwe59YtUsSotlIOAmns+mnfN5fZ1nP1Cbe12nGmTYdOs3OzBJeHdOdG7tHqV2OEM1Kwkk4lcPHKpj36hEA/HzdmftkZ3QOvj2RoihoNBpu7B5F1+ggElo6zzR4IS5EJkQIp1FYVMsTc/dTV2fGzQ3+ObMzMZGOvRP3ydIabnjjF/ZkW7ooJZiEq5CWk3AKtbUmnnzhAIXFZyZATG5Dn54hKlfVNBmFVdy5dDvlegMms2xHJFyLhJNweCaTwnP/OcyhoxUAjBwWwa0jHXtM5kheBePe2Y7RZGbNlH4kRTnfeVNC/B0JJ+HQFEXh9XdO8OPWQgB6dg3ikWltHXoCRHphFWMWb8XT3Y0Pp11Gu1b+apckhM1JOAmHtvbTHD7+/CQACbE+zHsyEQ8HP249OljHDd0imTwwgbgWMsYkXJOEk3BY32w+zcJlaQC0DPFk/pwu+Ps57o/01hNFtAnzJczfm3/emKR2OUKoyrH/xBQu65cdRcx7xXIEhk7nzr+f7UKrUMc9ivzrA3ncvWwH8748pHYpQtgFCSfhcHbvK2X2SwcxmcHTQ8NLs5No19pP7bIabcPuk9z3/i4SowL45w3SYrK1qhNZapcgzsNx+0CESzp0tJzHn7esZXJ3g38+0ZmeXYLULqvRVm/P4ukN++iX0IKldyfj6yW/ks1NURTKUw9z+tNvydvwLRX7j3K94YjaZYk/kd8E4TAOH6vg/57ZS3WNCY0Gnv6/jgzs47hHRdQaTSz/JZ2rO4Tx5p098fZw7J0s7JmiKJTt3M+pdf8jb/3XVKdlq12SuAgJJ+EQDh+v4OHZe6msMgHw2H3tGHpVK5WrahxFUTCZFby07qyZ2o8Abw88nejIeHuhKAplv+0j9+OvyFv/NTWZJ//ynIDunQm/abAK1YmL0SiKYo3rWOUiwrl98MEHLFy4kNTUVKqrqzEajQ16v6QuPYlJep2KSsvzH7uvHTdd55i7jCuKwgsbD5FbqmfBHT1wl5NrrUpRFMr3HOLURxvJ/fgratJz/vKc4Mt6EH7zMMJvHoJPfPTZu+UbYWek5SRsJjg4mPvuu4+amhqmTp3aoPdJPVBK1slqguItwfTI9LYOG0wms8KsDftZsyOLCf3j5dXQiiqPppP7wRfkrv2CqqMZ5z6o0RAyoBfho4YRfvNQdNHhqtQoLo2Ek7CZYcOGAbB58+YGPX/HrmKenHcAs9nSMH/03nbc7KDnMhlMZmZ8lMqGPbncf3UbHhvawaF3sbAH+tzT5H74FblrPqNs14G/PB7UrweRo68j4tZr8Y50zC5gVybhJOzS03NWszmlFRqNFg3w9P914LprHPcv3ifW7WPDnlxmXtuB+65qq3Y5DstYWUXeJ9+Ss2oDRT9sgz8NSwT0SCRyzHAibr0OnzjH3l/R1Uk4Cbuz7ouT/LgrCo0GtFoNkeE6hw4mgLF9Y+geG8T4fnFql+JwFJOJoi07yFm5gbxPvsFUVX3O4z5tYom8fQRRt4/Ar2MblaoU1ibhJOyGoigsei+dVR9bpvnqvN2Y+2Qi9091zB/Tcr2BHw7nc2P3KHrFhdArzrGP8LC1qhNZ5Ly3npz3PkGfk3fOY54tg4kcM4LIO0YS1KerdJE6Icf8rRdOp7bOzIsLjvDtlnwAgoM8+PezXejY1jF35C6uquPuZTs4nFdOj5hgYls49qGHtmKsqiZv/Tdkv7uO4i07znnMzdODsJHXED3uJkKHXY6bh4dKVQpbkHASNmMymTAYDNTVWQ4E1Ov1AFRVa3jqhQMcOGI5jyk6Qsd/nutCVIROtVqbIr9cz51Lt5NVXM3i8ckSTA1QtucQWUvXkrvmc4zllec8FtS7K9F3jyLytuF4BMu5Vq5CwknYzMqVK5k4cWL9/3U6HT7+bRh80wqKSiyLa7slBjLvyUSCAh3zr+KckmruXLqdwopaVkzsw2VtWqhdkt0yVlaRu3YjWUvXUpay75zHPENDiBp3EzF3j8I/sZ1KFQo1ySJcoZpvNp/mpdePUltnOYJ8+KBWPHZ/ezz/dB5TcnIyKSkpapR4ydbtzOGfXxxkxcTe9IgNVrscu1Rx4BiZb6/m5PufYqyo+v0BjYbQa68gdtJowq6/ytbddjJoZWek5SRszmg08+byND78zLKdjEYD0+9OYOyoGIcd2K41mvDSunNLr2iu6RhGsK+n2iXZFbPBwOlPvyPjrdUU/3juWJJXZBixk0YTPeEWmf4t6kk4CZvKL6zlufmHSD1QBkCAv5Y5j3WiT0/Hncm2O6uEe1ft4vWxPegdHyLB9Ae1+UVkLfmAzEVrqD1VcM5jocMuJ3bq7YQNvwo3rbwUiXPJT4Swma0pRcz972HKKixbEbVr7ce8JzsTGe6YEx/AcnrtPe/+Rgs/L8IDHPewQ2srTz1M+uvvkvvBF5hr6+rv9wgJImbCKGKn3oFvm1gVKxT2TsJJNLs6g5nF76XzwYbfN+EcOTSch6a0xdvbcY+J+OFIPtNX7iQ2xIdV9/SllYuHk2I2k//lD6S9tuIv08ADuncm/v5xRI65Hneda3+dRMNIOIlmdSKjkn/+5zAnMiwD3zqdOzMc+LiLs1KzS5n6Xgodwv15b1JfQly4K8+kr+Xk+5+R9uoyqg6n/f6AmxvhNw4m4R93Ezygl8OOJwp1SDiJZmEyKXz4WQ6L30vHYLRM5uzY1p9nHutIbJTjr/tJigrkH9e0467+8QTqHHPae1MZSsrIXLSGjDdWUnu6sP5+bYAfMZNvI/6+O/94JIUQl0TCSVhdZnY1L7x2uH5RrZsbjB8dy8Tb49A6+KF6H6VkM7BdSyICdTw4yDXX39SeLiTt1eVkvr0aU+Xv+9x5R4eT8I+7iZl8Gx4BfipWKJyBhJOwGqPRzJpPcli+JoM6g6W1FBXhzaz/60iXTo6/sv+tzSd46X+HmTggnmdHJqpdjs3VZOVy4j/vkL3sI8z62vr7/ZPa0/rRyUSOuV62FBJWI+EkrGLvwTL+vfAo6VmWv6Q1GrjthiimjEtw6EkPYNmQdv43R1j4wwlu6BbJU8M7qV2STVVn5HD8hbfIWbkB5Q+nFwf370nbJ6cTOuwKGU8SVifhJJqkpKyOxe+l8/k3v+8aHRftwxP/aO8UrSWzWbHs+PBrBnf0iWHuTV1c5mj16syTHP/X2+S8u/6cUGo5ZCBtn5xOi8t7q1idcHYSTqJRjEYzn2zM5Z3VGVRWWfbF8/R04+7bYhk7KgYPD8ceWzqr2mBie3ox9wxM4OnrO7lEC6EmJ4/jL75N9rKPUQyG+vtbjbyGtk/eS1DvripWJ1yFhJO4JIqisDWlmDeXp5GR/ftgeN+ewTwyvZ3D7iT+ZwaTGZNZwc9Ly8fTL8PH093pg6muqITjLy0m881V5yycDbv+atrPfoDAXkkqVidcjYSTaLDDxypYuPwEu/eV1d8XHaHjwXva0L93iNO8eOsNJu5/fxcaDSy5KxlfL+f+NTFWVZO+4F3S5i8957iK0GuvoP3sBwnqIy0lYXvO/VsnrCIts4plqzPY/Ovva1n8fN0ZPzqW0TdE/2UXcUdWVWtkynspbE0r4vkbk5wmcM/HbDSS/c5HHJu7kNq83/e9C7m8Nx3mPkJI/54qVidcnYSTuKDM7Gre+zCTb7bkc/ZkFa1Ww83DI5kwJo7AAOeaNlxWbWDCih3szSnjv7d14+YezruANP/rHzk08yUqDx6vv8+/Swc6znuU0Gtl9p1Qn4ST+IsTGZW8uzaLH34pqA8lNzcYdlUrJtwRR5QDb9T6d+5fvYv9J8tYOLYn1yaFq11Os6g4eJxDM1+k4Ouf6u/TxUfRfs5DRN0xEo2b87SChWOTcBKAZaLDnv1lrFmfza8pxfX3azRw9YBQJo2NIz7GV8UKm9/j13akuLqOK9uHql2K1dUVl3L02dfIWrIWxWSZXan196XtU/cS/8BduHt7qVyhEOeScHJxRqOZzb8WsvbTHA4drai/380NBl8RxvjRsSTEOm8oZRVVs+nwaSYOSKBLtOOvy/ozxWwme8U6Dj81H0NRqeVONzdiJ4+m/ZyH8AqTY+SFfZJwclHFJXV89s0pNmzMpbD492nDnp5uDB/UittviiE60jm77846nl/BnUu3U2s0M6JrJKH+ztV6KNt1gP0PPkfpjtT6+1oO6k+nfz9BQJcOKlYmxMVJOLkQRVHYtbeUDV+d4sdthZhMSv1jQQEe3HRdBLeMiCI4yPmPf9h/soy7lu3ATaNh7dTLnCqYDGUVHJn9CpmL1oDZDFg2Ze38n6cIv3moTHYQDkHCyQWcLtDz1fen+WpTHidP6c95rG2CL6NviGbwFWF4ebrGYPjOzGImLP+NAG8PVt3Tl4SWztNtmffpd+x/cE79kegarZaEhyfQ7un70Po5z+cpnJ+Ek5OqrjayZWshX28+zc7U0vpZdwCeHhquuTyMG6+NIKljgMv9JZ1VXE2ovxcrJ/clKsg5ui71eQUcePh58tZ9XX9fi6v6krjgWfw7tVGxMiEaR8LJidTWmtiaUsz3Pxfwy44iauvM5zzeJt6XEUPCGXpVK6dbo9QQxVV1hPh6cnOPaIZ3icBL69i7pYOlqzbn3fUcmvkShhLLzh3aoAA6//sJou8e5XJ/eAjnIeHk4KqrjWzdWcyP2wr59bdiampM5zweFOjB4MvDuG5QK9q38XPZF6vPUnN5ct1e3p3Uh+T4EKcIJn3uafZOffqcNUvhtwwj8dXZeIc733R44VoknBxQfmEtv/5WxK87ikhJLak/2O8snc6dgX1aMPSqMHp3D3b402ebau1vWTyxfh+940PoEO6vdjlWcerjr9h3/xwMxZbp4V4RoSS9PofwGwerXJkQ1iHh5ACMRjP7D5ezbWcx23eVcCyt8i/P8dG5c1lyCNcMDKVfrxC8vBy/ZWAN7/yczvNfHOTK9qG8Pa4XOk/H/roYSsvZ/49/krvm8/r7osbeQOJrs/EIClCxMiGsS8LJDimKQnpWNSmpJaTsKWH3/rK/dNcBhAR50L93C67s35Je3YKdagNWa9hytIDnvzjIdUnhvHp7d4fvyivcvI3UiY+jz7Ec7OjRIoguC58j4pZrVa5MCOuTcLIDJpNCWmYVew+WsXt/KXv2l1FaZjjvczu29advr2AG9GlBx7b+uLnIqayNcXnblrx8S1dG9YxC6+64wa2YTByb9ybH5i7k7LTL0OuupOuiuXhHhKlcnRDNQ8JJBZVVRg4eKefAkXL2H7bcqqr/2jICCG3hSXK3YJK7B9OnR7BLLJBtCrNZ4ZXvjnJbcgwxIT7c1jtG7ZKaRH8qnz13PUbR5u0AuOm86Tz/SWKnjHHZyS3CNUg4NbMavYljaZUcPVHJoWMVHDpaTtbJmgs+PyTIgx5dgiy3pCBio3XyItRARpOZmev2sn7XSXy9tEy/0rHX9xRu+pXddz1GXX4RAH6d2tBzzWv4J7ZTuTIhmp+Ek5UoikJhcR0nMqo4nl7JsfRKTqRXkZlTfc4C2D+LidKR1CGAbomBdE0MJCZSwqgxao0mHv5gD1/tz+PRIe2ZdkVrtUtqNMVs5tjched040XfNYrEBbPR+vqoXJ0QtiHhdIkURaGouI6MnGoysqrJyK4iPauatMwqKiqNf/u+gf5aOrbzp3P7ADp3sPzriothra2mzsT0VTvZcrSA2SM6M3lggtolNZqhvJLUCTM4/fn3ALj76Eh6/Vmi77pZ5cqEsC0Jp/NQFNWIzt8AACAASURBVIXScgO5eXpOnqoh+2QN2bk1ZOdWk3Wy5rwz5/6sZYgnbRP8aN/Gjw5t/GjX2o+IVt7SKmoGJkWhXG/gpVu6MKZ3rNrlNFrVsQxSbrmPykMnAPDtkECvD9/Av3NblSsTwvZcMpwURaGi0sjpglry8vX1t9zTek6d1pObp6e6AQEElvVFcTE+tInzpXW8L63jfGkb70dQoLSImltpdR2eWjf8vLR8PL0/7g48czH/6x/ZPe5RjKXlAIQNv4ru783HI9A5Fg0LcamcLpyMRjMlZQYKi2opKqmjsLiOgqJaCgpryS+qJb+glvzCWvS15otf7A/CWnoRE6kjNtqH2CgdcdE+xMf6EtrCU1pDKsiv0DN+6Q5iQnxYeneywwaToiikvbKMw0/Orz/eos0T0+kw5x9o3B17XZYQTWH34WQ2K1RWGykrN1BWbqC03EBpqYGSMgOlZXUUlxooKaujpNRAUUkdZeWGv52AcCGenm5EhHkT0cpyi4rwJipcR1SEjqhwb9lxwY6cLK1h3NLtnC7X88zIzmqX02iKycSBh+eS+fZqwDK+1HXpC0SOHq5yZUKozybhZDIpVNeYqKo2UlllpKraSEWVkcpKExWVBiqqjFRUWm7lFYYz/xopqzBQUWHAdGmNnL9wc4OQIE9CW3gR2tKLsDO3VqFehId5Ex7mTXCghyxodQAZhVXcuXQ75XoDKyf3oVdciNolNYqpRs/u8Y9y+tPvAMthgMmfvE1g904qVyaEfbBKOM1/8yjVNSZqakxnQshEdY2RqhoT1dVGavRNTJcL8PZyIzjIk5AgjzP/etIixPJvyxDLrUWIFyHBnmjdJXgcnaIo3Pv+LmoMJtZM6UdSVKDaJTVKXVEJKTffS8nW3QD4J7Wn9+dL0EWHq1yZEPbDKuG04atTTb6GTueOv6+WwAAPAvy0+PtrCfT3IMDfg8AALUEBngQGWB4PDvQkKNADnbd0tTkak8nEE088wYoVK9Dr9QwdOpRFixbRsmXLi76vRqPhlTHd0LppaBvmmBMFqtOz2TFyClVH0gHLgYC9Pl4oEx+E+BOrhFNQoAc+Ond03u746M7etPj5uuPjo8VH546vjzt+Plp8fbX4+bjj7+eBn68WX19LKLn6sQ6u4sUXX+TTTz9l+/bttGjRgkmTJjF+/Hi++uqrC75PVa2RV749ysOD29Ex3HF33q44eJztwyZQm2c5Qj3y9hF0Xfov3L1kSyoh/kyjNGb2wF9Z5SLC+cXFxfHMM88wefJkAE6cOEHbtm1JT08nPj7+L8//8WgBQ6/qz4BHl7Dh/gH4ezvmFP3yvYfZfu1E6gqKAWj96GQ6vvAYGjf5o8xOSL+/nbFKOCUmJio6nc4K5aivoKCA0FDHP0XUHj8Pk8nEnj176NSpEz4+v2/Ds3v3bhISEggKCqq/r6CggPzSSsy6YAxF2XTtkoTWQSesmKr1VB1LRzFa1s55R7XCywlOqrXHn7HG2rlz5wFFUZLUrkP8zirhlJycrKSkpFihHPUlJyfjDJ+LPX4e2dnZxMbGkpaWRkLC71sMxcXFMW/ePMaNG1d/34bdJ3n0o1S6Rgfy9cxhVFf+9YBFR1Caso8dwydjKCkD4H+RXryeuVflqqzDHn/GGkuj0exUFCVZ7TrE76RPQdiMv79l0L+srOyc+0tLSwkIOHcsSeuuoX+bFqya3BeNdbqeba5keyrbr51YH0yd//s0W0NlfEmIhpBwEjYTFBREbGwsu3btqr8vLS2N8vJyunbtavl/gaWFNKJrJO9N6oOvl92vEz+v0pR97LhuIsayCgCSXn+WhAfvUrkqIRyHVcJp6tSp1riMXXCWz8VeP4+pU6fy0ksvkZ6eTnl5OY8//jjDhg0jLi6O/357lKGv/EhqdilA/bZQDZlmbk8qj6Tx28gpGCuqQKOhy1vPEzd9LGC/35fGcKbPBVisdgHiXDJbT9iUyWTi8ccfZ8WKFdTW1jJkyBAWLVrE29sLeOfndG5LjuZfo7qes1eeI41t6E+e5tcrbqcmKxeAxAXPEH/vnSpXJRrAMWfbODEJJ6Eqk1lh1oZ9rNmRzYT+8TwzovNftpFylHCqKy5l6zXjqDxwDIB2zzxI+9kPqFyVaCAJJzvjmB36wml8ue8Ua3Zkc//VbXhsaAeH3eHdVF1Dyk3T64MpbvpY2s26X+WqhHBcEk5CVSO7RhCk8+CK9o67XsZsMLDrjofr98qLuPVaEl+d5bBBK4Q9sOpsvUmTJhETE0NAQAARERFMmjSJkpISa36IZldbW8u0adNo164d/v7+xMbGMmPGDPR6vdqlNcqCBQvo27cvPj4+tG1rHyeqVtcZeeiD3aQVVKLRaC4YTCaTiRkzZpCamoq/vz+33HILhYWFNq724g4++i/yN24GoMU1l9Ftxb/POYvpgw8+4PLLLycgIACt1rH/Hnz88cdJTEwkICCAyMhIpkyZQnFxsdplNdrTTz9NQkICGo2mXKPR5Gs0mo81Go3jHqfsRKw65rR//34SEhLw9fWltLSU6dOn4+bmxurVq63xMWyiqqqKF154gfHjx9OuXTtycnIYNWoUAwcO5LXXXlO7vEv28ccfo9FoOHz4MMuXL+f48eOq1lOuNzBp+W/syirhtdt7MLJb5AWfO2/ePN599128vLz4+eefmTRpEtXV1X+7D5+tZS37iH3TZgEQ0L0zl32/Eq2/3znP+frrrykuLqampoapU6diNBrVKNUqnnrqKUaPHk1SUhKlpaXcddddeHp68umnn6pdWqMcPnyYiIgIAgMDNRqNxgeYC/RTFKW/Na6/a9euYVqt9llFUcKRpTt/ZAJ+NhqNU3r16lV3vic024SI0tJS7rvvPo4cOcLOnTut8TFUs3DhQhYvXkxqaqrapTTaihUrmDt3rqrhVFxVx13LtnMkr4LXbu/B8C4Rf/v8s/vwvfXWW6SkpFx0Hz5bK9m6m22Dx2OuM+AZ1oKB29ahi7nw57R582YGDx7s0OH0Z19++SVjx479y8JqB6TRaDS+wHPAREVRWjT1grt27Rrm5eX1Rnx8fJ1Op9O7ubnJxLEzzGazJjMzM6i0tHRBjx49zvtXv9WT/MUXX8Tf35/g4GA2bNjA008/be0PYXObNm2qXyQqGie/Qs+YRVs5drqSxXclXzSYysrKyMrKolevXvX3tWnThoCAAPbuVX/7H33uaXaOeRBznQGNVkuvtQv+NpiclTP8bqxevRqNRlMGVAIPAXOscV2tVvtsfHx8na+vb40E07nc3NyUyMjISnd39wkXfE5DLqTRaFZoNBrlQrdZs2bVP/eJJ56goqKCtLQ0Hn30UbsZ5wCYMGECGo3mgrc/fh5nvfrqq/z888/MmzdPhYovrDGfi5r8vLREBet4d1Ifru4QdtHnl5eXAxAYeO6BgkFBQfWPqcWkr2Xn6AepPWU5+iLxlacJGeh627KtW7eOJUuWOGR39x+NHTsWRVECgQgswbTPGtdVFCVcp9M55mC1DXh6ehrOfN3Pq6Gjsw8Aj13owaeeeqrgz/clJCQwcuRIhg8fTlZWFm52cDTAG2+8wfz58y/4+B93ygZ45ZVXeOmll/j++++JjbWvMdJL/VzUkl5YRUs/T/y9PVgxsU+D3+9S9uGzJUVR2P/gc5TusHTxxky+jdhpd6hWj1o++ugjpk2bxmeffUbPnj3VLscqFEXJ02g0S4A0jUYTqyhKU2d6uEmL6cLOzGa9YDA0KJwURanE0uS9JEajkZMnT1JVVVX/YqMmPz8//Pz8Lv5E4Pnnn2fRokVs2bKFDh06NHNll+5SPhe1HMwtZ/w72+nbOoQ37+x18Xf4g4bsw6eGrCVryVmxDoCgfj1IfG22y00ZX758OY8++iiff/45AwYMULsca9MCvkAk4LjTEJ2A1Zoz+fn5vPfee5SWWvZFO3r0KDNnzmTgwIF2EUyXYsaMGSxdutRug+lSGI1G9Ho9BoMBRVHQ6/U2mRa/K6uE2xdvxVPrxqNDG/c1PLsPX21t7Tn78Kk1GaLy8AkOPvYvALwiw+j14YIGnWJrMpnQ6/XU1VkmJZ39HlhpMpJNLViwgMcee4yvv/7a4YPJbDbzxhtvkJ+fD4BGo4kGFgIZwGEVSxNYMZw0Gg0rVqygdevW+Pr6MmTIEJKSkvj444+t9SFsIjMzk/nz55OXl0e3bt3qWyiJiYlql9Yoc+fORafTMXXqVNLS0tDpdDT3wZBbTxQxbul2gn09+XDaZbQJbVwL74knnmDkyJEcOnSIqKgoTCYTq1atsnK1DWOuq2PP3TMw1+hBo6HHe/Pxjrj42BnAypUr0el0DBs2DJPJVP89yMzMbOaqre+hhx6ivLycq6++uv53w95b8H9n48aNJCUlodFoqoDtQDUwWFEU55lSaQUmk4lu3bp1PHLkiOeJEyc8kpOTO5w4caJZj6WWvfWEVRlNZoa88iMe7hpWTe5LWIB3k69pD3vrHXnmFY7/623AcsR6pxdnqlqPsDqr982mpqZmdOvWzf5WjTfSZ5995v/GG2+ElZWVuS9dujSzS5cutU29Zmpqastu3brFn+8xx16uLuyO1t2NZRN6E6jzIMTXOQ7WK/5lJ8dfspyoENC1I+2fe1jlioSwvT59+lRv377d/+WXX86yRjBdjPpT6IRT+Cglm+c+P4CiKCS09HWaYDKUV7Jn4kwwm3Hz8qT7u/9u0DiTEI5o/fr1Ab169eoQGBjYPSoqqstrr73WAqCsrMxt1KhRrV944YWsFStW2OSANWk5iSZ799cMnv3sAJe3a0mdyYyX1v3i7+QgDj4yj5r0HAA6znsU/6T2KlckHNkLrx2JSc+sstk6j4Q43+qnHuqQ3ZDnvvXWWyFPP/10zPLly9NGjBhRsXnzZt8bbrihfZ8+faqffPLJqNmzZ5+67rrrKjdt2hSwevXqwLFjxzbrtiASTqJJFv5wnH9/fYShnVvx+tgeThVMeRu+Jefd9YBlQ9d4OWZdNFF6ZpXPoWMVdjeDpLy83O2JJ56IXbBgQcaNN95YATBo0KCqrl27Vu3YscNn8+bN9fuerV+/PsMWNUk4iUb777dHWbDpGDd2j2T+6G54uDtPL7GxopL9/3gOAG1QAN3eeRGNHSwkF44tIc632h4/3ldffeVfUVHh/sADD8Q/8MDvB2QajUbNtGnT8putwL8h4SQarUtUIOP6xfLcDUnnHKvuDI7Ne7N+e6LO859EFx2uckXCGTS0i83WCgoKtJ06dao+cODAIbVrOUv+FBSXxGRW+C3DsnB+SOdWzL2pi9MFU+XhE6S/9i5g2QUievxNKlckRPO67LLLqtLS0rzff//9QJPJRG1trea3337z/vHHH1XbB03CSTRYndHMP9bsZsyirRzPr1C7nGahKAoHHp6LYjSCRkPSa7OlO084vV69eulfffXVzDlz5kQHBAT0CAsL6zZ9+vS4uro61f7ylG490SB6g4n73t/F94fzmXV9J9qGOdaWVA2Vt+FbCjf9CkDslDEE9nTMnUGEuFTTpk0rnjZtmt3sJyjhJC6qstbIlHdT2JZexAs3d2FsX/vaod1aTNU1HDqzd55HiyA6PP9/KlckhOuScBIX9UVqLjsyinnltu7c1CNK7XKazYl/L6EmKxeAjs8/gmdIkMoVCeG6JJzEBSmKgkajYUzvGLrHBtExXL1zlJpbdVo2J/69BIDAnonETLpV5YqEcG0y0ivO61RZDaPf3sqRvAo0Go1TBxPAwZkvYq61HGmR+NozaNydZzGxEI5IWk7iLzKLqrhz6XbKqg1U6A1ql9PsynYf5PSn3wEQPf5mgvt1V7kiIYSEkzjHsdMV3Ll0OwaTmdVT+tElOlDtkprd8RctR2Fo3N1p98wDF3m2EMIWpFtP1DueX8lti7aiAGunXeYSwVRx4Bh5678GIOrOG/GJj1a5IiHUM3DgwHZz5sxppXYdIOEk/iA6WMc1HVvx0bTLaN/KOdcx/dnZVhNubrR5fKq6xQihsoMHD/r06tXLpvv/XYh06wl2pBfToZU/gT4e/Oe2bmqXYzOVR9PJ/XAjAJG3DcevfYLKFQmhnvT0dI+SkhJtv3797CKcpOXk4r45kMe4pduZ++VBtUuxuRMvLQazGYC2T0xXuRoh1LVt2zafyMjIutDQUNP69esDwsLCur788suh8fHxSTqdrseQIUPa1NbWXnA7o3nz5oVFRUV1qamp0QDs2rXLu2XLlt2WLVsW3Jh6pOXkwj7dc5JHPkwlKSqQp6/vpHY5NlWdns3J9z8FIHzUMPwT26lckXAFqVOeiqnYf9Rmm6n6J7Wv7rbkhQbthL5z506fxMTEaoCUlBSf0tJSbXV1tdvBgwcPnD59Wtu9e/fETz75JOD2228/7yGDM2fOzF+6dGnoyy+/HDpmzJjS4cOHt589e3bOpEmTShpTu4STi/pgRxZPfrKPPvEhvDOhN35ervWjcOLlJSgmEyCtJmE7FfuP+pSl7LO7wwYBUlNTfXr06FENsGfPHp9hw4aVzJkz5zRAQkKCwc/Pz6QoygXf38PDg3nz5uXcf//98UuXLg174IEH8h566KGixtbjWq9IAoCaOhMLNx/nyvahvHVnL3SerrXgtCYnj+wzJ9yGXX81gT06q1yRcBX+Se1tOp5zKR/v4MGDPpMnTy4E2L9/v8+sWbNOnn2ssLDQ/dSpU569evWq+btr9OzZs6ampsY9MTGxetasWU06pFDCyYUoioKigM7TnQ+nXUYLXy88ta437Jg2fymKwbK4uO2T96pcjXAlDe1is7WCggL33Nxcz379+lUXFRW55+TkeA0cOLDq7OO//PKLT1BQkLF9+/Z1F7pGRkaGx5AhQ9qPHTu2YPXq1aG7d+/27tGjh76xNbneK5OLUhSFF786zJPr92E2K0QE6lwymAyl5WS98yEALQcPILiv68xOFOJCtm7d6hMSEmKMj483/Prrrz4BAQGmzp071wfRjh07fLt06VJ1offPzc3VDho0qP2YMWOKli1bln3jjTcWPfLII01aNOh6r04uyGxWmLVhP4t+TMPLw7W/5blrv8SsrwWg9f9NVLkaIezDzp07fTp37lw/GSIxMfGcINq9e3f9eNSfFRUVuQ8ePLjdoEGDyubPn38K4MUXX8zdtm2b/4YNGxq9YFLzdwNcl8AqFxHWZzSZmfnxXtbvPsm9V7Vh5rAOaDSOdax6cnIyKSkpVrnWL/1HU/rbXryjw7nm+Peywas4y+q/FKmpqRndunUrtPZ1nUlqamrLbt26xZ/vMdf+M9oFzDgTTDOGdeDxazuqFkwLFiygb9+++Pj40LZtW1VqqDhwjNLf9gIQfdfNEkxC2DGZEOHkbu4RRbfoQCYMUHf3g8jISGbOnMnhw4dZvny5KjWcnaEHEH3XKFVqEEI0jLScnFCF3sDXB/IAuKJ9qOrBBHDrrbdyyy23EBWlzkm6ZoOhftFtyOW98W3jnEfNC+EsJJycTElVHXcu3c6Dq3eTW/q3SxLs2uLFi0lOTiY5OZmCgoImX6/gfz9Sl29ZDxgz4ZYmX08I0bwknJxIfrmeMYu3cjivgjfv7ElkkK7ZP+aECRPQaDQXvM2aNatR1506dSopKSmkpKQQGhra5DqzV6wDwN3Ph/BbhjX5ekI0gMlsNjvW7CMbOvO1MV/ocRlzchInS2u4c8k28itqWT6hNwPatrTJx33jjTeYP3/+BR/38bHZNmIXVJtfRP7GLQBEjh6O1lf9moRL+DkzM3NgZGRkpaenp8HRZsk2J7PZrCkoKAgE9l/oORJOTuL7Q6cpqqpj5eS+9Ipr1CbAjeLn54efn11uFVbv5PufoRiNAETfLRMhhG0YjcYppaWl91ZUVExQFCUE6an6IzOw32g03nOhJ0g4OTiDyYyHuxvjL4tnWGI4YQHeapd0XkajEaPRiMFgQFEU9HrLribe3s1br6IoZL9r6dLzbR9PcP+ezfrxhDirV69edcBrZ27iEkmSO7C9OaUM+s8W9p+07GBvr8EEMHfuXHQ6HVOnTiUtLQ2dTodO1/xjYmUp+6g8cAyA6LtvcbgFyEK4KgknB7U9rYixS7ajoBDg7aF2ORc1Z86cMxvPnntrbvVrm9zciB53Y7N/PCGEdUi3ngPafCSf6at2EhWk4/17+hEeaL8tJjWZ9LXkrv0SgNChA/GObKVyRUKIhpJwcjA7M0uY8l4K7cL8WTm5Dy38vNQuyW4V/5yCsbQcgOjxN6tcjRDiUkg4OZguUYHcc3lrpl/ZhkCd/Xfnqanw258tb7i5ETp0oLrFCCEuiYw5OYhP95ykqLIWT60bj1/bUYKpAQq/+xWAoN5d8QgKULkaIcSlkHByAIu2nOChD/aw6Mc0tUtxGPq8Asr3HgYgdMgAlasRQlwq6dazY4qi8Mq3R1nw/XFGdI1gxrAOapfkMAo3/Vr/dsvBEk5COBoJJzulKArPf3GIZb+kc1tyNP8a1RV3N1mj01CF3/4CgDbAj6A+XVWuRghxqSSc7FR5jZEfjuQzcUA8s6/vjJsEU4MpikLhd5ZwanF1P9w8ZHxOCEcj4WRnDCbLJr2BPh5suG8AATqt7GpwiSr2HaH2tOV07JaD+qtcjRCiMWRChB3RG0zcu2onMz5KRVEUAn08JJgaoeBMqwkgdIhMIRfCEUk42YnqOiP3vJvCd4fy6RUXLKHUBGfHm3QJ0fjIibdCOCTp1rMDZTUGJq34jd1ZJcwf3Y1be0WrXZLDMtXoKf7pNwBCBw+QkBfCQUk4qUxRFKa+l8LenFIWju3JdV0i1C7JoRX/shNzbR0ALaVLTwiHJeGkMo1Gw/8NaU+NwcTVHcLULsfh/XHLopZX91O3GCFEo0k4qSS7uJqtJ4q4rXcM/Vq3ULscp1FwZrxJtiwSwrFJOKngeH4l45ZuR280MaRzK4J9PdUuySnoT+VTse8IIFsWCeHoJJxs7GBuOePf2Y5Go2HNlH4STFYkWxYJ4TxkKrkN7coq4fbFW/HSuvHhtH50ipBuJ2uSLYuEcB7ScrKhA7nlhPh6suqevkQH+6hdjlNRzOb6lpNsWSSE45NwsoFyvYEAbw/G94vjlp5R+HjKl93aKg4cq9+yKFS69IRweNKt18y+2JvL5S/9wP6TZQASTM2kfM+h+rdDBiarWIkQwhoknJrRhynZ/GPNbtq38iOuhXTjNafKQ8cB0Li749s+Xt1ihBBNJn/GN5MVv6Qz5/ODXN6uJYvG95IWUzOrOBNOPu3icPOUGZBCODp5xWwG3x08zZzPDzK0cyteH9sDL6272iU5vcpDJwDw79RW5UqEENYg4dQMruoQynM3JDK2bywe7tJz2txMNXqq07IB8OvURuVqhBDWIK+cVmI2KyzYdIz8cj1adzfu7h8vwWQjlUfSQVEA8JOWkxBOQV49rcBoMjNz3V7+++1RPkvNVbscl3N2MgRIt54QzkK69Zqozmjm4bW72bgvj/8b3J7JAxPULsnl1IeTm5vM1BPCSUg4NYHeYGL6qp1sPlLArOs7cc/lrdUuySVVnJkM4dM6Bnedt8rVCCGsQcKpCWrqTOSV6Xnh5i6M7SvHgavlbMvJv7N06QnhLCScGqGsxoC3hxvBvp589sBAPLUydKcWU20d1cezAJkMIYQzkVfVS1RQUcuYRVuZ8dFeAAkmlVUdTUcxmQCZRi6EM5FX1kuQW1rDmEVbySyqZnRytNrlCH5ffAsyU08IZyLdeg2UWVTF2CXbKa8xsHJyH5LjQ9QuSfCHmXoaDX4dZUKKEM5CwqkBTGaFe95NobrOyOop/egSHah2SeKMs3vq6eKjcPfRqVyNEMJaJJwawN1Nw0u3dsXPS0v7Vv5qlyP+QPbUE8I5yZjT30jJKGbpT2kA9IwNlmCyM2aDgaqjGYBMhhDC2Ug4XcBPxwoY/84OVm/PorrOqHY54jyqjmehGC3fG5lGLoRzkXA6j28O5DF5RQpxLXxYO+0yOYupiWpra5k2bRrt2rXD39+f2NhYZsyYgV6vb9J1ZU89IZyXhNOffLrnJPe+v4vOkQF8MLUfof5eapfk8IxGIy1btuTzzz+ntLSUn376ie+//57HH3+8Sdf9Yzj5dZKZekI4E2kS/IneYKJPfAhL7k7Gz0u+PNbg6+vLvHnz6v8fFxfHpEmTWLx4cZOue3YyhHdMBFp/vyZdSwhhX+TV94yckmqig30Y0zuWW3vF4O6mUbskp7Zp0ya6du3apGtUHDyzp55MhhDC6bh8t56iKLz23TEG/WcLh06VA0gwXYIJEyag0WgueJs1a9Zf3ufVV1/l559/Pqc19WeLFy8mOTmZ5ORkCgoK/vK42Wik6mg6IJMhhHBGGuXMCaJNZJWL2JqiKLyw8RBLfkrnlp7RvHRLF7Ryeu0lqays/NuJDT4+Pvj4+NT//5VXXuGll17iu+++IykpqUEfIzk5mZSUlHPuqzqWwebOwwDo8vZcYiePbkT1QtSTv0jtjMt265nNCrM+3c/q7VncdVkcc0Ym4iYtpkvm5+eHn1/Dxnuef/55Fi1axJYtW+jQoUOTPu7ZLj2QozKEcEYuG07rduWwensW917VhpnDOqDRSDA1pxkzZvDhhx+yZcsW2rRp+hjRuTP1ZMxJCGfjsuE0qmc0gToPhiaGq12K08vMzGT+/Pl4enrSrVu3+vvj4uI4cOBAo655tuXkFRmGR1CAVeoUQtgPlwqnmjoTz3y6n4eHtCcqSCfBZCNxcXFYaWyznuypJ4Rzc5nR/3K9gbuX7WDdrhx2Z5WoXY5oAsVkovKwJZykS08I5+QSLaeSqjruWraDQ6fKee32HozoGql2SaIJajJzMetrAZlGLoSzcvpwyq/QM27pdjKKqlk0vheDOrVSuyTRRBWHZKaeEM7O6cPJS+tOkM6TFRMS6d+2pdrlCCvQ5+TVv+2TEKNiJUKI5uK0K+MtSwAABfdJREFU4ZRdXE2ovxeBOg/WTusnU8WdiLGquv5td39fFSsRQjQXp5wQcehUOTe/+StPfbIPQILJyZiqaurf1vrK0exCOCOnC6c92aXcvngbWjcN910l4xHOyHSm5eTm5YnG3V3laoQQzcGpuvW2pRUxecVvtPDz4v17+hIT4nPxdxIO52zLyd1Pvr9COCunCadao4lH1u4hIkjHqsl9CQ/0Vrsk0UyMlZaWk9ZXwkkIZ+U04eSldeedCb0J8/eihZ+cXuvMTNVnWk4y3iSE03L4Mad1O3P477dHAegUESDB5ALOjjm5S8tJCKfl0OG0cmsGj36Uys7MYgwms9rlCBs5260nLSchnJfDduu9veUEL351mMGdwnhjbE885JBAl3F2QoSMOQnhvBwynP777VEWbDrGyG6R/Pe2bhJMLsZUfablJLP1hHBaDhlOrVv6ckefWObelIS7nF7rcuq79XykW08IZ+Uw4WQyKxzILaNrdBA3do/kph5RapckVCLdekI4P4foDzOYzDz0wW5ufWsrGYVVapcjVKQoiizCFcIF2H3LSW8w8cDqXXx3KJ8nr+tIfEvZ6NOVmesMKEYjILP1hHBmdh1OVbVGpq5M4ZfjRTx/UxLj+8WpXZJQmekPO5JrZcxJCKdl1+H0YUo2W08U8d/bujGqZ7Ta5Qg78Mcdyd39pBUthLOy63Ca0D+eXnHBdI0OUrsUYSfOztQD6dYTwpnZ3YSIvDI9dy7dRmZRFRqNRoJJnOOcbj0JJyGcll2FU3ZxNaMX/Upqdhn5FbVqlyPs0NlNX0H21hPCmdlNt97x/AruXLodvcHM+/f0pVuMtJjEX53brSfhJISzsotwOp5fwW2LtuGm0bB2Wj86hgeoXZKwU9KtJ4RrsItwCg/U0a91CDOGdSRB1jGJv3HObD0JJyGclqrhtCurhA6t/PHz0vLmnb3ULEU4CJlKLoRrUG1CxKZDp7l98Tb+9dUhtUoQDsgo3XpCuARVwunz1FymrdxJx3B/Hh3SQY0ShIOqH3PSaHDTeatbjBCi2di8W+/D37J5fP1eeseF8M6EZPy9PWxdgnBg9Zu++urQaOS4FCGclU3DqbLWyPxvjjCwbUsWj09G5+luyw8vnMDZqeRyXIYQzs0m4aQoCgB+Xlo+mn4Z4YHeeGklmMSlO9utJzP1hHBuzR5OiqLw0v+OYDSZefr6TsS1kBlWovHO7hAhC3CFcG7NOiHCbFZ45tMDvL3lBNUGE2caUEI0Wv0R7RJOQji1Zms5GU1mHl+3j3W7cph6RWuevK6jDGCLJvv9iHbp1hPCmTVbOD36USqf7snlkSHtefCathJMwirqx5zkiHYhnFqzhdO1ieF0jQ5i8sCE5voQwgWdXYTrLqfgCuHUrBpOFXoDu7NKuaJ9KNd1ibDmpYUA/titJy0nIZyZ1SZElFbXMe6dHUx5L4X8Cr21LivEOeoX4Uq3nhBOzSrhVFBRy+2Lt3Eot5w3xvYkzF+2lRHnmjRpEjExMQQEBBAREcGkSZMoKSm5pGsoZrOscxLCRVglnMYs2kpmUTXLJvRmSOdW1rikcDKPPPIIhw8fpry8nEOH/r+9u0dpKIjCMPyeYCGaQixsBNHGKqW4EktRUbDJBgRtbVyFYCm4AVtXIIKkcQOWiTdY6LEJiL+gScwkvk83l5nhdB/zc5lbqqqi2Wz+aI6n7uuK3Nt60mQbyJnTffuRs7111pbnBzGdJlCj0XjTrtVqtFqtH83x5rmMGbf1pEkW6Z+x+iMRcQAcAnWgC2xm5sUXffeB/V5zOjMbn/WTNJkMJ/UlIk6B7W+6HGfm0bsxK8AucJ6Z10MsT9KYMpzUl4ioA9/dgKkys3r/MSLWgQtgKTOfh1WfpPE00mfaNf4yswN0fjF0ClgEZoH2QIuSNPZG9ky7/o+IWIiIrYiY67VXgRPgKjMNJkkfGE76CwnsAHcR8QBcAjfAxiiLklQuz5wkScVx5SRJKo7hJEkqjuEkSSqO4SRJKo7hJEkqjuEkSSqO4SRJKo7hJEkqzgt/msEnIfUzngAAAABJRU5ErkJggg==",
+      "text/plain": [
+       "<Figure size 432x324 with 1 Axes>"
+      ]
+     },
+     "metadata": {
+      "needs_background": "light"
+     },
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "import matplotlib as mpl\n",
+    "import matplotlib.pyplot as plt\n",
+    "import numpy as np\n",
+    "from math import log\n",
+    "import matplotlib.ticker as plticker\n",
+    "\n",
+    "\n",
+    "# Create figure and add axes\n",
+    "fig = plt.figure(figsize=(6, 4.5))\n",
+    "ax = fig.add_subplot(111)\n",
+    "\n",
+    "\n",
+    "colors = plt.get_cmap('coolwarm', 2)\n",
+    "\n",
+    "\n",
+    "labels = [r\"$e^x$\",r\"$ln\\ x$\"]\n",
+    "\n",
+    "plt.ylim(-3,3)\n",
+    "plt.xlim(-3,3)\n",
+    "\n",
+    "ax.spines['left'].set_position('zero')\n",
+    "ax.spines['bottom'].set_position('zero')\n",
+    "\n",
+    "# Eliminate upper and right axes\n",
+    "ax.spines['right'].set_color('none')\n",
+    "ax.spines['top'].set_color('none')\n",
+    "\n",
+    "# Show ticks in the left and lower axes only\n",
+    "ax.xaxis.set_ticks_position('bottom')\n",
+    "ax.yaxis.set_ticks_position('left')\n",
+    "\n",
+    "#loc = plticker.MultipleLocator(base=1.0) # this locator puts ticks at regular intervals\n",
+    "#ax.yaxis.set_major_locator(loc)\n",
+    "\n",
+    "xe = np.linspace(-3,3, 100) \n",
+    "y = np.exp(xe)\n",
+    "ax.plot(xe, y, color=colors(0), linewidth=2.5, label=labels[0])\n",
+    "\n",
+    "xl = np.linspace(0.01, 10, 100)\n",
+    "y = np.log(xl)\n",
+    "ax.plot(xl, y, color=colors(1), linewidth=2.5, label=labels[1])\n",
+    "    \n",
+    "ax.plot(xe,xe,ls='dashed',lw=1.5)\n",
+    "\n",
+    "ax.tick_params(axis='both', which='major', labelsize=13)\n",
+    "\n",
+    "    \n",
+    "# Add legend\n",
+    "\n",
+    "leg = ax.legend(bbox_to_anchor=(1.05, 0.3),loc='lower left', \n",
+    "          frameon=True, labelspacing=0.2, fontsize=13)\n",
+    "\n",
+    "plt.tight_layout()\n",
+    "plt.savefig('log_base_02.png')"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Maths for Biologists\n",
+    "  \n",
+    "## Making mathematical statements"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Sets and relations  \n",
+    "  \n",
+    "Start by watching the [video](https://web.microsoftstream.com/video/a9ad1a8c-3d80-4910-b9f2-fab98c01b40b)."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Polynomial functions  \n",
+    "  \n",
+    "One of the simplest elemntary functions is given by:  \n",
+    "  \n",
+    "  $$ f(x) = a_n x^n + a_{n-1} x^{n-1}+ \\cdots + a_1 x^1 + a_0, $$  \n",
+    "  \n",
+    "where $n$ is a *non-negative integer*, and the coefficients $a_n$, $a_{n-1}$, ..., $a_0$. The highest value of $n$ (for which the coefficient $a_n$ is different from $0$) defines the *degree* of the polynomial.  \n",
+    "  \n",
+    "```{admonition} Examples\n",
+    "$f(x) = 3x^5 + 12x^4 - 6x^3 - x^2 + x - 16$ (Polynomial function of degree 5)  \n",
+    "$g(x) = \\frac{3}{4}x^2 - \\frac{5}{2} $ (Polynomial function of degree 2)  \n",
+    "$h(x) = 1.6x^8 - 0.576x^4 + 1.89x^2 - 0.7$ (Polynomial function of degree 8)  \n",
+    "```  \n",
+    "  \n",
+    "\n",
+    "```{tip}\n",
+    "When the domain of the polynomial function is not explicitly given, it is frequently assumed to be all real numbers, ${\\rm I\\!R}$.\n",
+    "```\n",
+    "```{tip}\n",
+    "When no application is specified, the function name \\'$f$\\' and variable name \\'$x$\\' are usually chosen. For application purposes, the function and variable are often named for what they represent; for example, $A(r)=\\pi r^2$ for the area of a circle dependent on its radius.\n",
+    "```\n",
+    "\n",
+    "Let us discuss three of the most common forms of polynomial functions with degrees $0$, $1$, and $2$.\n",
+    "\n",
+    "### Degree 0  \n",
+    "  \n",
+    "Polynomial functions of degree 0 assume the form:  \n",
+    "  \n",
+    "  $$ f(x) = c, $$  \n",
+    "  \n",
+    "where $c$ is a real number. This function is graphically represented by:  \n",
+    "  \n",
+    "```{image} ./poly_deg0.png\n",
+    "\n",
+    "``` \n",
+    "\n",
+    "```{admonition} Example  \n",
+    "For most mobile phone contracts nowadays a fixed fee is paid for the month with unlimited usage. So the cost as a function of used minutes would be a polynomial of degree zero, since the cost does not depend on the usage.\n",
+    "```\n",
+    "  \n",
+    "  \n",
+    "### Degree 1\n",
+    "\n",
+    "Polynomial functions of degree 1 are also called *linear functions* and are given by:  \n",
+    "  \n",
+    "$$ f(x) = mx + b,$$  \n",
+    "  \n",
+    "where again $m$ and $b$ are real numbers (with $m \\ne 0$).  \n",
+    "  \n",
+    "The constants $m$ and $b$ are frequently referred to as *slope* and *intercept*. The intercept $b$ shows where the function crosses the $y$-axis, while the slope $m$ is related to the inclination of the function.\n",
+    "  \n",
+    "````{panels}  \n",
+    "  \n",
+    "```{image} ./poly_deg1.png\n",
+    "\n",
+    "```  \n",
+    "\n",
+    "---  \n",
+    "  \n",
+    "If we consider two arbitrary points, $(x_1,y_1)$ and $(x_2,y_2)$ on the line, their projections define two intervals on the $y$ and $x$ axes. These intervals have sizes $\\Delta y = y_2 - y_1$ and $\\Delta x = x_2 - x_1$ respectively.  \n",
+    "  \n",
+    "The slope $m$ will be given by the ratio between the changes in $y$ and $x$, for any chosen two points on the line. Thus:  \n",
+    "\n",
+    "  $$m = \\frac{y_2 - y_1}{x_2 - x_1} = \\frac{\\Delta y}{\\Delta x}. $$  \n",
+    "  \n",
+    "Also note that the triangle defined by the sides $\\Delta y$, $\\Delta x$ and this fragment of the function line forms a right triangle. Thus, there is a relationship between the slope $m$ and the angle $\\theta$ between the function line and the $x$-axis (or any horizontal line parallel to it) which is given by:  \n",
+    "  \n",
+    "  $$m =  \\frac{\\Delta y}{\\Delta x} = tan(\\theta).$$\n",
+    "  \n",
+    "````  \n",
+    "  \n",
+    "The sign of the slope $m$ also determines if the function increases or decreases as a function of $x$. We have:  \n",
+    "  \n",
+    "  $$ \\begin{align}\n",
+    "  m>0 &\\rightarrow \\mbox{Increasing function of } x \\\\\n",
+    "  m<0 &\\rightarrow \\mbox{Decreasing function of } x\n",
+    "  \\end{align} $$  \n",
+    "    \n",
+    "    \n",
+    " \n",
+    "````{panels}  \n",
+    "  \n",
+    "```{image} ./slope_01.png  \n",
+    "\n",
+    "```  \n",
+    "\n",
+    "---  \n",
+    "  \n",
+    "The graph on the left show the functions $y = mx$ (intercept $b=0$), for different values of $m$ (see legend colours).  \n",
+    "  \n",
+    "Note that positive values of $m$ give **increasing** functions, while negative values of $m$ return **decreasing** functions.\n",
+    "      \n",
+    "```` \n",
+    "  \n",
+    "One can also ask what is the value of $x$ where the function $f(x)$ crosses de $x$-axis, or, in mathematical notation, what is $x\\ \\mbox{so that}\\ f(x) = 0$. This value of x is called a **root of $f$** and it can be obtained by:  \n",
+    "\n",
+    "$$\\begin{align}\n",
+    "    f(x) &= 0 \\\\\n",
+    "    mx + b &= 0 \\\\\n",
+    "    mx &= -b \\\\\n",
+    "    x &= -\\frac{b}{m}\n",
+    "    \\end{align} $$\n",
+    "    \n",
+    "`````{admonition} Example  \n",
+    "\n",
+    "````{panels}  \n",
+    "  \n",
+    "```{image} ./poly_deg1_02.png\n",
+    "\n",
+    "```  \n",
+    "\n",
+    "---  \n",
+    "  \n",
+    "  Consider the function $y = -2x + 1$.  \n",
+    "  We have:  $m=-2$ and $b=1$. The root of this function will be $x = -\\displaystyle\\frac{b}{m} = -\\displaystyle\\frac{1}{(-2)} = \\displaystyle\\frac{1}{2}$.  \n",
+    "  Additionally, the function crosses the $y$-axis at the intercept $b=1$.\n",
+    "  \n",
+    "````\n",
+    "`````  \n",
+    "\n",
+    "```{admonition} Examples  \n",
+    "The metabolic rate of some organisms depends (approximately) linearly on the body's temperature, with higher rates at higher temperatures.\n",
+    "```\n",
+    "  \n",
+    "### Degree 2\n",
+    "\n",
+    "Now we look at polynomial functions of degree 2, given by the general form:  \n",
+    "\n",
+    "$$f(x) = ax^2 + bx +c,$$  \n",
+    "  \n",
+    "the coefficients $a$, $b$, and $c$ are real numbers (with $a \\ne 0$). The graph of a polynomial function of degree 2 is a *parabola* and many of its properties can be readily known by analysing the coefficients.\n",
+    "\n",
+    "#### Concavity\n",
+    "\n",
+    "````{panels}  \n",
+    "Concave up (positive concavity)\n",
+    "^^^\n",
+    "  \n",
+    "```{image} ./poly_deg2_01.png  \n",
+    "\n",
+    "```  \n",
+    "\n",
+    "---  \n",
+    "  \n",
+    "Concave down (negative concavity)\n",
+    "^^^\n",
+    "  \n",
+    "```{image} ./poly_deg2_02.png  \n",
+    "\n",
+    "```  \n",
+    "  \n",
+    "````\n",
+    "\n",
+    "#### Intercept & Roots  \n",
+    "\n",
+    "````{panels}  \n",
+    "  \n",
+    "```{image} ./poly_deg2_roots.png  \n",
+    "\n",
+    "```  \n",
+    "\n",
+    "---  \n",
+    "  \n",
+    "For the function $f(x) = ax^2 +bx + c$ the coefficient $c$ represents the **intercept**, the point where the function curve crosses the $y$-axis.  \n",
+    "  \n",
+    "The **roots** of the function $f$ are the values of $x$ for which $ax^2 +bx + c = 0$, where the curve crosses the $x$-axis. A polynomial function of degree 2 has a maximum number of two real roots*, given by:  \n",
+    "\n",
+    "$$\\begin{align}\n",
+    "x_1 &= \\frac{-b-\\sqrt{b^2-4ac}}{2a} \\\\\n",
+    "x_2 &= \\frac{-b+\\sqrt{b^2-4ac}}{2a}\n",
+    "\\end{align}$$\n",
+    "\n",
+    "+++\n",
+    "\\* A polynomial function of degree $n$ will always have $n$ roots, but these can be imaginary or multiple roots. It therefore can (but does not have to) have a maximum of $n$ real roots.\n",
+    "  \n",
+    "````  \n",
+    "  \n",
+    "The expression inside the square root distinguish between different properties of this function and therefore receives the name of *discriminant*:  \n",
+    "\n",
+    "$$ \\Delta = b^2 -4ac.$$  \n",
+    "  \n",
+    "Depending on the value of the discriminant a quick conclusion can be reached regarding the roots of the function.\n",
+    "\n",
+    "`````{tabbed} Case 1  -  $\\Delta > 0$  \n",
+    "\n",
+    "````{panels}  \n",
+    "  \n",
+    "```{image} ./poly_deg2_delta01.png  \n",
+    "\n",
+    "```  \n",
+    "\n",
+    "---  \n",
+    "  \n",
+    "When $\\Delta > 0$ there are two real **distinct** roots for the polynomial function.  \n",
+    "  \n",
+    "```{admonition} Examples\n",
+    "Consider the function  $y = 0.5 x^2 - x -1.5$. We have:  \n",
+    "  \n",
+    "  $$\\Delta = (-1)^2 - 4(0.5)(-1.5) = 1 + 3 = 4.$$  \n",
+    "  \n",
+    "  The roots will then be given by:  \n",
+    "    \n",
+    "  $$\\begin{align}\n",
+    "  x_1 &= \\frac{-(-1)-\\sqrt{4}}{2(0.5)} = \\frac{1-2}{1} = -1 \\\\\n",
+    "  x_2 &= \\frac{-(-1)+\\sqrt{4}}{2(0.5)} = \\frac{1+2}{1} = 3 \\end{align}$$  \n",
+    "  \n",
+    "``` \n",
+    "\n",
+    "  \n",
+    "````  \n",
+    "`````  \n",
+    "  \n",
+    "`````{tabbed} Case 2  -  $\\Delta = 0$  \n",
+    "\n",
+    "````{panels}  \n",
+    "  \n",
+    "```{image} ./poly_deg2_delta02.png  \n",
+    "\n",
+    "```  \n",
+    "\n",
+    "---  \n",
+    "  \n",
+    "When $\\Delta = 0$ there are two real **identical** roots for the polynomial function.  \n",
+    "  \n",
+    "```{admonition} Examples\n",
+    "Consider the function  $y = 0.5 x^2 - x +0.5$. We have:  \n",
+    "  \n",
+    "  $$\\Delta = (-1)^2 - 4(0.5)(0.5) = 1 -1 = 0.$$  \n",
+    "  \n",
+    "  The roots will then be given by:  \n",
+    "    \n",
+    "  $$\\begin{align}\n",
+    "  x_1 = x_2 = \\frac{-(-1)\\pm\\sqrt{0}}{2(0.5)} = 1\\end{align}$$  \n",
+    "  \n",
+    "``` \n",
+    "  \n",
+    "````  \n",
+    "`````  \n",
+    "  \n",
+    "`````{tabbed} Case 3  -  $\\Delta < 0$\n",
+    "\n",
+    "````{panels}  \n",
+    "  \n",
+    "```{image} ./poly_deg2_delta03.png  \n",
+    "\n",
+    "```  \n",
+    "\n",
+    "---  \n",
+    "When $\\Delta < 0$ there are no real roots for the polynomial function, since the square root of a negative number is not defined for real numbers.  \n",
+    "  \n",
+    "  ```{admonition} Examples\n",
+    "The graph on the left corresponds to the function  $y = 0.5 x^2 - x + 1$. We see that the curve does not cross the $x$-axis at any point, showing there is no real root for this function.  \n",
+    "\n",
+    "``` \n",
+    "  \n",
+    "````  \n",
+    "`````  \n",
+    "```{admonition} Example\n",
+    "If we throw a ball that is heavy enough so that air resistance is negligible (alternatively we can travel to the moon) the height of the ball as a function of time follows a quadratic equation. Knowing the roots we can calculate when and where the ball will land.\n",
+    "```\n",
+    "\n",
+    "#### Minimum/ maximum  \n",
+    "  \n",
+    "  Depending on the concavity of the parabola (up or down), there will be a point where the function assumes a minimum or a maximum value. This point is called the *vertex* of the parabola. This extreme point can be easily determined by the symmetry properties of the function curve.  \n",
+    "   \n",
+    "   ````{panels}  \n",
+    "  \n",
+    "```{image} ./poly_deg2_vertex.png  \n",
+    "\n",
+    "```  \n",
+    "\n",
+    "---  \n",
+    "  \n",
+    "The coordinate $(x_v,y_v)$ of the vertex is given by:  \n",
+    "  \n",
+    "  $$\\begin{align}\n",
+    "  x_v &= -\\frac{b}{2a} \\\\\n",
+    "  y_v &= -\\frac{(b^2-4ac)}{4a} = -\\frac{\\Delta}{4a}\n",
+    "  \\end{align}$$  \n",
+    "    \n",
+    "  ```{tip}  \n",
+    "  Since the parabola is symmetric in relation to a vertical line passing through the vertex, the $x$ coordinate of the vertex is given by the middle point between the two roots $x_v = (x_1 + x_2)/2$. The $y$ coordinate is then given by substituting $x_v$ in the general form of the polynomial of degree 2, as $y_v = f(x_v) = ax_v^2 + bx_v + c$. Try to obtain the expressions above as an exercise.\n",
+    "  ```\n",
+    " ```{tip}  \n",
+    "The extrema is the point of the parabola where the derivative becomes zero $f'(x)=0$.\n",
+    "  ```\n",
+    "  \n",
+    "````\n",
+    "```{admonition} Example\n",
+    "Imagine the population of a specific region having a negative quadratic relationship - it undergoes initial growth, reaches its peak, and then declines. To determine both the maximum population size and the time when it peaked, we need to identify the extrema of this parabolic curve.\n",
+    "```\n",
+    "\n",
+    "  \n",
+    "Now watch this [video](https://web.microsoftstream.com/video/f64abd8a-9997-4b5e-b11f-0ca09b7603fe) for a short introduction to complex numbers."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Rational functions  \n",
+    "  \n",
+    "  \n",
+    "Rational functions are defined as quotients of polynomial functions:  \n",
+    "  \n",
+    "  \n",
+    "$$f(x) = \\frac{p(x)}{q(x)},\\ \\ \\ q(x) \\ne 0$$  \n",
+    "    \n",
+    "      \n",
+    "````{panels}  \n",
+    "  \n",
+    "```{image} ./rational_01.png  \n",
+    "\n",
+    "```  \n",
+    "\n",
+    "---  \n",
+    "  \n",
+    "The plot on the left shows the graph for the function $y = \\displaystyle\\frac{2x-5}{x-3}$.  \n",
+    "  \n",
+    "Note that the graph approaches a vertical line at $x=3$ when $x$ is close to $3$, and it approaches a horizontal line at $y=2$ when $x$ approaches large values (positive or negative). These lines are called *vertical asymptote* and *horizontal asymptote*, respectively.  \n",
+    "    \n",
+    "Vertical asymptotes appear for the values of $x$ corresponding to the roots of the denominator function ($q(x)$) of the rational function $f(x)$, while horizontal asymptotes to the limits of the function $f(x)$ (if it exists) for large $x$*.  \n",
+    "  \n",
+    "Asymptotes give us a lot of information about the limiting behaviours of the function, and they are especially relevant when modelling biological phenomena.\n",
+    "\n",
+    "+++\n",
+    "\\* We will be talking more about limits on the following lectures.\n",
+    "      \n",
+    "````  \n",
+    "\n",
+    "```{admonition} Example\n",
+    "Rational functions are often used to express ratios, for example of chemicals. Lets assume chemical A increases quadratically $A(t) = 2 t^2$ and chemical B increases linearly $B(t)=t+5$, the ratio will be:\n",
+    "\n",
+    "\n",
+    "$$r(t) = \\frac{A(t)}{B(t)} = 2\\frac{t^2}{t+5}$$\n",
+    "```\n",
+    "        "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Power functions  \n",
+    "  \n",
+    "Another type of function used in many applications are power functions (sometimes also called power laws). These functions have the functional form:  \n",
+    "  \n",
+    "  $$f(x) = Cx^{\\alpha}$$  \n",
+    "    \n",
+    "where $C$ and $\\alpha$ are real constants (with $C \\ne 0$).\n",
+    "  \n",
+    "  \n",
+    "````{panels}  \n",
+    "  \n",
+    "```{image} ./powerlaw_01.png  \n",
+    "\n",
+    "```  \n",
+    "\n",
+    "---  \n",
+    "  \n",
+    "The plot on the left shows the graphs for the functions $y = x^{\\alpha}$ (with $C=1$) for different values of the exponent $\\alpha$ (see the legend colours).  \n",
+    "  \n",
+    "There are a few interesting aspects to note in this plot. First, the graphs were presented only for positive values of $x$. Some of these functions are also defined (on the set of real numbers) for negative values of $x$ which could be presented on the plot. This is the case for $x^{2}$ or $x^{-2}$, but not the case for $x^{1/2}$ or $x^{-1/2}$.  \n",
+    "  \n",
+    "When analysing positive values of $x$ (as it is frequently the case for models in Biology), for $C > 0$, the exponent $\\alpha$ determines the behaviour of the functions, with **increasing** functions of x for $\\alpha > 0$ and **decreasing** functions of x for $\\alpha < 0$.  \n",
+    "      \n",
+    "````  \n",
+    "Many commonly used functions are examples of power functions, e.g. the square root $\\sqrt{x}=x^{\\frac 12}$ or the inverse function $x^{-1}$.\n",
+    "\n",
+    "Watch this [video](https://web.microsoftstream.com/video/bf7415e1-823f-449e-ace1-e379fac8fe0a) about additional properties of power law functions."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Exponential functions  \n",
+    "  \n",
+    "Similar to power functions, but with very different properties are the exponential functions:  \n",
+    "  \n",
+    "  $$f(x) = a^x, $$  \n",
+    "    \n",
+    "where $a$ is a positive constant. The parameter $a$ is called *base* and the variable $x$ is the exponent (remember that for power functions, the variable $x$ is the base, raised to the power of the parameter $\\alpha$).  \n",
+    "  \n",
+    "  Exponential functions are used to model many different real-world applications, from bacterial growth to radioactive decay.  \n",
+    "    \n",
+    "```{admonition} Example  \n",
+    "\n",
+    "Suppose we start a bacterial colony with a single individual on a Petri dish. At each time step, counted as an average reproduction interval, each individual bacterium duplicates, originating two other individuals. The number of bacteria in the colony as a function of time, $N(t)$, can be counted as:  \n",
+    "    \n",
+    "    \n",
+    "| $t$      |      $N(t)$   |\n",
+    "|----------|:-------------:|\n",
+    "| $0$      |  $1\\ (= 2^0)$ |\n",
+    "| $1$      |  $2\\ (= 2^1)$ |\n",
+    "| $2$      |  $4\\ (= 2^2)$ |\n",
+    "| $3$      |  $8\\ (= 2^3)$ |\n",
+    "| $4$      |  $16\\ (= 2^4)$ |\n",
+    "| $5$      |  $32\\ (= 2^5)$ |  \n",
+    "  \n",
+    "If no limitation (on space or resources) is imposed to the colony, the number of bacteria at each time step $t$ can be well modelled by an exponential function of the form:  \n",
+    "  \n",
+    "$$ N(t) = 2^t. $$\n",
+    "  \n",
+    "```  \n",
+    "  \n",
+    "Exponential functions of the form $f(x) = a^x$ (with $a>0$) can never assume negative values and their behaviour depends on the value of $a$.  \n",
+    "  \n",
+    "```{image} ./exponential_01.png  \n",
+    ":align: center\n",
+    "```  \n",
+    "  \n",
+    "```{tip}  \n",
+    "Note that the function *increases* with $x$ if $a>1$ (growth) and *decreases* with $x$ if $0<a<1$ (decay). To remember this, one can directly apply the properties of exponentiation. If $f(x) = (1/3)^x$ (thus $a = (1/3) < 1$), we have:  \n",
+    "  \n",
+    "$$f(x) = \\left(\\frac{1}{3}\\right)^x = \\frac{1}{3^x},$$  \n",
+    "  \n",
+    "and since $3^x$ is an increasing function of $x$, then $1/3^x$ should be a decreasing function of $x$ (as the denominator increases, the fraction decreases).\n",
+    "```\n",
+    "```{admonition} Example  \n",
+    "In the early stages of a new desease, we can approximate the spread as exponential. In this case $a$ is the number of people every infected person infects themselves. For $a<1$ every person infects less than one other person on average and the desease will slow down, for $a>1$ the desease will spread. At the beginning of the COVID-19 pandemic, an estimate for $a$ (or in this case $R$ for reproductive value) was $a=4$. If we start with 1 infected person, how many do we have after 10 infection cycles?\n",
+    "```\n",
+    "  \n",
+    "Two of the most used bases are $a=10$ and $a=e$, where $e=2.71828$ (called *Euler's number* or *Napier's constant*). Exponential functions with $a=e$ are commonly written as:  \n",
+    "  \n",
+    "$$f(x) = e^x = \\mbox{exp}(x) $$  \n",
+    "\n",
+    "The unique charachteristic of the exponential of base $e$ is, that it is its own derivative\n",
+    "\n",
+    "$$\\frac{\\text{d}}{\\text{d}x} e^x = e^x $$\n",
+    "  \n",
+    "````{panels}  \n",
+    "  \n",
+    "```{image} ./exp_lamb_01.png  \n",
+    "\n",
+    "```  \n",
+    "\n",
+    "---  \n",
+    "  \n",
+    "Another common representation of the exponential function is:  \n",
+    "  \n",
+    "$$ f(x) = e^{\\lambda x}, $$  \n",
+    "  \n",
+    "where $\\lambda$ is a real constant, usually called *rate constant*. The absolute value of the parameter $\\lambda$ controls how fast is the growth (in case $\\lambda > 0$) or the decay (for $\\lambda < 0$) of the function $f(x)$ for increasing values of $x$.  \n",
+    "  \n",
+    "For a fixed value of $x$, which $\\lambda$ corresponds to the highest and the lowest values among these exponential functions? Does it change if $x$ is positive or negative? How would it change if, instead of $a=e$, it was a different base $a<1$?\n",
+    "      \n",
+    "````  \n",
+    "The both formulations can be changed into one another via\n",
+    "\n",
+    "\n",
+    "$$f(x) = e^{\\lambda x} = ({\\underbrace{e^\\lambda}_a})^x.$$\n",
+    "\n",
+    "\n",
+    "We have exponential growth when $a>1$ and therefore when $\\lambda>0$ and decay when $0<a<1$ and therefore $\\lambda<0$."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Logarithmic function  \n",
+    "  \n",
+    "Let us start this section by understanding what the expression  \n",
+    "  \n",
+    "$$ \\mbox{log}_a b = x $$  \n",
+    "  \n",
+    "actually means. One possible way to read it is: \"$\\mbox{log}_a b$ is the number with which I have to exponentiate the base $a$ so that I find the number $b$. Since \"$\\mbox{log}_a b = x$, the above expression is equivalent to:  \n",
+    "  \n",
+    "$$ a^x = b.$$  \n",
+    "  \n",
+    "The expression $\\mbox{log}_a b$ is called the *logarithm of b in the base a* and it is also possible to define a logarithmic function given by:  \n",
+    "  \n",
+    "$$f(x) = \\mbox{log}_a x,$$  \n",
+    "  \n",
+    "where $a$ is the base of the logarithm and it is assumed $a>0$ and $a \\ne 1$.  \n",
+    "  \n",
+    "By defining a function like this, for each $x$ we will have a number $y = f(x)$ for which $a^y = x$. Therefore, the logarithmic function is only defined for $x>0$. Like the exponential function, the behaviour of the logarithmic function for increasing $x$ will depend on the value of the base $a$.  \n",
+    "  \n",
+    "````{panels}  \n",
+    "  \n",
+    "```{image} ./log_base_01.png  \n",
+    "\n",
+    "```  \n",
+    "\n",
+    "---  \n",
+    "Note that the logarithmic functions are *increasing* if $a>1$ and *decreasing* if $0<a<1$. For a fixed value of $x$ the functions showing the highest and lowest values $f(x)$ will depend both on the value of the base and on which interval $x$ belongs to ($0<x<1$ or $x>1$). Can you determine these values for each case?  \n",
+    "  \n",
+    "        \n",
+    "````\n",
+    "\n",
+    "````{tip}  \n",
+    "The similarities between the graphs for exponential and logarithmic functions are not coincidental. In fact, the graph for a logarithmic function with base $a$ can be obtained by drawing the graph for an exponential function with base $a$ and reflecting it in relation to the line $y = x$.  [MAKE COMMENT ON INVERSE FUNCTIONS]\n",
+    "  \n",
+    "```{image} ./log_base_02.png\n",
+    "```  \n",
+    "    \n",
+    "(here we use the common nomenclature $\\mbox{log}_e x = \\mbox{ln}\\ x$).  \n",
+    "````"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 13,
+   "metadata": {
+    "tags": [
+     "remove-cell"
+    ]
+   },
+   "outputs": [
+    {
+     "data": {
+      "image/png": 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",
+      "text/plain": [
+       "<Figure size 864x216 with 1 Axes>"
+      ]
+     },
+     "metadata": {
+      "needs_background": "light"
+     },
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "image/png": 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",
+      "text/plain": [
+       "<Figure size 432x324 with 1 Axes>"
+      ]
+     },
+     "metadata": {
+      "needs_background": "light"
+     },
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "#Codes for figures\n",
+    "\n",
+    "import numpy as np\n",
+    "import pandas as pd\n",
+    "import matplotlib.pyplot as plt\n",
+    "\n",
+    "def multiple_formatter(denominator=2, number=np.pi, latex='\\pi'):\n",
+    "    def gcd(a, b):\n",
+    "        while b:\n",
+    "            a, b = b, a%b\n",
+    "        return a\n",
+    "    def _multiple_formatter(x, pos):\n",
+    "        den = denominator\n",
+    "        num = np.int(np.rint(den*x/number))\n",
+    "        com = gcd(num,den)\n",
+    "        (num,den) = (int(num/com),int(den/com))\n",
+    "        if den==1:\n",
+    "            if num==0:\n",
+    "                return r'$0$'\n",
+    "            if num==1:\n",
+    "                return r'$%s$'%latex\n",
+    "            elif num==-1:\n",
+    "                return r'$-%s$'%latex\n",
+    "            else:\n",
+    "                return r'$%s%s$'%(num,latex)\n",
+    "        else:\n",
+    "            if num==1:\n",
+    "                return r'$\\frac{%s}{%s}$'%(latex,den)\n",
+    "            elif num==-1:\n",
+    "                return r'$\\frac{-%s}{%s}$'%(latex,den)\n",
+    "            else:\n",
+    "                return r'$\\frac{%s%s}{%s}$'%(num,latex,den)\n",
+    "    return _multiple_formatter\n",
+    "\n",
+    "class Multiple:\n",
+    "    def __init__(self, denominator=2, number=np.pi, latex='\\pi'):\n",
+    "        self.denominator = denominator\n",
+    "        self.number = number\n",
+    "        self.latex = latex\n",
+    "\n",
+    "    def locator(self):\n",
+    "        return plt.MultipleLocator(self.number / self.denominator)\n",
+    "\n",
+    "    def formatter(self):\n",
+    "        return plt.FuncFormatter(multiple_formatter(self.denominator, self.number, self.latex))\n",
+    "\n",
+    "xt=np.linspace(-4*np.pi,4*np.pi,num=500)\n",
+    "##\n",
+    "fig, ax = plt.subplots(figsize=(12, 3))\n",
+    "ax.plot(xt, np.sin(xt), color='b', lw=2, label='$sin(x)$')\n",
+    "ax.plot(xt, np.cos(xt), color='g', lw=2, label='$cos(x)$')\n",
+    "ax.tick_params(axis='both', labelsize= 15)\n",
+    "ax.set_xlabel('$x$', fontsize = 16)\n",
+    "ax.legend(fontsize=14)\n",
+    "\n",
+    "ax.axhline(0, color='black', lw=1)\n",
+    "ax.axvline(0, color='black', lw=1)\n",
+    "\n",
+    "ax.xaxis.set_major_locator(plt.MultipleLocator(np.pi / 2))\n",
+    "ax.xaxis.set_minor_locator(plt.MultipleLocator(np.pi / 12))\n",
+    "ax.xaxis.set_major_formatter(plt.FuncFormatter(multiple_formatter()))\n",
+    "\n",
+    "fig.tight_layout()\n",
+    "fig.savefig('sin_cos.png')\n",
+    "\n",
+    "##\n",
+    "xt=np.linspace(-2*np.pi,2*np.pi,num=500)\n",
+    "fig, ax = plt.subplots(figsize=(6, 4.5))\n",
+    "y = np.tan(xt)\n",
+    "y[:-1][np.diff(y) < 0] = np.nan\n",
+    "\n",
+    "ax.plot(xt,y, color ='r', lw=2)\n",
+    "ax.tick_params(axis='both', labelsize= 15)\n",
+    "\n",
+    "ax.set_xlabel('$x$', fontsize = 16)\n",
+    "ax.set_ylabel(\"$tan(x)$\", fontsize = 16)\n",
+    "\n",
+    "plt.ylim(-10,10)\n",
+    "ax.axhline(0, color='black', lw=1)\n",
+    "ax.axvline(0, color='black', lw=1)\n",
+    "\n",
+    "ax.xaxis.set_major_locator(plt.MultipleLocator(np.pi / 2))\n",
+    "ax.xaxis.set_minor_locator(plt.MultipleLocator(np.pi / 12))\n",
+    "ax.xaxis.set_major_formatter(plt.FuncFormatter(multiple_formatter()))\n",
+    "\n",
+    "fig.tight_layout()\n",
+    "fig.savefig('tan.png')"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 14,
+   "metadata": {
+    "tags": [
+     "remove-cell"
+    ]
+   },
+   "outputs": [
+    {
+     "data": {
+      "image/png": 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",
+      "text/plain": [
+       "<Figure size 864x216 with 1 Axes>"
+      ]
+     },
+     "metadata": {
+      "needs_background": "light"
+     },
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "image/png": 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",
+      "text/plain": [
+       "<Figure size 864x216 with 1 Axes>"
+      ]
+     },
+     "metadata": {
+      "needs_background": "light"
+     },
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "image/png": 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",
+      "text/plain": [
+       "<Figure size 864x216 with 1 Axes>"
+      ]
+     },
+     "metadata": {
+      "needs_background": "light"
+     },
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "import numpy as np\n",
+    "import pandas as pd\n",
+    "import matplotlib.pyplot as plt\n",
+    "\n",
+    "\n",
+    "#xt=np.linspace(0,15*np.pi,num=500)\n",
+    "xt=np.linspace(0,50,num=1000)\n",
+    "##\n",
+    "fig, ax = plt.subplots(figsize=(12, 3))\n",
+    "\n",
+    "y1=0.62*np.cos(0.52*xt) + 1.09*np.sin(0.52*xt)\n",
+    "y2=3.75 + 4.33*np.cos(0.26*xt) + 2.5*np.sin(0.26*xt)\n",
+    "\n",
+    "ax.plot(xt, y1+y2, color='g', lw=2)\n",
+    "ax.tick_params(axis='both', labelsize= 15)\n",
+    "ax.set_xlabel('$t$', fontsize = 16)\n",
+    "ax.set_ylabel('$f\\,(t)$', fontsize = 16)\n",
+    "\n",
+    "ax.set_xlim(0,50)\n",
+    "fig.tight_layout()\n",
+    "fig.savefig('comp_signal.png')\n",
+    "\n",
+    "fig, ax = plt.subplots(figsize=(12, 3))\n",
+    "\n",
+    "ax.plot(xt, y1, color='b', lw=2)\n",
+    "ax.tick_params(axis='both', labelsize= 15)\n",
+    "ax.set_xlabel('$t$', fontsize = 16)\n",
+    "ax.set_ylabel('$y_1\\,(t)$', fontsize = 16)\n",
+    "\n",
+    "ax.set_xlim(0,50)\n",
+    "fig.tight_layout()\n",
+    "fig.savefig('y1_signal.png')\n",
+    "\n",
+    "fig, ax = plt.subplots(figsize=(12, 3))\n",
+    "\n",
+    "ax.plot(xt, y2, color='y', lw=2)\n",
+    "ax.tick_params(axis='both', labelsize= 15)\n",
+    "ax.set_xlabel('$t$', fontsize = 16)\n",
+    "ax.set_ylabel('$y_2\\,(t)$', fontsize = 16)\n",
+    "\n",
+    "ax.set_xlim(0,50)\n",
+    "fig.tight_layout()\n",
+    "fig.savefig('y2_signal.png')"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 3,
+   "metadata": {
+    "tags": [
+     "remove-cell"
+    ]
+   },
+   "outputs": [
+    {
+     "data": {
+      "application/vnd.jupyter.widget-view+json": {
+       "model_id": "a785b675781e44dcb130fe4aba0b9568",
+       "version_major": 2,
+       "version_minor": 0
+      },
+      "text/plain": [
+       "interactive(children=(FloatSlider(value=0.0, description='c', max=2.0, min=-2.0), Output()), _dom_classes=('wi…"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "application/vnd.jupyter.widget-view+json": {
+       "model_id": "b9af681dfe384b9fbd3b14793a2feede",
+       "version_major": 2,
+       "version_minor": 0
+      },
+      "text/plain": [
+       "interactive(children=(FloatSlider(value=0.0, description='c', max=2.0, min=-2.0), Output()), _dom_classes=('wi…"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "text/plain": [
+       "<function __main__.vtrans(c)>"
+      ]
+     },
+     "execution_count": 3,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "from ipywidgets import interact\n",
+    "import matplotlib.pyplot as plt\n",
+    "import numpy as np\n",
+    "\n",
+    "def func(x):\n",
+    "    return x**3\n",
+    "\n",
+    "def htrans(c):\n",
+    "    x = np.linspace(-3,3)\n",
+    "    \n",
+    "    y = func(x+c) #constant c in the argument of the function\n",
+    "    \n",
+    "    plt.plot(x,y, label='$f\\, (x+c)$')\n",
+    "    plt.title('Horizontal translation')\n",
+    "        \n",
+    "    plt.xlabel('x')\n",
+    "    plt.ylabel('y')\n",
+    "    plt.legend(fontsize=12)\n",
+    "    \n",
+    "    plt.ylim(-8, 8) #remember to change the limits for better visualisation if necessary\n",
+    "    \n",
+    "    plt.axhline(0, color='black', lw=1)\n",
+    "    plt.axvline(0, color='black', lw=1)\n",
+    "    \n",
+    "    \n",
+    "\n",
+    "    plt.show()\n",
+    "\n",
+    "def vtrans(c):\n",
+    "    x = np.linspace(-3,3)\n",
+    "    \n",
+    "    y = func(x)+c #constant c summing the whole function\n",
+    "    \n",
+    "    plt.plot(x,y,  label='$f\\, (x)+c$')\n",
+    "    plt.title('Vertical translation')\n",
+    "    \n",
+    "    plt.xlabel('x')\n",
+    "    plt.ylabel('y')\n",
+    "    plt.legend(fontsize=12)\n",
+    "    \n",
+    "    plt.ylim(-8, 8) #remember to change the limits for better visualisation if necessary\n",
+    "        \n",
+    "    plt.axhline(0, color='black', lw=1)\n",
+    "    plt.axvline(0, color='black', lw=1)\n",
+    "    \n",
+    "    plt.show()\n",
+    "    \n",
+    "# create a slider\n",
+    "\n",
+    "interact(htrans, c=(-2.0,2.0))\n",
+    "interact(vtrans, c=(-2.0,2.0))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 6,
+   "metadata": {
+    "tags": [
+     "remove-cell"
+    ]
+   },
+   "outputs": [
+    {
+     "data": {
+      "application/vnd.jupyter.widget-view+json": {
+       "model_id": "12d5a3f30eab4d5395e543cb666f5e70",
+       "version_major": 2,
+       "version_minor": 0
+      },
+      "text/plain": [
+       "interactive(children=(FloatSlider(value=1.05, description='c', max=2.0, min=0.1), Output()), _dom_classes=('wi…"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "application/vnd.jupyter.widget-view+json": {
+       "model_id": "bd6164c6fecf4c72b93b1a39f7947633",
+       "version_major": 2,
+       "version_minor": 0
+      },
+      "text/plain": [
+       "interactive(children=(FloatSlider(value=1.05, description='c', max=2.0, min=0.1), Output()), _dom_classes=('wi…"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "text/plain": [
+       "<function __main__.vstetcom(c)>"
+      ]
+     },
+     "execution_count": 6,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "from ipywidgets import interact\n",
+    "import matplotlib.pyplot as plt\n",
+    "import numpy as np\n",
+    "\n",
+    "def func(x):\n",
+    "    return x**3-2*x\n",
+    "\n",
+    "def hstetcom(c):\n",
+    "    x = np.linspace(-3,3)\n",
+    "    \n",
+    "    y = func(c*x) #constant c multiplying the argument of the function\n",
+    "    \n",
+    "    plt.plot(x,y, label='$f\\, (cx)$')\n",
+    "    plt.title('Horizontal stretching/compression')\n",
+    "        \n",
+    "    plt.xlabel('x')\n",
+    "    plt.ylabel('y')\n",
+    "    plt.legend(fontsize=12)\n",
+    "    \n",
+    "    plt.ylim(-4, 4) #remember to change the limits for better visualisation if necessary\n",
+    "    \n",
+    "    plt.axhline(0, color='black', lw=1)\n",
+    "    plt.axvline(0, color='black', lw=1)\n",
+    "    \n",
+    "    \n",
+    "\n",
+    "    plt.show()\n",
+    "\n",
+    "def vstetcom(c):\n",
+    "    x = np.linspace(-3,3)\n",
+    "    \n",
+    "    y = c*func(x) #constant c multiplying the whole function\n",
+    "    \n",
+    "    plt.plot(x,y,  label='$c\\, f\\, (x)$')\n",
+    "    plt.title('Vertical stretching/compression')\n",
+    "    \n",
+    "    plt.xlabel('x')\n",
+    "    plt.ylabel('y')\n",
+    "    plt.legend(fontsize=12)\n",
+    "    \n",
+    "    plt.ylim(-4, 4) #remember to change the limits for better visualisation if necessary\n",
+    "        \n",
+    "    plt.axhline(0, color='black', lw=1)\n",
+    "    plt.axvline(0, color='black', lw=1)\n",
+    "    \n",
+    "    plt.show()\n",
+    "    \n",
+    "# create a slider\n",
+    "\n",
+    "interact(hstetcom, c=(0.1,2.0))\n",
+    "interact(vstetcom, c=(0.1,2.0))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 48,
+   "metadata": {
+    "tags": [
+     "remove-cell"
+    ]
+   },
+   "outputs": [
+    {
+     "data": {
+      "image/png": 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",
+      "text/plain": [
+       "<Figure size 720x360 with 2 Axes>"
+      ]
+     },
+     "metadata": {
+      "needs_background": "light"
+     },
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "fig, ax = plt.subplots(1,2,figsize=(10, 5))\n",
+    "\n",
+    "xt = np.linspace(-3,5)\n",
+    "\n",
+    "# x-axis reflection\n",
+    "ax[0].plot(xt, (xt-1)**2, color ='r', lw=2, label = '$(x-1)^2$')\n",
+    "ax[0].plot(xt, -(xt-1)**2, color ='b', lw=2, label = '$-(x-1)^2$')\n",
+    "ax[0].tick_params(axis='both', labelsize= 12)\n",
+    "\n",
+    "ax[0].set_xlabel('$x$', fontsize = 14)\n",
+    "ax[0].set_ylabel(\"$y$\", fontsize = 14)\n",
+    "\n",
+    "ax[0].set_title('x-axis reflection')\n",
+    "\n",
+    "ax[0].set_ylim(-10,10)\n",
+    "\n",
+    "ax[0].legend()\n",
+    "\n",
+    "ax[0].axhline(0, color='black', lw=1)\n",
+    "ax[0].axvline(0, color='black', lw=1)\n",
+    "\n",
+    "# y-axis reflection\n",
+    "\n",
+    "xt = np.linspace(-4,4)\n",
+    "\n",
+    "ax[1].plot(xt, (xt-1)**2, color ='r', lw=2, label = '$(x-1)^2$')\n",
+    "ax[1].plot(xt, (-xt-1)**2, color ='b', lw=2,  label = '$[(-x)-1]^2$')\n",
+    "ax[1].tick_params(axis='both', labelsize= 12)\n",
+    "\n",
+    "ax[1].set_xlabel('$x$', fontsize = 14)\n",
+    "#ax[1].set_ylabel(\"$y$\", fontsize = 14)\n",
+    "\n",
+    "ax[1].set_title('y-axis reflection')\n",
+    "\n",
+    "ax[1].set_ylim(-1,10)\n",
+    "\n",
+    "ax[1].legend()\n",
+    "\n",
+    "ax[1].axhline(0, color='black', lw=1)\n",
+    "ax[1].axvline(0, color='black', lw=1)\n",
+    "\n",
+    "fig.tight_layout()\n",
+    "fig.savefig('reflection.png')"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Describing shapes and patterns\n",
+    "  "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Measuring angles  \n",
+    "  \n",
+    "An **angle** is a primitive geometrical element, such as the *point*, the *curve* or the *plane*. It is defined as the figure formed by two line segments that extend from a common point.  \n",
+    "    \n",
+    "<br/>\n",
+    "    \n",
+    "```{image} angle.png  \n",
+    "\n",
+    "```  \n",
+    "  \n",
+    "<br/>\n",
+    "<br/>\n",
+    "      \n",
+    "The two most common units for measuring angles are the *degree* and the *radian*.  \n",
+    "  \n",
+    "  \n",
+    "`````{tabbed} Degrees \n",
+    "\n",
+    "````{panels}  \n",
+    "  \n",
+    "```{image} ./degree.png  \n",
+    "\n",
+    "```  \n",
+    "\n",
+    "---  \n",
+    "  \n",
+    "If we divide a full rotation by 360 equal parts, each of these parts is called a *degree* (noted as $1^{\\circ}$). Thus, the degree corresponds to $1/360$ of one full rotation. Each degree can also be divided in 60 equal parts called *minutes* ($1^{\\circ} = 60'$) and each minute further divided in 60 *seconds* ($1' = 60''$).  \n",
+    "  \n",
+    "+++  \n",
+    "In the figure, one degree (shown in red) and eighty nine degrees (shown in blue).\n",
+    "\n",
+    "  \n",
+    "````  \n",
+    "`````  \n",
+    "  \n",
+    "`````{tabbed} Radians  \n",
+    "\n",
+    "````{panels}  \n",
+    "  \n",
+    "```{image} ./Circle_radians.gif  \n",
+    "\n",
+    "```  \n",
+    "\n",
+    "---  \n",
+    "  \n",
+    "One *radian* is the angle comprised in a circunference when the size of the arc is equal to the radius of the circunference. If $C$ is the size of the arc and $R$ the radius of the circunference, the corresponding angle in radians $\\phi$ will be given by:  \n",
+    "  \n",
+    "$$ \\phi = \\frac{C}{R} $$\n",
+    "  \n",
+    "Thus, if we are considering the arc for the full circunference, $C = 2\\pi R$, and we will have:  \n",
+    "  \n",
+    "$$\\phi = \\frac{2\\pi R}{R} = 2\\pi, $$  \n",
+    "  \n",
+    "and the angle for a full rotation corresponds to $2\\pi$ radians.  \n",
+    "  \n",
+    "```{tip}  \n",
+    "When the angle is written in radians, the unit (radians or rad) is frequently omitted.\n",
+    "\n",
+    "```\n",
+    "  \n",
+    "````  \n",
+    "`````  \n",
+    "  \n",
+    "For a given angle, transforming between units of degrees and radians can be simply done with:  \n",
+    "  \n",
+    "$$\\frac{360^{\\circ}}{\\theta} = \\frac{2\\pi}{\\phi},$$  \n",
+    "  \n",
+    "where $\\theta$ is the angle measured in degrees and $\\phi$, measured in radians. In this lecture, we will consider the angles measured in radians (independent of the symbol used for the variable), unless stated otherwise. \n",
+    "  \n",
+    "```{admonition} Example  \n",
+    "If $\\theta = 90^{\\circ}$, then using the previous expression we have $\\phi = \\displaystyle\\frac{\\pi}{2}$. Since $\\theta$ and $\\phi are directly proportional, we also have:  \n",
+    "  \n",
+    "  $$\\theta = 180^{\\circ} \\longrightarrow \\phi = 2 \\cdot \\frac{\\pi}{2} = \\pi $$  \n",
+    "  $$\\theta = 270^{\\circ} \\longrightarrow \\phi = 3 \\cdot \\frac{\\pi}{2} = \\frac{3\\pi}{2} $$  \n",
+    "  $$\\theta = 360^{\\circ} \\longrightarrow \\phi = 4 \\cdot \\frac{\\pi}{2} = 2\\pi $$\n",
+    "  \n",
+    "```"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Trigonometric circle  \n",
+    "  \n",
+    "  \n",
+    "We obtain the trigonometric circle by superimposing a circle of radius $1$ to the cartesian system of coordinates:   \n",
+    "\n",
+    "```{image} trig_circ.png\n",
+    "```  \n",
+    "\n",
+    "  \n",
+    "The projections on the $x$ and $y$-axes have special meanings, if we consider the angle $\\theta$ between the line segment in the radial direction and the $x$-axis.  \n",
+    "  \n",
+    "From the geometric relations on the right triangle (triangle in which one internal angle is a right angle):  \n",
+    "  \n",
+    "````{panels}  \n",
+    "  \n",
+    "```{image} ./right_trig.png\n",
+    "\n",
+    "```  \n",
+    "\n",
+    "---  \n",
+    "  \n",
+    "The larger side of the triangle, opposing the right angle, is called *hypothenuse*, while the other two smaller side are called *catheti*. We define the *sine* and *cosine* of the angle $\\alpha$ as the ratios of the opposing and adjacent catheti, respectively, with the hypothenuse:  \n",
+    "  \n",
+    "$$\\mbox{sin}(\\alpha) = \\frac{a}{c}$$  \n",
+    "$$\\mbox{cos}(\\alpha) = \\frac{b}{c}$$\n",
+    "  \n",
+    "````  \n",
+    "  \n",
+    "Back to the trigonometric circle, it is possible to construct a right triangle where the hypothenuse corresponds to the radius of the superimposed circle. In this case, since this radius equals to $1$, we have:  \n",
+    "  \n",
+    "$$\\mbox{sin}(\\theta) = y$$  \n",
+    "$$\\mbox{cos}(\\theta) = x.$$  \n",
+    "  \n",
+    "In other words, the projections of the end of the radial line segment on the $y$ and $x$-axes correspond to the sine and cosine of the angle $\\theta$. Sine and cosine for this angle can be obtained by the projections even if $\\theta > \\displaystyle\\frac{\\pi}{2}$.  \n",
+    "  \n",
+    "### Trigonometric identities  \n",
+    "  \n",
+    "  With the aid of the projections on the trigonometric circle, some trigonometric identities become imediately clear. Consider $\\theta > 0$ for counterclockwise angle measurements starting from the $x$-axis and $\\theta < 0$ for clockwise measurements. We have:  \n",
+    "    \n",
+    "$$\\mbox{sin}(0) = 0,\\ \\ \\ \\mbox{sin}\\left(\\frac{\\pi}{2}\\right) = 1,\\ \\ \\ \\mbox{sin}(\\pi) = 0,\\ \\ \\ \\mbox{sin}\\left(\\frac{3\\pi}{2}\\right) = -1,\\ \\ \\ \\mbox{sin}(2\\pi) = 0$$  \n",
+    "$$\\mbox{cos}(0) = 1,\\ \\ \\ \\mbox{cos}\\left(\\frac{\\pi}{2}\\right) = 0,\\ \\ \\ \\mbox{cos}(\\pi) = -1,\\ \\ \\ \\mbox{cos}\\left(\\frac{3\\pi}{2}\\right) = 0,\\ \\ \\ \\mbox{cos}(2\\pi) = 1$$\n",
+    "  \n",
+    "From the fact that a full rotation brings you back to the same point:  \n",
+    "  \n",
+    "$$\\mbox{sin}(\\theta) = \\mbox{sin}(\\theta + 2\\pi)$$  \n",
+    "$$\\mbox{cos}(\\theta) = \\mbox{cos}(\\theta + 2\\pi)$$  \n",
+    "  \n",
+    "or from Pythagoras' theorem on the right triangle in the trigonometric circle:  \n",
+    "  \n",
+    "$$\\begin{align}\n",
+    "[\\mbox{sin}(\\theta)]^2 + [\\mbox{cos}(\\theta)]^2 =& 1^2 \\\\\n",
+    "\\mbox{sin}^2(\\theta) + \\mbox{cos}^2(\\theta) =& 1\n",
+    "\\end{align}$$  \n",
+    "  \n",
+    "```{tip}\n",
+    "The notation $\\mbox{sin}^2(\\theta)$ is frequently used to express the square of $\\mbox{sin}(\\theta)$, or:  \n",
+    "  \n",
+    "$$ \\mbox{sin}^2(\\theta) = [\\mbox{sin}(\\theta)]^2 = [\\mbox{sin}(\\theta)]\\cdot[\\mbox{sin}(\\theta)],$$  \n",
+    "  \n",
+    "which is different from $\\mbox{sin}(\\theta^2)$.\n",
+    "```  \n",
+    "  \n",
+    "  \n",
+    "Now watch this [video](https://web.microsoftstream.com/video/b3023026-7d45-4378-a8ee-c166a43eb2e7) for a different application of the same trigonometric construct.\n",
+    "  \n",
+    "  "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Trigonometric functions  \n",
+    "  \n",
+    "Since sine and cosine are defined for any real value of a given angle $x$ ($x \\in {\\rm I\\!R}$), we can define the functions:  \n",
+    "  \n",
+    "$$f(x) = \\mbox{sin}(x)$$  \n",
+    "$$f(x) = \\mbox{cos}(x).$$  \n",
+    "  \n",
+    "Thus, the association of sine and cosine with pure geometric ratios gives place for a more generic definition of functions of real variables. The graphs of these functions are given by:  \n",
+    "  \n",
+    "```{image} ./sin_cos.png\n",
+    "\n",
+    "```  \n",
+    "  \n",
+    "All the other known trigonometric relations can also be defined as functions in a similar way. Take for example the tangent of an angle $\\theta$, which is:  \n",
+    "  \n",
+    "$$tan(\\theta) = \\displaystyle\\frac{\\mbox{sin}(\\theta)}{\\mbox{cos}(\\theta)}.$$\n",
+    "\n",
+    "If $x \\in {\\rm I\\!R}$, than we can define a function  \n",
+    "  \n",
+    "$$ f(x) = \\mbox{tan}(x) = \\displaystyle\\frac{\\mbox{sin}(x)}{cos(x)}, \\ \\ \\ (\\mbox{for }\\mbox{cos}(x) \\ne 0).$$  \n",
+    "  \n",
+    "The graph from the tangent function will be given by:  \n",
+    "  \n",
+    "```{image} ./tan.png\n",
+    "\n",
+    "```  \n",
+    "  \n",
+    "Note by the above graphs that the functions sine and cosine have patterns that repeat themselves every interval of $2\\pi$, while the tangent repeats itself every interval of $\\pi$. We say that these three functions are **periodic**.\n",
+    "\n",
+    "\n",
+    "A function $f(x)$ is periodic if there is a positive constant $a$, so that:  \n",
+    "  \n",
+    "$$f(x + a) = f(x),$$  \n",
+    "  \n",
+    "for all values of $x$ for which the function f(x) is defined.  \n",
+    "  \n",
+    "If $a$ is the smallest number with this property, we call it the *period* of f(x).\n",
+    "\n",
+    "```{admonition} Example  \n",
+    "  \n",
+    "Since $\\mbox{sin}(x + 2\\pi) = \\mbox{sin}(x)$ for any value of $x$ and since $2\\pi$ is the smallest number for which this property holds, then sine is a periodic function with period $2\\pi$. With the same reasoning, it can be shown that the tangent is periodic with period $\\pi$.\n",
+    "```  \n",
+    "```{admonition} Example  \n",
+    "  \n",
+    "Many natural processes can be modelled as periodic. Think for example of a pendulum, the tides, the seasons. Often periodicity can be a good first order approximation, even if it does not strictly hold. For example when describimg the shape of the cloud cover in the picture below.\n",
+    "```\n",
+    "```{image} ./cloud.png\n",
+    "```  \n",
+    "  \n",
+    "### Amplitude and Period  \n",
+    "  \n",
+    "  \n",
+    "From the basic functions $\\mbox{sin}(x)$ or $\\mbox{cos}(x)$ we can write a more general function $f(x)$ given by  \n",
+    "  \n",
+    "$$f(x) = A\\mbox{sin}(\\omega x),\\ \\ \\ \\ \\ \\ A, \\omega \\in {\\rm I\\!R}\\ \\ (A, \\omega \\ne 0)$$  \n",
+    "  \n",
+    "Since the sine ranges from $-1$ to $1$ (see the graph for $sin(x)$ for reference), by its definition, $f(x)$ will range from $-A$ to $A$. We call $A$ the **amplitude** of $f(x)$ and it gives the extent of the variation of the oscillation of the function.  \n",
+    "  \n",
+    "Being sine a periodic function, $f(x)$ will also be, possibly with a different period. The period $T$ of $f(x)$, as discussed before, will be determined by the equation $f(x + T) = f(x)$, that should be valid for every $x$. Thus:  \n",
+    "  \n",
+    "$$\\begin{align}\n",
+    "    f(x + T) &= f(x) \\\\\n",
+    "    A\\mbox{sin}(\\omega(x+T)) &= A\\mbox{sin}(\\omega x) \\\\\n",
+    "    \\mbox{sin}(\\omega(x+T)) &= \\mbox{sin}(\\omega x) \\\\\n",
+    "    \\mbox{sin}(\\omega x+\\omega T)) &= \\mbox{sin}(\\omega x) \\\\\n",
+    "    \\mbox{sin}(z+\\omega T)) &= \\mbox{sin}(z)\\ \\ \\ \\ \\ \\ (\\mbox{where we change the variable to } z=\\omega x)\n",
+    "  \\end{align}$$\n",
+    "  \n",
+    "But since we know that sine is periodic with period $2\\pi$, we should have $|\\omega|T = 2\\pi$. Thus, the **period** of the function $f(x)$ will be:  \n",
+    "  \n",
+    "$$T = \\frac{2\\pi}{\\omega}.$$  \n",
+    "  \n",
+    "The constant $\\omega = \\displaystyle\\frac{2\\pi}{T}$ is called **angular frequency** and measures the number of full rotations the function $f(x)$ oscillates in one interval of $T$.  \n",
+    "  \n",
+    "```{admonition} Example  \n",
+    "  \n",
+    "The function $f(x) = 3.6\\, \\mbox{cos}\\left(\\displaystyle\\frac{\\pi}{3}x\\right)$ has an amplitude $A=3.6$, an angular frequency $\\omega = \\displaystyle\\frac{\\pi}{3}$, and a period $T = \\displaystyle\\frac{2\\pi}{\\omega} = \\displaystyle\\frac{2 \\pi}{\\frac{\\pi}{3}} = 6$.  \n",
+    "```"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Modelling complex periodic phenomena  \n",
+    "  \n",
+    "  \n",
+    "Watch this [video](https://web.microsoftstream.com/video/02f373a2-621b-4313-a18a-265f5fb65656) about periodic functions decomposition."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Transformation on graphs  \n",
+    "  \n",
+    "  \n",
+    "Starting from the elementary functions we have discussed  \n",
+    "      \n",
+    "$$ax+b,\\ \\ ax^2+bx+c,\\ \\ \\sqrt{x},\\ \\ \\mbox{e}^x,\\ \\ \\mbox{log}(x),\\ \\ \\mbox{sin}(x),\\ \\ \\mbox{cos}(x)$$  \n",
+    "  \n",
+    "we can introduce several transformations to obtain new functions, by adding or multiplying constants to the argument (function variable) or to the whole function. These are the simplest cases of what is know as *function composition* with the resulting function called *composite function*.  \n",
+    "  \n",
+    "When working with mathematical models, this will help you to understand how the shape of a given function might change by changing one of the parameters, or how to construct a set of functions (that might be useful for your particular model) with small modifications of basic functions.  \n",
+    "  \n",
+    "  \n",
+    "### Translations  \n",
+    "  \n",
+    "A given function can be dislocated, either vertically or horiontally, by *adding* a constant $c$ to the whole function or to its argument. In general, starting from a function $f(x)$, translations are built following the rules:  \n",
+    "  \n",
+    "- **Horizontal** (Constant $c$ summed to the argument of the function)  \n",
+    "\n",
+    "  $y = f(x + c)\\ \\ \\ \\  -  \\left\\{\n",
+    "\\begin{array}{ll}\n",
+    "      c > 0 & \\rightarrow \\ \\  \\mbox{function moves to the left} \\\\\n",
+    "      c < 0 & \\rightarrow \\ \\  \\mbox{function moves to the right} \\\\\n",
+    "\\end{array} \n",
+    "\\right. $  \n",
+    "  \n",
+    "- **Vertical** (Constant $c$ summed to the whole function)  \n",
+    "\n",
+    "  $y = f(x) + c\\ \\ \\ \\  -  \\left\\{\n",
+    "\\begin{array}{ll}\n",
+    "      c > 0 & \\rightarrow \\ \\  \\mbox{function moves up} \\\\\n",
+    "      c < 0 & \\rightarrow \\ \\  \\mbox{function moves down} \\\\\n",
+    "\\end{array} \n",
+    "\\right. $  \n",
+    "  \n",
+    "```{admonition} Example  \n",
+    "If we start with the function $f(x) = x^3$, the function $y = f(x + 2) = (x+2)^3$ will represent a horizontal translation, while the function $y = f(x) + 2 = x^3+2$ will represent a vertical translation.  \n",
+    "```  \n",
+    "  \n",
+    "To see how this works, try to implement the following code in your own jupyter notebook. Try to use different functions $f(x)$ (defined in 'func(x)') to see how different graphs behave to horizontal and vertical translations.\n",
+    "  \n",
+    "```{code}\n",
+    "from ipywidgets import interact\n",
+    "import matplotlib.pyplot as plt\n",
+    "import numpy as np\n",
+    "\n",
+    "def func(x):\n",
+    "    return x**3\n",
+    "\n",
+    "def htrans(c):\n",
+    "    x = np.linspace(-3,3)\n",
+    "    \n",
+    "    y = func(x+c) #constant c in the argument of the function\n",
+    "    \n",
+    "    plt.plot(x,y, label='$f(x+c)$')\n",
+    "    plt.title('Horizontal translation')\n",
+    "    plt.legend(fontsize=12)\n",
+    "    \n",
+    "    plt.ylim(-8, 8) #remember to change the limits for better visualisation if necessary\n",
+    "    \n",
+    "    plt.axhline(0, color='black', lw=1)\n",
+    "    plt.axvline(0, color='black', lw=1)\n",
+    "    \n",
+    "    \n",
+    "\n",
+    "    plt.show()\n",
+    "\n",
+    "def vtrans(c):\n",
+    "    x = np.linspace(-3,3)\n",
+    "    \n",
+    "    y = func(x)+c #constant c summing the whole function\n",
+    "    \n",
+    "    plt.plot(x,y,  label='$f(x)+c$')\n",
+    "    plt.title('Vertical translation')\n",
+    "    \n",
+    "    plt.xlabel('x')\n",
+    "    plt.ylabel('y')\n",
+    "    plt.legend(fontsize=12)\n",
+    "    \n",
+    "    plt.ylim(-8, 8) #remember to change the limits for better visualisation if necessary\n",
+    "        \n",
+    "    plt.axhline(0, color='black', lw=1)\n",
+    "    plt.axvline(0, color='black', lw=1)\n",
+    "    \n",
+    "    plt.show()\n",
+    "    \n",
+    "# create a slider\n",
+    "\n",
+    "interact(htrans, c=(-2.0,2.0))\n",
+    "interact(vtrans, c=(-2.0,2.0))\n",
+    "```  \n",
+    "  \n",
+    "  \n",
+    "### Stretching/ Compression  \n",
+    "  \n",
+    "A given function can also be stretched or compressed, either vertically or horiontally, by *multiplying* a constant $c$ to the whole function or to its argument. Starting from a function $f(x)$, stretchings/compressions are built following the rules:  \n",
+    "  \n",
+    "- **Horizontal** (Constant $c$ multiplied to the argument of the function)  \n",
+    "\n",
+    "  $y = f(cx)\\ \\ \\ \\  -  \\left\\{\n",
+    "\\begin{array}{ll}\n",
+    "      c > 1 & \\rightarrow \\ \\  \\mbox{function is horizontally compressed} \\\\\n",
+    "      0 < c < 1 & \\rightarrow \\ \\  \\mbox{function is horizontally stretched} \\\\\n",
+    "\\end{array} \n",
+    "\\right. $  \n",
+    "  \n",
+    "- **Vertical** (Constant $c$ multiplied to the whole function)  \n",
+    "\n",
+    "  $y = cf(x)\\ \\ \\ \\  -  \\left\\{\n",
+    "\\begin{array}{ll}\n",
+    "      c > 1 & \\rightarrow \\ \\  \\mbox{function is vertically stretched} \\\\\n",
+    "      0 < c < 1 & \\rightarrow \\ \\  \\mbox{function is vertically compressed} \\\\\n",
+    "\\end{array} \n",
+    "\\right. $  \n",
+    "  \n",
+    "```{admonition} Example  \n",
+    "For the function $f(x) = x^3-2x$, the function $y = f(2x) = (2x)^3-2(2x) = 8x^3-4x$ will represent a horizontal compression, while the function $y = 2f(x) = 2x^3 - 4x$ will represent a vertical stretching.  \n",
+    "```  \n",
+    "```{admonition} Example  \n",
+    "One example where we stretch or compress functions comes from chemical processes. Many chemical processes happen faster when the temperature is higher. If we describe the concentration of a certain chemical as a function of time $c(t)$, increasing the temperature will lead to a horizontal compression of the function.\n",
+    "```   \n",
+    "\n",
+    "Now play with the code below to see how different graphs behave to horizontal and vertical stretching/compression.\n",
+    "  \n",
+    "```{code}  \n",
+    "from ipywidgets import interact\n",
+    "import matplotlib.pyplot as plt\n",
+    "import numpy as np\n",
+    "\n",
+    "def func(x):\n",
+    "    return x**3-2*x\n",
+    "\n",
+    "def hstetcom(c):\n",
+    "    x = np.linspace(-3,3)\n",
+    "    \n",
+    "    y = func(c*x) #constant c multiplying the argument of the function\n",
+    "    \n",
+    "    plt.plot(x,y, label='$f\\, (cx)$')\n",
+    "    plt.title('Horizontal stretching/compression')\n",
+    "        \n",
+    "    plt.xlabel('x')\n",
+    "    plt.ylabel('y')\n",
+    "    plt.legend(fontsize=12)\n",
+    "    \n",
+    "    plt.ylim(-4, 4) #remember to change the limits for better visualisation if necessary\n",
+    "    \n",
+    "    plt.axhline(0, color='black', lw=1)\n",
+    "    plt.axvline(0, color='black', lw=1)\n",
+    "    \n",
+    "    \n",
+    "\n",
+    "    plt.show()\n",
+    "\n",
+    "def vstetcom(c):\n",
+    "    x = np.linspace(-3,3)\n",
+    "    \n",
+    "    y = c*func(x) #constant c multiplying the whole function\n",
+    "    \n",
+    "    plt.plot(x,y,  label='$c\\, f\\, (x)$')\n",
+    "    plt.title('Vertical stretching/compression')\n",
+    "    \n",
+    "    plt.xlabel('x')\n",
+    "    plt.ylabel('y')\n",
+    "    plt.legend(fontsize=12)\n",
+    "    \n",
+    "    plt.ylim(-4, 4) #remember to change the limits for better visualisation if necessary\n",
+    "        \n",
+    "    plt.axhline(0, color='black', lw=1)\n",
+    "    plt.axvline(0, color='black', lw=1)\n",
+    "    \n",
+    "    plt.show()\n",
+    "    \n",
+    "# create a slider\n",
+    "\n",
+    "interact(hstetcom, c=(0.1,3.0))\n",
+    "interact(vstetcom, c=(0.1,3.0))\n",
+    "```  \n",
+    "  \n",
+    "  \n",
+    "### Reflection  \n",
+    "  \n",
+    "If the constant $c$ multiplying the whole function or its argument has the particular value $c = -1$, we will have a *reflection*. Reflections can be done in relation to the $x$ or $y$-axes and they are built, starting from a function $f(x)$, following the rules:  \n",
+    "  \n",
+    "  \n",
+    "- **About the x-axis** - $y = -f(x)$  \n",
+    "  \n",
+    "- **About the y-axis** - $y = f(-x)$  \n",
+    "\n",
+    "The figure below shows these two reflections using the function $f(x) = (x-1)^2$.  \n",
+    "  \n",
+    "```{image} ./reflection.png\n",
+    "\n",
+    "```  "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 2,
+   "metadata": {
+    "tags": [
+     "remove-cell"
+    ]
+   },
+   "outputs": [
+    {
+     "data": {
+      "image/png": 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fv3PHmJQeEGA5b/yVK8xTqh9XxjqJTo6mcoHlCCpQ843NzW4k6YlKiqIKyysYPFhpmqxZ09Ons3PLwYHobNamRLbR6YgmTiRDmJ+lxpXJOn/d+cuQk7Tx6kbeckyCbChxICktieqtq2eIRxabOxIcbEwG5lFgITCQjW1ryxKTZayLiIQIQxx6p+2dzBZu9yHOhJ0xeLJWXlop+nuTJxs9O7dumVFgBiQlGXOi6tWTE+GtkV+v/WrYnfzz1p/ZPk5WDSUiZizpc5YKzS9Ez2LFWfGvXhGVLMnOq06dzBdu9yHOnDF6slatsuzYMpmj1qqp1eZWhpyhuJQ4i47/OPqxISdqwJ4BojfUfvmFDGFwu3ebV+P76HREQ4aw8QsVstyGmox4rr64Ss4znQkq0E8nfuItx2TIhhInXsa/pGKLihFUoJ67emZ6oXr1ipX+Boi++IJfrtDo0WRwuT/MeQE1GRORpkmjRhsaGXokJaQmcNGx8epGQ7jSv4/+zfTzGzaQoe/GkaylnJiMFy+IihblP7dk/pfLzy8bjO/AC4E5OlZ2DCUiNreabWxGUIFqr6mdaYVHtZqoUSN2PtWqRZTAZyrSxo1kCFf67z8+GmQyZvTB0QQVyGuu1wf7fpmbC88uGIr9zD8zP9PPBwUZje/lyy0gMAPS0oiaNWMa6tSRN7asifC4cENPu75/9pVs4YaMkA0ljgS/DDaUbv7YIkCjMfbLqFOHb8NMjYaoQwempXZtOWHYWvjm6DcEFajwgsIUHhfOVcvEfyYSVCDPOZ4fTRgODjZWAlu92oICM+D6daO3NjBn63EZExGdHG3wkA7dOzTHx8uuoUREFJkYaSjd3G93v48uAqZMYedR4cL8G2aOH8+05M9PFBbGV4sMY9O1TQQVyHa6LZ0KPcVVyx+3/jBsbH0s7y862tj/a2jOp2KOiIgwbhp/9ZW8sWUNaLQaar6xOUEFari+oUkq4FoTsqHEmZ03dxr6dlx7cS3Dz8yaxf4a3t5E4XzXwETESizrL1TjxvFWI3Po/iFDha4TT07wlkMarYbabGlDUIGabWyWYQhgQgJR+fLsHBowgIPIDNi5kwx9O65lPBVlLIROpzNUuKu+qrpJ+nTlxFAiYkVLnGY6EVSgX69lnKh58CAZQpNOnszRcCZBoyFq3ZppatbM8iGAMu/yIOqBYXN09WXOu0NvGXtoLEEF8lvkR2+S3vzP+zqdMZezRg3r8OJcvcp67AGm67Mnk330OaTe87zpRTzn3SEzIBtKVsDgvwcTVKBygeX+J2Tq/Hljee6DBzkJzICzZ4269u7lrSbv8jzuORWYW8DqYoJfJbwyxMDPOjnrf94fOJCdO/7+/EKTMkJfzcnadOU1lpxfYqgO+iDqgUmOmVNDiYho/ZX1BBXIdZYrPXrzbv36Z8+M5blnzszxUCbj1SvWYBkgmj2bt5q8S5omjeqsrUNQgT7f+bnVhCalalKp9praBBWo847O/6Nr0SIy5JBaU7j9+vVMl6ur+VpJyGTOmbAzhubzhx8c5i3HLMiGkhWQmJZIFZdXNCRW6omNNbq7x47lKPADzJnDtHl6WoenK6+h0+kMjYxbbGph8eINmXHw/kGCCqScpqTzT88bXv/tN6Pn5vp1jgIzIDHR6OkaOJC3mrzJrYhbZD/DnqBClhrKZoYpDCWdTkfdfutGUIEarG9gKIWv0xkbGbdqxSo5WhN6T5eNDdHFnLXwk8km3x79lqACFV1YNEPPDU8evXlkaB6evhDPrVvsOg2wNiHWhE5nrJbasKHsLeVBdHK0Idd+4j8TecsxG7KhZCXceHWDHH5yIKhA++/tJyKi/v3ZX6F6devstK7VskUBwFzzVrJBlmdYc3mNISHY1J3cTYU+rKPUklKUkJpAr16xfAmeCcGZcf26cXGwfz9vNXkLjVZDddfW/Z9NI1NgCkOJiOh14msqvKAwQQWafnw6EbEcO4B5lKy1zLy+EE/p0mxDQMZynAo9RYJKIMU0BZ18YgUxmRmw48YOggrkMsuFnkQ/IbWa5URbU3j0+7x+TeTjwzT+ZD0BFXmG/nv6G4rcpGpyb8K6bChZEfPPzCeoQL4LfWnPwVhDzwBLl0zOCqGhzPUNME+BjGUIiwkj11muBBVox40dvOV8kBR1ClVeUZmgAk04PIF69mTnSosW1m1Yz5/PdBYtyjy7MpZh7um5hmtgdpphfgxTGUpEREceHjEkwR++Gmy4Bu6w3qlIyclElSsznRNz7+av1ZGsTjb0S/r26Le85XwUvbe01eZWNHu2jgAiX1+iGNNORZNy+LDRWyo3Drcc/zz4x5BffyfyDm85ZuVjhpICMhZlTL0xqFOkDp7FPUPfX6cAAH78EShfnrOwj1CsGDBvHnseEAC8fs1XT16AiDBk3xDEp8Wjs39n9KjYg7ekD2JvY491ndZBISiw8NxC/HYyCE5OwJo1gCDwVvdhRo8GatcGnj4FvvmGt5q8we3I2/jhvx8AAGs6rIG7gztnRR+mRckWGFZrGDQ6DXpsGYz4BB26dAF6WO9UhIMDsG4doFAACxcCV67wVpQ3mHlyJu5G3YW/lz9+bPojbzkfZXm75fB09MQ/D//B93/8AoCdM+7WOxXRqhUwbBig0QBDhgA6HW9FuZ+EtAQM3jcYAKBqpkI5r3KcFfFDNpQsjFKhxPpO66EgW8T7r0TJ5icxaRJvVZkzaBDQvDkQGQmMGsVbTe5nc/BmHHxwEB4OHljRbgUEa7Y4ANQpUgdDq42GDjqg00DMmKlGiRK8VX0cGxtg/Xr274oVwKlTvBXlbnSkw6C9g5CqTcWAagPQtkxb3pIyZfans5FP6YNY1/NwarIKK1ZYt/EPAHXqsGu0VgsMHMgWlzLm48arG/j5zM8AgLUd18Lexp6zoo9T0KUgFrdeCgDQthiH3kPC0bo1Z1EimD0b8PEBzp0DVq/mrSb38/2/3+NJzBNUL1Qd4+uP5y2HK7KhxIGUsErQnfwWAKBuOxA6IZWzosxRKNiuk5MTsH07cPQob0W5l+jkaIz/h12YFrVeBB9XH86KxJF6aAYQXRzwuYbUmgt5yxFF5cpGb9LAgUCq9U9FybLp+iaceXoGBZ0LYkHrBbzliEKX7A7t3mXs+SdToHV6zlmROGbMAPz8gKtXgUWLeKvJvWh1WgzaOwganQbDag1Do2KNeEsShTqoN3C3A+AQi9Sm0lgEu7sDgYHs+ZQpQHg4Xz25mYvPL2LphaVQCmxj31Zpy1sSV2RDycLodMCIEQBOfgtPrT+eJt3H4vOLecsSRcmSwA8sagYjRwJpaXz15FZ++O8HvE56jaZ+TdGvaj/eckRx5QqwYZUzFAfZVt+Mk9MQFhvGWZU4vvsO8PcH7t0DFktjKkqO6ORoTDrCXOfzW81HPod8nBWJ44cfgPiLXZA/8jOkUDxGHZKGO93FBVi1ij2fOpWFl8qYnnVX1uHC8wso4loEsz+dzVuOKKKjgcmTBeDAMtgKDvjj3g4cf3KctyxRdO0KdOwIxMXJkS3mQkc6BBwIAIEwvv54VPepzlsSd2RDycJs2gRcvAj4eNthQw/m/p5xcgaex0ljp3LsWKBMGeDOHePujozpuPbyGlZeXgmloMSydsusPuQOYMZ/QABABIzp0AqfV/gcyZpkw8LY2rG3B5ayqYgZM+SdSnPw/b/fIzIpEo2LNUafyn14yxHF1avAypWAUilgR79lcLZ1xp+3/8SxR8d4SxNFmzbA558DycmQRHi31IhOjsZ3/34HgHn+rTnfLj3ffcfyjJtW88P3TVlkS8CBAKi1as7KMkcQgOXLAWdn4I8/gGPSmIqSYuO1jbgUfgmFXQvjh6Y/8JZjFciGkgWJi2MuYwCYMwf4rFJLdPHvgkR1IiYdlcadzN7eaCCpVMCLF1zl5CqICAEHAqAjHUbWGYlK3pV4SxLFli0sbrxgQbZ7Pa/lPDjaOOK3kN9wMvQkb3miaNkS6NIFSEyUF5WmJig8yGD8L2+3XDLG/4gR7N+RI4EWdXzxXWO2KB5zeAw0Omkk/sybxwo87Ngh5+CZGtVxFaKSo9DUrym6V+jOW44ogoKYp1GpZAbHpIYTUcqjFEIiQ7D80nLe8kRRtCjwLbPvMGaMnINnSmJSYjDlKFukzms5Dy52LpwVWQeyoWRBZswAXr0C6tcH+rzdVF3YeiEcbByw7cY2nA47zVegSFq3Bjp3BhISgIkTeavJPWy9sdWQw6FqpuItRxRxcUbDYs4cwM0N8Mvnh8kNJwMARh0cBa1Oy1GheBYsYBsBW7cCZ87wVpM7ICKMOTwGBMKouqNQuWBl3pJEkd74V6nYa2Prj0WJfCVwM+Im1gat5apPLH5+wGQ2FQ0FHmRyzs2Im1h+aTkUggJL2y6VhPFPxAwLIlbxs2JFwMHGAUvaLAEATD0+FS8TXnJWKY5x44DixYGbN4G10piKkkB1XGXw/H9R6QvecqwG2VCyEHfvsvwHQWBhPoq3v3zxfMXfWVTqSBp1LxcuZDuVW7eyUEKZnJGsTsY3x1hVgTkt5kgmjGP2bKPx37ev8fWJDSeimHsxXH91HWuvSONOVqKE0egbOVIuQWsK9tzZg9Nhp+Hl5IWpTafyliOKpCTjjvWcOcayyQ42Dpjfaj4Alkf4JvkNJ4VZY9Iktgt/7Rqr8iiTM4gIYw6NgZa0GFpzKKoUrMJbkij27AFOnwa8vFhLEj3ty7ZHh7IdEJcaB9VxFTd9WcHBAZjPpiJ++IHlXcnkjNuRt7Hs4jIoBAUC2wZKwvi3FLKhZCG++Ya5iAcMAGrVeve9yQ0nw9fNF1dfXsX2G9v5CMwiJUqw3SmA3YiJ+OqROovPL8azuGeoXqg6+lbtm/kXrICnT43FDxYvNhr/AOBk64T5LY2LyrjUOA4Ks86UKYCvL8tP2baNtxppk6ZNM4QUq5qqJGP8L14MPH8OVK/+rvEPAF38u6B58eaISo7CtOPT+AjMIk5Oxj54338PxMfz1SN1Dtw/gGOPj8HDwQPTm0/nLUcUaWnGTSCV6n97Js1rOQ9KQYl1V9bhduRti+vLDl27Ak2bAlFRwDRpTEWr5ptj30BLWgysPhBVC1XlLceqkA0lC3D2LLB7N7thzZjxv+872jpiRnP2xnf/focUTYqFFWaPKVOA/PmBEyeAfft4q5EukYmRmH2aVUya13IeFII0puWPPwIpKawBZ506//t+9wrd0aBoA7xOeo0FZ6VRDtrJCZj+du3z/ffs/08me6y+vBoP3jxA2fxlMbjmYN5yRBERAfzMWuJg3rx3jX8AEAQBi9sshgABKy+vxKPoR5YXmQ169GBe38hIFmIqkz20Oi2mHHvbKL7pj8jvlJ+zInGsWgU8eACUKwcMzmAq+nv5Y1CNQdCS8f/P2hEEYMkS9u+KFcAjaUxFq+R02Gn8dfcvONs6Y1pz2ep8H2msyCQMkTFGfOxY1jAtI/pW6YvK3pURGhuKFZdWWE5gDnB3N5YLnzxZTqrMLjNOzkB8WjzalG6DT0t+yluOKIKDgV9/BWxtgVmzMv6MIAiY02IOAGDBuQWSiX/v1w+oVAkIDWU3YJmsE5MSg2kn2A13bou5kunDMX0687i0bQt8+oGpWKVgFfSr2g9qnRo//vdjxh+yMgSBhRECLGTp1Su+eqTKluAtuBlxE37ufhhWaxhvOaKIiTF6XObOZdfsjJjabCqcbZ3x992/JVOEp2pV5vVVq98NJ5QRDxFh4hGWbD6+/ngUcinEWZH1IRtKZmbvXmNc8MeqaSkVSvzcgm1l/nTyJ0QnSyPodtgw1l/p9m1gwwbeaqTH/aj7WHl5JRSCAnNbzOUtRzSTJ7NNgGHDgFKlPvy5RsUaoVO5TkhUJ2LGiQzcqVaIUmn0KsycyRYaMllj7pm5iEqOQhO/JuhUrhNvOaK4dw9YvZp5keZmMhWnNZsGO6Udtt3Yhmsvr1lGYA5p3Jj1oElMzDiyQebjpGhS8MN/bGfwp09+gr2NPWdF4pgzB3jzhoWpdez44c8VcimESQ3ZImXCPxMkky89bRpgZ8dCpa9f561Geuy+sxvnn52Ht7M3JjSYwFuOVSIbSmZEo2G5SQDzvLi5ffzzbUu3RbPizTy/r08AACAASURBVBCdEo2fT/9sfoEmwM6OJfQDLPY5OZmrHMkx9fhUaHQafFn1S8lUBPvvP+DQIcDVlYWnZcbsT2dDISiw5soa3I+6b36BJqBdO7awePPGaDTJiCMiMQJLLrBKWnNbzJVMUvDUqeya3b8/8yh+DL98fhheazgIZCjCIgVmzWKG4OrVLBRLRjzLLi7D07inqFqwKnpX7s1bjigiIow94ubMYZ7FjzGu/jgUcimES+GXsPv2bvMLNAHFi7MNOyLjektGHBqdxnD9+rHJj3C1d+WsyDqRDSUzsmULcOsWK3wwZEjmnxcEweBVCLwYKJlQpc8/B2rUYD2VVq/mrUY6hESEYMfNHbBV2EqmHDiRMdxy0iSgQIHMv1OhQAX0r9YfGp0G3/8nwrKyAgTB6FVYulQOVcoKc07PQZI6CR3KdkBd37q85YgiOJj1GrKzYwaTGL5r8h1c7Vxx6MEhHH9y3Kz6TEWlSsCXXzKD8Ae5l6RoYlNiMesUizH+ucXPkskjnTOHVXHs0AGoK2Iquti54Icm7MSYenyqZFo7fPcd4OICHDzIcqZlxLH5+mbci7qHUh6lJJNHygNpzHYJotEAP/3Enk+bxvqziKF2kdroVK4TkjXJmHtGGqFYgmCMgZ49m4V2yGTOtBPTQCAMqjEIxdyL8ZYjiqNHWY8hT0/Wi0MsqmYq2CntsCtkF25G3DSfQBNSpw7QqRPzkmYWiiXDCI8Px4rLLLFrejNpVAQDjMbR0KGslLYYvJy8MLEBi+3/7t/vQBIp/akPVfrtNyAkhLcaabD0wlJEp0SjiV8TtC7VmrccUYSHG3Msp2dhKn5d/Wv4ufshJDIEv4X8Zh5xJqZAAWDC26ixb7+Vq/CKQa1VY8ZJFoM7telUyeSR8kA2lMzE5s3Aw4dA2bLAF1ns26VqqgIArLy8Ei/iX5henBlo3x6oXZu5+leu5K3G+gl+FYxdt3bBXmmPbxpLI16AyJgwO3EiC70Ti6+bLwbVGAQCGS7OUkDfbHTlSuClNBy8XJl9ajZSNCnoWr4rqvtU5y1HFJcvsx4zjo5ZD90ZU28MPB09cfbpWRx7fMw8Ak1M0aLAoEFsPsu5SpkTmxKLhecXAmC5aVIJJZ01i1Xt7NqVlboXi72NvcGrpDqugkYnjSpN48axDbyzZ4Fj0piKXNl0fRMexzxG2fxl8UVlubnsx5ANJTOgVhu9ST/8ANjYZO371X2qo7N/Z6RoUjDnzBzTCzQD6b1Kc+cCCQl89Vg7+sZ+Q2oOga+bL18xIjl8GDh/nhUmCQjI+venNJoiOa9S9epA587MqzRHGlORG2GxYVhzZQ0ECIbNHimgD0EbORIolMWCT672rphQn21lTzsxTTJepSlTmFdp507Zq5QZSy8sRUxKDJr6NUWz4s14yxFFWBiwdu279+Ws0K9qP5T2LI37b+5j8/XNphdoBlxdgfHj2XOVSvYqfYw0bRp+OsUWqT82+RE2iiwuUvMYsqFkBrZsYTX9s+NN0qNfaKy6vArh8eGmE2dG2rQB6tVjvTqWL+etxnq5+uIqdt/ZDQcbB0xpJI2eFUTG8KRJk1g8eFbxdfPF4BqDJedV0v9/r1rF8vBkMmbWqVlI06ahR8UekilMcvYsK0zi4sK8pNkhoE4APB09cTrsNP59/K9pBZoJX1/ZqySG9N4kqeSRAmyjNi0N6NUr88IkGWGrtDWsQaadmIY0bZppBZqJgADAw4OFh/8rjanIhV+v/YonMU/g7+WPXpV68ZZj9ciGkolRq403nh9/ZKWGs0PVQlXRtXxXpGpTJVMB732vktwBPmNUJ1QAgOG1hsPH9QONtayMgweBixcBb29g+PDsH0eKXqVq1YAuXVgYi+xVypjH0Y+x/up6KASF5BaUAMu38/LK3jFc7V0xvj7bypa9SrkLvTepWfFmkvEmPXoE/PILq24otjBJRvSq1AvlvcojNDYUW4K3mE6gGXFzYyF4AFuLSGQqWpT3vUlKRTYXqXkI2VAyMZs3A48fsw7YvXJoqOt3dNYErcHzuOc5F2cBWrYEGjZkZZUDA3mrsT4uh1/G33f/hpOtk6FnhbXzvjfJ2Tn7xyriVsTgVZp+QjrJ/vpcpVWrWJK0zLvMODkDGp0GfSr3gb+XP285oggKYhsAzs6sGXhOCKgTAA8HD5wKOyWZCniyV+njxKTEGL1JEgolnT6dFZPq25etQ7KLUqHEd42/AwDMPj1bMrlKI0cyr9KpU8Dx47zVWB+/XP0FYbFhqFCgAnpU7MFbjiSQDSUTkj43KSfeJD2VC1bG5xU+R6o2FbNPz865QAsgCMYKO/PnA7GxfPVYG1OPM4sjoHYACroU5KxGHPv2sYT3ggVZv4qcMqXRFNgr7bHrlnS8SlWqAN26Aampcl+l93kS8wSbrm+CUlDix6Y/8pYjmpkz2b/DhgH58+fsWG72bgavkt5jLAVkr9KH0XuTmhdvjqbFm/KWI4rHj1nov1JpLLyTE3pW6olSHqXw4M0D7ArZlfMDWgB3d+PGR3bys3IzqZpUzDzFLnxTm06VvUkikQ0lE7JpE7tQ+fsDPXua5phTm06FAAFrr6yVjFepeXOgSRMgOlqugJeeqy+u4sD9A3CydZJMB2wio+E7ZQrg5JTzYxZxK2Lo2SAlr5Leq7ZmDfBcGlPRIsw7Mw9a0qJ35d4o7VmatxxR3LwJ7N7N2jboE8Bzysi6I+Hh4IGToSdlr5LEiUmJwaLziwCwe7BUmDcP0GqBPn2AkiVzfjwbhQ2+acRKQc48NRM60uX8oBZg1CggXz7WU0n2KhnZeG0jnsY9RSXvSuheoTtvOZJBNpRMhEbDynECrIpSTr1Jeip6V0T3Ct2Rpk3DwnMLTXNQMyMIrAEcACxaxCqGyQA/n2GuiCE1h6CAs4hOrVbA0aPMm1SgADDYhP3oJjecbPAq3Yq8ZboDm5HKlYHu3ZlXadEi3mqsg5cJL7H+6noAkExhEsB4rR40KOuV7j6Em70bxtZjW9n6qpZSIL1X6fZt3mqsg+UXl0vOm/TiBbBhA7v/TjHhVOxbtS+KuhVFSGQI/rrzl+kObEbc3YExY9hz2avE0Og0hirK3zf+XjJNk60B+ZcyEX/8wZIoS5UynTdJj34BsjpoNaKSokx7cDPRsiVQowbrq7RhA281/LkXdQ+7QnbBVmGLcfXH8ZYjmtlvIz7HjDGNN0lPEbci+Lr61wAgmcbKgLHPzqpVLA8vr7Po3CKkalPRxb8LKhSowFuOKO7dY81WbW2zX+nuQ4yqOwr5HPLhROgJnAk7Y9qDmwlfX2DAAOZVkhsrA0nqJCy+sBgA8H2T7zmrEc+iRWwTp0sXoHx50x3XTmlnyKf96dRPkilWMno0M5iOH2dV8PI6O0N24nHMY5TxLCN7k7KIbCiZACLjgnLSJNN5k/TU8KmBVqVaIVGdiOWXpFF3WxBYh2yAhQOo1Xz18GbumbkgEL6s+qVk+iZduAD89x/rT5GTSncfYkKDCVAKSmy9sRVhsWGmH8AM1KgBtGoFJCYCy5bxVsOX6ORorLi8AgAM4TlS4OefAZ0O+PJLoFgx0x7b3cEdAbVZkzGp9MADmMGoULD8lqdPeavhy4arG/A66TXqFKmD5sWb85YjivRh7lltmiyGr6t/jYLOBXHlxRUcenDI9AOYgXz5jP3+8nq1UiIyVE+e1HCSnJuURWRDyQQcPgxcvw74+LCbrznQL0SWXliKxLRE8wxiYrp0YVV3QkOBHTt4q+HHs7hn2HR9ExSCQjKV7gBj0YLhw9lNx9SU8CiBnpV6QqPTSCasFDAuRJYuZQZTXmXZxWVISEtAy5ItUbtIbd5yRPHkCatMqlCYNjwpPaPqjoKjjSP23tsrmWIlJUuySAiNBlgonaloctRaNeadnQcAmNJwCgRB4KxIHMuWsSbvrVoBtWqZ/viOto6GvNoZJ2dIxqs0ahTg4ADs3cvyEvMq++/vx42IGyjsWhh9q/TlLUdyyIaSCdB7k8aOZcnB5qCpX1PU862HqOQorL2y1jyDmBiFApg8mT3X7+LmRRacXQC1To3uFbqjTP4yvOWI4tYtYM8edj7rY73NweSG7ARZe2UtXie9Nt9AJqRpU9ZYOSoKWLeOtxo+JKQlGMKTpORNmjePGQO9e7MwaXNQwLkABlQfwMZ7u+iWAvpr9Zo1wGtpTEWTs+PmDoTFhqFc/nL4zP8z3nJEkZAALGZT0RDFYQ6G1hoKT0dPnHt2TjLFSry9WVgpkHfDSonIUDV5XL1xsLcx0yI1FyMbSjnk7Fng5Em24z5kiPnGEQTBsCBZcG6BZDpl9+nDYuBv3QL+/pu3GsvzOuk11lxZA0BaC0p9qMKAAaZLds+IKgWroF2ZdkhSJyHwgjQab6VPlp4/H0iTxlQ0KWuD1uJN8hvU860nmUackZHGfElzeZP0jK8/HkpBiW03tiE0JtS8g5mIqlWBtm2BpKS8GVaqI50hXHJyw8mSSXZfu5blSzZowKrNmgsXOxeMqct2zaQUVjphAkuH2L6dRbfkNU6HncbZp2fh4eBhqDYrkzWkcSWwYvThSSNGsK7Q5qRD2Q6oUKACnsU9w9bgreYdzETY2bELFcA8bxLx2JuMpReWIkmdhLal26JaoWq85YgiNBTYto3dXEyd7J4RUxqyVWvgxUAkpCWYf0AT0LEjUKEC8OwZ+63yEqmaVMw/Nx8A8G2jbyUTnrRyJZCSArRrB1SsaN6xSniUQK9KvSQXVqo3IAMDmaciL7Hv3j6ERIbA180Xfar04S1HFKmpbLMGYN4kc0/F4bWHw8nWCYcfHkbwq2DzDmYiSpTI22Glem/SyDoj4WrvylmNNJENpRxw8yaLfXVwYLGw5kYhKAyLyjln5kCr05p/UBMwcCDg5QVcvMiKA+QV4lPjEXiReUm+bWzGmAgTM38+u6n06sVuMuamsV9jNCzaENEp0VgbJJ2wUv2ics6cvBVWuvXGVoTHh6OSdyW0L9uetxxRpKQYvSQTLNTCTIphpY0bA/XrMw9FXgorTR+eNL7+eNgp7TgrEsfWrUB4OGuI3a6d+cfL75TfUK10/tn55h/QREx6mxq8bl3eCiu99vIaDj44CCdbJ4yqa4FFai5FNpRygD7m9euvWSysJehVqRf83P1wN+ou9tzZY5lBc4izMyvVCRg9cHmBdVfWISYlBg2LNkSjYo14yxFFZCSwnrXFMXt4Unr0JfClFFbaqxfg5wfcucPyufICRGTwkExqMEky4UmbN7Nzu3p1oFkzy4xZuWBltC/THsmaZCy7KI1YtvRhpQsW5J2w0lNhp3D+2Xl4OnpiYI2BvOWIgsjoIZkwwfzeJD1j642FQlBg+83teBorjRKJeTWsVB8iObjGYOR3ys9ZjXSRxl3OCgkLM4YnWWqHEgBslbaG6jP68BcpMHw468Nz5AgQLA2PfY7Q6DSGZHcpVbpbsYI1CG7fHqhUyXLjtivTDpW8K+F5/HPJhJXa2hrn/jzp5OzniMMPDyMkMgSFXQujZyUTN4wzEzodnwUlYPQqSSmstEMHY1jpVmlMxRyjL508ss5IuNi5cFYjjsOHgZAQoHBh0/du/BglPErg8wqfQ6PTYOmFpZYbOIekDyvNC9VKn8Q8wc6QnbBR2Eiqd6M1IhtK2WTJEkCrZReo4sUtO3b/av3h4eCB88/O4+zTs5YdPJt4ejLPG5A34oR/v/U7wmLDUDZ/WXQo24G3HFEkJwPL37bpsqTxD7Cw0kkNmEG54NwCyZSf7d8f8PAAzp9nhV1yO/pwm1F1RkkmPOnAAeb18/UFPv/csmM3KtYIDYo2wJvkN1h/Zb1lB88mCoUxVGnBgtyfVxoSEYKDDw7C0cYRAXUCeMsRzYIF7N/Ro1kusCWZ2IAlr64OWo3YlFjLDp5NGjdm1UrzSljpkvNLoCMdelXqhaLuRXnLkTSyoZQNYmNZpRkAGD/e8uM72zljWK1hAKQVJzxmDLsJb9sGPH/OW435ICLD32VcvXGSCU/asoWFJ9WowUpgW5qelXqisGthhESG4PDDw5YXkA2cnYFhbCoakqpzK9deXsOxx8fgbOssqepJ+gXlmDHMC2hJBEEwbAAsvrAYGp3GsgKyyRdfME9FSAjzXORm9KGkX1X7Cl5OXpzViOPaNeDoUcDFBRjMYSrWLFwTzYs3R3xaPNYErbG8gGwgCMYS+IsXszzc3EpMSgzWXWXW4Pj6HBapuQxprOCsjHXrgPh4FuteowYfDQF1AmCntMOeO3vw4M0DPiKySMmSQLdugFrN3N+5lZOhJxH0IgheTl7oV7Ufbzmi0OmMC0pLhyfpsVPaYVQdlnC64NwCywvIJgEBbEd3zx7ggTSmYrbQLygH1hgID0cPzmrEERQEHD8OuLqyojI86FC2A0p7lsaTmCfYfXs3HxFZxM4OGDmSPV8gnamYZV4mvMSWG1sgQMDYemN5yxGNPirj66/N0wxcDHqv0pILSySTV9qxI1C6NGs8vVsaUzFbrAlag4S0BHxS4hPJVNu1ZmRDKYuo1SzsDuDjTdLj4+qDPpX7gEBYfH4xPyFZRP+brV7NjM3ciH6RP6L2CDjaOnJWI44DB4C7d4GiRYHu3fnpGFxzMJxtnXH00VFcf3mdn5As4OPD+oURGRs/5jaexz3H9pvboRAUGF13NG85otEv8gcPBtzd+WhQKpSGRbiUwkqHDGEe06NHgevSmIpZZvnF5UjTpqFTuU6SaQb+/DnrCaRQGIsk8aBN6TaGvNLtN7bzE5IFlEpjA/XcGlaapk0z5I5NqG/hGPpcimwoZZHffweePgXKlbNMOc6PoU/Q23B1A6KSoviKEUndukCjRkBMjLH5Y27i7uu72HtvL+yV9hheezhvOaJJH+9u6fCk9Hg4ehjKzy48L51ktnFvc2V/+YXFwOc2Ai8GQqPToFv5bijhYYGa8SYgLAzYuZMtjizRvuFjfFXtK3g6euLC8wuSySv18GANp4HcmVeapE7CyssrAUgrPCkwkIWNdetmmfYNH0IQBMNCfP65+ZLZAPjqK3ZuX7iQO/NKd4bsxPP456hQoALalG7DW06uQDaUsgCRMQ9h3Di2o8OTSt6V0LpUayRrkrHq8iq+YrKAvlDAokW5L05YH57Ur2o/eDtbqGZ8DtGHJ7m5AYMG8VYDjKk3hpWfvbEdz+OkkcxWqRLQujUrP7tKOlNRFAlpCVgdtBqAtBaU6QvuFCvGV4uTrZMhr1RKYaX6vNLt21m/ntzEr9d+RVRyFGoXri2Z9g3x8cbrC8+IFj1fVP4CRVyL4GbETRx6cIi3HFGkzyvNbWGl7+dHS6UZuLUjG0pZ4MQJ4MoVoEABoG9f3moY+lLhyy4tQ6omlbMacXTsCJQpA4SGAn/8wVuN6YhIjMCm4E0AIKlynPqbxaBBzFjiTQmPEuhavivUOrWhYa8U0C9cAgOBVGlMRVFsuLrB0A+srm9d3nJEwbvgTkZINa+0S5fcl1eqIx0WnV8EgBn/UllQbtjAzu1GjVh0Bm/slHaGUNx5Z6XTIyF9XunDh7zVmI5/H/+L66+uw9vZG32q9OEtJ9cgG0pZQL+gHDECcLSS1JNPS3yKKgWr4GXCS2y7sY23HFEoFMbFS26KE15xaQVSNCnoULYD/L38ecsRhTWFJ6VH77lYHbRaMv1nWrQAqlQBXr5klR1zAxqdxrCg1G/KSIG1a/kX3HmfQi6FJJ1XumoVkCCNqZgpe+/uxf039+Hn7oduFbrxliMKjcaYA2ktxj/A8kpd7Vzx35P/cDn8Mm85ovDxAXr3zn15pXpv9cg6I+Fg48BZTe5BNpREcucOsG8f4ODAmqdaC4IgGBaVUkoU7tcP8PICLl0CTp3irSbnJKuTsfwSa0IkpQRKawpPSk8933poULQBYlJisOGqNJLZBMG4gFm4MHdsAOy+vRtPYp6gtGdpdCzbkbccUaQvuGPpfmCZofc0/3LtF7xJlkYyW/367BETw3LwcgP6BeXouqNho7DhrEYcu3ezam2lS7OoDGvB3cHd0C5ASmGl+rzSDRtyR17prchbhn5g+jBfGdMgG0oiWcQ2VdGvHwu9syZ6Veoluf4zjo7MMwfkjv4zm65vwuuk16jpUxNN/JrwliMKawxPSo/e4Fx8fjG0Oi1nNeLo1Yv1n7l5E/jnH95qcgYRGRY+Y+uNhVKh5KxIHDt3As+eAf7+QNu2vNW8iz6vNEmdJKm8Uv31YfFitrEiZS49v4RTYafgbu+OgTU41YzPIunzo8eOZREA1oTe4NwVsguhMaG85YiicmWgVSuWV7p6NW81OSd9P7D8Tvk5q8ldyIaSCCIjgU0s9QRjrbDVglT7zwwfzjx0e/cyj51U0ZHOUKFtQoMJkol3t8bwpPR0KtcJpTxK4XHMY+y+I42mF7mp/8zZp2dx4fkFeDp64qtqX/GWIxr97z5+PP+COxmhjwAIvBgombzSzp1ZvtKjRyyvQ8ro75GDaw6Gq70rZzXiOHMGuHgRyJ+fVW2zNoq6F0WPij2gJS2WXFjCW45o0ueVpkmjFVSGvEx4ic3BmyXXD0wqmO02IghCBUEQjgmCkCQIQrggCNMFQbCyfRBxrFgBpKQAHTqwXUprJH3/mWsvr/GWIwpvb+DLL9lzvcdOiuy/tx/3ou6hmHsxdK/AsQlRFrDm8CQ97/efkQr6/jNHjgDBwbzVZB/9bz6s1jA42TpxViOeq1fZteX//o+3koxpUbKFIa90+01p9p+RKqExofj91u+wUdhgVF0rSsrMBP1vPmwY4GSlU1G/AbD2ylrEpMRwViOOli1ZxdIXL1hlR6mi7wf2mf9nkukHJiXMYigJguAB4CgAAvAZgOkAxgOYZo7xzElyMrBsGXtujeFJejwcPQxhBHoXrBTQe+h+/RWIiOCrJbvMP8diIsbUHSOZePddu6w3PCk9X1X7Ch4OHjj/7Lzcf8aCPHjzAHvu7IGd0g4BdQJ4y8kyAQHMW22NCIKAcfVYgsTCcwslk1favz87t8+dYw8psuTCEmhJi54Ve8LXzZe3HFHcvw/89RfzVuvD1a2RGj410Lx4cySkJWBt0FreckQhCMZcJakWlpJqPzApYS6P0lAAjgC6EtERIloFZiSNEwTBCgoQi2fzZuD1axaa1LQpbzUfZ3Td0az/zM3teBb3jLccUZQrxxJTU1OZ505qXHp+CSdDT8LN3g1f1/iatxxRWFs/sI/hbOcs6f4z27ZJs//MonOLQCD8X+X/QyGXQrzliOLWLfavo6OxT4q18kXlL+Dj4oMbETdw5NER3nJE4eICDB3KnkvRqxSbEot1V9YBkNaCctEids3+v/8DCln5VNRXxlxyYQnStNKIZevdm/2uN24AR4/yVpN19P3A6hSpg4ZFG/KWkysx1xKpLYDDRBSX7rUdYMaTlZsbRnQ6447whAls98GaKeFRAt3Kd4NGp0HgBek0vdB76pYvZx48KWGId68xGG720tgDOH7cGJ5kLf3APoa+/8zu27vx8I00ml6ULAl07SrN/jNRSVH45RorbyalfmD6a/VXX7GKmtaMndIOI+uwZDYpbQAEBAC2tqwC26NHvNVkjbVX1iI+LR7NizdHdZ/qvOWIIioK2LiRPR8nganYpnQblPcqj+fxz7EzZCdvOaKwt2fnNSC9CACtTmvIj5ZSP7Bz59i5LRXMZSj5A3gnPZ+IwgAkvX1PEhw4ANy9CxQtCnSXRuqJYUdnddBqxKfGc1YjjiZNgJo1medu82beasQj9Xj3ESOsNzwpPT6uPuhduTcIZOjpIwWk2n9m1eVVSNYko03pNqjoXZG3HFG8fGm8dlhjwZ2MGFJrCJxsnfDPw39w49UN3nJEUbgw24HX6aTVf0atVRuKDEjJm7RyJds8bNsWqCiBqagQFJJsVzJ0KPNEHzoEhITwViOevff24sGbB/Bz90PX8l15yxFFWhrw+eesHYk+CsDaEcxxIguCoAYwkYgWv/f6MwCbiOjb914fDGDw2/+saXJBMjIyMjIyMjIyMjIy/0sQEdXK6A1zZidkZIEJGb1ORGuIqBYR1apZsyaIyGoeajUhOZm/jqw89tzeA6gAv0V+UGvV3PWIeaSlEYoWJQCEvXv568nsEZMcA9dZroAKCAoP4q5H7GPQIPYbDxnCX0tWH602twJUwMyTM7lrEfv44w/2e5csSdBo+OvJ7LHhygZABVRdWRU6nY67HjGPxESCpycZbi289WTl8SDqAQSVANvptgiPC+euR+yjZUv2e8+ezV9LZg+dTofqq6oDKmDN5TXc9Yh9rFvHfuOqVQk6HX89WXnMODEDUAFtt7TlrkXs4/59giAQ7OwIL17w15PZ48KzC4AKcJ/tjriUOO56xDx0OkK1auy8XreOv570j49hLkMpGkC+DF53ByCNupFvsbGRRnhSejqW64gynmUQGhuKP2//yVuOKGxtjeVnpdCAdt2VdYhPi0ez4s1Qw8cKmxBlQESEdfcDywx9A1op9Z/57DOgVClp9J8hIknGu2/cCLx5A9Sty1tJ1inlWQpdyneBWqfGsovLeMsRjT6sdOlS6+8/c/zJcVx9eRXezt7oW1UCSZlgxRv0+TLjx1t/fvT7DKs1DI42jjj44CBCIqQRy1a6NLtep6UZKx1bM1LsB/bff8C1ayw/uk8f3mrEYy5D6Q7ey0USBKEoAGe8l7skY3oUgsKQhD3/7PxMrWVrYeBAwM0NOHECCArirebDSDXefcUKVl2wY0dWbVBqyP1nzMs/D//BzYibKOxaGD0r9eQtRxRarbEHmzW3b/gY+mvIyssrkZiWyFmNOFq1kk7/Gf2CckTtEXCwkcau56FDLH+jSBGgpzSm4jvkd8qP/tX6A5BWuxJ9T8GVK4FEK56KT2KeSDo/2prbN2SEuQylgwBaC4KQ3sztCSAZwAkzjSmTjn5V+yG/QYB1bwAAIABJREFUY35cCr+E02GnecsRhZsbMGgQe27Ni8rfb/2Op3FPUS5/ObQr0463HFEkJ7OqgoB0F5Ry/xnzou8HNqrOKNgp7TirEcfffwMPHgDFiwNduvBWkz0aFG2A+r71EZ0Sbag2aO0IgvE6Ys39Z25H3sb++/vhYONgaDMgBfRRFaNGsf5JUmRs/bEQIGDLjS14mfCStxxRNGjAPNNv3hirDVojSy8shY50kuoHdusWK5AmhfYN72MuQ2kVgFQAfwqC0OJtsQYVgIX0bslwGTPhZOuEEbVZdzr9AkgKjBrFduF37gTCwnir+V/ShyeNrTcWCsGKmxClI30/sCZNeKvJPlLsP+PsbP39Z669vIajj47Cxc4FQ2oN4S1HNPrfc+xYFiYtVfRepUXnF0Gr03JWI44vvrD+/jP6KplfVv0SBZwLcFYjjqtXgX//ZX2rBg/O/PPWSmnP0ujs3xlp2jTJhJUKgtGrtGgR81hbG1LtB6YPJf3yS+tv3/A+ZlnlEVE0gE8BKAHsBWs2uwjAVHOMJ5MxI+qMgL3SHnvv7sW9qHu85YiiWDGgRw92gVq6lLea/+VU2ClcDr8MLycv9Kvaj7ccUeh074YnSS3ePT1y/xnzoA+PGVh9IPI5ZJRean2cPw+cOQPkywcMGMBbTc7o7N8ZJT1K4lH0I/x19y/eckRhbw+MZFPRKjcAIhIjsOk6S8ocW086SZn633LgQHZuSxn9Qn7FpRWSCSvt0gUoUQJ4+JB5rK0NKfYDe/WKbdYKgjTzo822HU5Et4joEyJyJCIfIvqBiKzQPs+9eDt7o2+Vvqz/zDnp9Z9ZswaIjeWr5X30i/PhtYbD0daRsxpxHDwI3LkD+Pqy/gVSJ33/meBXwbzliMKa+888i3uG7Te3QykoMbreaN5yRKNfUA4dynbfpYxSoTQs5ueflU4EwNChgJMTcPgw8yxZEysurUCqNhUdy3ZEOS9pJGU+fQrs2MGiKkZLZyp+kAZFG6Cebz1Ep0Rj47WNvOWIIn1eqbUVlpJqfvTy5axIRqdOQNmyvNVkHWnEDclkG31Rh43XNyIyMZKzGnHUrAk0bQrExwPr1vFWY+Re1D3svbsX9kp7jKgzgrcc0egXlKNHM6+G1PF09MSAasyFIKVE4XFsKmLDBiA6mq+W9Cy9sBQanQbdK3RH8XzFecsRxcOHwJ9/svNZ79WQOv2r9YeHgwfOPTuHc0+tOJktHZ6eRm/eQiuaisnqZCy/xJIypbSgXLqURVN8/jnLu5M6giAYqpUuPL9QMmGlAwYwb97Zs8xzbS3surULz+Kewd/LH23LtOUtRxRJSayQFCDd/GjZUMrllC9QHu3LtEeKJgUrL6/kLUc0+gm1ZAmgVvPVomfRuUUgEPpW6QtvZ2/eckRx9SoryenqaiyUkRsYU28MFIIC225sQ3h8OG85oqhSBWjZklVTWr2atxpGXGocVgcxMRMaTOCsRjwLFzLvXJ8+zFuXG3C2c8bQWiyZTUphpWPGsJCarVtZFTxrYNP1TXid9Bq1CtdCEz9pJGXGxhqvC1JdUGaEFMNKXVysL6+UiAzXhXH1xkkmP3rTJiAqCqhdG2jUiLea7CGNX1omR+gXQMsuLkOyOpmzGnG0b89KWD99Cvz+O281wOuk19h4fSMAVs1HKqSPd3d356vFlJTyLIUu/tLtPxMYaB39Z9ZfWY+41Dg09WuKWoUzbEpudbx+DfzytjjcBOnYdqIYWWckbBW2+PP2n3j45iFvOaIoVYrldajV1tF/Rkc6SfYDW7eORVE0bQrUksZUFIVUw0pHjmQe6z//tI680hOhJ3DlxRUUcCogmX5guSU/WjaU8gBN/Zqihk8NRCZFYkvwFt5yRKFQGEOVrKH87KrLq5CiSUHb0m1RoUAFvmJE8uwZ8NtvuSfe/X30ITWrLq+STKKwvv9MeDjLReCJWqvG4gssYUpK4UnLl7Ny9+3bAxUr8lZjWnxcfdCnSh8QCIvPW1ky20fQbwBYQ/+Z/ff2417UPRRzL4buFbrzFSMStZpFTwC5z/gH3g0rPfv0LG85orC2vFIp9gPbtw+4dw/w8wO6deOtJvvIhlIeIH2c8IJzC6AjHWdF4ujbFyhQgDWfPXmSn44UTYrBayGlBeXSpYBGA3Tvzi5UuY36RetLsv+MtWwA/H7rd4TFhqFc/nJoX7Y9PyFZICnJ6LWYOJGvFnOh7xW24doGvEl+w1mNOBo0AOrVY7l3vPvP6BeUo+uOho1CGjXjd+1i0RPlygHtpNGaL0s42zkb+lhJKaxUf61ev571VuLFndd3sO/ePjjYOGB47eH8hGSR9PnRUm7fIBtKeYTuFbqjqFtR3I26iwP3D/CWIwpHR2D422sCzzjhbTe24VXiK1QtWBWflPiEn5AsEB/PqgYCuSve/X2k2H+md2/WfyY4GDh2jI8GIjL0Vxtff7xk4t03bmShd7VrS7sf2MeoXLAyWpdqjSR1ElZdXsVbjmj0npCFC/n1nwkKD8KJ0BNws3fDwBoD+YjIIkTG6mrjx7NoitxIQJ0A2CntsPv2bsmElVapwqIAkpL45pXqqxb3q9JPMv3ALl9mG9xubsDXX/NWkzMEa+tuX6tWLbp8+fIH34+Li0NERATU1pLhLyHiUuMQnRwNext7FHIplOnnbW1t4e3tDTc3Nwuoy5iICNZbKTUVuH0b8Pe37Pg60qHyysq4FXkLmzpvkkxs8OLFrF9B48Z8vXHmRqvTouyysngU/Qh/9PgDXct35S1JFDNnAt9/D7Rpw8q3W5rjT46j+a/NUcCpAMLGhkkilEOrZTvuDx+yhtQZlboXBAHWdk/LDkceHkGrLa1QyKUQnox+Ansbe96SMkWrZaV/Hz0C/vgD6MphKvb+oze239yO8fXHY34raeTD/Pcf8MkngLc3EBoKOFj/VMw2A/4agF+u/YIRtUdgWTsrSGgTwT//AK1bAz4+wOPHrH+YJYlMjESxxcWQoknB7RG34e9l4UVQNvniCxZePmECMG8ebzWZIwhCEBFlnB1IRFb1qFmzJn2I2NhYunfvHiUmJpJOp/vg52QyRqPV0JXwK3Tp+SVKSE346Gd1Oh0lJibSvXv3KDY21kIKM2bQICKAaPBgy4/9152/CCqQ70JfStWkWl5ANkhLIypalP1me/bwVmN+lp5fSlCB6q+rz1uKaF6/JnJ0ZH+j4GDLj99+a3uCCjTt+DTLD55Ndu1iv1fJkkQaTcafYbc06aPT6ajKyioEFeiXq7/wliOapUvZ36hBA8uPHRoTSsppSlJOU1JoTKjlBWSTdu3YbzZNOlMx29x4dYOgAjnNdKLXia95yxGFTkdUuTL7G/3yi+XH//HfHwkqUIdtHSw/eDZ58oRIqSSysSEKC+OtRhwALtMH7BJJOXkjIiJQpEgRODk5SaaSjTWhVCjh5eQFAHiV+OqjnxUEAU5OTihSpAgiIiIsIe+D6OOEN20CIi3cCmrOmTlMQ71xsFPaWXbwbLJ9O4t39/cHOnbkrcb89K9uTBQ+HXaatxxR5M/PKhECwNy5lh37VuQt7L+/X1Lx7kTGXclx41iBktyMIAiGsNL5Z+dLxkvWvz/g4cH6z5w5Y9mxF55bCC1p0aNiDxRzL2bZwbNJSAhw4ADzIg2XxlTMEZW8K6FN6TaSCisVBGP4uqXzShPSEhB4MRAAMKnBJMsNnEMWLGAe5p49gaJFeavJOZIylNRqNRwdHXnLkDQFnQtCgIA3yW+QqknN9POOjo7cwxz9/YEOHYCUFFbxylKcDjuNs0/PwsPBA4NqSqMJkU5nXHhPnpx7493T42LngoA6AQCA2adnc1YjnvHj2YJ/+3bgyRPLjTv3DDtBvqr6lWHjxNo5dQq4eJEZmP3781ZjGXpV6oUirkUQEhmC/ff385YjChcXYMTbXtyzLTgVXye9xtorawEAkxtOttzAOWQO24fDgAGAlzSmYo7RbwAEXgwUtQaxBr74glXBu3mTGbaWYm3QWkSnRKNB0QZoVEwaTYgiI1mpe4CtQXIDkltGyZ6knGFnYwcPRw8AmXuVAOv5vfWJwoGBQEKCZcb8+fTPAFgSqoudi2UGzSH797NdSl9fVjQgrzCq7ig42TrhwP0DCH4VzFuOKPz82N9IqzUmc5ubsNgwbL2xFQpBgYkNpVM2Tu9NGjECcHLiq8VS2CntMK4+c6fPPj1bMl6lUaNYIZ79+1nBEksQeCEQSeoktCvTDlULVbXMoDnkyRNg2za2WZJbKzhmxKclPkXVglXxKvEVtt7YyluOKOzsWM4vYLkNgDRtmqFC4JSGU6xmLZYZS5ey9g0dOgCVK/NWYxokZyjJ5Bx9IYfXSa+h1kqjKEaTJkD9+qxEp76amzm58eoG9t/fD0cbR4ysM9L8A5oAIuNFfPx4dnHPK3g5eWFQDeb10xu4UmDS22iK9etZ4RJzM//sfGh0GvSq1AslPUqaf0ATcOsW68fh4AAEBPBWY1kG1xwMT0dPnH16FqfC/r+98w6L6tra+LvpRbChBCt2o5LEqIglscWeYox605uJJrGXJOqNOsTesBu75ouJMRpjvPYSeywxCvaGFUUURKSXmfX9sWEGkDLAzJxzYP2eZx5hTntZnrPPXnutvfYhpeWYRYUKprTSjIiJNYlNjjWmJ41uPdr6F7QQM2fKQZJ33gF8fZVWYzsyp5XOOjpLMwMA/fvLtNIjR2SE29r8fOZn3I29i4YVGmpm+YbYWNPyDaNGKavFkrCjVAJxc3RDaefSMJABD+KVnX9kLkIAY8bIn2fNklXwrMn0v2V6Ut/GfTVTjvPwYeDoUaBcOVNHpSQxvMVwONg5YN35dZopP9uokZxHlpQkR+KsycP4h1h+SuZEjGqlnbdYRrTt449lJ7wkUcqplHGgRmtppQ4OsurV9evWvdbSf5ciOikaraq20kx6UkSEHBwBik96UkH4T6P/oLJHZVx4eEEzy5V4eACD0sdMrR1VMpDBOD96VOtRmlm+YckS4PFjoHVroFUrpdVYDm1Yn7E4Ph4+AIAH8Q+QZkhTWI15dO8u1zW4d08WdrAWtx7fwtqza2Ev7DGipXYWIZqaHkgZNEjOFShpVCtdDe8/9z4MZMDMv7VRGhgARqcPgi9cCDx5Yr3rzD0+F4lpiXit7mvw89ZGTsTNm8BPP8m5dhnptyWNQf6D4O7ojh3XduB0+Gml5ZhFRlqpwWDdtNLktGQEHQsCoK1o0ty5cnDktdfkYElJw8neCUMDhgIAJh2apJmo0uDBMvV3+3YgONh61/nz0p+4HHUZ1UtXx38a/sd6F7IgyclyDTXA9E4rLrCjVEIp5VQKHk4e0JMeD+NtXEqukAhhegCnTQPSrOTfzTo6C3rS4x2/d+Bbxtc6F7EwZ87ISaZubiUvPSkz37T8BgICq4JX4X7cfaXlmEWLFjK19PFj66WVPkl+ggUnZE6EljqU06fL5/zdd4FatZRWowzl3cqjX5N+AICpR7STVpoRKVm5ErhvpUfxpzM/4V7sPfhV9EO3Ot2scxELExNjKkpU3DqUBeGLpl+gnGs5HA07iv039ystxyzKl5cpeIBpYNLSEJHxOR/ZciQc7R2tcyEL83//B4SHy8Hsrl2VVmNZ2FFSCQMGDEDPAq7QFx0dDW9vb4SGmpdm1KtXLwRluPwwRZUi4iOgNyi0lHoB6d0bqF1bLjq5fr3lz585PUlL5Tgz5gJ8/nnJqZ6UE89WeBY96vdAsj4Zc47NUVqO2WTkcwcFWSetdPHJxYhJjkGb6m3QomoLy1/ACty7J9OTMg+QlFSGtxgORztHbLiwAVejriotxywaNAB69JD38xwrPIp6g95YwXFUa+1Mdl+8WEaO27SRgyQllVJOpTAsQFZImHhoosJqzGf4cMDRUfY/rlrhUdx/cz9O3D0BLzcvfNr4U8tfwAro9aZqu6NGyTa7OMGOkkqYOHEi/q+A+WSTJ09Gt27dUMvModbx48dj4sSJiImJAQB4OHnA3dEdaYY0RCZEFlizEtjbmybAT5li+TUNZh2dhcS0RHSv010z6UnXrgHr1sk5ARlrTpVkRrWWXseifxbhcdJjhdWYR5cuciQuPBxYvdqy505KS8LsY7MBaCuaNHMmkJICvPWW7HSXZKp4VsGHz38IAxmMzoEWyBgA+OEHGUmxJBsvbsTVR1dRo0wN9GnYx7IntxKJicBs+SiWeOcfkBVlPZ098deNv/D3nb+VlmMWVaoAH36YdSkOSzL58GQAwJDmQ+DmqI0Sn7//LvshNWrIweziBjtKKqFs2bIoVYCJJQkJCVi+fDn69u1r9jF+fn6oWbMm1qxZA0BWn8mogBcRHwEDGQomWiE+/FCuaXD2rCxBaykiEyKN6Unj24y33ImtzKRJckTnww+BatpYZ9Gq+Ff2R/sa7RGbEov5x+crLccsMkdNpkyRDoKlWB28Gvfj7qPxM43RqVYny53Yijx8KEfeAeC//1VWi1r4ppVMK/0x5EfcfXJXaTlm0bw50K6djKDMt+CjaCADJh2aBECmJznYOVju5FZk9WpZyKFxY6CTNh5Fq1LGpYyxWEnG/6cW+OYb2Wb/+CNw14KP4pHbR7Dn+h54OntiQLMBljuxFTEYgInpAcGvv5YDtsUNzTtKQijzKShEhOnTp6NevXpwdXVFxYoV8dZbbwEAwsLCIITA5cuXs/z+559/okuXLnB3d0etWrWwb98+4/m2bdsGOzs7tMpUWmT9+vVwdnbGrVu3jN8NGTIEtWrVQkSEXDPp9ddfx9q1a43by7iUgYuDC1L0KXiU+Kjgf5gCODubJnZPmmS5qFLQ0SDEp8aja+2uaFa5mWVOamWuXZOT3e3tuUOZmbEvjwUABB0L0kxUqXdv4NlngVu35AvYEqToU4zV0ka3Hq2Z9KTZs01rcbzwgtJq1EHd8nXRu2FvpBpSNVUC/7vv5L9BQZaLKm26tAkhESGo5FFJM+lJycnAZBkswJgxxS89qbAMDRhqXAPvVPgppeWYRd26sr1OTbVsCfzAA4EAZDQpY71LtfPHH3LQunJluXBycUTzjpJWmDFjBlatWoVFixbh0qVL2Lx5Mzp27AgACA4OhpubG+rUqWP8HQCCgoIwcuRIhISEoFGjRhieKa/q0KFDaNKkSZaOT69eveDn54eJ6e79zJkzsXbtWuzYsQPe3t4AAH9/f5w4cQKJiYkAZFTJp5ScqxQeG66Z6jP9+smJlceOAbt3F/18UQlRxrU4tBpNqqmNZXFsQlvftmhTvQ0eJz3GvONWrrttIeztgXHj5M+TJlkmqrTy9ErcjrmNhhUa4q0GbxX9hDYgOtq0Fgc7/1kZ+/JYCAgsPbUUYU/ClJZjFu3aAS+9JP9fLVEC30AG6PbrAEjn38XBpegntQErVgBhYXIRzgJORy7WeLl54cumXwLQVlQpYwBg6VLLRJWO3D6C3dd3w9PZ01gRUO0YDIBOJ38eM0YOYhdHNO8oESnzKSg7duxAt27d0KFDB1SvXh0BAQH44osvAAAhISF47rnnYGdnZ/zd09MT69atwyuvvILatWujV69eePjQVJ3u1q1b8PHxyXINIQQmT56M1atXY+rUqQgMDMTWrVuNDhgAVKpUCampqbh3757xu3Ku5eBs74xkfTKiEqMK/scpgLu7aa7S2LFFjyrNPjYbcSlx6FyrM5pXaV50gTYgNJSjSXmha6sDICOFJTGqlJyWbOx46NrqNLMWx/z5cuHCV14BAgKUVqMuGlVshD4N+yBFn4LJhyYrLccshAAC5UC5RaJKf1z8A2cfnEVlj8r47EVtLBiXlCQHPwBg/HhZ7p4xMaLFCDjbO2PjxY04/+C80nLMws8P6NNHRgotsa5SRjRpsP9glHMtV/QT2oCNG4Fz5+S8rQLMAtEc/LjaiNdffx1z5szBK6+8giVLliAy0lQ8ITg4GC9kyi8JDg5G9+7d8cwzzxi/u3btGmrXrm38PTExES4uT4+kderUCc2aNcN3332H3377Dc2aZU0hc3V1NR6fgRAClTwqAQDuxd7TzFylAQOAihWBEyeKNlfpUeIjY9RBq9Gkklo6OS/a+rZFW9+2iEmOwdxjc5WWYxaWjCotP7UcYU/C8Jz3c+j5rDaGsGNiTBXSMkZsmayMazMOAgLLTy3H7ZjbSssxi3btZJW3x4/lGkKFxUAG6A7oAABjXhqjmWjSsmWyiuPzzwNvvqm0GvXh4+FjdHq1VAFv/Hg5ELBsGXC7CI9iRjTJw8kDw1oMs5xAK2IwmAZAinM0CWBHyWYMHToUly9fRpcuXbBo0SLUqlULFy9eBCAjSJkdpZCQELTIVjf09OnTWfbx8vJCdHT0U9f566+/EBISAiIypttl5tEjOQ+pQrYl7su5ljPOVdJKBTx3d1NVpXHjCh9VmnNsDmJTYtGxZkfNlE6+fl2uW8DRpLwJbCtb8tnHZpeoqFJiaqKxepKujXaiSXPmyBStl1+WH+ZpGlRogHf83kGqIRWTDmonVSlzVOlxIR/F3y/8jnMPzqGKZxX0bayNIezERFPEQafjaFJufNvqWzjZO2HduXU4G3FWaTlm0aAB8PbbckBrchECvJnnJmklmvT77zKaVLVq8Z2blAE/sjakdu3aGDlyJE6ePAkiwpkzZxAfH4/Q0FCjE5Txe+PGjbMcm91Raty4MS5cuJBln5CQEPTs2RPz589Hjx49MDqH+qPnzp1DpUqVnnKiMkeVwmPDNRNV+uILwMcHOH0a2LSp4MdHJ0Zj7nE5xKnFaNIHH3A0KS9erv4y2tdoj5jkGM2sq2SJqNLSf5fiXuw9NH6mMXrU72FZgVYiKsq0svuECTzZPS/GvTwOdsIOK4NX4kb0DaXlmEWbNjKylDlqWBAMZDB2KMe0HgNnB20MYS9ZIsv+N24MvPGG0mrUS9XSVdG/SX8QCOP3a+ddPG6cdH5XrABu3iz48X/f+ZujSSqHHSUbMG3aNKxevRoXLlzAlStXoNPp4OTkhLZt2+LMmTMAZOluAMbfMztFUVFRCAsLy/Jd586dcfHiRURFyTlFt27dQrdu3TB8+HB8+umnCAwMxO7du7F///4sWg4dOoQuXbrkqLOsS1m4Orgi1ZCKh/EPc9xHbbi6ygcVkA2WoYD+3exjs/Ek+Qk61OiAVtVa5X+ACrh+XUYaOJpkHro2OgDy/zo68ekorBopSlQpITXBuLK7rq1OM5XuZs6UZaQ7deJoUn7U86qH9/zeQ5ohTVMT4DM6V3PmFDyqtOHCBpx/eB5VPatqptJdQgIwNb1AYWAgO//5Mbr1aLg6uOKPS3/g33v/Ki3HLOrXB957D0hLM81DKwhajSadPy+jSZ98orQa68OOkg1ITk7GtGnT0LRpU7Rs2RIhISHYu3cvvL29ERISgrp168LNTS4sFhISgjp16mRZU+n06dNwdHREg0yrLvr5+cHf3x+//vorHj16hC5duuDVV1/FuPSh6EaNGqF3795ZokpJSUn4448/8Pnnn+eoUwiByh6VAQD34+5Db9Bb3BbW4PPP5WTCc+fkatnmEhEXgaCjcgg7I0VLC4wfL6NJ770HZJq2xuTCS9VfQocaHfAk+Ylx4VW1kzmqNGGCnAxuLotPLsb9uPto4tMEr9V9zToCLUxEhKki2oQJymrRCmNfHgt7YY/VwasR+ihUaTlm8dJLQIcOMqo0uwCPot6gN3Yo//vSfzUTTVq8WN7bTZvKUvdM3vh4+GCg/0AAwNh9YxVWYz5jx8o2e9UqWWTJXA7eOohdobs0FU3S60tWNAmAXN9HTZ8mTZpQbly4cCHXbSWR7du3U926dSktLc2s/RcsWEAdO3bMcx+DwUDnH5ynf+7+Q+Gx4USkDbsvXizrEdavT2SmOWjQtkEEHei1X16zrjgLEhJCJASRoyPR9etKq9EOh28dJuhA7pPc6X7sfaXlmEVaGpGfn7yvZ80y75gnSU+o4oyKBB1oy+Ut1hVoQYYOlX/n668X7TzylVZy+HjTxwQd6IONHygtxWwOH5b/16VKEUVEmHfM6tOrCTpQ9dnVKTkt2boCLcSTJ0QVKsi/detWpdVoh4fxD6nU5FIEHejwrcNKyzGbjz+W/9cffWTe/gaDgQKWBxB0IN0+nVW1WZJVq+TfWb06UbI2HkWzAHCScvFLOKKkYbp06YIBAwYgLMy89TQcHR0xP5/l0bUaVfrkE8DXF7h0CVizJv/9r0dfx+KTiyEgMKm9dlJXRo+WRSu+/BKoUUNpNdqhVbVWeK3ua4hPjcf3B75XWo5Z2Nub0nYmTTKvrPKso7PwIP4BAqoEoFudbtYVaCFu3AAWLZI/f6+N/xrVMO7lcXC0c8SaM2sQcj9EaTlm0aoV0L07EBdnXvQwKS3JGF2Y0G4CnOydrKzQMsyYATx8CLRoAXTtqrQa7eDl5oVhATK6MmrvKM2s7Th2LODoKIssnTWjFsWmS5twLOwYKrpXxPAWw/M/QAUkJsq/E5DPrpM2HsUiw46Sxhk8eDCqV69u1r79+vVDvXr18t3P09kTpZxKIc2Qhvtx94sq0SY4OZnCwd99Jx/ovBi/fzxSDan44PkP4OftZ32BFuDgQWDbNsDDg0snF4YpHabATthh6amluBp1VWk5ZtG1q5wE/+gRMH163vuGx4Zj5t8zAQAzOs7QzNyk//5XFqx47z1ZPpkxnxpla2BAswEgEL7d863Scsxm6lQ5AX7xYuDatbz3XXBiAe48uYPnvJ/Du37v2kZgEQkPB2bNkj/PmMFzkwrKiBYj4OXmhcO3D+PPy38qLccsataUA5hEwLf5PIpphjSM3iunRYx9eSw8nD1soLDoLFggF01+/nnZXpcU2FFinkIIgSqeVQAA9+PvI82QprAi83j/feCFF+SDnFdVpTMRZ/DzmZ/hZO+kmblJmRvfkSOBbNXdGTNoWLEhPn7+Y6ToWtcSAAAgAElEQVQZ0jDmrzFKyzELIYBp0+TPs2fLtVhyI/BAIOJT49Gjfg+0rtbaNgKLyMmTwNq1Ms+9MBOhGeC7l79DaefS2Bm6E7tDdystxywaNQI++khOgB+Tx6MYnRhtXFh32ivTYG9nbyOFRUOnk4Uc3nxTRtCYglHapbSxCu23e75Fqj5VYUXm8d13gKcnsH07sHdv7vutDl6Ny1GXUbNsTfRr0s92AotAdLSpBPq0aSWrzH0J+lOZglDKqRTKupQFEWlm/Rk7O1k5C5DrVjx48PQ+RIThO4eDQPiy6ZfwLeNrU42FZeNG4NgxucDucG1E6VVJYLtAuDi4YMOFDTgedlxpOWbRvDnw1lsySqrT5bzPxYcXsfzUctgLe0zpYIFl4m0AkXT6AWDwYMDMwDiTjfJu5TGqtVxQ7ps932hmaYfvvwdcXGQBnhMnct5nyuEpiE6KRjvfduhcq7NtBRaSixeB5ctl6uwUbTyKqqR/k/6oU64OrkRdwbJTy5SWYxYVKpjWdvz665yr8ManxBvLn09qP0kzqaRTpshKle3by8qkJQl2lJhcqexZGQIC8SnxOBV+Smk5ZtGhA9CtGxAbm/N8h82XN2Pvjb0o61IWY1/WRlWdxERTh1KnAzIVRGQKSBXPKhjafCgAYPiu4ZrJf580CXBwkB2w4OCnt4/aOwp60uOzFz9Dfa/6thdYCLZsAQ4cAMqVyzuqwOTPkOZDUMWzCoLvB+PnMz8rLccsqlQBhgyRP48Y8fSC4dejr2PecVkKcXrH6ZpJJR01SnaQP/8cMCPTnckFR3tHTH0lfZmD/To8SX6isCLzGDIEqFxZru34yy9Pb596eCruxd5DE58m6NOwj+0FFoLQUGCuXG4S06aVvFRSdpSYXHFxcEFF94oAgBG7RmimUzl9uin//eJF0/fJackYsWsEAFkOvLxbeYUUFoygILmQ3XPPAf20EaVXNaNaj0JF94r4+87f+OVsDm8yFVKvHjBwoOxMDh6ctVO5K3QXNl/eDHdHd80smpySIkdcATk5uEwZZfVoHVdHV0xsNxGAdJpjk2MVVmQeo0bJUfjDh4Fff826bcSuEUjWJ+M9v/fQtFJTZQQWkJ07gc2b5WDWeG08iqrmzfpvolXVVniY8NCYgql23NxMRUpGjZJFSzK4+fgmZvw9AwAwt8tc2AltdMFHjJBt9ocfylL3JQ1t/C8xiuHj4QM7YYf9N/dj48WNSssxi4YNgc8+k/X+M3cq5x2fh9DoUDzr9Sy+aPqFsiLN5O5dU17wnDkynYMpGqVdSmNqBzlS+fXurzXTqRw/XnYqDx0CfvtNfpeiT8Hg7YMBAOPajIOPh4+CCs1nzhzg8mWgTh05AZopOh88/wH8K/vjXuw9TDw4UWk5ZlGmjCk9beRIU6dyV+gubLq0Ce6O7pjeMZ8qJiohJUW+bwC5BtozzyirpzgghEBQZ7nWYdDRIFyOvKywIvPIcCju3s0693LkrpFI1ifjXb93NbPA/a5dwJ9/Suc/owprSYMdJSZPHOwcUMZFDvcO2TEEcSlx+RyhDiZNkik9e/YAGzbIxWUnHJTDPEGdg+Bo76iwQvMYNUpOCn7rLaBdO6XVFB8+euEjNKvUDOFx4ZoZqSxTxvTS/fpreV/MOz4Pl6Muo275uhgaMFRZgWYSFmZKi50/v4QsWGgD7IQdFnRdAAGB2cdma6ZT+cknslN5754cFErVp2LIDpmTN/blsajkUUlhheYxdy5w5YqM/makFDJFx7+yP/o27otUQyoG7xisicwWe3tZIQ6Q1Q+vXAH23diH3y/+DjdHN0x7ZZqyAs0kNdV0L48dC/hoYxzO4rCjxORLKadSaFqpKe7G3tXMGjReXqaRymHDgCHbRiA2JRbd6nRDl9pdlBVnJocOyTWhnJ1liVnGctgJO8zvKtcUCzoWpJly4Z9+CjRuDNy5A/x36j0EHpBVG+d1maeZScEjRwLx8bIiWGdtzM/XDM0qN8OnjT/VVKfSzk46zIDsVOq2LcClyEuoXa62Zpz/e/dMzv/cuSVnfRlbMaXDFJRxKYNdobs0k9nSvLlsr1NTgYFDTM7/6NajjVWF1c6CBXJtyjp1Srbzz44Sky9CCCzqtsg4Unn+wXmlJZlF377p4W/n3Vh38We4OLhgXpd5Sssyi+Rk03ykb7/lxWWtQfMqzfHxCx8jRZ+CAdsGaKJTaW9v6lTOvfg14lLi0KN+D3SurQ2PY+9eYN06wNVVljtnLE/mTuUfl/5QWo5ZBATIcuEpLmGYdkJO7pnTeQ6cHbQRbsxIG+zRg51/a1DBvQImt5eR/2E7hyE+JV5hReYxdarMBNgdNxNnH5xFjTI1MKLFCKVlmcXt2zKFFJBtdUmO/LOjxJhFs8rN0K9JP6QZ0jTVqQyanwC8Kucj9a8/HrXK1VJYlXlMmSJHcurVA0aPVlpN8WXaK9NQzrUcdl/fjf8L+T+l5ZhFq1ZA54HbQI1+gZ3eBTNfCVJaklkkJgJffSV/HjOGy4FbiwruFTChnUwzHrhtoGaWd5g6leDY40voHWLRxK0HutftrrQks9i6Va4F5uoqC+8w1qFfk3540edF3HlyB7r9OqXlmEWFCsAg3VUgfb3GWW2XwNXRVWFV+UMk547Gxcm0/+7aeBStBjtKxYTo6Gh4e3sjNDTU7GN69eqFoAK07JM7TIaXmxcO3DqAJf8uKYxMm7MtbgJQ7joQ0QiHpo5AmgbWzr140VTAYckSudYIYx0qulfE7M4ytDFs5zBExEUorCh/niQ/wVnf/gAAw56J2LlOG+HGceNkrn6DBqaKd4x1+LLpl2hRpQXC48IxYqc2RrD3R65Das0tQFJphM5fiAj1P4qIiQH6y0cREydy5N+a2NvZY3H3xbATdgg6FoRjYceUlpQvRIRDpfsDDslAyAfYtqCj0pLM4tdfgW3bZDQsY65VSYYdpWLC5MmT0a1bN9SqZX7EZPz48Zg4cSJiYmLM2r+cazks7LYQgKzeciP6RqG02org+8GYeXQmBAS8TyzFqZOOqp/rYzDIlLvUVFm5r00bpRUVfz547gN0qtUJ0UnRGLR9kNJy8uWb3d/gXlwYarv6A8eGYtQoWSBBzRw/Lkfb7eyAlStLdhqHLbC3s8fKN1bC2d4ZK4NXYnfobqUl5UlkQqTx2WsQNgOP71QyVpBTM998IyubNW9esudw2IpmlZthZIuRMJABn/z5CZLSkpSWlCergldh/619KOvsBce/grB8ObBb3Y8iIiNN1RtnzuTqjQA7SsWChIQELF++HH379i3QcX5+fqhZsybWrFlj9jF9GvZBn4Z9EJ8aj76b+6p2FfjE1ES8v/F9pBnS8FWzr7BmagsAcsHW8yqeYhUUJNcU8faW60Ex1kcIgSWvLoG7ozvWX1iv6snC+27sw5J/l8DRzhF/fLQSPd6wR2ysdKpzWgVeDSQlycpmBgMwfLjsVDLWp75XfeO6Wp//73NVVywdsmMIIhMi0da3LbZ8/xnc3WUJ/D9UPMVq3z5g6VLA0RFYsYKXbrAVge0CUd+rPi5FXkLg/kCl5eTKzcc3MWznMADAvG6zofvGC4BciDhOpY9ixjp9kZFA+/ayGAUDGRpU06dJkyaUGxcuXHjqO+igyKcwREZG0hdffEHe3t7k7OxMDRs2pJ07dxr/ttdee408PT2pQoUKNGDAAEpISDAeazAYaNq0aVS3bl1ycXGhChUqUM+ePYmIaP369VSuXDkyGAxZrvfbb7+Rk5MT3bx50/jd4MGDqWbNmnT//n0iIgoMDKRWrVrlqTu73R/GP6SKMyoSdKAFxxcUyhbWZsj2IQQdqN78ehSfEk9ERP36EQFETZsSpaQoLDAHTp0icnSUGv/3P6XVlDzmHZtH0IHKTStHd2LuKC3nKR4lPKJqs6sRdKDv939PRER37xKVLy/vmdmzFRaYC19/LfXVrUuUqUmzCvKVxmSQkpZCLy55kaADfbLpE6Xl5Mj/Bf8fQQdym+RGVyKvEBHR3LnynilfnigsTGGBORAVRVS1qtT4/fdKqyl5HL1zlOwC7cgu0I6O3jmqtJynSNWnUuuVrQk60OtrXyeDwUApKUSNG8t75rPPlFaYMz/+KPW5uxNdvaq0GtsC4CTl4pdwRMlGhIWF4fnnn0d4eDg2btyIc+fOYeTIkfD09MSZM2fQokUL1K9fH//88w82btyILVu2YFxGyREAM2bMwKpVq7Bo0SJcunQJmzdvRseOMt/10KFDaNKkCYQQWa7Zq1cv+Pn5YeJEufjgzJkzsXbtWuzYsQPe3t4AAH9/f5w4cQKJiYlm/y1ebl5Y3H0xALlg59mIs0WyjaXZFboLc4/PhYOdA9b0XAM3RzcAssR2tWrAyZNyMrmaSEgA3ntPptx99RXw6qtKKyp5DPAfgK61u+JR4iO8v/F96A16pSUZISL03dwXt2Nuo1mlZvi29bcAgEqV5Gg2IKsjBgcrKDIHtm+Xz11Gyp2r+ucxFysc7R3xY48f4eLgglXBq/DL2V+UlpSFa4+u4attssLHvC7zUKd8HQDAwIGyelxUFPD++3LxcLVAJCuq3rkjo6OjRimtqOQRUCUAI1qMgIEMeHvD24hOjFZaUhamHp6Kw7cPw6eUD1a8vgJCCDg6Aj/+KNOOly+X84DUxLVrwIAB8uf584HatZXVoypy86CU+hQ0oqQVunXrRl27dn0q6kNE5O/vT59++mmW7zKiRxm0a9eOhg8fnuO533jjDfrwww9z3LZz505ycHCgKVOmUKlSpejEiRNZtoeEhBAAunbtWq7ac7P7J5s+MUZtniQ9yfV4WxIRF0E+M30IOtDEAxOf2n74MJG9vRw12bxZAYG58OWXUtOzzxLFxyutpuQSERdB3jO8c71/lGLhiYUEHchziieFPgp9anv//uq7f8LCiLy8pK6JNjIlOKKUI0tOLiHoQKUml6KrUeoYKk5OS6ZmS5sRdKDev/V+6t14/z5RxYq2vX/MYcECqal0aaLr15VWU3LJfP+8sfaNHPtWSnDszjGyD7Qn6EC7ru16avsPP8j7x8NDPVGb5GSZaQMQ/ec/RCoxpU1BHhElxR2j7J/i6CjdunWLANA///zz1LZLly4RADpz5kyW7+fMmUPVq1c3/j579myys7OjDh060OLFi+nhw4fGbZ06daJ+/frlev0WLVqQvb09bdu27altV65cIQB09uzZXI/Pze7xKfHkt8iPoAP1Wd9H8YYqJS2FXlr5EkEHar2yNaXp03Lcb/p0eeeXLUuUKStRMTLC3Y6ORKdPK62G2XltJ0EHsg+0pwM3Dygth4LDg8l5gjNBB1p3bl2O+8THE9WvL++jDz5Q/kWXmkr00ktST8eORHq9ba7LjlLOGAwG6v1bb4IO9OKSFykpNUlpSTR8x3CCDlR9dnWKTozOcZ+dO+U9ZG9PdPCgjQXmQHAwkbOz1PTbb0qrYa4/uk6lp5Qm6ECzjyqfe3w/9j5VDapK0IFG7ByR4z4GA1GvXvIeevFFoiTlH0UaOlTqqV6dKDrnR7HYk5ejxKl3NuD06dNwcHBAkyZNntp27tw52Nvb49lnn83y/YULF+Dn52f8fejQobh8+TK6dOmCRYsWoVatWrh48SIAwMvLC9HROYee//rrL4SEhICIjOl2mXn06BEAoEKFCgX+u9wc3bC+93qUciqF387/hgUnlK0jOWznMBy6fQg+pXzwW6/fYG+X8+zaESNkalt0NNC7t1zfRSlOnDAtLDtvHvDCC8ppYSSdanXCNy2/gZ706LmuJ65HX1dMS0RcBHqs64FkfTL6vdgPfRr2yXE/Nze5kKubG/DTT7JakZKMGgUcOgT4+ABr1sjUO0Y5hBBY9toy1ChTA6fCT6Hfln5ypFQhVp1ehaBjQXCwc8DPPX9GGZcyOe7XqZMsJa/Xy/VcbihYaPX+feD11+Vi4P37y3cHoyw1ytbAqjdWAZDTAA7dOqSYlhR9Ct767S3ceXIHAVUCMKn9pBz3EwJYtkyWkj91St5LCj6KWLECmDMHcHAAfvlFlgRnspGbB6XUpzhGlLZu3UoAKCYm5qltO3fuJCEExWfKl7l//z65u7vT2rVrczxfSkoKeXh40K+//kpERDNmzKCGDRs+tV9wcDCVLl2aVqxYQT179qROnTo9tc/y5cupUqVKeerPz+7rzq0zjsBvvbI1z32txfJ/lxN0IKcJTnTszrF894+MlKMnAFHPnkRpOQefrMrdu0Q+PlLDF1/Y/vpM7qTqU6nrmq4EHajBwgb0OPGxzTUkpCRQ82XNCTqQ/zJ/SkjJvxLC77/L+0kIoi1bbCAyBxYulBocHIj277fttcERpTw5de8UuU1yI+hAEw5MUETDgZsHyPF7R4IOtPTk0nz3T00l6txZ3lMNGxLl8Bq1OgkJRP7+UkNAgPWLkjAFY9iOYQQdqOzUsnTp4SWbX99gMFDfP/sSdKAqQVUoPDY832NOniRydVU2tfTAAVMBqeXLldGgFsCpd8oSGRlJZcuWpXfeeYfOnTtHFy9epGXLllFwcDA9fvyYvLy8aNCgQXTt2jU6cOAANW7cmN54w5RzO3XqVFq1ahWdP3+eLl++TGPGjKHy5csbK9edOXOG7OzsKDIy0njNmzdvUqVKlSgwMJCIiM6ePUtCCNq3b18WbR999NFT86OyY47d/7v3v8bKRcfDjhfEPEVm+9XtxhfvylMrzT7u3DmZZw7I0LMtiYkx5QS3aSNzhBl18TjxMTVY2ICgA3VZ04VS9ak2u7beoDemSlWbXc2sF28GgYFkzIEPCbGiyBzYsoXIzk5ef9Uq216biB0lc9h0cRMJnSDoQL+c+cWm174WdY3KTytP0IGGbje/0X38WM6/A4i6dpXOk63Q64l695bX9vWVc6cYdZGqT6XXfnmNoAPVmFODIuIibHr9GUdmEHQgl4kudPLuSbOP27RJDmoBRLmMi1uNa9dMFVOHDbPttdUIO0oq4PDhw9SyZUsqVaoUlS5dml555RUKD5ednyNHjlDTpk3J1dWVfH19SafTUXKmnnNgYCDVr1+fXF1dqXz58tS9e3cKDg7Ocv6AgABasECW6o6KiqL69es/NW+pT58+FBAQYPw9MTGRPD096ejRvMtrmmN3g8FAH/3xEUEH8pruZbMJw/tu7COXiS4EHWj4jpyLXeTFX3+ZRlRmzbKCwByIjSVq3Vpes0YNogcPbHNdpuCEPgolr+leBB3o7Q1v28RZMhgMNHjbYGPxhrMRuc8fzPl4oj595P1VoYIcELAFx4/LsrIA0bhxtrlmdthRMo+gv4MIOpDzBGfacXWHTa55I/oGVZ9dnaADdV3TNdc5pLkRGmrq2L37rm2cJYOBaOBAeU1PT9s9S0zBiUuOo6ZLmxoj8LYqMLXg+ALjsjFrzxbc25k9W95fzs5Eu56u/WAVbtwwZdR07apMRo3aYEepBLB9+3aqW7cupRXgjl+wYAF17Ngx3/3MtXtKWgp1WdPFGH6+8MC6/19H7xwl90nuBB3osz8/K3QxiZ9/lk8CQDR1qoVFZiM+nqhdO3mtKlW4apIWOHrnKHlM9rCJs6Q36OmL/31hTCPdeW1noc6TmGhKV/L2Jrp40cJCs3HgAFGpUqR4MQl2lMzDYDDQwK0DjffZ/y5bd+G2zE6S/zJ/ikkqXP7c33+b7rO337aus6TXm6pJOjnZrhPLFJ77sffJd46v8T6LSoiy6vUyqklCB1p4YmGhzpHZGXd2tv4aipmdpBYtlEllVSPsKJUQ5s6dm2Vx2fxYsmQJXbqUfz5vQewemxxrXGit/LTyVkvD++v6X+Q5xZOgA737+7sFHp3Mzg8/mELgo0dbp6MXE0PUvr28ho8P0ZUrlr8GYx3+vv230Vnqs74PJadZPlcyTZ9mLLnvMtGFtl/dXqTzJSQQvfKKvN+eeYYoj8KWRWLHDlOu/bvvKruYMztK5qM36I3OksP3DvT7hd+tcp3QR6FGJ6n5suZFnu935IhMKwVk5NQaactpaUSffiqv4eIi73FGG4Q+CqUac2oQdKBGixoVKG25IPzwzw9GJ2nO0TlFOpdeT/TVV2Ssfrthg4VEZuPaNXaScoMdJaZIFNTu8Snx1P3n7gQdyH2SO2278nRZ8qLwY/CP5DTBybj+RkqaZXpma9aY1lj67DPLlu28cYPouedMI/x8K2uPzM5Sm1VtKDI+Mv+DzCQmKcaYY+860ZX2hO6xyHkzRzA9PS3f4Vu1ypS6+tlnyqdwsKNUMAwGA43YOYKgA9kF2tHso7MtuszDoVuHjKmrAcsDLFYU5ehReT9nzPGMtNyjSDExRK++Ks/t6kq0xzKPImNDwmLCqP6C+gQdqNbcWnT+wXmLnTtVn0ojd440OkkzjsywyHkNBqIRI8hYDn/OHMsO2B46ZFrXLiCAnaTs2NxRAvAfABsBhAMgAB+beyw7SuqjMHZPSUuhD//4kKADCZ2g7/Z+V2SHJi45jvr/r7+xgRq0bVCRI0nZ2bzZtE5G8+ZyBKao/PmnXLMJIKpXj9PttMzJuyeNCxpXDapKh24dKvI5/733L9WbX89YtengTcsuGJOQYJqMLgTR2LFFj/rExRH160fGlNXhw5Vfu4mIHaXCYDAYSLdPZ2xX397wdq7rGpmL3qCn6Yenk8P3DgQdqPNPnQudbpcbJ0+aqoZWrSo7gpY4Z9268pzlysmUUkabPIh7QI0XNzYutPxTyE9FHgS4++QutV3d1hiFXXFqhYXUSgwGIp3O1K6+807R1zXS6+XakQ4OZJyTxE7S0yjhKK0HcArAMnaUtE9h7a436On7/d8bKyy9sPiFQqfibb+6nWrPq23MqV/277JCnccc/vmHqFo1+XS4uckiD4VJ77h7l+i990yN3quvEj16ZHm9jG25E3PHWLZb6AQN3ja4UCPlcclxNGbPGGPFxoYLG9K1KAt45jmg18tqeBnppY0by+ILhWH7dqLatck4d0NNZWXZUSo8686tM875rBJUhTac31CojuWZ+2eo5YqWRsdr+I7hVpvXd+eOqWy3nR3R4MGyQl5BiYuTKdcZnclGjSwzSMYoS1xyHL2z4R3jvdjj1x50/VHBRyrT9Gm05OQS4+K23jO86a/rf1lBsWTdOtn3yJjLvGFD4QaiQkJkil1GH2TECNtWjNQSSjhKdun/lrK0o6S31RLvDBER6fX6Ijuo+27sM06whA7Uc11POnL7SL4vYb1BTzuv7aT2P7Y3Huu3yI9Oh58ukh5ziIyUozkZDUytWkRLlsgXan7cvEk0cqSpoXN1lZVt+NYtPqSkpdCo3aPIPtDeGAmacGCCWWVpoxKiaOaRmeQ9w9t4Xw/YOsCsdZKKyl9/yRLHGff1W2/JOR/5vYTT0mTaXsYcO4DIz8/25cfzgx2lonEl8gr5L/M33pcBywNo08VNZjk6J++epPd+f884MPbMzGesXiSCSEZHR482pU2XLUs0YYJ5ZbwjI4lmzpTp0Bn39cCBvE5SccJgMNCKUyuMadPOE5zpqy1f0ZXI/CcJJ6Um0Y/BPxqXiYAO9Oovr9K9J/esrvvyZaJmzUz3ZYsWMjvFHEfn5Ek5SJsxMObjY/0iEVonL0dJyO3WQQhRCkAsgE+IaLU5xzRt2pROnjyZ47bbt29DCAFvb284OjpCCGE5sUwWiAipqamIiIgAEaFatWpFOl9cShwmH5qM2cdmIyktCQBQu1xtdKvdDc0qN0O10tXg6uCK+NR43Ii+gWNhx7D16lbcjb0LAPBw8sDYl8dicPPBcHZwLvLfZy5btwIjRgCXL8vf3dyArl2Bl14C6tUDypeXq8bfuweEhAB79wJHjpiOf/NNYPp0oHZtm0lmbEjw/WAM2TEEB28dBAA42DmgrW9btPdtj0YVG6GCewUICEQmROLCwwvYf2s/9lzfgxR9CgCgaaWmmNdlHlpUbWEzzXFxwMSJwNy5QJJ8FFG7NtCtG9C0KVC9urzP4+KA69eBY8eA7duBsDC5r6cnMHYsMHgw4ORkM9lmIYSANd9pJQG9QY+l/y7F2H1jEZUYBQDwdvdG1zpd0bJKS9QqVwseTh5ISktC2JMw/Bv+L3Zc24HzD88DABztHNG/SX9MaD8BZVzK2Ex3cDAwZAhwUD6KcHAA2rYF2rcHGjUCKlaU30dGAhcuAPv2yfY6RT6KaNoUmD8fCAiwmWTGhoQ9CcOoPaPw89mfjd81q9QMHWt2xAvPvIAqnlXgYOeA6KRoXI26iiN3jmDr1a14kvwEAOBbxhdTO0xFn4Z9bNb31OuBJUuAceOAKPkowttb9kFatpTttocHkJgI3LkDnDol2+oLF+S+jo7Al18CgYFAGds9ippECPEvETXNcZuWHCWDwYDIyEjExMQgLS3NckKZHHFwcEDp0qXh5eUFOzs7i5zz7pO7WPjPQqw4vQIP4h/ku79vGV981vgzfNXsK5R1LWsRDQUlLQ3YsAGYNw84ejT//V1cpIM0bBjQrJn19THKQkTYe2Mv5p+Yjy1XtsBAhjz3FxDoVKsTBvkPQrc63RQb8Ll7F1i4EFi+HHj4MP/9fX2Bzz8HvvpKvS9ddpQsR1xKHJb+uxRL/12Ky1GX892/jEsZfPz8xxjcfDBqlK1hA4VPQySdn/nzgS1bAEPejyKEADp1AgYNkgMFPPZa/Dn34ByCjgZh/YX1iEuJy3f/xs80xkD/gXjP7z2bDtJmJi5OOkzLlpkGbfOibFng44/lYJavr7XVFQ9U7ygJIfoB6AcA1apVa3Lr1i2raWLUgd6gx5E7R3Dw1kGERITgXuw9JKclw8XBBdXLVMfz3s+jrW9bNKvUTFWRwxs35Iv42DH585Mn8uXq7S0jTC1ayBevh4fSShkliEqIws7QnThx9wQuRV5CVGIUBATKuJRBnXJ10LxKc3Ss2RE+Hj5KSzWi18so6MGDcgxBaSAAAA7dSURBVFQ+PFxGmlxdZXTpueeAdu2k06+iRzFH2FGyPESEMxFnsOf6Hpx5cAY3om8gLiUOro6u8Hb3hl9FP7TxbYPW1VrDyV49IcbISGDXLuD4cdm5jIqS92+ZMkDduoC/v2yrn3lGaaWMEiSkJmDv9b04fPswLkddRnhcONIMaSjjUgbVSldDU5+m6FSrE+qUr6O0VCNEpsyVM2dkHyQ+Xg7OenvLtvqll4CXX5bRJMZ8iuwoCSFKA8j3zU5El7IdZ9GIEsMwDMPkBjtKDMMwTEHJy1FyMPMcvSEr2OV7LbNVMQzDMAzDMAzDqBSzJp4Q0XIiEvl9rC2WYRiGYRiGYRjGFlhmhj7DMAzDMAzDMEwxwtzUuwIhhGgAoAEAl/Svmgoh4gA8JKID1rgmwzAMwzAMwzCMpbBK1TshhA7A+Bw2HSCitvkc+xBATmXvSgOIyePQomzP71gvAJEKXNdampXUldf2omi2pi6l7o+i6rKWrflZLNix/Cza7lh+Fi23nZ9F2+niZ9F2utT4LBbVHvwsFv3Y6kRUIccjcluJVm0fAEuttd2MY3NdsdfK17WKZoV15XVsoTWr1dYK37dWsTU/i5bTrLAufhbV8Tfxs6gCzQrr4mdRHX+T5u4Pa9q6JD6LOX20NEfpf1bcnt+xSl3XWpqLem5rHZsfWrS1kvettXTzs1iwY/ODn0XLHVvY8xZ1Oz+LtrsuP4uWO7cW74/8tmvxWeT7w7Lntqguqy44W1wQQpykXOqrqxXWbDu0qFuLmgFt6mbNtkOLurWoGdCmbtZsO7SoW4uaAW3q1pJmLUWUlGSp0gIKAWu2HVrUrUXNgDZ1s2bboUXdWtQMaFM3a7YdWtStRc2ANnVrRjNHlBiGYRiGYRiGYbLBESWGYRiGYRiGYZhssKPEMAzDMAzDMAyTDXaUGIZhGIZhGIZhssGOkhkIISoLIeKEECSEKKW0nuKGEKKXEOJvIUSUECJJCHFZCPGdEMJJaW3FDSFEbyHEZiHE3fR7+l8hxDtK62LUhxCiihDifvq9EiyECBNCRAghXlNaW3aEEA2EEHuFEAlCiHtCiO+FEPZK68oNLdmWsS7cJtsO7msog9b70A5KC9AIMwDEAXBXWkheCCFqAfg901c+AP4mojcVkmQu5QHsg7TzYwD+AHQAngEwUDlZeSOEuAkgAUBK+lfvEtEF5RSZxXAANwAMg1wVuxuAX4QQXkQ0X1FleSCEOACgDAAB4AqAT4noibKqijdEFCaE2AJgDxH9KoSYDuA2ERV1fQyLIoQoC2APgAsA3gBQC8AsyIHA7xSUlitasa1W0Vh7ock2OQONvQc12dcANN2/AzTSh84NrnqXD0KIlwD8CWAy5H+2BxHFKavKPIQQ+wEsIaK1SmspKEKISQAGAChLKr1J018QbYnopsJSzCb95RuZ7btfALQgohoKycoXIURpIopJ/zkIQDwRjVVYVoEQQuwB0CGPXXyJ6Jat9ACAEOIwgCo5bNpLRH2FECcAfExEF4QQuwBMJKKDttSYH0KI0QC+AVA9ozMshPgG6R0gtXaQtWDbzKjx/s0NLbUXWm2TM9DiezAzWuhr5IRW+nda7kNnwKl3eZCeujEfwPeQIz057bMnPZyY26e6TUWbdFUD8AKATem/ay3VIwpAlnC4Wm2dHTXbOvsLOZ3TACpm/kJtts7U6bGDHJWi9N9Va+scGAkgGsBCAC3SPysB3AMQoEQnk4haE5FvDp++6bauDTkiD8j25IytNZpBVwA7szlEvwJwBdBGGUl5oyHbZkZ1929u5NReqLWtMKdNVlt7nB9qtXUuZOlraMHWmft3arZ1fn1oLdgaYEcpP74A4AL5YsgNtb483gfwOxElAjLVA8AWACOI6AUAvwCYoKZUDyGEvRDCTQjRGsBgAD9kG+FRo603CSFChBCThBCOgDZsnY2WkGlLmVGdrYUQ2wBEAKgHYDqgOVvfBFAWwG4iOkZExwB4AjhDRMcVVZYzdQDcIqI0IURFAKlE9FhpUTlQH8ClzF8Q0W3IdKD6iijKH63YNjM3oaH7N3t7obG2InubrLr2OBtZ3oNqt3U+fQ212xrI1L9Tua3z60NrwdbsKOWGEKI8gAkAhhNRah673oQNXx5CiMNCiJs5fFZk2/V9AD9l++45mEYt1TiCGZ/+OQTgAICvs22/CXXZunV6w9QKQAPIhz4DtdsaACCE6AA5ryN7Q3YT6rI1iKgbZC75CQBfZTpcEVsLIUoLIern98l0SKP0f89n037OFnoLwXMAzmb63UEI0UspMXlQFnK+QXai07epEa3YNjOaun9zaS9U3y7n0ibfhHqd1Nzeg2q2dV59jZtQr60zyN6/U52tzexD34T6bV1yijkIIUpDTn7LEyLKGJmcBOA4EW3L55DcXh6bCyzSDIiodX77CCGaAnCDbAAyvrNZqkchbJ1BS0jd/gDGAViArB1iVdk6fSQHRBSX3qHvD2jG1hBC+EKOPv1JRKuzbVaVrTPtpxdC/AhgHYDpCqcw9QawzIz9RPq/jSCjHNcBQAjhCqldrR3N9QDWp//8AIC3soryJKe5BSKX7xVHY7bNQFP3L5C1vRBCzITK0x3zaJNt2h4XhJzegxpILc2rr6FaWwNP9+9UbGtz+tCqtnUGJcZRQgE6NUKIhgA+BfCyEKJM+vdu6f+WFkLoM1LaoM6XxwcA1mRLW7NlqkdBO5AAACI6lf7jYSFEJIAfhRCziCg0/XvV2FoI4Q7AnoieCCEcALwFU+OkelsLIcoB2A7gNuToVHbUZOuyAJyIKCL9q7cy6VAshYmIlgNYXoBDGgK4SESGTL/bQcUdTY0QDVnhLDulkXOkiSkcmrh/82gvVJ3umE+brJr2ODN5vAdVbet8+hqqtHUmsvfvVGfrAvSh1W5rACUo9Y6IlhORyO+TvnsdAI4AjkK+hDNyKAEgDHJyWgaqenmkN1ZvI+e0O5ukehTQ1rmR0ZBlrvqjJlt7AzgohDgDIASAHnIEBVC5rYUQbpA5zU4AuhNRfA6nVpOtywLYKoQ4I4Q4C9m4DknfpqUUpkbIOnLWCIABT88PYwrGJWSbiySEqAo5if+pSCpTaLRy/+bWXqi2rTCjTVZTe5yZ3N6DqrV1DmTva6jV1rn179Roa3P70Kq1dWZKUkSpIBwG0C7bd10AfAu5xsH1TN+r6uVBRGnIIY1Dg6kerdL/vZHpO9XYmoiuQ4a4c9qmWlunN7TrIRuyVun6ckJttm6ayzbV2joHGgLInIZQBTJPXpH0MCFEka9rxoCHLdgO4GshhAcRxaZ/9x8AiciUfqwm8rO9SuyaHVXdv7mRR3uhyrbCzDZZNe1xZvJ4D6rS1rmQva+hSlsDOffvVPoONLcPrVpbZ4YdpRwgWa5zf+bv0nOHAeAQZa0Br4mXh5oRQuyAXDDyPOSIVCsAIwCsy5R2B7CtLcEiyIZqCIByQoiATNtOE1Fy+s9sawuSnhJRAVlHyk5BpkT+A8DP1ppU2hkvDIshK1dtFEJMA1ATcg2lIFLpGkpas70a799ihDltMrfHFsDMvgbbuogUoA+tCVuzo1QE+OVhMf4B8DEAXwBpkKMNoyE7QADY1hakU/q/c3PYVgPATba15Ukf6cs+J28bAA9lFD1NerraagCVIEf1tgL4NttcR9VBRNHplcIWAPgf5Lyk2ZDOkirRmq21cP9qmDzbZCFEArg9thR59jX43Wc7tGRrodJ2mckBS6TK5IbWRjitDdvadrCt1YEQwgdAZSI6KYRwArAbwDwi+l1hacUOtrX1KEp7wu1FwWBb2w62tXJwRElD8M1uO9jWtoNtrQ6IKBxAePrPKekTtKsqq6p4wra2Htye2A62te1gWytHial6VxIQQlQVQuwVQlwUQpwXQkwXQvDDZQXY1raDbW170hcL7AFgp9Jaijtsa+vBbYftYFvbDra1bWFHqXiRBpnn/iyAxgCaA+iprKRiC9vadrCtbYgQwhnABgBziOii0nqKM2xrq8Nth+1gW9sOtrUN4dS7YgSnc9gOtrXtYFvbDiGEPYCfIattzVJaT3GGbW19uO2wHWxr28G2ti3sKBVTMqVzdMpvX6ZosK1tB9va6iwBEAtZMpexLmxrG8Jth+1gW9sOtrX14dS7Yginc9gOtrXtYFtbFyFEKwB9IRfrPC2ECBZCDFZYVrGEbW1buO2wHWxr28G2tg1cHryYkZ7OsQ7AbSIarrSe4gzb2nawrRmGKQzcdtgOtrXtYFvbDnaUihlCiOUA7AF8qtbFC4sLbGvbwbZmGKYwcNthO9jWtoNtbTvYUSpGpKdzHIZc6Vif/vVKIpqnnKriCdvadrCtGYYpDNx22A62te1gW9sWdpQYhmEYhmEYhmGywcUcGIZhGIZhGIZhssGOEsMwDMMwDMMwTDbYUWIYhmEYhmEYhskGO0oMwzAMwzAMwzDZYEeJYRiGYRiGYRgmG+woMQzDMAzDMAzDZIMdJYZhGIZhGIZhmGywo8QwDMMwDMMwDJMNdpQYhmEYhmEYhmGywY4SwzAMo0mEELWFEKlCiMBs3/8ghIgVQjRVShvDMAyjfdhRYhiGYTQJEV0DsBzAMCGEFwAIIcYB+BTAm0R0Ukl9DMMwjLYRRKS0BoZhGIYpFEKIZwCEAlgE4BKApQDeIaLfFBXGMAzDaB4HpQUwDMMwTGEhovtCiDkARkC+0wazk8QwDMNYAk69YxiGYbTOVQDOAI4S0UKlxTAMwzDFA3aUGIZhGM0ihGgPYAmAowBaCSGeV1gSwzAMU0xgR4lhGIbRJEKIFwFsgizo0BbAbQCTldTEMAzDFB/YUWIYhmE0hxCiNoDtAHYBGEREKQACAXQTQrysqDiGYRimWMBV7xiGYRhNkV7p7m/ICFJnIkpO/94ewDkA0UTUUkGJDMMwTDGAHSWGYRiGYRiGYZhscOodwzAMwzAMwzBMNthRYhiGYRiGYRiGyQY7SgzDMAzDMAzDMNlgR4lhGIZhGIZhGCYb7CgxDMMwDMMwDMNkgx0lhmEYhmEYhmGYbLCjxDAMwzAMwzAMkw12lBiGYRiGYRiGYbLx/+j47BGUC9FWAAAAAElFTkSuQmCC",
+      "text/plain": [
+       "<Figure size 864x216 with 1 Axes>"
+      ]
+     },
+     "metadata": {
+      "needs_background": "light"
+     },
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "image/png": 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",
+      "text/plain": [
+       "<Figure size 288x216 with 1 Axes>"
+      ]
+     },
+     "metadata": {
+      "needs_background": "light"
+     },
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "#Codes for figures\n",
+    "\n",
+    "import numpy as np\n",
+    "import pandas as pd\n",
+    "import matplotlib.pyplot as plt\n",
+    "\n",
+    "def multiple_formatter(denominator=2, number=np.pi, latex='\\pi'):\n",
+    "    def gcd(a, b):\n",
+    "        while b:\n",
+    "            a, b = b, a%b\n",
+    "        return a\n",
+    "    def _multiple_formatter(x, pos):\n",
+    "        den = denominator\n",
+    "        num = np.int(np.rint(den*x/number))\n",
+    "        com = gcd(num,den)\n",
+    "        (num,den) = (int(num/com),int(den/com))\n",
+    "        if den==1:\n",
+    "            if num==0:\n",
+    "                return r'$0$'\n",
+    "            if num==1:\n",
+    "                return r'$%s$'%latex\n",
+    "            elif num==-1:\n",
+    "                return r'$-%s$'%latex\n",
+    "            else:\n",
+    "                return r'$%s%s$'%(num,latex)\n",
+    "        else:\n",
+    "            if num==1:\n",
+    "                return r'$\\frac{%s}{%s}$'%(latex,den)\n",
+    "            elif num==-1:\n",
+    "                return r'$\\frac{-%s}{%s}$'%(latex,den)\n",
+    "            else:\n",
+    "                return r'$\\frac{%s%s}{%s}$'%(num,latex,den)\n",
+    "    return _multiple_formatter\n",
+    "\n",
+    "class Multiple:\n",
+    "    def __init__(self, denominator=2, number=np.pi, latex='\\pi'):\n",
+    "        self.denominator = denominator\n",
+    "        self.number = number\n",
+    "        self.latex = latex\n",
+    "\n",
+    "    def locator(self):\n",
+    "        return plt.MultipleLocator(self.number / self.denominator)\n",
+    "\n",
+    "    def formatter(self):\n",
+    "        return plt.FuncFormatter(multiple_formatter(self.denominator, self.number, self.latex))\n",
+    "\n",
+    "xt=np.linspace(-4*np.pi,4*np.pi,num=500)\n",
+    "##\n",
+    "fig, ax = plt.subplots(figsize=(12, 3))\n",
+    "ax.plot(xt, np.sin(xt), color='b', lw=2, label='$sin(x)$')\n",
+    "ax.plot(xt, np.cos(xt), color='g', lw=2, label='$cos(x)$')\n",
+    "ax.tick_params(axis='both', labelsize= 15)\n",
+    "ax.set_xlabel('$x$', fontsize = 16)\n",
+    "ax.legend(fontsize=14)\n",
+    "\n",
+    "ax.axhline(0, color='black', lw=1)\n",
+    "ax.axvline(0, color='black', lw=1)\n",
+    "\n",
+    "ax.xaxis.set_major_locator(plt.MultipleLocator(np.pi / 2))\n",
+    "ax.xaxis.set_minor_locator(plt.MultipleLocator(np.pi / 12))\n",
+    "ax.xaxis.set_major_formatter(plt.FuncFormatter(multiple_formatter()))\n",
+    "\n",
+    "fig.tight_layout()\n",
+    "fig.savefig('sin_cos.png')\n",
+    "\n",
+    "##\n",
+    "xt=np.linspace(-2*np.pi,2*np.pi,num=500)\n",
+    "fig, ax = plt.subplots(figsize=(4, 3))\n",
+    "y = np.tan(xt)\n",
+    "y[:-1][np.diff(y) < 0] = np.nan\n",
+    "\n",
+    "ax.plot(xt,y, color ='r', lw=2)\n",
+    "ax.tick_params(axis='both', labelsize= 15)\n",
+    "\n",
+    "ax.set_xlabel('$x$', fontsize = 16)\n",
+    "ax.set_ylabel(\"$tan(x)$\", fontsize = 16)\n",
+    "\n",
+    "plt.ylim(-10,10)\n",
+    "ax.axhline(0, color='black', lw=1)\n",
+    "ax.axvline(0, color='black', lw=1)\n",
+    "\n",
+    "ax.xaxis.set_major_locator(plt.MultipleLocator(np.pi / 2))\n",
+    "ax.xaxis.set_minor_locator(plt.MultipleLocator(np.pi / 12))\n",
+    "ax.xaxis.set_major_formatter(plt.FuncFormatter(multiple_formatter()))\n",
+    "\n",
+    "fig.tight_layout()\n",
+    "fig.savefig('tan.png')"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 3,
+   "metadata": {
+    "tags": [
+     "remove-cell"
+    ]
+   },
+   "outputs": [
+    {
+     "data": {
+      "application/vnd.jupyter.widget-view+json": {
+       "model_id": "a785b675781e44dcb130fe4aba0b9568",
+       "version_major": 2,
+       "version_minor": 0
+      },
+      "text/plain": [
+       "interactive(children=(FloatSlider(value=0.0, description='c', max=2.0, min=-2.0), Output()), _dom_classes=('wi…"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "application/vnd.jupyter.widget-view+json": {
+       "model_id": "b9af681dfe384b9fbd3b14793a2feede",
+       "version_major": 2,
+       "version_minor": 0
+      },
+      "text/plain": [
+       "interactive(children=(FloatSlider(value=0.0, description='c', max=2.0, min=-2.0), Output()), _dom_classes=('wi…"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "text/plain": [
+       "<function __main__.vtrans(c)>"
+      ]
+     },
+     "execution_count": 3,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "from ipywidgets import interact\n",
+    "import matplotlib.pyplot as plt\n",
+    "import numpy as np\n",
+    "\n",
+    "def func(x):\n",
+    "    return x**3\n",
+    "\n",
+    "def htrans(c):\n",
+    "    x = np.linspace(-3,3)\n",
+    "    \n",
+    "    y = func(x+c) #constant c in the argument of the function\n",
+    "    \n",
+    "    plt.plot(x,y, label='$f\\, (x+c)$')\n",
+    "    plt.title('Horizontal translation')\n",
+    "        \n",
+    "    plt.xlabel('x')\n",
+    "    plt.ylabel('y')\n",
+    "    plt.legend(fontsize=12)\n",
+    "    \n",
+    "    plt.ylim(-8, 8) #remember to change the limits for better visualisation if necessary\n",
+    "    \n",
+    "    plt.axhline(0, color='black', lw=1)\n",
+    "    plt.axvline(0, color='black', lw=1)\n",
+    "    \n",
+    "    \n",
+    "\n",
+    "    plt.show()\n",
+    "\n",
+    "def vtrans(c):\n",
+    "    x = np.linspace(-3,3)\n",
+    "    \n",
+    "    y = func(x)+c #constant c summing the whole function\n",
+    "    \n",
+    "    plt.plot(x,y,  label='$f\\, (x)+c$')\n",
+    "    plt.title('Vertical translation')\n",
+    "    \n",
+    "    plt.xlabel('x')\n",
+    "    plt.ylabel('y')\n",
+    "    plt.legend(fontsize=12)\n",
+    "    \n",
+    "    plt.ylim(-8, 8) #remember to change the limits for better visualisation if necessary\n",
+    "        \n",
+    "    plt.axhline(0, color='black', lw=1)\n",
+    "    plt.axvline(0, color='black', lw=1)\n",
+    "    \n",
+    "    plt.show()\n",
+    "    \n",
+    "# create a slider\n",
+    "\n",
+    "interact(htrans, c=(-2.0,2.0))\n",
+    "interact(vtrans, c=(-2.0,2.0))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 6,
+   "metadata": {
+    "tags": [
+     "remove-cell"
+    ]
+   },
+   "outputs": [
+    {
+     "data": {
+      "application/vnd.jupyter.widget-view+json": {
+       "model_id": "12d5a3f30eab4d5395e543cb666f5e70",
+       "version_major": 2,
+       "version_minor": 0
+      },
+      "text/plain": [
+       "interactive(children=(FloatSlider(value=1.05, description='c', max=2.0, min=0.1), Output()), _dom_classes=('wi…"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "application/vnd.jupyter.widget-view+json": {
+       "model_id": "bd6164c6fecf4c72b93b1a39f7947633",
+       "version_major": 2,
+       "version_minor": 0
+      },
+      "text/plain": [
+       "interactive(children=(FloatSlider(value=1.05, description='c', max=2.0, min=0.1), Output()), _dom_classes=('wi…"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "text/plain": [
+       "<function __main__.vstetcom(c)>"
+      ]
+     },
+     "execution_count": 6,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "from ipywidgets import interact\n",
+    "import matplotlib.pyplot as plt\n",
+    "import numpy as np\n",
+    "\n",
+    "def func(x):\n",
+    "    return x**3-2*x\n",
+    "\n",
+    "def hstetcom(c):\n",
+    "    x = np.linspace(-3,3)\n",
+    "    \n",
+    "    y = func(c*x) #constant c multiplying the argument of the function\n",
+    "    \n",
+    "    plt.plot(x,y, label='$f\\, (cx)$')\n",
+    "    plt.title('Horizontal stretching/compression')\n",
+    "        \n",
+    "    plt.xlabel('x')\n",
+    "    plt.ylabel('y')\n",
+    "    plt.legend(fontsize=12)\n",
+    "    \n",
+    "    plt.ylim(-4, 4) #remember to change the limits for better visualisation if necessary\n",
+    "    \n",
+    "    plt.axhline(0, color='black', lw=1)\n",
+    "    plt.axvline(0, color='black', lw=1)\n",
+    "    \n",
+    "    \n",
+    "\n",
+    "    plt.show()\n",
+    "\n",
+    "def vstetcom(c):\n",
+    "    x = np.linspace(-3,3)\n",
+    "    \n",
+    "    y = c*func(x) #constant c multiplying the whole function\n",
+    "    \n",
+    "    plt.plot(x,y,  label='$c\\, f\\, (x)$')\n",
+    "    plt.title('Vertical stretching/compression')\n",
+    "    \n",
+    "    plt.xlabel('x')\n",
+    "    plt.ylabel('y')\n",
+    "    plt.legend(fontsize=12)\n",
+    "    \n",
+    "    plt.ylim(-4, 4) #remember to change the limits for better visualisation if necessary\n",
+    "        \n",
+    "    plt.axhline(0, color='black', lw=1)\n",
+    "    plt.axvline(0, color='black', lw=1)\n",
+    "    \n",
+    "    plt.show()\n",
+    "    \n",
+    "# create a slider\n",
+    "\n",
+    "interact(hstetcom, c=(0.1,2.0))\n",
+    "interact(vstetcom, c=(0.1,2.0))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 48,
+   "metadata": {
+    "scrolled": true,
+    "tags": [
+     "remove-cell"
+    ]
+   },
+   "outputs": [
+    {
+     "data": {
+      "image/png": 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",
+      "text/plain": [
+       "<Figure size 720x360 with 2 Axes>"
+      ]
+     },
+     "metadata": {
+      "needs_background": "light"
+     },
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "fig, ax = plt.subplots(1,2,figsize=(10, 5))\n",
+    "\n",
+    "xt = np.linspace(-3,5)\n",
+    "\n",
+    "# x-axis reflection\n",
+    "ax[0].plot(xt, (xt-1)**2, color ='r', lw=2, label = '$(x-1)^2$')\n",
+    "ax[0].plot(xt, -(xt-1)**2, color ='b', lw=2, label = '$-(x-1)^2$')\n",
+    "ax[0].tick_params(axis='both', labelsize= 12)\n",
+    "\n",
+    "ax[0].set_xlabel('$x$', fontsize = 14)\n",
+    "ax[0].set_ylabel(\"$y$\", fontsize = 14)\n",
+    "\n",
+    "ax[0].set_title('x-axis reflection')\n",
+    "\n",
+    "ax[0].set_ylim(-10,10)\n",
+    "\n",
+    "ax[0].legend()\n",
+    "\n",
+    "ax[0].axhline(0, color='black', lw=1)\n",
+    "ax[0].axvline(0, color='black', lw=1)\n",
+    "\n",
+    "# y-axis reflection\n",
+    "\n",
+    "xt = np.linspace(-4,4)\n",
+    "\n",
+    "ax[1].plot(xt, (xt-1)**2, color ='r', lw=2, label = '$(x-1)^2$')\n",
+    "ax[1].plot(xt, (-xt-1)**2, color ='b', lw=2,  label = '$[(-x)-1]^2$')\n",
+    "ax[1].tick_params(axis='both', labelsize= 12)\n",
+    "\n",
+    "ax[1].set_xlabel('$x$', fontsize = 14)\n",
+    "#ax[1].set_ylabel(\"$y$\", fontsize = 14)\n",
+    "\n",
+    "ax[1].set_title('y-axis reflection')\n",
+    "\n",
+    "ax[1].set_ylim(-1,10)\n",
+    "\n",
+    "ax[1].legend()\n",
+    "\n",
+    "ax[1].axhline(0, color='black', lw=1)\n",
+    "ax[1].axvline(0, color='black', lw=1)\n",
+    "\n",
+    "fig.tight_layout()\n",
+    "fig.savefig('reflection.png')"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 91,
+   "metadata": {
+    "tags": [
+     "remove-cell"
+    ]
+   },
+   "outputs": [
+    {
+     "data": {
+      "image/png": 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",
+      "text/plain": [
+       "<Figure size 432x288 with 1 Axes>"
+      ]
+     },
+     "metadata": {
+      "needs_background": "light"
+     },
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "import matplotlib.pyplot as plt\n",
+    "import numpy as np\n",
+    "    \n",
+    "time = np.linspace(0,15)\n",
+    "rat = 0.5 + ((1-0.5)/10)*time\n",
+    "\n",
+    "fig, ax = plt.subplots(figsize=(6, 4))\n",
+    "\n",
+    "\n",
+    "ax.plot(time,rat)\n",
+    "    \n",
+    "ax.set_xlabel('$N_t$',fontsize = 16)\n",
+    "ax.set_ylabel(r'$\\frac{N_t}{N_{t+1}}$',fontsize = 22)\n",
+    "\n",
+    "ax.set_yticks([0.5,1])\n",
+    "ax.set_yticklabels(['$1/R$','1'],fontsize=14)\n",
+    "\n",
+    "ax.set_xticks([0,10])\n",
+    "ax.set_xticklabels(['0','K'],fontsize=14)\n",
+    "\n",
+    "ax.axhline(1, ls='--', color='black', lw=0.5)\n",
+    "ax.axvline(10, ls='--', color='black', lw=0.5)\n",
+    "\n",
+    "ax.set_ylim(0, 1.5)\n",
+    "ax.set_xlim(0, 15)\n",
+    "\n",
+    "fig.tight_layout()\n",
+    "fig.savefig('parent-offspring.png')"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "metadata": {
+    "tags": [
+     "remove-cell"
+    ]
+   },
+   "outputs": [
+    {
+     "data": {
+      "image/png": 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",
+      "text/plain": [
+       "<Figure size 600x400 with 1 Axes>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "import matplotlib.pyplot as plt\n",
+    "import numpy as np\n",
+    "\n",
+    "lisg1=[]\n",
+    "lisg2=[]\n",
+    "lisg3=[]\n",
+    "\n",
+    "n0g1=1\n",
+    "n0g2=1\n",
+    "n0g3=1\n",
+    "\n",
+    "lisg1.append(n0g1)\n",
+    "lisg2.append(n0g2)\n",
+    "lisg3.append(n0g3)\n",
+    "\n",
+    "vect = range(1,25)\n",
+    "for t in vect:\n",
+    "    n0g1=1.1*n0g1\n",
+    "    lisg1.append(n0g1)\n",
+    "    \n",
+    "    n0g2=1*n0g2\n",
+    "    lisg2.append(n0g2)\n",
+    "    \n",
+    "    n0g3=0.9*n0g3\n",
+    "    lisg3.append(n0g3)\n",
+    "\n",
+    "fig, ax = plt.subplots(figsize=(6, 4))\n",
+    "time = range(25)\n",
+    "\n",
+    "ax.scatter(time,lisg1,  label='$R>1$')\n",
+    "ax.scatter(time,lisg2,  label='$R=1$')\n",
+    "ax.scatter(time,lisg3,  label='$0<R<1$')\n",
+    "    \n",
+    "ax.set_xlabel('t',fontsize = 16)\n",
+    "ax.set_ylabel('$N_t$',fontsize = 16)\n",
+    "\n",
+    "ax.tick_params(\n",
+    "    axis='both',          \n",
+    "    which='both',      \n",
+    "    bottom=False,      \n",
+    "    left=False,\n",
+    "    labelbottom=False,labelleft=False)\n",
+    "\n",
+    "\n",
+    "\n",
+    "#plt.yticks([0,1],['0','K'],fontsize = 13)\n",
+    "#plt.axhline(1, ls='--', color='black', lw=0.5)\n",
+    "\n",
+    "ax.set_ylim(0, 5)\n",
+    "ax.grid()\n",
+    "ax.legend(fontsize=12)\n",
+    "\n",
+    "fig.tight_layout()\n",
+    "fig.savefig('expgrowth.png')"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 3,
+   "metadata": {
+    "tags": [
+     "remove-cell"
+    ]
+   },
+   "outputs": [
+    {
+     "data": {
+      "image/png": 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",
+      "text/plain": [
+       "<Figure size 600x400 with 1 Axes>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "import matplotlib.pyplot as plt\n",
+    "import numpy as np\n",
+    "\n",
+    "lisg1=[]\n",
+    "lisg2=[]\n",
+    "lisg3=[]\n",
+    "\n",
+    "n0g1=0.1\n",
+    "n0g2=0.8\n",
+    "n0g3=1.4\n",
+    "\n",
+    "lisg1.append(n0g1)\n",
+    "lisg2.append(n0g2)\n",
+    "lisg3.append(n0g3)\n",
+    "\n",
+    "vect = range(1,16)\n",
+    "for t in vect:\n",
+    "    n0g1=1.5*n0g1/(1+((1.5-1)/1)*n0g1)\n",
+    "    n0g2=1.5*n0g2/(1+((1.5-1)/1)*n0g2)\n",
+    "    n0g3=1.5*n0g3/(1+((1.5-1)/1)*n0g3)\n",
+    "    \n",
+    "    lisg1.append(n0g1)\n",
+    "    lisg2.append(n0g2)\n",
+    "    lisg3.append(n0g3)\n",
+    "    \n",
+    "time = range(16)\n",
+    "\n",
+    "fig, ax = plt.subplots(figsize=(6, 4))\n",
+    "\n",
+    "ax.scatter(time,lisg1)\n",
+    "ax.scatter(time,lisg2)\n",
+    "ax.scatter(time,lisg3)\n",
+    "    \n",
+    "ax.set_xlabel('t',fontsize = 16)\n",
+    "ax.set_ylabel('$N_t$',fontsize = 16)\n",
+    "\n",
+    "ax.tick_params(\n",
+    "    axis='x',          \n",
+    "    which='both',      \n",
+    "    bottom=False,      \n",
+    "    top=False,\n",
+    "    labelbottom=False)\n",
+    "\n",
+    "\n",
+    "ax.set_yticks([0,1])\n",
+    "ax.set_yticklabels(['0','K'],fontsize=12)\n",
+    "ax.axhline(1, ls='--', color='black', lw=0.5)\n",
+    "\n",
+    "ax.set_ylim(0, 1.5)\n",
+    "\n",
+    "fig.tight_layout()\n",
+    "fig.savefig('logistic.png')"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Analysing change, one step at a time\n",
+    "  "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Imagine a colony of wasps is founded by a queen with the following pattern of eggs laid:  \n",
+    "\n",
+    "<br />  \n",
+    "    \n",
+    "| Days     |  Eggs  |\n",
+    "|----------|:------:|\n",
+    "| $0$      |  $100$ |\n",
+    "| $1$      |  $135$ |\n",
+    "| $2$      |  $170$ |\n",
+    "| $3$      |  $205$ |\n",
+    "| $4$      |  $240$ |  \n",
+    "    \n",
+    "\n",
+    "Since there is an additive pattern for the increase in the total number of eggs laid, with the number of eggs in any given day, we can determine the number of eggs the colony will have in the following day. If we index the days by a sequence of natural numbers *n*, we have for the above colony:  \n",
+    "  \n",
+    "$$a_{n+1} = a_{n} + 35,$$  \n",
+    "  \n",
+    "where $a_n$ represents the total amount of eggs in a given day $n$ and $a_{n+1}$ the same variable on the day following day $n$. The constant $35$ represents the daily increment (or decrement) on this total number. A sequence of number built with constant additive increments receives the name of *arithmetic progression*.  \n",
+    "  \n",
+    "The above expression defines an iterative process (also known as a *recursion*) with which we can construct the whole sequence starting from a single initial value. If we want to know how many eggs will have been laid on a given day $n$, this shall be given by the expression:  \n",
+    "  \n",
+    "$$a_n = 100 + 35n.$$  \n",
+    "  \n",
+    "Thus, the total amount of eggs laid on day $n=100$ is $a_{100} = 100 + 35 \\cdot 100 = 3600$.  \n",
+    "  \n",
+    "<br />\n",
+    "<br />\n",
+    "Now imagine a single *Escherichia coli* cell put on a Petri dish with a favorable growth medium. If given the right conditions, *E. coli* cells divide, generating two other cells, in a time of approximately $30$ minutes. For our experiment, we would have:  \n",
+    "\n",
+    "<br />  \n",
+    "    \n",
+    "| Minutes  |  # cells  |\n",
+    "|----------|:------:|\n",
+    "| $0$      |  $1$ |\n",
+    "| $30$      |  $2$ |\n",
+    "| $60$      |  $4$ |\n",
+    "| $90$      |  $8$ |\n",
+    "| $120$      |  $16$ |  \n",
+    "  \n",
+    "    \n",
+    "If $t$ represents the number of reproduction intervals since the start of the experiment (assuming values $t=0, 1, 2, \\ldots$), then we can construct the recursion:  \n",
+    "  \n",
+    "$$N_{t+1} = 2N_t,$$  \n",
+    "  \n",
+    "where $N_t$ represents the number of *E. coli* cells after $t$ intervals of reproduction (each interval corresponding to $30$ minutes) and $N_{t+1}$ represents the number of cells on the interval following $t$. See that this time the constant $2$ provides a multiplicative increment, so that the sequence formed by this multiplicative iteration is called *geometric progression*.  \n",
+    "  \n",
+    "If we want $N_t$ for a given interval $t$, we then have:  \n",
+    "  \n",
+    "$$N_t = 2^t,$$\n",
+    "    \n",
+    "leading to an exponential growth in discrete steps.  \n",
+    "  \n",
+    "In general, if we have a population growth described by a geometric progression, starting with a number of individuals $N_0$, the number of individuals on the subsequent generations can be described by the recursion:  \n",
+    "  \n",
+    "$$N_{t+1} = RN_t,$$  \n",
+    "  \n",
+    "where $t$ indicates the number of generations and $R$ is a positive constant ($R>0$).  \n",
+    "  \n",
+    "With this recursion, starting from $N_0$, we can obtain the full sequence by construction:  \n",
+    "  \n",
+    "$$\\begin{align}  \n",
+    "N_1 &= RN_0 \\\\\n",
+    "N_2 &= RN_1 = R(RN_0) = R^2N_0 \\\\\n",
+    "N_3 &= RN_2 = R(RN_1) = R[R(RN_0)] = R^3N_0 \\\\\n",
+    "    &\\vdots \\end{align}$$  \n",
+    "  \n",
+    "From this process of construction, we can obtain what we call the *solution* of the above recursion, given by:  \n",
+    "  \n",
+    "$$N_t = N_0R^t.$$  \n",
+    "  \n",
+    "In problems of population dynamics, $R$ is called **growth constant** and its value determines the long-term behaviour of the function:  \n",
+    "  \n",
+    "```{image} expgrowth.png\n",
+    "```"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Sequences  \n",
+    "  \n",
+    "The two last problems are examples of *sequences* for which a formal definition is given as:  \n",
+    "  \n",
+    "$$\\begin{align}\n",
+    "    f\\!:\\ &{\\rm I\\!N} \\rightarrow {\\rm I\\!R} \\\\\n",
+    "       & n \\rightarrow f(n),\\end{align}$$  \n",
+    "         \n",
+    "in other words, a sequence is a function $f$ from the set of natural numbers to the set of real numbers. For each value of the independent variable $n$ (a natural number), the sequence determines its image $f(n)$.  \n",
+    "  \n",
+    "```{admonition} Example  \n",
+    "The sequence defined by the rule $f(n) = \\displaystyle\\frac{1}{n+1}$ defines the sequence of numbers (starting from $n=0$):  \n",
+    "  \n",
+    "$$1,\\frac{1}{2},\\frac{1}{3},\\frac{1}{4},\\frac{1}{5},\\ldots,$$  \n",
+    "  \n",
+    "corresponding to the set of values $n=0,1,2,3,4,\\ldots$.\n",
+    "```  \n",
+    "  \n",
+    "  \n",
+    "\n",
+    "Sequences are usually written as ordered lists of numbers $a_0, a_1, a_2, a_3, \\ldots$ where $a_n=f(n)$ and referred to with the notation $\\{a_n\\}$ (which is short for $\\{a_n: n \\in {\\rm I\\!N}\\}$, making it explicit that $n$ is a natural number).  \n",
+    "  \n",
+    "As we have seen before, two forms of representation are possible:  \n",
+    "  \n",
+    "````{panels}\n",
+    "**Explicit description**\n",
+    "^^^\n",
+    "$$a_n=f(n)$$  \n",
+    "  \n",
+    "```{admonition} Examples  \n",
+    "$$a_n = 100 + 35n$$  \n",
+    "$$a_n = 100\\cdot2^n$$  \n",
+    "$$a_n = \\frac{4n^2-1}{n^2}$$  \n",
+    "$$a_n = \\mbox{cos}\\left(\\displaystyle\\frac{3\\pi n}{2}\\right)$$\n",
+    "```\n",
+    "+++\n",
+    "In this case, any term $a_n$ of arbitrary index $n$ can be obtained by direct calculation if we know the function $f(n)$. \n",
+    "---\n",
+    "\n",
+    "**Recursive description**\n",
+    "^^^\n",
+    "$$a_{n+1}=g(a_n)$$  \n",
+    "  \n",
+    "```{admonition} Examples  \n",
+    "$$a_{n+1} = a_n + 35,\\ \\ \\ \\mbox{with}\\ a_0 = 100$$  \n",
+    "$$a_{n+1} = 2a_n,\\ \\ \\ \\mbox{with}\\ a_0=100$$  \n",
+    "$$a_{n+1} = \\frac{3}{a_n},\\ \\ \\ \\mbox{with}\\ a_0=2$$ \n",
+    "```  \n",
+    "  \n",
+    "The recursive description is also called *difference equation*.\n",
+    "+++\n",
+    "Note that with the form $a_{n+1}=g(a_n)$ the following term $a_{n+1}$ can be determined only with the term one step back $a_n$, knowing the function $g$. In general, recursions can be dependent on several other backward terms. When it depends only on the immediate previous term it is called a **first-order** recursion. \n",
+    "````"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Limits  \n",
+    "  \n",
+    "  \n",
+    "In many applications, we are interested to know what is the *long-term behaviour* of a given sequence $\\{a_n\\}$. For example, is a given population tending towards a stable number of individuals as time passes? In other words, we want to know if there is a number $L$ for which the sequence tends to as $n$ increases. To represent this we use the following notation:  \n",
+    "  \n",
+    "$$\\lim_{n\\rightarrow \\infty} a_n = L\\ \\ (\\mbox{or simply}\\ \\lim a_n =L)$$  \n",
+    "  \n",
+    "If such a number exists, we say that the sequence $\\{a_n\\}$ is **convergent**.  \n",
+    "  \n",
+    "```{admonition} Examples  \n",
+    "1. $a_n = \\displaystyle\\frac{1}{n+1}: \\ \\ \\ \\left\\{1,\\displaystyle\\frac{1}{2},\\displaystyle\\frac{1}{3},\\displaystyle\\frac{1}{4},\\displaystyle\\frac{1}{5},\\ldots\\right\\}$  \n",
+    "    <br />\n",
+    "    Note that each term is smaller than the previous and they are all positive. We can conclude that:  \n",
+    "  \n",
+    "    $$\\lim a_n = 0.$$  \n",
+    "\n",
+    "2. $b_n = (-1)^n: \\ \\ \\ \\left\\{1,-1,1,-1,1,-1,\\ldots\\right\\}$  \n",
+    "    <br />\n",
+    "    The terms in this sequence are alternating between $1$ and $-1$, never convergeing for a specific value. Thus, the limit of sequence $\\{b_n\\}$ does not exist.  \n",
+    "    <br />\n",
+    "3. $c_n = 2^n: \\ \\ \\ \\left\\{1,2,4,8,16,32,\\ldots\\right\\}$  \n",
+    "    <br />\n",
+    "    For this sequence, the terms are ever increasing. We frequently say that the sequence tends to infinity (or tends to $\\infty$). However, $\\infty$ represents an abstraction (\"ever increasing\"), not a specific number. Therefore, the limit for this sequence also does not exist.  \n",
+    "    <br />\n",
+    "4. $d_n = \\displaystyle\\frac{n+2}{n+1}: \\ \\ \\ \\left\\{2,\\displaystyle\\frac{3}{2},\\displaystyle\\frac{4}{3},\\displaystyle\\frac{5}{4},\\displaystyle\\frac{6}{5},\\ldots\\right\\}$  \n",
+    "    <br />\n",
+    "    For this sequence, each term is smaller than the previous (look at the decimal forms of the fractions: $2, 1.5, 1.333\\cdots, 1.25, 1.2 \\ldots$) and they are greater than $1$. We can conclude that:  \n",
+    "  \n",
+    "    $$\\lim a_n = 1.$$  \n",
+    "  \n",
+    "```  \n",
+    "  \n",
+    "Although there is a formal way to *prove* that a sequence $\\{a_n\\}$ converges to a certain limit, in practice we use a set of rules to apply limits to sequences and composition of sequences.  \n",
+    "  \n",
+    "```{tip}  \n",
+    "Consider two sequences $\\{a_n\\}$ and $\\{b_n\\}$, and $C$ as a real constant. If these two sequences are convergent with  $\\lim a_n = L$ and $\\lim b_n = M$, we have:  \n",
+    "  \n",
+    "$$\\lim (a_n + b_n) = \\lim a_n + \\lim b_n = L + M$$  \n",
+    "$$\\lim (Ca_n) = C\\lim a_n = CL$$  \n",
+    "$$\\lim (a_nb_n) = (\\lim a_n)(\\lim b_n) = LM$$  \n",
+    "$$\\lim \\left(\\frac{a_n}{b_n}\\right) = \\frac{\\lim a_n}{\\lim b_n} = \\frac{L}{M},\\ \\ (\\mbox{if}\\ M \\ne 0)$$\n",
+    "  \n",
+    "In other other words, if you can decompose a given sequence into the sum, product or division of two other convergent sequences, the limit of this sequence will be the composition of the limits of the two others.\n",
+    "```  \n",
+    "  \n",
+    "```{admonition} Examples  \n",
+    "If $a_n = \\displaystyle\\frac{4n^2-1}{n^2}:$  \n",
+    "  \n",
+    "$$\\lim\\left(\\displaystyle\\frac{4n^2-1}{n^2}\\right) = \\lim\\left(4-\\displaystyle\\frac{1}{n^2}\\right) = \\lim 4 - \\left(\\lim\\displaystyle\\frac{1}{n}\\right)\\left(\\lim\\displaystyle\\frac{1}{n}\\right) = 4,$$  \n",
+    "  \n",
+    "since as we know: $\\ \\lim 4 = 4\\ $ and $\\ \\lim \\displaystyle\\frac{1}{n} = 0$.\n",
+    "\n",
+    "```  \n",
+    "  \n",
+    "For recursions, one way to find the limit is to find the solution (corresponding explicit description) and then calculate the limit of the sequence with the limit rules. However, finding the explicit form of a given sequence is not always possible.  \n",
+    "  \n",
+    "There is a procedure to find *candidates* for the long-term behaviour of a given sequence. We do this by calculating **fixed points**.  \n",
+    "  \n",
+    "Fixed points are specific values on a given sequence for which all subsequent values equal the first. That is, if $a$ is a fixed point of the sequence $\\{a_n\\}$, and if $a_0 = a$, then $a_1 = a$, $a_2 = a$, and all the other values will be $a$. Thus, given the recursion $a_{n+1} = g(a_n)$, if $a$ is a fixed point we will have:  \n",
+    "  \n",
+    "$$a = g(a)$$  \n",
+    "  \n",
+    "and the last equation provides a way of find possible values of $a$.  \n",
+    "  \n",
+    "```{admonition} Example  \n",
+    "  \n",
+    "Consider the following recursion  \n",
+    "  \n",
+    "$$a_{n+1} = \\sqrt{3a_n},\\ \\ \\ a_0 = 2.$$  \n",
+    "  \n",
+    "To obtain the fixed points we calculate:  \n",
+    "  \n",
+    "$$\\begin{align}\n",
+    "  a &= \\sqrt{3a} \\\\\n",
+    "  a^2 &= 3a \\ \\ \\longrightarrow a = 0\\ \\mbox{ or }\\ a = 3 \\end{align}$$  \n",
+    "  \n",
+    "Starting with $a_0 = 2$, we can use the recursion to construct the sequence as:  \n",
+    "  \n",
+    "  $$\\left\\{\\sqrt{6},\\sqrt{3\\sqrt{6}},\\sqrt{3\\sqrt{3\\sqrt{6}}},\\sqrt{3\\sqrt{3\\sqrt{3\\sqrt{6}}}},\\ldots \\right\\}$$  \n",
+    "    \n",
+    "or also: $\\{2.449\\cdots,2.711\\cdots,2.852\\cdots,2.925\\cdots, \\ldots\\}$. We see that starting with $a_0=2$, we have $a_n>2$ for all $n$, with increasing values. Since $a=3$ is a fixed point, we conclude that $\\lim a_n = 3$.  \n",
+    "```  \n",
+    "\n",
+    "```{admonition} Example\n",
+    "Whenever we are asking questions like 'where does this process stabilise' or 'how high/much will it be in the end,' we are interested in the limit of some series. This could be a variety of things, like population size, ground water levels, global mean temperature, number of trees, etc.\n",
+    "```  \n",
+    "  \n",
+    "```{warning}  \n",
+    "  \n",
+    "Remember that fixed points are only **candidates** for long-term behaviour. See for example the recursion:  \n",
+    "  \n",
+    "$$a_{n+1} = \\displaystyle\\frac{3}{a_n}$$  \n",
+    "  \n",
+    "For the fixed points, we calculate:  \n",
+    "  \n",
+    "$$\\begin{align}\n",
+    "  a &= \\displaystyle\\frac{3}{a} \\\\\n",
+    "  a^2 &= 3 \\\\\n",
+    "  &\\longrightarrow a = -\\sqrt{3}\\ \\mbox{ or }\\ a = \\sqrt{3}. \\end{align}$$  \n",
+    "    \n",
+    "Indeed, if we construct the sequence from the recursion starting with any of these two values, all the subsequent elements of the sequence will have the same value. However:  \n",
+    "  \n",
+    "$$\\mbox{if } a_0 = 2 \\ \\mbox{the sequence will be given by: }\\ \\left\\{2,\\displaystyle\\frac{3}{2},2,\\displaystyle\\frac{3}{2},\\ldots \\right\\}$$  \n",
+    "  \n",
+    "or  \n",
+    "  \n",
+    "$$\\mbox{If } a_0 = -3 \\ \\mbox{the sequence will be given by: }\\ \\left\\{-3,-1,-3,-1,\\ldots \\right\\}$$  \n",
+    "  \n",
+    "showing that for any initial value chosen (different from the fixed points), the sequence will oscillate between two values, without a clear tendency to any limit.\n",
+    "```\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Discrete-time population models  \n",
+    "  \n",
+    "Sequences are generally used in many applications in Biology. Important examples are models for **seasonally breeding** populations.  \n",
+    "  \n",
+    "  \n",
+    "### Density-dependent population growth  \n",
+    "  \n",
+    "As we have seen before, the recursion relation for the exponential growth model is given by $N_{t+1} = RN_t$. Rearranging the terms of this equation we have:  \n",
+    "  \n",
+    "$$\\displaystyle\\frac{N_t}{N_{t+1}}=\\frac{1}{R}.$$  \n",
+    "  \n",
+    "The left-hand term of the last equation is known as *parent-offspring ratio* as it denotes the ratio between the population size at time $t$ (parents) and the population size at the following time $t+1$ (offspring). Here we are considering a population that breeds without overlapping generations. Each generation breeds, dies, and is replaced by their offspring on the following generation.  \n",
+    "  \n",
+    "On the equation we also see that the parent-offspring ratio is equal to $\\displaystyle\\frac{1}{R}$, which is a constant. Therefore, we say that this growth is *density-independent* as the parent-offspring ratio does not depend on the density of individuals at any time step. Relating the parent-offspring ratio to the value of $R$ allows us to predict the behaviour of this population in terms of growth or decline:  \n",
+    "  \n",
+    "$$\\begin{align}\n",
+    "R>1 \\ &- \\ \\mbox{there will be more offspring than parents (population grows)} \\\\\n",
+    "R<1 \\ &- \\ \\mbox{there will be more parents than offspring (population declines)} \\\\\n",
+    "R=1 \\ &- \\ \\mbox{there will be no change in population size} \\end{align}$$\n",
+    "  \n",
+    "We know, however, than in real scenarios, that the growth factor of a given population should depend on the current size of that population, due to space or resources limitation. These are called *density-dependent* effects. One of the simplest methods to include density-dependent effects in this population model is to consider the parent-offspring ratio as a linear function of $N_t$, instead of a constant. Let us assume that the parent-offspring ratio show the following linear behaviour:  \n",
+    "  \n",
+    "```{image} parent-offspring.png\n",
+    "```  \n",
+    "  \n",
+    "This means that the parent-offspring ratio is smaller than $1$ (population growth) for values of $N_t$ smaller than a given population threshold $K$, and greater than $1$ (population decline) if the population size is greater than this threshold. The graph above leads to the following expression:  \n",
+    "  \n",
+    "$$\\begin{align}\n",
+    "\\displaystyle\\frac{N_t}{N_{t+1}} = \\frac{1}{R} + \\frac{1-\\frac{1}{R}}{K}N_t \\\\\n",
+    "N_{t+1} = \\frac{N_t}{\\frac{1}{R} + \\frac{1-\\frac{1}{R}}{K}N_t}, \\end{align}$$  \n",
+    "  \n",
+    "which leads, after rearranging the terms, to:  \n",
+    "  \n",
+    "$$N_{t+1} = \\frac{RN_t}{1 + \\frac{(R-1)}{K}N_t}.$$  \n",
+    "  \n",
+    "The previous expression is known as **Beverton-Holt recruitment curve** (or Beverton-Holt model) and it describes a type of population growth called *logistic growth*.  \n",
+    "  \n",
+    "The fixed points $\\bar{N}$ for this recursion can be calculated as:  \n",
+    "  \n",
+    "$$\\bar{N} = \\frac{R\\bar{N}}{1 + \\frac{(R-1)}{K}\\bar{N}}\\ \\longrightarrow \\ \\bar{N}\\left(1 + \\frac{(R-1)}{K}\\bar{N}\\right) = R\\bar{N}$$  \n",
+    "  \n",
+    "From the last term we can show (try it as an exercise) that $\\bar{N}=0$ and $\\bar{N}=K$ are the two fixed points. Starting from different values for the initial population size $N_0$ it can be shown that the population tends to $K$ as time increases as:  \n",
+    "  \n",
+    "```{image} logistic.png\n",
+    "```  \n",
+    "  \n",
+    "The constant $K$ is know as *carrying capacity* and represents the maximum population size a given population can reach at a given place, due to intra-specific competition for resources or space.  \n",
+    "  \n",
+    "  \n",
+    "Watch this [video](https://www.youtube.com/watch?v=Kas0tIxDvrg) about application of growth models to epidemics.  \n",
+    "  \n",
+    "  \n",
+    "### Annual plants  \n",
+    "  \n",
+    "  \n",
+    "Suppose a given population of plants produces, on average $f$ seeds per individual in the reproductive period. Seeds survive winter with a probability $\\sigma$ and germinate on the next season with probability $\\alpha$ (these seeds reach reproductive maturity on the same season). If $p_n$ is the plant population size on season $n$ and $p_{n+1}$ the population size on the season following $n$, we shall have:  \n",
+    "  \n",
+    "$$p_{n+1} = (\\alpha\\sigma f)p_n$$  \n",
+    "  \n",
+    "which correspond to the density-independent model since $R=\\alpha\\sigma f$ is a constant.  \n",
+    "  \n",
+    "Now suppose that some of the seeds that were produced on the previous season and survived the previous winter, but did not germinate on the current season, might have a second chance of germinating. They might do so on the following season, provided they survive the winter and germinate on the following season. From the average number of seeds produced last season $fp_{n-1}$, a number $\\sigma fp_{n-1}$ of them survived last winter. From this number, a fraction $(1-\\alpha)$ did not germinate this season (1 minus the probability of germinating). These seeds will have a second chance provided they survive next winter (with probability $\\sigma$) and germinate on the next season (with probability $\\alpha$). Thus, the number plant population size on the season following $n$ will be:  \n",
+    "  \n",
+    "$$\\begin{align}\n",
+    "p_{n+1} = \\alpha\\sigma fp_n + \\alpha\\sigma(1-\\alpha)\\sigma fp_{n-1} \\\\  \n",
+    "p_{n+1} = (\\alpha\\sigma f)p_n + [\\alpha(1-\\alpha)\\sigma^2 f]p_{n-1} \\\\  \n",
+    "p_{n+1} = \\beta p_n + \\gamma p_{n-1}\\end{align}$$  \n",
+    "  \n",
+    "where $\\beta=\\alpha\\sigma f$ and $\\gamma = \\alpha(1-\\alpha)\\sigma^2 f$.  \n",
+    "  \n",
+    "The previous model is an example of a discrete population model with overlapping populations. Since the term $n+1$ depends not only on the term $n$ but also on $n-1$ this is called a *second-order difference equation* or *delay (or lagged) difference equation*.  \n",
+    "  \n",
+    "  \n",
+    "### Fibonacci sequence  \n",
+    "  \n",
+    "  \n",
+    "The Fibonacci sequence is a sequence defined by the following rule: each number is equal to the sum of the previous two numbers (starting with $0$ and $1$). Thus the following sequence might be formed:  \n",
+    "  \n",
+    "$$\\left\\{ 0,1,1,2,3,5,8,13,21,\\ldots \\right\\}. $$  \n",
+    "  \n",
+    "It can also be defined by the second-order difference equation:  \n",
+    "  \n",
+    "$$F_{n+1} = F_{n} + F_{n-1}.$$  \n",
+    "  \n",
+    "Although this is clearly not a convergent sequence (with sustained growth for each element), if we define a new sequence given by the ratio between successive elements of the Fibonacci sequence as $b_n = \\displaystyle\\frac{F_{n+1}}{F_n}$, it can be show that this sequence is convergent.  \n",
+    "  \n",
+    "Assuming that $\\lim \\displaystyle\\frac{F_{n+1}}{F_n} = \\lambda$, we have:  \n",
+    "  \n",
+    "$$\\begin{align}\n",
+    "F_{n+1} &= F_{n} + F_{n-1}\\quad \\mbox{(Fibonacci seq.)} \\\\  \n",
+    "\\displaystyle\\frac{F_{n+1}}{F_{n}} &= 1 + \\frac{F_{n-1}}{F_{n}}\\quad \\mbox{(divide both sides by}\\ F_{n}\\mbox{)} \\\\\n",
+    "\\lim \\left(\\displaystyle\\frac{F_{n+1}}{F_{n}}\\right) &= \\lim\\left(1 + \\frac{F_{n-1}}{F_{n}}\\right) \\\\\n",
+    "\\lambda &= 1 + \\frac{1}{\\lambda} \\\\\n",
+    "\\lambda^2 &- \\lambda -1 = 0 \\end{align}$$\n",
+    "  \n",
+    "The previous equation is solved finding the roots of the second-degree polynomial, which gives $\\lambda = \\displaystyle\\frac{1+\\sqrt{5}}{2}$ or $\\lambda = \\displaystyle\\frac{1-\\sqrt{5}}{2}$. Since the solution $\\displaystyle\\frac{1-\\sqrt{5}}{2}$ is negative, it can be the limit of the ratio of Fibonacci terms (all of them positive). Hence we conclude:  \n",
+    "  \n",
+    "$$\\lim \\left(\\displaystyle\\frac{F_{n+1}}{F_{n}}\\right) = \\displaystyle\\frac{1+\\sqrt{5}}{2} \\approx 1.618.$$  \n",
+    "  \n",
+    "The previous number is known as the *Golden ratio* and it has fascinated mathematicians, phylosophers, and artists since ancient times. Fibonacci numbers and the Golden ratio can be found in several interesting natural patterns, as you can see in this [video](https://www.youtube.com/watch?v=ahXIMUkSXX0)."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## From table arrangements to powerful tools\n",
+    "  "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Systems of linear equations  \n",
+    "  \n",
+    "As we saw previously, the equation $ax + by = c$, with $a$, $b$, and $c$ constants, defines a straight line on the $xy$-plane (note that it can also be written as $y = mx + k$, $m$ and $k$ constants, by a suitable change of the coeffients). Since the coefficients do not depend on the variables $x$ and $y$, we call it a *linear equation*. A possible plot for this equation would be:  \n",
+    "  \n",
+    "```{image} lineq.png\n",
+    "```  \n",
+    "  \n",
+    "Note that all the points $(x,y)$ on the blue line *satisfy* or *solve* the lienar equation $ax + by = c$.  \n",
+    "  \n",
+    "Now consider that we have the following:  \n",
+    "  \n",
+    "$$\\begin{cases} ax + by = c \\\\ dx +ey = f \\end{cases}$$  \n",
+    "  \n",
+    "where $a, b, c, d, e,\\ \\mbox{and}\\ f$ are all constants.In this case we have a *system of linear equations*. Solving this system means to find all possible pairs $(x,y)$ that solve both equations **simultaneously**. For the previous system, we would have 3 options:\n",
+    "  \n",
+    "```{image} syslineq.png\n",
+    "```  \n",
+    "  \n",
+    "On the left panel, the two lines intersect at a *single point*. This point is the only one that solves both linear equations at the same time, and, therefore, constitutes the **only solution** for the system of two equations. On the central panel, the two lines are *parallel*, so there is no point of intersection. In this case there is **no solution** and the system of equations is said to be *inconsistent*. On the right panel, the two lines coincide. Thus, every point in those lines correspond to a possible solution for the system, so that it has **infinite solutions**.  \n",
+    "  \n",
+    "   \n",
+    "```{admonition} Example  \n",
+    "  \n",
+    "Given the system of linear equations  \n",
+    "  \n",
+    "$$\\begin{cases} 2x + 3y = 6 \\\\ 2x + y = 4 \\end{cases}$$\n",
+    "\n",
+    "we can obtain the solution by isolating one of the variables in one of the equations and substituting on the other equation. Isolating $x$ on the first equation gives:  \n",
+    "  \n",
+    "$$x = 3 - \\displaystyle\\frac{3}{2}y$$  \n",
+    "  \n",
+    "Then by substituting on the second we have:  \n",
+    "  \n",
+    "$$\\begin{align} \n",
+    "2\\left(3 - \\displaystyle\\frac{3}{2}y \\right) + y &= 4 \\\\\n",
+    "6 - 3y + y &= 4 \\\\\n",
+    "-2y = -2 \\\\\n",
+    "y = 1 \\end{align}$$  \n",
+    "  \n",
+    "The value of $x$ shall then be given by:  \n",
+    "  \n",
+    "$$x = 3 - \\displaystyle\\frac{3}{2}\\cdot 1 = 3 - \\frac{3}{2} = \\frac{3}{2}$$  \n",
+    "  \n",
+    "We see that the system has a single solution, given by $\\left(\\displaystyle\\frac{3}{2},1\\right)$.  \n",
+    "  \n",
+    "```  \n",
+    "  \n",
+    "````{admonition} Example  \n",
+    "  \n",
+    "Now consider the sytem  \n",
+    "  \n",
+    "$$\\begin{cases} 2x + 3y &= 6 \\\\ -4x + -6y &= -12 \\end{cases}$$\n",
+    "\n",
+    "Isolating $x$ on the first equation gives the same as before:  \n",
+    "  \n",
+    "$$x = 3 - \\displaystyle\\frac{3}{2}y$$  \n",
+    "  \n",
+    "Then by substituting on the second we have:  \n",
+    "  \n",
+    "$$\\begin{align} \n",
+    "-4\\left(3 - \\displaystyle\\frac{3}{2}y \\right) - 6y &= -12 \\\\\n",
+    "-12 + 6y - 6y &= -12 \\\\\n",
+    "0 = 0 \\end{align}$$  \n",
+    "  \n",
+    "which is just a trivial equation.  \n",
+    "  \n",
+    "When we look at the equations in our system we see that the second equation is simply a constant multiplied by the first $(\\text{Eqn.} 2) = -2\\cdot(\\text{Eqn.} 1)$, and thus they should represent the same straight line.  \n",
+    "  \n",
+    "This system has then infinite solutions which are represented by the set of points $\\left(3 - \\displaystyle\\frac{3}{2}y,y\\right)$\n",
+    "  \n",
+    "```{tip}\n",
+    "Note that writing the solution as $\\left(3 - \\displaystyle\\frac{3}{2}y,y\\right)$ represents an infinite set of points, since for every real value of $y$ we would have a different ordered pair. This set of points forms the straight line corresponding to $2x+3y = 6$.\n",
+    "\n",
+    "```\n",
+    "  \n",
+    "````  \n",
+    "  \n",
+    "Solving the system by this method of substitutions, however, can get much more complicated if we have more than two variables. For these cases, we need a simpler method.  \n",
+    "  \n",
+    "One strategy is to find equivalent systems, which have the same solution as the original one, but are much easier to solve. We can apply the following set of operations without changing the solutions of a given system:  \n",
+    "  \n",
+    "- Multiplication of an entire row by a non-zero constant  \n",
+    "- Adding/subtracting one row to another  \n",
+    "- Rearranging the order of the rows  \n",
+    "  \n",
+    "The examples below show this procedure applied to two systems.  \n",
+    "  \n",
+    "```{admonition} Examples  \n",
+    "  \n",
+    "(i) $\\begin{cases} 3x + 5y - z = 10 \\\\ 2x –  y + 3z = 9 \\\\ 4x + 2y - 3z = -1 \\end{cases}$  \n",
+    "  \n",
+    "See the solution on this [video](https://web.microsoftstream.com/video/194b0152-903a-4614-b608-306ce71d9a39)\n",
+    "  \n",
+    "(ii) $\\begin{cases} x - 3y + z = 4 \\\\ x – 2y + 3z = 6 \\\\ 2x - 6y + 2z = 8 \\end{cases}$  \n",
+    "  \n",
+    "See the solution on this [video](https://web.microsoftstream.com/video/067c8938-5e45-4f08-bf90-783cb125c6f5)\n",
+    "```  \n",
+    "  \n",
+    "Note that by applying the rules of matrix multiplication, the system *(ii)* can also be written in the form:  \n",
+    "  \n",
+    "$$\\begin{pmatrix} 1 & -3 & 1\\\\ 1 & -2 & 3 \\\\ 2 & -6 & 2  \\end{pmatrix} \\begin{pmatrix} x \\\\ y \\\\ z   \\end{pmatrix} = \\begin{pmatrix} 4 \\\\ 6 \\\\ 8   \\end{pmatrix} \\quad \\longrightarrow \\quad A\\mathbf{x}=\\mathbf{b}$$  \n",
+    "  \n",
+    "where $A$ is the matrix of coefficients and $\\mathbf{x}$ is the vector of variables.  \n",
+    "  \n",
+    "```{tip}  \n",
+    "We will be using the common notation of uppercase letters ($A$) to represent matrices and lowercase bold letters ($\\mathbf{x}$) to represent vectors.\n",
+    "```\n",
+    "\n",
+    "In general, the system with $n$ variables $\\{x_1, x_2, \\ldots, x_n \\}$ and $m$ equations can be written as  \n",
+    "  \n",
+    "$$\\begin{cases} a_{11}x_1 + a_{12}x_2 + \\ldots + a_{1n}x_n = b_1 \\\\ a_{21}x_1 + a_{22}x_2 + \\ldots + a_{2n}x_n = b_2 \\\\ \\dots \\\\ a_{m1}x_1 + a_{m2}x_2 + \\ldots + a_{mn}x_n = b_m \\end{cases}$$  \n",
+    "  \n",
+    "can be written as $A\\mathbf{x}=\\mathbf{b}$, where  \n",
+    "  \n",
+    "$$A=\\begin{pmatrix} a_{11} & a_{12} & \\cdots & a_{1n} \\\\ a_{21} & a_{22} & \\cdots & a_{2n} \\\\ \\vdots & & \\ddots & \\vdots \\\\ a_{m1} & a_{m2} & \\cdots & a_{mn} \\end{pmatrix}$$  \n",
+    "  \n",
+    "and  \n",
+    "  \n",
+    "$$\\mathbf{x} = \\begin{pmatrix} x_1 \\\\ x_2 \\\\ \\vdots \\\\ x_n   \\end{pmatrix}\\quad \\mbox{and} \\quad \\mathbf{b} = \\begin{pmatrix} b_1 \\\\ b_2 \\\\ \\vdots \\\\ b_m   \\end{pmatrix}$$  \n",
+    "  \n",
+    "The matrix $A$ has size $m \\times n$ (since the number of equations can, in principle, be different from the number of variables), while the vectors $\\mathbf{x}$ and $\\mathbf{b}$ have sizes $n \\times 1$ and $m \\times 1$, respectively.  \n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Matrix operations  \n",
+    "  \n",
+    "Let us quickly remind the basic operations that can be performed with matrices. If $A$ and $B$ are both matrices of size $m \\times n$ ($m$ rows and $n$ columns), we have:  \n",
+    "  \n",
+    "- $A=B\\ $ if and only if $a_{ij} = b_{ij}$ for every $1 \\le i \\le m,  1 \\le j \\le n$ (elements at the same position have to be equal).  \n",
+    "  \n",
+    "- $C = A + B$ is the matrix for which $c_{ij} = a_{ij} + b_{ij}$ (sum is performed summing the elements at the same position)  \n",
+    "  \n",
+    "- If $k \\in \\rm I\\!R$ than $kA$ is the $m \\times n$ matrix with elements $ka_{ij}$  \n",
+    "  \n",
+    "- The following properties are valid:  \n",
+    "    \n",
+    "    $$\\begin{align}\n",
+    "    A+B &= B+A \\\\\n",
+    "    (A+B)+C &= A+(B+C) \\\\\n",
+    "    A + 0 &= A\\ (0\\ \\mbox{here represents the matrix with null elements}) \\end{align}$$  \n",
+    "  \n",
+    "    ```{admonition} Example  \n",
+    "    If $A=\\begin{pmatrix} 1 & 0  \\\\ -1 & 2  \\end{pmatrix}$ and $B=\\begin{pmatrix} 2 & 3  \\\\ -1 & 1  \\end{pmatrix}$, then:  \n",
+    "\n",
+    "    $$C = 2A+B = \\begin{pmatrix} 2 & 0  \\\\ -2 & 4  \\end{pmatrix} + \\begin{pmatrix} 2 & 3  \\\\ -1 & 1  \\end{pmatrix} = \\begin{pmatrix} 4 & 3  \\\\ -3 & 5  \\end{pmatrix}$$\n",
+    "    ```  \n",
+    "      \n",
+    "- If $A$ is the $m \\times n$ matrix with elements $a_{ij}$, the transpose of $A$ (denoted by $A'$) is the $n \\times m$ matrix with elements $a_{ij}' = a_{ji}$.  \n",
+    "  \n",
+    "    ```{admonition} Example  \n",
+    "    $$A=\\begin{pmatrix} 1 & 2 & 3 \\\\ 4 & 5 & 6  \\end{pmatrix} \\longrightarrow A'=\\begin{pmatrix} 1 & 4 \\\\ 2 & 5 \\\\ 3 & 6   \\end{pmatrix}$$\n",
+    "      \n",
+    "    $$B=\\begin{pmatrix} 1 & 2 \\end{pmatrix} \\longrightarrow B' = \\begin{pmatrix} 1 \\\\ 2 \\end{pmatrix}$$\n",
+    "    ```  \n",
+    "      \n",
+    "- **(Matrix multiplication)** Consider now that $A$ and $B$ are matrices with sizes $m \\times l$ and $l \\times n$ (number of columns of $A$ is equal to the number of rows of $B$). Then we can define $C = AB$  as the $m \\times n$ matrix with elements $c_{ij}$ formed by multiplying every element of the $i$-th row of $A$ with the corresponding element of the $j$-th column of $B$.  \n",
+    "    \n",
+    "    ```{admonition} Example  \n",
+    "      If $A=\\begin{pmatrix} 1 & 2  \\\\ -1 & 0  \\end{pmatrix}$ and $B=\\begin{pmatrix} 1 & 2 & 3  \\\\ 0 & -1 & 4  \\end{pmatrix}$, then:  \n",
+    "        \n",
+    "      $$C = AB = \\begin{pmatrix} 1 & 0 & 11  \\\\ -1 & -2 & -3  \\end{pmatrix}.$$  \n",
+    "        \n",
+    "      Note that the matrix $BA$ cannot be defined here since the number of columns of $B$ is different from the number of rows of $A$.\n",
+    "    ```  \n",
+    "      \n",
+    "    ```{warning}  \n",
+    "    If matrices $A$ and $B$ are *square matrices* (same number of rows and columns) then we can define both $AB$ and $BA$. But, in general, we **cannot assume** that $AB=BA$. For example, if  $A=\\begin{pmatrix} 2 & -1  \\\\ 4 & -2  \\end{pmatrix}$ and $B=\\begin{pmatrix} 1 & -1  \\\\ 2 & -2  \\end{pmatrix}$, then:  \n",
+    "      \n",
+    "    $$AB = \\begin{pmatrix} 0 & 0  \\\\ 0 & 0  \\end{pmatrix} \\quad and \\quad BA = \\begin{pmatrix} -2 & 1  \\\\ -4 & 2  \\end{pmatrix}$$\n",
+    "    ```  \n",
+    "      \n",
+    "- For matrix multiplication, the following properties are valid:  \n",
+    "    \n",
+    "    $$\\begin{align}\n",
+    "    (A+B)C&=AC+BC \\\\\n",
+    "    A(B+C)&=AB+AC \\\\\n",
+    "    (AB)C &= A(BC) \\\\\n",
+    "    A0 &= 0A = 0 \\end{align}$$  \n",
+    "  \n",
+    "- We can also define powers of $A$. If $k$ is a positive integer:  \n",
+    "    \n",
+    "    $$A^k = A^{k-1}A = AA^{k-1} = A\\cdot A\\cdot A \\cdots A\\ (\\mbox{product of}\\ k\\ \\mbox{matrices})$$\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Inverse matrix  \n",
+    "  \n",
+    "Suppose that $A$ is a $n \\times n$ matrix. If there exists a $n \\times n$ matrix $B$ so that  \n",
+    "  \n",
+    "$$ AB = BA = I_n,$$  \n",
+    "  \n",
+    "where $I=\\begin{pmatrix} 1 & 0 & \\cdots & 0 \\\\ 0 & 1 & \\cdots & 0 \\\\ \\vdots & & \\ddots & \\vdots \\\\ 0 & 0 & \\cdots & 1 \\end{pmatrix}$ is the $n \\times n$ *identity matrix* (with elements $1$ on the main diagonal and $0$ otherwise). \n",
+    "  \n",
+    "In this case we call the matrix $B$ the *inverse* of $A$ and denote $B=A^{-1}$. If $A$ has an inverse than it is called *invertible* or *non-singular*. If it does not have an inverse, $A$ is *singular*.\n",
+    "\n",
+    "Finding the inverse of $A$ automatically gives us the solution to the system of linear equations $A\\mathbf x = \\mathbf x$\n",
+    "\n",
+    "$$\\begin{align}  \n",
+    "A\\mathbf{x}&=\\mathbf{b} \\\\\n",
+    "A^{-1} A\\mathbf{x} &= A^{-1}\\mathbf{b} \\\\\n",
+    "\\mathbf{x} &= A^{-1}\\mathbf{b} \\end{align}$$  \n",
+    "\n",
+    "Finding ways of inverting matrices is therefore needed for a wide range of applications. A whole research field is dealing with optimising matrix inversions. \n",
+    "  \n",
+    "```{admonition} Example  \n",
+    "  \n",
+    "To find the inverse of matrix $A= \\begin{pmatrix} 2 & 5  \\\\ 1 & 3  \\end{pmatrix}$, we have to find the matrix $B = \\begin{pmatrix} b_{11} & b_{12}  \\\\ b_{21} & b_{22}  \\end{pmatrix}$ so that  \n",
+    "  \n",
+    "$$ AB = I \\quad (\\mbox{then check that }\\ BA = I.$$  \n",
+    "  \n",
+    "Thus  \n",
+    "  \n",
+    "$$\\begin{align}  \n",
+    "  \\begin{pmatrix} 2 & 5  \\\\ 1 & 3  \\end{pmatrix}\\begin{pmatrix} b_{11} & b_{12}  \\\\ b_{21} & b_{22}  \\end{pmatrix} = \\begin{pmatrix} 1 & 0  \\\\ 0 & 1  \\end{pmatrix} \\\\  \n",
+    "  \\begin{pmatrix} 2b_{11} + 5b_{21}  & 2b_{12} + 5b_{22} \\\\ b_{11} + 3b_{21} & b_{12} + 3b_{22}  \\end{pmatrix} = \\begin{pmatrix} 1 & 0  \\\\ 0 & 1  \\end{pmatrix}, \\\\\n",
+    "\\end{align}$$  \n",
+    "  \n",
+    "which leads to two systems of equations:  \n",
+    "\n",
+    "$$\\begin{cases} 2b_{11} + 5b_{21} &= 1 \\\\ b_{11} + 3b_{21} &= 0 \\end{cases}\\quad \\mbox{and}\\quad \\begin{cases} 2b_{12} + 5b_{22} &= 0 \\\\ b_{12} + 3b_{22} &= 1 \\end{cases}$$  \n",
+    "  \n",
+    "The solutions of these two systems, leads to $B= \\begin{pmatrix} 3 & -5  \\\\ -1 & 2  \\end{pmatrix}$\n",
+    "```  \n",
+    "  "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Determinant  \n",
+    "  \n",
+    "If $A = \\begin{pmatrix} a_{11} & a_{12}  \\\\ a_{21} & a_{22}  \\end{pmatrix}$ then the expression:  \n",
+    "  \n",
+    "$$\\mbox{det}A = a_{11}a_{22} - a_{21}a_{12}$$   \n",
+    "  \n",
+    "given by the product of the elements on the main diagonal minus the product of the elements on secondary diagonal. The term $\\mbox{det}A$ is called the *determinant* of $A$. The determinant can be calculated for matrices of any size, but the expression gets increasingly more complicated for larger matrices.  \n",
+    "  \n",
+    "The determinant establishes a clear criterium for the existence of $A^{-1}$. The matrix $A$ will be invertible if and only if $\\mbox{det}A \\ne 0$.  \n",
+    "  \n",
+    "```{note}  \n",
+    "The previous criterium has an important implication. Consider the system of linear equations defined by $A\\mathbf{x}=\\mathbf{b}$ with equal number of equations and variables. If $\\mbox{det}A \\ne 0$ then the inverse of $A$ exists. Thus, if we multiply $A^{-1}$ to the left of each side of the equation defining the linear system, we have:  \n",
+    "  \n",
+    "$$\\begin{align}  \n",
+    "A\\mathbf{x}&=\\mathbf{b} \\\\\n",
+    "A^{-1}(A\\mathbf{x}) &= A^{-1}\\mathbf{b} \\\\\n",
+    "(A^{-1}A)\\mathbf{x} &= A^{-1}\\mathbf{b} \\\\\n",
+    "\\mathbf{x} &= A^{-1}\\mathbf{b} \\end{align}$$  \n",
+    "  \n",
+    "which means that if $\\mbox{det}A \\ne 0$ the system will have a **single solution** with the variables assuming the values $\\mathbf{x} = A^{-1}\\mathbf{b}$.\n",
+    "```"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Structured models  \n",
+    "  \n",
+    "In the models we have seen until now, all the individuals in a given population are identical. However, real populations have importance differences between individuals according to their life stages. For example, some insects have stages of eggs, larvae, pupae, and adults, while many plants pass through stages of seeds, seedlings, saplings, and mature individuals. Thus, understanding the differences in the dynamics of each life stage is important to add realism to our models (especially in applied models, such as pest control or harvesting management).  \n",
+    "  \n",
+    "Suppose our population is composed by two classes: *immature* and *mature* individuals. Let us assume non-overlapping breeding seasons and that the timescales for reproduction and maturation are similar. Thus, time can be counted in discrete steps corresponding to this reproduction/maturation interval. At time $t$ we have $X_t$ immature individuals and $Y_t$ mature individuals. Immature individuals survive until maturity with a probability $p$, while mature individuals reproduce and generate, on average, $m$ immature individuals for the next generation. Mature individuals die after reproduction. With these assumptions, we can calculate the sizes of each population class on the step $t+1$ as:\n",
+    "  \n",
+    "$$\\begin{align}\n",
+    "X_{t+1} = m Y_t \\\\\n",
+    "Y_{t+1} = p X_t \\end{align}$$  \n",
+    "  \n",
+    "See that now we have a model with two equations describing the variable class sizes in our population. Moreover, the dynamics of each class depends on the other, so that we have a system of *coupled* difference equations.  \n",
+    "\n",
+    "Interestingly, we can also write this system in matrix form, as:  \n",
+    "  \n",
+    "$$\\begin{pmatrix} X_{t+1}  \\\\ Y_{t+1} \\end{pmatrix} = \\begin{pmatrix} 0 & m  \\\\ p & 0  \\end{pmatrix} \\begin{pmatrix} X_{t}  \\\\ Y_{t} \\end{pmatrix}, \\quad \\mbox{or}$$\n",
+    "\n",
+    "$$\\mathbf{n_{t+1}} = P\\mathbf{n_{t}}$$\n",
+    "  \n",
+    "where the sum of the components of the population vector $\\mathbf{n_{t}}$, gives the total population (immature plus mature individuals) at time $t$: $N_t = X_t + Y_t$. The matrix $P=\\begin{pmatrix} 0 & m  \\\\ p & 0  \\end{pmatrix}$ is called *projection matrix* or *Leslie matrix*. Multiplying matrix $P$ by the population vector at time $t$, $\\mathbf{n_{t}}$, we can project the next population size at time $t+1$. Thus, from an initial population state, say $\\mathbf{n_{0}}$, the projection matrix governs the dynamics of the model.  \n",
+    "  \n",
+    "<br />\n",
+    "\n",
+    "This model can be easily generalised for a larger number of age classes. Suppose we now have $q+1$ classes, $X_0, X_1, X_2, \\ldots, X_q$, from the youngest (meaningful) age $X_0$ until the oldest age $X_q$. At each time step $t$, individuals from class $X_i$ have a chance of surviving to the next age class $X_{i+1}$, with probability $p_{i+1\\ i}$ (imagine an arrow for direction of age class progession $p_{i+1\\leftarrow i}$. We are going to omit the arrow for simplicity). Additionally, suppose that each age class $X_i$ excluding the youngest, will, in principle, be able to reproduce, with average fecundity $m_i$ generating new individuals for class $X_0$. Thus, with the age class sizes at time $t$, and the parameters for survival probabilities and average fecundities, we have:  \n",
+    "  \n",
+    "$$\\begin{align}\n",
+    "X_0(t+1) &= m_1X_1(t) +  m_2X_2(t) + m_3X_3(t) + \\ldots + m_qX_q(t) \\\\\n",
+    "X_1(t+1) &= p_{10}X_0(t) \\\\\n",
+    "X_2(t+1) &= p_{21}X_1(t) \\\\\n",
+    "X_3(t+1) &= p_{32}X_2(t) \\\\\n",
+    "         &\\vdots \\\\\n",
+    "X_q(t+1) &= p_{q\\ q-1}X_{q-1}(t)\n",
+    "    \\end{align}$$  \n",
+    "    \n",
+    "This can also be written in matrix form as:  \n",
+    "  \n",
+    "$$\\begin{pmatrix} X_0(t+1)  \\\\ X_1(t+1) \\\\X_2(t+1)  \\\\X_3(t+1) \\\\ \\vdots \\\\X_q(t+1)  \\\\ \\end{pmatrix} = \\begin{pmatrix} 0 & m_1 & m_2 & m_3 & \\cdots & m_q \\\\ p_{10} & 0 & 0 & 0 & \\cdots &0 \\\\ 0 & p_{21} & 0 & 0 & \\cdots & 0 \\\\ 0 & 0 & p_{32} & 0 & \\cdots & 0 \\\\ \\vdots & \\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\ 0 & 0 & 0 & \\cdots & p_{q\\ q-1} & 0\\end{pmatrix} \\begin{pmatrix} X_0(t)  \\\\ X_1(t) \\\\X_2(t) \\\\X_3(t) \\\\ \\vdots \\\\X_q(t)  \\\\ \\end{pmatrix}$$  \n",
+    "  \n",
+    "$$\\mathbf{n_{t+1}} = P\\mathbf{n_{t}}$$\n",
+    "  \n",
+    "The generalised projection matrix $P$ has then all the surviving probabilities just below the main diagonal, and the fecundities for each class in its $0$-th row.  \n",
+    "  \n",
+    "```{note}  \n",
+    "Two important differences here. First, the rows and columns for the matrix are indexed starting from $0$ until $q$, not starting from $1$ as usual. Second, the time steps are written in parentheses for each age class, instead of a subscript as we have seen before for other population models, so that it does not conflict with the index for the age classes (going from $0$ to $q$).\n",
+    "```  \n",
+    "  \n",
+    "```{admonition} Example  \n",
+    "Consider a population with three age classes $X_0, X_1,\\mbox{ and}\\ X_2$, and the following projection matrix  \n",
+    "  \n",
+    "$$P=\\begin{pmatrix} 0 & 5 & 10\\\\ 0.5 & 0 & 0  \\\\ 0 & 0.2 & 0 \\end{pmatrix}$$\n",
+    "\n",
+    "  \n",
+    "If the population starts with $10$ individuals in the oldest category at time $t=0$, the population vector at time $t=1$ will be:  \n",
+    "  \n",
+    "$$\\mathbf{n_{1}} = P\\mathbf{n_{0}} \\longrightarrow \\begin{pmatrix} X_0(1)  \\\\ X_1(1) \\\\X_2(1) \\end{pmatrix}=\\begin{pmatrix} 0 & 5 & 10\\\\ 0.5 & 0 & 0  \\\\ 0 & 0.2 & 0 \\end{pmatrix}\\begin{pmatrix} 0  \\\\ 0 \\\\ 10 \\end{pmatrix} \\longrightarrow \\begin{pmatrix} 100  \\\\ 0 \\\\ 0 \\end{pmatrix}$$  \n",
+    "  \n",
+    "The population age distribution at each time step will then be given be the successive application of the matrix $P$ to the previous population distribution. For the first 10 time steps we have:  \n",
+    "  \n",
+    "$$\\begin{pmatrix}0 \\\\ 0 \\\\ 10 \\end{pmatrix} \\rightarrow \\begin{pmatrix} 100  \\\\ 0 \\\\ 0 \\end{pmatrix} \\rightarrow \\begin{pmatrix} 0  \\\\ 50 \\\\ 0 \\end{pmatrix} \\rightarrow \\begin{pmatrix} 250  \\\\ 0 \\\\ 10 \\end{pmatrix} \\rightarrow \\begin{pmatrix} 100  \\\\ 125 \\\\ 0 \\end{pmatrix} \\rightarrow \\begin{pmatrix} 625  \\\\ 50 \\\\ 25 \\end{pmatrix} \\rightarrow \\begin{pmatrix} 500  \\\\ 312 \\\\ 10 \\end{pmatrix} \\rightarrow \\begin{pmatrix} 1662  \\\\ 250 \\\\ 62 \\end{pmatrix} \\rightarrow \\begin{pmatrix} 1875  \\\\ 831 \\\\ 50 \\end{pmatrix} \\rightarrow \\begin{pmatrix} 4656  \\\\ 937 \\\\ 166 \\end{pmatrix}$$  \n",
+    "  \n",
+    "Two important things are worth noting here. First, the total population size (sum accros all age classes), follows the sequence $N_t = \\left\\{ 10,100,50,260,225,700,822,1975,2756,5759,\\ldots \\right \\}$. The reproduction process does not always lead to increasing population sizes in the beginning, but is subject to size fluctuations due to the demographic distribution of this population. After a few steps increases in population size become more consistent. Second, as the population size continues to increase, the population age distribution tends to a given proportion of the population classes given by: $X_0 \\approx 76\\%, X_1 \\approx 22\\%, \\mbox{and}\\ X_2 \\approx 2\\%$.  \n",
+    "  \n",
+    "Now consider that, instead of $10$ individuals in the oldest class, we initiate the population with the following distribution $\\mathbf{n_{0}} = \\begin{pmatrix} 7.6  \\\\ 2.2 \\\\ 0.2 \\end{pmatrix}$. The population progression by successive application of the projection matrix will be  \n",
+    "  \n",
+    "$$\\begin{pmatrix} 7.6  \\\\ 2.2 \\\\ 0.2 \\end{pmatrix} \\longrightarrow \\begin{pmatrix} 13  \\\\ 4 \\\\ 0.4 \\end{pmatrix} \\longrightarrow \\begin{pmatrix} 23  \\\\ 6 \\\\ 0.76 \\end{pmatrix} \\longrightarrow \\begin{pmatrix} 40  \\\\ 12 \\\\ 1 \\end{pmatrix} \\longrightarrow \\begin{pmatrix} 72  \\\\ 20 \\\\ 2 \\end{pmatrix} \\longrightarrow \\begin{pmatrix} 124  \\\\ 36 \\\\ 4 \\end{pmatrix} \\longrightarrow \\begin{pmatrix} 219  \\\\ 62 \\\\ 7 \\end{pmatrix} \\longrightarrow \\begin{pmatrix} 381  \\\\ 109 \\\\ 12 \\end{pmatrix} $$  \n",
+    "  \n",
+    "You can check that although each age cathegory is increasing, the percentage contribution of each class to the total population remains nearly constant. To this distribution of classes we give the name of **stable age distribution**.  \n",
+    "  \n",
+    "Moreover, at each time step, each age class increases with approximately the same factor in relation to the previous step, around $\\lambda = 1.75$. If for each age class $X_i(t+1) = \\lambda X_i(t)$ is valid,then we have:  \n",
+    "  \n",
+    "$$\\mathbf{n_{t+1}}=\\lambda\\mathbf{n_{t}},$$  \n",
+    "  \n",
+    "so that we recover the usual model for exponential growth for the total population size $N_{t+1} = \\lambda N_t$.  \n",
+    "\n",
+    "```  "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 40,
+   "metadata": {
+    "tags": [
+     "remove-cell"
+    ]
+   },
+   "outputs": [
+    {
+     "data": {
+      "image/png": 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",
+      "text/plain": [
+       "<Figure size 360x360 with 1 Axes>"
+      ]
+     },
+     "metadata": {
+      "needs_background": "light"
+     },
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "import matplotlib.pyplot as plt\n",
+    "import matplotlib.ticker as plticker\n",
+    "import numpy as np\n",
+    "\n",
+    "X = np.array((0,0,0,0))\n",
+    "Y= np.array((0,0,0,0))\n",
+    "U = np.array((3,2,1,8))\n",
+    "V = np.array((-1,3,7,12))\n",
+    "\n",
+    "fig, ax = plt.subplots(figsize=(5, 5))\n",
+    "q = ax.quiver(X, Y, U, V,units='xy', scale=1)\n",
+    "\n",
+    "ax.spines['left'].set_position('zero')\n",
+    "ax.spines['bottom'].set_position('zero')\n",
+    "\n",
+    "# Eliminate upper and right axes\n",
+    "ax.spines['right'].set_color('none')\n",
+    "ax.spines['top'].set_color('none')\n",
+    "\n",
+    "# Show ticks in the left and lower axes only\n",
+    "ax.xaxis.set_ticks_position('bottom')\n",
+    "ax.yaxis.set_ticks_position('left')\n",
+    "\n",
+    "ax.set_aspect('equal')\n",
+    "\n",
+    "ax.annotate('$\\mathbf{v_1}$', (3+0.2,-1-0.8),fontsize=14)\n",
+    "ax.annotate('$\\mathbf{v_2}$', (2+0.2,3-0.8),fontsize=14)\n",
+    "ax.annotate('$A\\mathbf{v_1}$', (1+0.2,7-0.8),fontsize=14)\n",
+    "ax.annotate('$A\\mathbf{v_2}$', (8+0.2,12-0.8),fontsize=14)\n",
+    "\n",
+    "ax.tick_params(axis='both', which='major', labelsize=13)\n",
+    "\n",
+    "loc = plticker.MultipleLocator(base=2) # this locator puts ticks at regular intervals\n",
+    "ax.xaxis.set_major_locator(loc)\n",
+    "\n",
+    "ax.set_ylim(-2,14)\n",
+    "ax.set_xlim(-2,10)\n",
+    "\n",
+    "plt.tight_layout()\n",
+    "plt.savefig('lin_trans.png')"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 46,
+   "metadata": {
+    "tags": [
+     "remove-cell"
+    ]
+   },
+   "outputs": [
+    {
+     "data": {
+      "image/png": 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",
+      "text/plain": [
+       "<Figure size 360x360 with 1 Axes>"
+      ]
+     },
+     "metadata": {
+      "needs_background": "light"
+     },
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "import matplotlib.pyplot as plt\n",
+    "import matplotlib.ticker as plticker\n",
+    "import numpy as np\n",
+    "\n",
+    "X = np.array((0,0))\n",
+    "Y= np.array((0,0))\n",
+    "U = np.array((1,2/3))\n",
+    "V = np.array((-1,1))\n",
+    "\n",
+    "fig, ax = plt.subplots(figsize=(5, 5))\n",
+    "q = ax.quiver(X, Y, U, V,units='xy', scale=1)\n",
+    "\n",
+    "ax.spines['left'].set_position('zero')\n",
+    "ax.spines['bottom'].set_position('zero')\n",
+    "\n",
+    "# Eliminate upper and right axes\n",
+    "ax.spines['right'].set_color('none')\n",
+    "ax.spines['top'].set_color('none')\n",
+    "\n",
+    "# Show ticks in the left and lower axes only\n",
+    "ax.xaxis.set_ticks_position('bottom')\n",
+    "ax.yaxis.set_ticks_position('left')\n",
+    "\n",
+    "ax.set_aspect('equal')\n",
+    "\n",
+    "ax.annotate('$\\mathbf{v_1}\\ (\\lambda=-1)$', (1+0.1,-1-0.1),fontsize=14)\n",
+    "ax.annotate('$\\mathbf{v_2}\\ (\\lambda=4)$', (2/3+0.1,1+0.1),fontsize=14)\n",
+    "\n",
+    "ax.tick_params(axis='both', which='major', labelsize=13)\n",
+    "\n",
+    "#loc = plticker.MultipleLocator(base=2) # this locator puts ticks at regular intervals\n",
+    "#ax.xaxis.set_major_locator(loc)\n",
+    "\n",
+    "ax.set_ylim(-2,2)\n",
+    "ax.set_xlim(-2,2)\n",
+    "\n",
+    "plt.tight_layout()\n",
+    "plt.savefig('eigenvec_01.png')"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## From table arrangements to powerful tools - part 2\n",
+    "  "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Linear transformations\n",
+    "\n",
+    "In general terms, a given matrix $A$ is a representation of what we call a *linear transformation*, a function from a special kind of set called *vector space* into another vector space. Vector spaces are generalised sets with particular properties, but for the moment you can think of two-dimensional and three-dimensional cartesian spaces as usual vector spaces. Then vectors are gonna be represented as arrays of real numbers as before: $\\mathbf{n_{0}} = \\begin{pmatrix} 7.6  \\\\ 2.2  \\end{pmatrix}$ or $\\mathbf{v} = \\begin{pmatrix} x \\\\ y \\\\ z \\end{pmatrix}$.  \n",
+    "  \n",
+    "Back to linear transformations, they are usually denoted as:  \n",
+    "  \n",
+    "$$\\begin{align}\n",
+    "A:V &\\longrightarrow W \\\\\n",
+    "  \\mathbf{v} &\\rightarrow A(\\mathbf{v}) \\end{align}$$\n",
+    "  \n",
+    "where $V$ and $W$ are the domain and codomain vector spaces, and the transformation $A$ is a function that projects vectors $\\mathbf{v}$ into the images $A(\\mathbf{v})$. The property that defines linear transformations is the following:  \n",
+    "  \n",
+    "$$A(a\\mathbf{v}+b\\mathbf{w})=aA(\\mathbf{v})+bA(\\mathbf{w}),$$  \n",
+    "  \n",
+    "where $a$ and $b$ are constants. In other words,for a linear transformation, the image of the transformation $A$ of a linear combination of two vectors ($\\mathbf{v}$ and $\\mathbf{w}$) is equal to the linear combination of the images of this vectors by $A$.  \n",
+    "  \n",
+    "If we represent vectors with their components in a cartesian system, the images of the transformation $A$ will be given be the multiplication of the matrix representation of $A$ on the vectors.  \n",
+    "  \n",
+    "````{admonition} Example\n",
+    "\n",
+    "Given the matrix $A = \\begin{pmatrix} 1 & 2 \\\\ 3 & 2 \\end{pmatrix}$ and the vectors $\\mathbf{v_1} = \\begin{pmatrix} 3  \\\\ -1  \\end{pmatrix}$ and $\\mathbf{v_2} = \\begin{pmatrix} 2  \\\\ 3  \\end{pmatrix}$, then we have:  \n",
+    "  \n",
+    "$$\\begin{align} A\\mathbf{v_1} &= \\begin{pmatrix} 1 & 2 \\\\ 3 & 2 \\end{pmatrix}\\begin{pmatrix} 3  \\\\ -1  \\end{pmatrix} = \\begin{pmatrix} 1  \\\\ 7  \\end{pmatrix} \\\\  \n",
+    "  A\\mathbf{v_2} &= \\begin{pmatrix} 1 & 2 \\\\ 3 & 2 \\end{pmatrix}\\begin{pmatrix} 2  \\\\ 3  \\end{pmatrix} = \\begin{pmatrix} 8  \\\\ 12  \\end{pmatrix} = 4\\begin{pmatrix} 2  \\\\ 3  \\end{pmatrix} \\end{align}$$  \n",
+    "    \n",
+    "The vectors and their respective transformations are given by:  \n",
+    "  \n",
+    "```{image} lin_trans.png\n",
+    "```  \n",
+    "  \n",
+    "We see that under the transformation $A$, vector $\\mathbf{v_1}$ gets rotated (counterclockwise) and stretched, while vector $\\mathbf{v_2}$ only gets stretched without changing the direction.\n",
+    "\n",
+    "````  \n",
+    "  \n",
+    "## Eigenvalues and eigenvectors  \n",
+    "\n",
+    "  \n",
+    "As seen on the previous example, there might be some vectors that upon application of $A$ do not have their direction changed, only their magnitude (or *module* of the vector). For these vectors, we have, in general:  \n",
+    "  \n",
+    "$$A\\mathbf{v} = \\lambda\\mathbf{v}, \\quad \\mbox{for some constant }\\ \\lambda$$  \n",
+    "  \n",
+    "But this is equivalent to:  \n",
+    "  \n",
+    "$$\\begin{align}\n",
+    "A\\mathbf{v} - \\lambda\\mathbf{v} = 0 \\\\  \n",
+    "A\\mathbf{v} - \\lambda I\\mathbf{v} = 0 \\\\\n",
+    "\\left(A - \\lambda I \\right)\\mathbf{v} = 0 \\end{align},$$  \n",
+    "  \n",
+    "where $I$ is the identity matrix. The equation $\\left(A - \\lambda I \\right)\\mathbf{v} = 0$ defines a system of linear equations where $A - \\lambda I$ is the coefficient matrix, $\\mathbf{v}$ is the vector of variables, and $\\mathbf{b}=0$. See that since $\\mathbf{b}=0$ we already know one possible solution for this system, which is $\\mathbf{v}=0$ (all the variables equal to $0$). Therefore, either this system has an unique solution (the null solution) or it has infinite solutions, but definitely it is not inconsistent.  Thus, if we want to know all possible non-zero vectors $\\mathbf{v}$ that satisfy $A\\mathbf{v} = \\lambda\\mathbf{v}$, we have to impose that the system assumes infinite solutions. This criterium should be defined by the determinant of the matrix of coefficients, as:  \n",
+    "  \n",
+    "$$\\mbox{det}(A-\\lambda I) = 0.$$  \n",
+    "  \n",
+    "  \n",
+    "The determinant then defines a polynomial in the variable $\\lambda$, $\\mbox{det}(A-\\lambda I)=p(\\lambda)$, called *characteristic polynomial*. Since we are interested in the condition $p(\\lambda)=0$, what we want to know are the roots $\\lambda$ of this polynomial.  \n",
+    "  \n",
+    "The solutions $\\lambda$ for these equations are called **eigenvalues** and their corresponding vectors are **eigenvectors**. In summary, the eigenvectors are all the vectors that, upon application of $A$, remain on the same direction, only changing magnitude, with the eigenvalue being the stretching/compressing factor. \n",
+    "  \n",
+    "````{admonition} Example  \n",
+    "\n",
+    "Let us calculate the eigenvalues and eigenvectors of $A = \\begin{pmatrix} 1 & 2 \\\\ 3 & 2 \\end{pmatrix}$. First we need to obtain matrix $(A-\\lambda I)$ and to calculate its determinant. We have:  \n",
+    "  \n",
+    "$$A-\\lambda I = \\begin{pmatrix} 1 & 2 \\\\ 3 & 2 \\end{pmatrix} - \\lambda \\begin{pmatrix} 1 & 0 \\\\ 0 & 1 \\end{pmatrix} = \\begin{pmatrix} 1 -\\lambda & 2 \\\\ 3 & 2 -\\lambda  \\end{pmatrix}$$  \n",
+    "  \n",
+    "The characteristic polynomial will be given by $p(\\lambda)=\\mbox{det}(A-\\lambda I)$, and applying the rule for the calculus of determinants of matrices size 2, we have:  \n",
+    "  \n",
+    "$$p(\\lambda)=\\mbox{det}(A-\\lambda I) = \\mbox{det}\\begin{pmatrix} 1 -\\lambda & 2 \\\\ 3 & 2 -\\lambda  \\end{pmatrix} = (1 -\\lambda)(2 -\\lambda) - 3\\cdot 2 $$  \n",
+    "  \n",
+    "$$p(\\lambda) = 2 -\\lambda -2\\lambda +\\lambda^2 - 6 = \\lambda^2 - 3\\lambda -4.$$  \n",
+    "  \n",
+    "Thus, calculating the roots of this 2nd-degree polynomial, we find $\\lambda_1=-1$ and $\\lambda_2=4$. For each of the eigenvalues, to obtain the eigenvectors we have to solve the linear system  $A\\mathbf{v} = \\lambda\\mathbf{v}$ for some general vector $\\mathbf{v} = \\begin{pmatrix} x  \\\\ y  \\end{pmatrix}$ (with coordinates $x$ and $y$ to be specified). Thus:  \n",
+    "  \n",
+    "- $\\mathbf{\\lambda_1 = -1}$:  \n",
+    "    \n",
+    "    $\\begin{align}\n",
+    "A\\mathbf{v} = \\lambda\\mathbf{v} \\longrightarrow \\begin{pmatrix} 1 & 2 \\\\ 3 & 2 \\end{pmatrix}\\begin{pmatrix} x  \\\\ y \\end{pmatrix}= -1\\begin{pmatrix} x  \\\\ y \\end{pmatrix}\\longrightarrow \\cases{x + 2y = -x \\\\ 3x + 2y = -y} \\longrightarrow \\cases{2x + 2y = 0 \\\\ 3x + 3y = 0}\n",
+    "\\end{align}$  \n",
+    "    \n",
+    "    Note that both equations correspond to the same linear equation $x + y = 0$, which is exactly what we would expect since we built the system to be underdetermined when we did $\\mbox{det}(A-\\lambda I) = 0$. The eigenvector corresponding to $\\lambda_1$ will be such that $x + y = 0$ or $x = -y$. Thus, we can choose $\\mathbf{v_1} = \\begin{pmatrix} 1  \\\\ -1  \\end{pmatrix}$.  \n",
+    "    \n",
+    "    ```{note}  \n",
+    "    Remember that infinite solutions are possible for the same system. The important is that the condition $x = -y$ holds. In fact, multiplying a non-zero constant for the vector $\\begin{pmatrix} 1  \\\\ -1  \\end{pmatrix}$ would still give a vector in the same direction. So if $\\mathbf{v_1}$ is an eigenvector, $k\\mathbf{v_1}\\ (k \\in {\\rm I\\!R})$ also is.  \n",
+    "    \n",
+    "    ```  \n",
+    "    <br />  \n",
+    "    \n",
+    "    \n",
+    "- $\\mathbf{\\lambda_1 = 4}$:  \n",
+    "    \n",
+    "    $\\begin{align}\n",
+    "A\\mathbf{v} = \\lambda\\mathbf{v} \\longrightarrow \\begin{pmatrix} 1 & 2 \\\\ 3 & 2 \\end{pmatrix}\\begin{pmatrix} x  \\\\ y \\end{pmatrix}= 4\\begin{pmatrix} x  \\\\ y \\end{pmatrix}\\longrightarrow \\cases{x + 2y = 4x \\\\ 3x + 2y = 4y} \\longrightarrow \\cases{-3x + 2y = 0 \\\\ 3x - 2y = 0}\n",
+    "\\end{align}$  \n",
+    "    \n",
+    "    Since the eigenvector needs to satisfy $3x=2y$, we choose $\\mathbf{v_2} = \\begin{pmatrix} 2/3  \\\\ 1  \\end{pmatrix}$.  \n",
+    "    \n",
+    "    The representation of these vectors on the cartesian plane will be:  \n",
+    "      \n",
+    "```{image} eigenvec_01.png\n",
+    "```\n",
+    "\n",
+    "````  \n",
+    "  \n",
+    "  \n",
+    "  \n",
+    "````{admonition} Example  \n",
+    "  \n",
+    "Let us calculate the eigenvalues and eigenvectors of matrix $A = \\begin{pmatrix} -2 & 1 \\\\ 0 & -1 \\end{pmatrix}$. We have:  \n",
+    "  \n",
+    "$$p(\\lambda)=\\mbox{det}(A-\\lambda I) = \\mbox{det}\\begin{pmatrix} -2 -\\lambda & 1 \\\\ 0 & -1 -\\lambda  \\end{pmatrix} = (-2 -\\lambda)(-1 -\\lambda) - 1\\cdot 0 $$  \n",
+    "  \n",
+    "$$p(\\lambda)=(-2 -\\lambda)(-1 -\\lambda)$$  \n",
+    "  \n",
+    "Since we need $p(\\lambda)=0$ either one of the two factors has to be zero. So, $\\lambda_1=-2$ and $\\lambda_2=-1$.   \n",
+    "For the eigenvectors:  \n",
+    "  \n",
+    "- $\\mathbf{\\lambda_1 = -2}$:  \n",
+    "    \n",
+    "    $\\begin{align}\n",
+    "A\\mathbf{v} = \\lambda\\mathbf{v} \\longrightarrow \\begin{pmatrix} -2 & 1 \\\\ 0 & -1 \\end{pmatrix}\\begin{pmatrix} x  \\\\ y \\end{pmatrix}= -2\\begin{pmatrix} x  \\\\ y \\end{pmatrix}\\longrightarrow \\cases{-2x + y = -2x \\\\ -y = -2y} \\longrightarrow \\cases{y = 0 \\\\ y = 0}\n",
+    "\\end{align}$  \n",
+    "    \n",
+    "    Since $y=0$, we choose $\\mathbf{v_1} = \\begin{pmatrix} 1  \\\\ 0  \\end{pmatrix}$.  \n",
+    "      \n",
+    "- $\\mathbf{\\lambda_2 = -1}$:  \n",
+    "    \n",
+    "    $\\begin{align}\n",
+    "A\\mathbf{v} = \\lambda\\mathbf{v} \\longrightarrow \\begin{pmatrix} -2 & 1 \\\\ 0 & -1 \\end{pmatrix}\\begin{pmatrix} x  \\\\ y \\end{pmatrix}= -1\\begin{pmatrix} x  \\\\ y \\end{pmatrix}\\longrightarrow \\cases{-2x + y = -x \\\\ -y = -y} \\longrightarrow \\cases{-x + y = 0 \\\\ y = y}\n",
+    "\\end{align}$  \n",
+    "    \n",
+    "    The second equation is trivial and provides no information. From the first equation, since $x=y$, we choose $\\mathbf{v_2} = \\begin{pmatrix} 1  \\\\ 1  \\end{pmatrix}$.  \n",
+    "      \n",
+    "```{tip}  \n",
+    "  \n",
+    "Note that the two eigenvalues $\\lambda_1 = -2$ and $\\lambda_2 = -1$ correspond to the elements of the main diagonal of the matrix $A$. Whenever the matrix $A$ is *triangular*, meaning all the elements *below* the main diagonal are $0$, the eigenvalues will correspond to the elements of the main diagonal.  \n",
+    "  \n",
+    "```\n",
+    "  \n",
+    "\n",
+    "````\n",
+    "                             "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Vector decomposition  \n",
+    "  \n",
+    "Now suppose matrix $A$ has eigenvalues $\\lambda_1$ and $\\lambda_2$, with corresponding eigenvectors $\\mathbf{u_1}$ and $\\mathbf{u_2}$. If $\\lambda_1 \\ne \\lambda_2$, then the vectors $\\mathbf{u_1}$ and $\\mathbf{u_2}$ are *linearly independent* meaning one can not be written as a multiple (multiplied by a real constant) of the other. Geometrically, this means that the directions corresponding to this two eigenvectors are not on the same line.  \n",
+    "  \n",
+    "In two dimensions, this is enough to say that any given vector $\\mathbf{v}$ can be written as a linear combination of $\\mathbf{u_1}$ and $\\mathbf{u_2}$ (or that $\\mathbf{u_1}$ and $\\mathbf{u_2}$ consitute a *basis* of this space). We have:  \n",
+    "  \n",
+    "$$\\mathbf{v}= a_1\\mathbf{u_1} + a_2\\mathbf{u_2},$$  \n",
+    "  \n",
+    "for constants $a_1$ and $a_2$ to be determined.  \n",
+    "  \n",
+    "But given that $A$ is a linear transformation, it should hold that:  \n",
+    "  \n",
+    "$$A\\mathbf{v} = A(a_1\\mathbf{u_1} + a_2\\mathbf{u_2}) = a_1A(\\mathbf{u_1}) + a_2A(\\mathbf{u_2}) = a_1\\lambda_1\\mathbf{u_1} + a_2\\lambda_2\\mathbf{u_2},$$  \n",
+    "  \n",
+    "with the last step coming from the fact that $\\mathbf{u_1}$ and $\\mathbf{u_2}$ are eigenvectors of $A$. Since $A\\mathbf{v}$ is also a vector in this space, we can go ahead and apply the transformation $A$ again, and we will have:  \n",
+    "  \n",
+    "$$A(A\\mathbf{v}) = A(a_1\\lambda_1\\mathbf{u_1} + a_2\\lambda_2\\mathbf{u_2})=a_1\\lambda_1 A(\\mathbf{u_1}) + a_2\\lambda_2 A(\\mathbf{u_2}) = a_1\\lambda_1^2\\mathbf{u_1} + a_2\\lambda_2^2\\mathbf{u_2}.$$\n",
+    "  \n",
+    "  \n",
+    "So, in general:  \n",
+    "  \n",
+    "  \n",
+    "$$A(A(A \\cdots A\\mathbf{v}) \\cdots )) = A^n\\mathbf{v} = a_1\\lambda_1^n\\mathbf{u_1} + a_2\\lambda_2^n\\mathbf{u_2},$$  \n",
+    "or, in other words, if we know the eigenvectors and eigenvalues of $A$ (with $\\lambda_1 \\ne \\lambda_2$), and we know the coeffiencients $a_1$ and $a_2$ of the expansion of $\\mathbf{v}$ into the eigenvectors, then it is easy to calculate what is going to be the image ov vector $\\mathbf{v}$ by any number of successive applications of $A$. This is also equivalent of applying the $n$-th power of matrix $A$ to $\\mathbf{v}$.  \n",
+    "  \n",
+    "This fact will have many important applications, as we will se next.  \n",
+    "  \n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Structured models revisited  \n",
+    "  \n",
+    "Let us considered a structured population model defined by the projection matrix $P=\\begin{pmatrix} 1.5 & 2  \\\\ 0.08 & 0  \\end{pmatrix}$ and an initial population vector given by $\\mathbf{n_0} = \\begin{pmatrix} 105  \\\\ 1 \\end{pmatrix}$. If we calculate the eigenvalues and eigenvectors, we will have:  \n",
+    "  \n",
+    "$$p(\\lambda)=\\mbox{det}(P-\\lambda I) = \\mbox{det}\\begin{pmatrix} 1.5 -\\lambda & 2 \\\\ 0.08 & -\\lambda  \\end{pmatrix} = (1.5 -\\lambda)(-\\lambda) - 0.08\\cdot 2 $$  \n",
+    "  \n",
+    "$$p(\\lambda)=\\lambda^2-1.5\\lambda-0.16=(\\lambda-1.6)(\\lambda+0.1)$$  \n",
+    "  \n",
+    "Thus, we have $\\lambda_1=1.6$ and $\\lambda_2=-0.1$.  \n",
+    "  \n",
+    "- $\\mathbf{\\lambda_1 = 1.6}$:  \n",
+    "    \n",
+    "    $\\begin{align}\n",
+    "A\\mathbf{v} = \\lambda\\mathbf{v} \\longrightarrow \\begin{pmatrix} 1.5 & 2  \\\\ 0.08 & 0 \\end{pmatrix}\\begin{pmatrix} x  \\\\ y \\end{pmatrix}= 1.6\\begin{pmatrix} x  \\\\ y \\end{pmatrix}\\longrightarrow \\cases{1.5x + 2y = 1.6x \\\\ 0.08x = 1.6y} \\longrightarrow \\cases{x - 20y = 0 \\\\ x - 20y = 0}\n",
+    "\\end{align}$  \n",
+    "    \n",
+    "    Since $x=20y$, we choose $\\mathbf{u_1} = \\begin{pmatrix} 20  \\\\ 1  \\end{pmatrix}$.  \n",
+    "  \n",
+    "  <br />\n",
+    "  \n",
+    "- $\\mathbf{\\lambda_2 = -0.1}$:  \n",
+    "    \n",
+    "    $\\begin{align}\n",
+    "A\\mathbf{v} = \\lambda\\mathbf{v} \\longrightarrow \\begin{pmatrix} 1.5 & 2  \\\\ 0.08 & 0 \\end{pmatrix}\\begin{pmatrix} x  \\\\ y \\end{pmatrix}= -0.1\\begin{pmatrix} x  \\\\ y \\end{pmatrix}\\longrightarrow \\cases{1.5x + 2y = -0.1x \\\\ 0.08x = -0.1y} \\longrightarrow \\cases{0.8x + y = 0 \\\\ 0.08x + 0.1y = 0}\n",
+    "\\end{align}$  \n",
+    "    \n",
+    "    Since $-\\displaystyle\\frac{4}{5}x=y$, we choose $\\mathbf{u_2} = \\begin{pmatrix} 5  \\\\ -4  \\end{pmatrix}$.  \n",
+    "\n",
+    "<br />\n",
+    "      \n",
+    "We can also note that the initial population vector can be decomposed into the eigenvectors $\\mathbf{u_1}$ and $\\mathbf{u_2}$ of the matrix $P$ according to:  \n",
+    "  \n",
+    "$$\\mathbf{n_0}=\\begin{pmatrix} 105  \\\\ 1 \\end{pmatrix} = 5\\begin{pmatrix} 20  \\\\ 1 \\end{pmatrix} + \\begin{pmatrix} 5  \\\\ -4 \\end{pmatrix} = 5\\mathbf{u_1} + \\mathbf{u_2}$$  \n",
+    "  \n",
+    "We know that to find the population vector for the following step, we have to apply the projection matrix on the current population vector. Thus:  \n",
+    "  \n",
+    "$$\\mathbf{n_1} = P\\mathbf{n_0}= P(5\\mathbf{u_1} + \\mathbf{u_2})=5P(\\mathbf{u_1}) + P(\\mathbf{u_2})=5(1.6)\\begin{pmatrix} 20  \\\\ 1 \\end{pmatrix} + (-0.1)\\begin{pmatrix} 5  \\\\ -4 \\end{pmatrix} = \\begin{pmatrix} 159.5  \\\\ 8.4 \\end{pmatrix}$$  \n",
+    "  \n",
+    "If we perform successive applications of $P$, we will finally have:  \n",
+    "  \n",
+    "$$\\mathbf{n_t} = P^t\\mathbf{n_0} = P^t(5\\mathbf{u_1} + \\mathbf{u_2})=5(1.6)^t\\begin{pmatrix} 20  \\\\ 1 \\end{pmatrix} + (-0.1)^t\\begin{pmatrix} 5  \\\\ -4 \\end{pmatrix}.$$  \n",
+    "  \n",
+    "<br />  \n",
+    "  \n",
+    "By decomposing into eigenvalues and eigenvectors, we know the behaviour of our population for any time $t$, which is much easier than applying $P$ a number $t$ of times, or than calculating $P^t$. Furthermore, as $t\\rightarrow\\infty$ the first term will dominate (with the second term dying out) and the proportions for each age class in this population will be dictated by the vector $\\begin{pmatrix} 20  \\\\ 1 \\end{pmatrix}$ (or, if we divide by the sum of the components: $\\displaystyle\\frac{1}{21}\\begin{pmatrix} 20  \\\\ 1 \\end{pmatrix} = \\begin{pmatrix} 95\\%  \\\\ 5\\% \\end{pmatrix}$.  \n",
+    "  \n",
+    "In general the eigenvector corresponding to the **largest eigenvalue** (also called *dominant eigenvalue*) of the projection matrix, which is always *real* and *positive*, will correspond to the stable age distribution, for which our population vector converges after we wait enough time.  \n",
+    "  \n",
+    "As we have seen, after enough time, the dominant eigenvalue will also correspond to the growth factor for each of the age classes. Thus, if $\\lambda_1>1$ the population size will increase, whereas if $0<\\lambda_1<1$ the population size will decrease.\n",
+    "  \n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 13,
+   "id": "c8c04f31",
+   "metadata": {
+    "tags": [
+     "remove-cell"
+    ]
+   },
+   "outputs": [
+    {
+     "data": {
+      "image/png": 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",
+      "text/plain": [
+       "<Figure size 864x216 with 1 Axes>"
+      ]
+     },
+     "metadata": {
+      "needs_background": "light"
+     },
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "import numpy as np\n",
+    "import pandas as pd\n",
+    "import matplotlib.pyplot as plt\n",
+    "\n",
+    "def multiple_formatter(denominator=2, number=np.pi, latex='\\pi'):\n",
+    "    def gcd(a, b):\n",
+    "        while b:\n",
+    "            a, b = b, a%b\n",
+    "        return a\n",
+    "    def _multiple_formatter(x, pos):\n",
+    "        den = denominator\n",
+    "        num = np.int(np.rint(den*x/number))\n",
+    "        com = gcd(num,den)\n",
+    "        (num,den) = (int(num/com),int(den/com))\n",
+    "        if den==1:\n",
+    "            if num==0:\n",
+    "                return r'$0$'\n",
+    "            if num==1:\n",
+    "                return r'$%s$'%latex\n",
+    "            elif num==-1:\n",
+    "                return r'$-%s$'%latex\n",
+    "            else:\n",
+    "                return r'$%s%s$'%(num,latex)\n",
+    "        else:\n",
+    "            if num==1:\n",
+    "                return r'$\\frac{%s}{%s}$'%(latex,den)\n",
+    "            elif num==-1:\n",
+    "                return r'$\\frac{-%s}{%s}$'%(latex,den)\n",
+    "            else:\n",
+    "                return r'$\\frac{%s%s}{%s}$'%(num,latex,den)\n",
+    "    return _multiple_formatter\n",
+    "\n",
+    "class Multiple:\n",
+    "    def __init__(self, denominator=2, number=np.pi, latex='\\pi'):\n",
+    "        self.denominator = denominator\n",
+    "        self.number = number\n",
+    "        self.latex = latex\n",
+    "\n",
+    "    def locator(self):\n",
+    "        return plt.MultipleLocator(self.number / self.denominator)\n",
+    "\n",
+    "    def formatter(self):\n",
+    "        return plt.FuncFormatter(multiple_formatter(self.denominator, self.number, self.latex))\n",
+    "\n",
+    "#xt=np.linspace(-4*np.pi,4*np.pi,num=500)\n",
+    "xt=np.linspace(0,52,num=1000)\n",
+    "##\n",
+    "fig, ax = plt.subplots(figsize=(12, 3))\n",
+    "\n",
+    "fun = 5 + 5*np.cos((np.pi/12)*(xt-3))\n",
+    "dfun = -(5*np.pi/12)*np.sin((np.pi/12)*(xt-3))\n",
+    "\n",
+    "ax.plot(xt, fun, color='b', lw=2, label='$f\\,(t)$')\n",
+    "ax.plot(xt, dfun, color='g', lw=2, label=\"$f\\,'\\,(t)$\")\n",
+    "ax.tick_params(axis='both', labelsize= 15)\n",
+    "ax.set_xlabel('$t$', fontsize = 16)\n",
+    "ax.legend(fontsize=14)  \n",
+    "\n",
+    "ax.axhline(0, color='black', lw=1)\n",
+    "ax.axvline(0, color='black', lw=1)\n",
+    "\n",
+    "ax.set_xlim(0,55)\n",
+    "ax.set_ylim(-2,12)\n",
+    "\n",
+    "#ax.xaxis.set_major_locator(plt.MultipleLocator(np.pi / 2))\n",
+    "#ax.xaxis.set_minor_locator(plt.MultipleLocator(np.pi / 12))\n",
+    "#ax.xaxis.set_major_formatter(plt.FuncFormatter(multiple_formatter()))\n",
+    "\n",
+    "fig.tight_layout()\n",
+    "fig.savefig('func_diff.png')"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "de6827dc",
+   "metadata": {},
+   "source": [
+    "## Analysing change, continuously\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "c8e845ed",
+   "metadata": {},
+   "source": [
+    "## Average rate of change\n",
+    "\n",
+    "Watch this [video](https://web.microsoftstream.com/video/2953e661-e1c1-43ff-ba78-ca52099ed1ec)\n",
+    "for a short introduction to this lecture."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "330c1e5e",
+   "metadata": {},
+   "source": [
+    "## Instantaneous rate of change  \n",
+    "  \n",
+    "  \n",
+    "\n",
+    "If we have a continuous function $f(x)$, for each two points $x_{0}$ and $x_{1}$, we can define the average rate of change between these points by the ratio between how much $f$ has changed and how much $x$ has changed:  \n",
+    "  \n",
+    "$$\\frac{\\Delta f(x)}{\\Delta x} = \\frac{f(x_{1}) - f(x_{0})}{x_{1} - x_{0}}$$  \n",
+    "  \n",
+    "For points $x_{0}$ and $x_{1}$ this can be graphically represented by:  \n",
+    "  \n",
+    "```{image} sec_tan.png\n",
+    "```  \n",
+    "  \n",
+    "The line that passes through the points $(x_0, f(x_0))$ and $(x_1, f(x_1))$ (yellow line) is named *secant line*. As we have seen before in our treatment of linear functions, the slope of the secant line will be given by the average rate of change between this points (remember that the slope for linear functions is defined as $m=\\displaystyle\\frac{\\Delta y}{\\Delta x})$.\n",
+    "  \n",
+    "Now consider that we change the value of $x_1$ so that it slowly approaches $x_0$ from the right. In this case, the secant lines also change (from yellow to red), as we change the image of $x_1$ the function, $f(x_1)$. In the limit where $x_1$ is very close to $x_0$, the secant line approaches the *tangent* (sark red line) of the function $f(x)$ at the point $x_0$ (the tangent is the straight line that \"just touches\" the curve at that point).  \n",
+    "  \n",
+    "If we represent the slope of the tangent line at the point $x_0$ by the term $f'(x_0)$, we will have:  \n",
+    "  \n",
+    "$$\\lim_{x_{1} \\to x_{0}} \\frac{\\Delta f(x)}{\\Delta x} = \\frac{f(x_{1}) - f(x_{0})}{x_{1} - x_{0}} = f'(x_{0})$$  \n",
+    "  \n",
+    "Since we can do the same process as we change the value $x_0$, we define a new function $f'(x_{0})$ which we call the **derivative** of the function $f$ at the point $x_0$, and its value tells us about the *instantaneous* rate of change of $f$ at the point $x_0$.  \n",
+    "  \n",
+    "If we look instead at the interval between $x_{0}$ and $x_{1}$ and define $h = x_1 - x_0$, if $x_1$ approaches $x_0$, then $h$ approaches $0$. and we would have for the derivative of $f(x)$:  \n",
+    "  \n",
+    "$$\\lim_{h \\to 0} \\frac{f(x + h) - f(x)}{h} = f'(x)$$  \n",
+    "\n",
+    "Visit this [link](https://www.demonstrations.wolfram.com/SecantAndTangentLines/) for an interactive demonstration of the limit of the tangent curve.\n",
+    "  \n",
+    "``````{note}  \n",
+    "Note that in the previous expression the index $0$ was omitted from $x_0$. We will assume that the derivative is defined for all the points in the domain of the function $f(x)$ for which the limit in the expression exists. \n",
+    "\n",
+    "It is good to have a graphical intuition for when the derivative is not well defined. Some cases are shown below.  \n",
+    "  \n",
+    "`````{tabbed} Case 1   \n",
+    "\n",
+    "````{panels}  \n",
+    "  \n",
+    "```{image} ./cases01a.png  \n",
+    "\n",
+    "```  \n",
+    "  \n",
+    "```{image} ./cases01b.png  \n",
+    "\n",
+    "``` \n",
+    "\n",
+    "---  \n",
+    "  \n",
+    "For the points where the function $f(x)$ presents discontinuities, or for the points where it is simply not defined, the derivative will also not be defined.\n",
+    "\n",
+    "  \n",
+    "````  \n",
+    "`````  \n",
+    "  \n",
+    "`````{tabbed} Case 2   \n",
+    "\n",
+    "````{panels}  \n",
+    "  \n",
+    "```{image} ./cases02.png  \n",
+    "\n",
+    "```  \n",
+    "\n",
+    "---  \n",
+    "  \n",
+    "For the points where the curve of the function $f(x)$ presents kinks, note that there are two tangent lines, one coming from the right (red line) and another coming from the left (yellow line). Since the function $f'(x)$ must have only one value for the image of $x$ (in other words, only one value for the slope of the tangent at point $f(x)$), then the derivative is not defined in this case.\n",
+    "  \n",
+    "````  \n",
+    "`````  \n",
+    "  \n",
+    "`````{tabbed} Case 3\n",
+    "\n",
+    "````{panels}  \n",
+    "  \n",
+    "```{image} ./cases03.png  \n",
+    "\n",
+    "```  \n",
+    "\n",
+    "---  \n",
+    "\n",
+    "For the points where the function $f(x)$ is locally vertical (in this case, for $x=0$), the slope of the tangent curve is infinite. Since infinite is not a real number, the derivative is not defined for those points.\n",
+    "  \n",
+    "````  \n",
+    "`````  \n",
+    "\n",
+    "``````  \n",
+    "  \n",
+    "  \n",
+    "````{admonition} Example  \n",
+    "  \n",
+    "By the definition of the derivative of $f(x)$:  \n",
+    "  \n",
+    "$$f'(x)=\\lim_{h \\to 0} \\frac{f(x + h) - f(x)}{h}$$  \n",
+    "  \n",
+    "Thus, if $f(x)=x^2$, we will have:  \n",
+    "  \n",
+    "$$f'(x)=\\lim_{h \\to 0} \\frac{(x + h)^{2} - x^{2}}{h} = \\lim_{h \\to 0} \\frac{x^{2} + 2xh + h^{2} - x^{2}}{h} = \\lim_{h \\to 0} (h + 2x) = 2x$$  \n",
+    "  \n",
+    "So that the derivative of $f(x)=x^2$ is the function $f'(x)=2x$.  \n",
+    "  \n",
+    "  \n",
+    "```{note}  \n",
+    "Although limits of functions can be defined in a more rigorous way, in this course we will treat them more intuitively, and use, if required, the known rules for limits. Thus, if $C$ is a constant, $\\lim_{x \\rightarrow a} f(x) = L$, and $\\lim_{x \\rightarrow a} g(x) = M$, it is generally valid that:  \n",
+    "  \n",
+    "$$\\lim_{x \\rightarrow a} [f(x) + g(x)] = \\lim_{x \\rightarrow a} f(x) + \\lim_{x \\rightarrow a} g(x) = L + M$$  \n",
+    "$$\\lim_{x \\rightarrow a} [Cf(x)] = C\\lim_{x \\rightarrow a} f(x) = CL$$  \n",
+    "$$\\lim_{x \\rightarrow a} [f(x)g(x)] = [\\lim_{x \\rightarrow a} f(x)][\\lim_{x \\rightarrow a} g(x)] = LM$$  \n",
+    "$$\\lim_{x \\rightarrow a} \\left[\\frac{f(x)}{g(x)}\\right] = \\frac{\\lim_{x \\rightarrow a} f(x)}{\\lim_{x \\rightarrow a} g(x)} = \\frac{L}{M},\\ \\ (\\mbox{if}\\ M \\ne 0)$$  \n",
+    "$$\\lim_{x \\rightarrow a} [C^{f(x)}] = C^{\\lim_{x \\rightarrow a} f(x)} = C^L$$  \n",
+    "\n",
+    "```\n",
+    "  \n",
+    "````\n",
+    "  \n",
+    "  \n",
+    "Other common notations for the derivative are given by:  \n",
+    "  \n",
+    "$$f'(x) = \\frac{df}{dx} = \\frac{d}{dx}f(x) = \\dot{f}(x)$$  \n",
+    "\n",
+    "```{Example}\n",
+    "Derivatives (or rate of change) are needed everywhere in science. Some common examples are the velocity as the derivative of position and acceleration as the derivative of velocity. When asking yourself 'how fast is the population growing?' or 'how steep is this mountain?' you can find the answer by calculating the derivative of the population number or the height profile.\n",
+    "```\n",
+    "\n",
+    "Visit this [link](https://the-learning-machine.com/article/calculus/derivatives) for examples that provide demonstrations of the intuitive interpretation of the derivative."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "49c8ba11",
+   "metadata": {},
+   "source": [
+    "## Calculating derivatives  \n",
+    "  \n",
+    "  \n",
+    "It is possible to generalise the previous example for $f(x)=x^2$ and obtain the derivative for any power function $f(x)=x^n$ ($n \\ne 0$):  \n",
+    "  \n",
+    "$$f(x) = x^{n} \\implies f'(x) = nx^{n-1}$$  \n",
+    "  \n",
+    "  \n",
+    "```{admonition} Example  \n",
+    "  \n",
+    "$$x^{2} \\implies 2x^{2-1} = 2x$$\n",
+    "$$\\ x^{3} \\implies 3x^{3-1} = 3x^{2}$$\n",
+    "$$\\ x^{4.2} \\implies 4.2x^{4.2-1} = 4.2x^{3.2}$$  \n",
+    "\n",
+    "```  \n",
+    "  \n",
+    "It is possible to show, from the definition of derivative and the rules for limits, that there are simple rules for derivatives when calculating them for a sum of functions or a product by a constant. If $C$ is a constant, we generally have:  \n",
+    "  \n",
+    "$$\\begin{align}\\frac{d}{dx}[f(x)+g(x)] &= \\frac{df}{dx} + \\frac{dg}{dx} \\\\\n",
+    "   \\frac{d}{dx}[Cf(x)] &= C\\frac{df}{dx}\\end{align}$$  \n",
+    "  \n",
+    "  \n",
+    "Thus, differentiating a given polynomial p(x) stands for differentiating each term separately and summing the result.  \n",
+    "  \n",
+    "```{admonition} Example  \n",
+    "  $$f(x) = 3x^{2} - 4x + 6 \\implies f'(x) = 3(2x) - 4(x^{0}) + 0 = 6x - 4$$  \n",
+    "  $$f(x) = x^{60} - 3x^{20} - 4x \\implies f'(x) = 60x^{59} - 60x^{19} - 4$$\n",
+    "```  \n",
+    "  \n",
+    "For products and quotients of functions, however, we have different rules:  \n",
+    "  \n",
+    "$$\\frac{d}{dx}[f(x)g(x)] = \\left(\\frac{df}{dx}\\right) g(x) + f(x)\\left(\\frac{dg}{dx}\\right)$$  \n",
+    "$$\\frac{d}{dx}\\left[\\frac{f(x)}{g(x)}\\right] = \\frac{\\left(\\displaystyle\\frac{df}{dx}\\right)g(x) - f(x)\\left(\\displaystyle\\frac{dg}{dx}\\right)}{[g(x)]^{2}}$$  \n",
+    "\n",
+    "```{tip}\n",
+    "If you do not want to remember the quotients rule, you can use the product rule on $f(x)j(x)$, with $j(x)=g(x)^{-1}$ and use the chain rule to evaluate $\\frac{d}{dx} j(x)$.\n",
+    "```\n",
+    "  \n",
+    "```{admonition} Examples  \n",
+    "  $$f(x) = (x + 2)(x - 4) \\implies f'(x) = (1)(x-4) + (x+2)(1) = x - 4 + x + 2 = 2x - 2$$  \n",
+    "  $$f(x) = \\frac{\\alpha x}{1 + \\beta x} \\implies f'(x) =  \\frac{(\\alpha)(1 + \\beta x) - (\\alpha x)(\\beta)}{(1 + \\beta x)^{2}} = \\frac{\\alpha + \\alpha\\beta x - \\alpha \\beta x}{(1 + \\beta x)^{2}} =\\frac{\\alpha}{(1 + \\beta x)^{2}}$$  \n",
+    "  \n",
+    "```  \n",
+    "  \n",
+    "Moreover, expressions for the derivatives of the other elementary functions can also be found:\n",
+    "\n",
+    "- $f(x) = a^{x} \\implies f'(x) = a^{x} \\cdot (\\ln a) ,\\ (a > 0, a \\neq 1)$  \n",
+    "    $\\left(\\mbox{in particular: }\\ (e^{x})' = e^{x}\\right)$  \n",
+    "    \n",
+    "- $f(x) = \\log_{a}x \\implies f'(x) = \\displaystyle\\frac{1}{(\\ln a)x}$  \n",
+    "    $\\left(\\mbox{in particular: }\\ (\\ln x)' = \\displaystyle\\frac{1}{x}\\right)$  \n",
+    "    \n",
+    "- $f(x) = \\mbox{sin}(x) \\implies f'(x) = \\mbox{cos}(x)$  \n",
+    "\n",
+    "- $f(x) = \\mbox{cos}(x) \\implies f'(x) = -\\mbox{sin}(x)$\n",
+    "  \n",
+    "  \n",
+    "```{admonition} Example  \n",
+    "If $f(x) = \\mbox{tan}(x) = \\displaystyle\\frac{\\mbox{sin}(x)}{\\mbox{cos}(x)}$, then:  \n",
+    "  \n",
+    "$$f'(x) = \\frac{(\\mbox{cos}(x))\\mbox{cos}(x) - \\mbox{sin}(x)(-\\mbox{sin}(x))}{(\\mbox{cos}(x))^{2}} = \\frac{\\mbox{sin}^{2}(x) + \\mbox{cos}^{2}(x)}{\\mbox{cos}^{2}(x)} = \\frac{1}{\\mbox{cos}^{2}(x)} = \\mbox{sec}^{2}(x)$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "42cb6b21",
+   "metadata": {},
+   "source": [
+    "## Chain rule  \n",
+    "  \n",
+    "The calculation of derivatives for more complicated functions can generally be made easier by breaking the function into components, for example:  \n",
+    "  \n",
+    "- $f(x) = (3x^{2} - 2x + 1)^{3} = (g(x))^{3}$, where $g(x) = 3x^{2} - 2x + 1$\n",
+    "- $f(x) = \\displaystyle\\frac{\\sqrt{2x - 1}}{1 + \\sqrt{2x - 1}} = \\displaystyle\\frac{g(x)}{1 + g(x)}$, where $g(x) = \\sqrt{h(x)}$, and $h(x) = 2x - 1$  \n",
+    "  \n",
+    "We have seen this before when discussing transformation of graphs.  When one function is nested within the other, we have a composite function, or as for the last example above:\n",
+    "\n",
+    "$$f(x) = f(g(h(x)))$$  \n",
+    "  \n",
+    "The derivative of composite functions in relation to the dependent variable is then given by the *chain rule*:  \n",
+    "  \n",
+    "- $\\displaystyle\\frac{d}{dx}f(x) = \\displaystyle\\frac{d}{dx}f(g(x)) = \\left(\\displaystyle\\frac{df}{dg}\\right)\\left(\\displaystyle\\frac{dg}{dx}\\right)$\n",
+    "- $\\displaystyle\\frac{d}{dx}f(x) = \\displaystyle\\frac{d}{dx}f(g(h(x))) = \\left(\\displaystyle\\frac{df}{dg}\\right) \\left(\\displaystyle\\frac{dg}{dh}\\right) \\left(\\displaystyle\\frac{dh}{dx}\\right)$  \n",
+    "  \n",
+    "The strategy then consists in identifying the nested functions, calculating their derivatives and multiplying the results.  \n",
+    "  \n",
+    "```{admonition} Examples  \n",
+    "  \n",
+    "- If $f(x) = (3x^{2} - 2x + 1)^{3}$, we have:  \n",
+    "      \n",
+    "    $$f(g) = g^{3} \\implies \\displaystyle\\frac{df}{dg} = 3g^2$$\n",
+    "    $$g(x) = 3x^{2} - 2x + 1 \\implies \\displaystyle\\frac{dg}{dx} = 6x - 2$$  \n",
+    "    \n",
+    "    Thus:  \n",
+    "      \n",
+    "    $$\\displaystyle\\frac{df}{dx} = \\left(\\displaystyle\\frac{df}{dg}\\right)\\left(\\displaystyle\\frac{dg}{dx}\\right) = 3g^{2} \\cdot g' = 3(3x^{2} - 2x + 1)^2(6x - 2)$$  \n",
+    "      \n",
+    "    <br />  \n",
+    "      \n",
+    "- If $f(x) = \\displaystyle\\frac{\\sqrt{2x - 1}}{1 + \\sqrt{2x - 1}}$, then:  \n",
+    "      \n",
+    "    $$f(g) = \\displaystyle\\frac{g}{1+g} \\implies \\displaystyle\\frac{df}{dg} = \\frac{(1+g) - g}{(1+g)^2} = \\frac{1}{(1+g)^2}$$  \n",
+    "    $$g(h) = \\sqrt{h} \\implies \\displaystyle\\frac{dg}{dh} = \\frac{1}{2}h^{-1/2}$$  \n",
+    "    $$h(x) = 2x-1 \\implies \\displaystyle\\frac{dh}{dx} = 2$$  \n",
+    "      \n",
+    "    Thus:  \n",
+    "      \n",
+    "    $$\\displaystyle\\frac{df}{dx} =  \\left(\\displaystyle\\frac{df}{dg}\\right) \\left(\\displaystyle\\frac{dg}{dh}\\right) \\left(\\displaystyle\\frac{dh}{dx}\\right) = \\frac{1}{(1+\\sqrt{2x - 1})^2}\\cdot \\frac{1}{2(\\sqrt{2x - 1})}\\cdot 2 = \\frac{1}{(\\sqrt{2x - 1})(1+\\sqrt{2x - 1})^2}$$  \n",
+    "      \n",
+    "    <br />\n",
+    "      \n",
+    "- If $f(t) = 5 + 5\\mbox{cos}\\left[\\displaystyle\\frac{\\pi}{12}(t - 3)\\right]$, we have:  \n",
+    "      \n",
+    "    $$f(g) = 5 + 5g \\implies \\displaystyle\\frac{df}{dg} = 5$$  \n",
+    "    $$g(h) = \\mbox{cos}(h) \\implies \\displaystyle\\frac{dg}{dh} = -\\mbox{sin}(h)$$  \n",
+    "    $$h(t) = \\frac{\\pi}{12}(t - 3) \\implies \\displaystyle\\frac{dh}{dt} = \\frac{\\pi}{12}$$  \n",
+    "      \n",
+    "    Thus:  \n",
+    "      \n",
+    "    $$\\displaystyle\\frac{df}{dx} =  \\left(\\displaystyle\\frac{df}{dg}\\right) \\left(\\displaystyle\\frac{dg}{dh}\\right) \\left(\\displaystyle\\frac{dh}{dx}\\right) = 5 \\cdot \\left(-\\mbox{sin}(h)\\right) \\cdot \\frac{\\pi}{12} = -\\frac{5\\pi}{12}\\mbox{sin}\\left[\\displaystyle\\frac{\\pi}{12}(t - 3)\\right]$$  \n",
+    "      \n",
+    "    For this last example if we plot the function $f(t)=5 + 5\\mbox{cos}\\left[\\displaystyle\\frac{\\pi}{12}(t - 3)\\right]$ together with its derivative $f'(t)=-\\displaystyle\\frac{5\\pi}{12}\\mbox{sin}\\left[\\displaystyle\\frac{\\pi}{12}(t - 3)\\right]$ we will have:  \n",
+    "      \n",
+    "    ```{image} func_diff.png  \n",
+    "    ```  \n",
+    "      \n",
+    "    Try to compare the two graphs. What happens with the function $f(t)$ on the intervals where the derivative $f'(t)$ is positive or negative? What happens at the points where the derivative is zero?"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "53d5ea2a",
+   "metadata": {},
+   "source": [
+    "## Higher order derivatives  \n",
+    "  \n",
+    "\n",
+    "\n",
+    "If the derivative of a given function is also a \"well-behaved\" function (without kinks or discontinuities), then we can also calculate the derivative of the derivative. Thus:  \n",
+    "  \n",
+    "$$\\frac{d}{dx}\\left(\\frac{df}{dx}\\right) = \\frac{d^{2}f}{dx^{2}} = f''(x)$$  \n",
+    "  \n",
+    "The function $f''(x)$ is called the *second derivative* with respect to $x$.  \n",
+    "  \n",
+    "We can continue this process and get higher-order derivatives:  \n",
+    "  \n",
+    "$$\\frac{d^{n}f}{dx^{n}} = f^{(n)}(x),$$  \n",
+    "  \n",
+    "assuming again that these functions are also well-behaved. Function $f^{(n)}(x)$ is called the $n$-th derivative of $f$ with respect to $x$ (note the parenthesis on $(n)$ to distinguish from powers of $f$).  \n",
+    "\n",
+    "```{admonition} Examples\n",
+    "The velocity is the derivative of the position, and the acceleration is the derivative of the velocity. Therefore, acceleration is the second-order derivative of the position\n",
+    "```\n",
+    "\n",
+    "```{admonition} Examples  \n",
+    "  \n",
+    "- $f(x) = 3x^{2} - 6x + 2$, then:  \n",
+    "      \n",
+    "    $$f'(x) = 6x - 6$$  \n",
+    "    $$f''(x) = 6$$  \n",
+    "      \n",
+    "    Polynomials are always well-behaved and they are infinitely differentiable. However, if $n$ is the degree of the polynomial (*i.e.* the largest exponent in the polynomial), we will have: $p^{(k)}(x) = 0,\\ \\mbox{for}\\ k \\ge n+1$.  \n",
+    "    \n",
+    "    <br />\n",
+    "      \n",
+    "- $f(x) = e^{-x^{2}}$, then:  \n",
+    "      \n",
+    "    $$f'(x) = e^{-x^{2}} \\cdot (-2x) = -2xe^{-x^{2}}$$  \n",
+    "      \n",
+    "    $$\\begin{align}f''(x)\n",
+    "&= -2[e^{-x^{2}} + x(-2x)(e^{-x^{2}})] \\\\\n",
+    "&= -2e^{-x^{2}} + 4x^{2}e^{-x^{2}} \\\\\n",
+    "&= -2e^{-x^{2}}(1 - 2x^{2})\\end{align}$$  \n",
+    "  \n",
+    "```  \n",
+    "  "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "c0be95ba",
+   "metadata": {},
+   "source": [
+    "## Functions of several variables  \n",
+    "  \n",
+    "A function $z = f(x,y)$ of two dependent variables $x$ and $y$ defines a surface on the three dimensional $xyz$-space. If this function is well-behave it will be smooth, like if we applied different deformations on a rubber sheet. For smooth functions, we can calculate derivatives with respect to each of the dependent variables. This is written as:  \n",
+    "  \n",
+    "$$ \\frac{\\partial f}{\\partial x} \\text{ or } \\frac{\\partial f}{\\partial x}(x, y) \\text{ or } \\frac{\\partial}{\\partial x}f(x,y)$$  \n",
+    "  \n",
+    "$$\\frac{\\partial f}{\\partial y}, \\frac{\\partial f}{\\partial{y}}(x,y), \\frac{\\partial}{\\partial y}f(x,y)$$  \n",
+    "  \n",
+    "The partial derivative informs us how the function $f$ varies along the $x$ (or the $y$) direction at a specific point.  \n",
+    "  \n",
+    "```{admonition} Example  \n",
+    "Consider $f(x,y) = 4x^2y-2xy^3$. To compute the partial derivative with respect to $x$ we consider all terms in $y$ constants, and calculate the derivative as usual. For the partial derivative with respect to $y$, we consider the $x$ terms constant. We have:  \n",
+    "  \n",
+    "$$\\frac{\\partial f}{\\partial x} = (4y) \\cdot (2x) - 2y^{3} \\cdot (1) = 8xy - 2y^{3}$$  \n",
+    "$$\\frac{\\partial f}{\\partial y} = 4x^{2} - 2x(3y^{2}) = 4x^{2} - 6xy^{2}$$\n",
+    "```  \n",
+    "  \n",
+    "Higher order partial derivatives can also be defined, and represented with the common notations:  \n",
+    "  \n",
+    "$$\\frac{\\partial}{\\partial x}\\left(\\frac{\\partial f}{\\partial x}\\right) = \\frac{\\partial^{2}f}{\\partial x^{2}} = \\partial_{xx} f = f_{xx}(x,y)$$\n",
+    "$$\\frac{\\partial}{\\partial y}\\left(\\frac{\\partial f}{\\partial y}\\right) = \\frac{\\partial^{2}f}{\\partial y^{2}} = \n",
+    "\\partial_{yy} f = f_{yy}(x,y)$$  \n",
+    "  \n",
+    "  \n",
+    "with subscripts representing the variable and the order of differentiation.  \n",
+    "  \n",
+    "\n",
+    "For well-behaved functions, the order of the variables chosen to differentiate does not matter (a special case of what is called Schwarz's theorem):  \n",
+    "  \n",
+    "$$f_{yx}(x,y) = \\frac{\\partial}{\\partial y}\\left(\\frac{\\partial f}{\\partial x}\\right) = \\frac{\\partial^{2}f}{\\partial y \\partial x} = \\frac{\\partial^{2} f}{\\partial x \\partial y}= \\frac{\\partial}{\\partial x}\\left(\\frac{\\partial f}{\\partial y}\\right) = f_{xy}(x,y)$$\n",
+    "\n",
+    "```{admonition} Example \n",
+    "Often, a quantity of interest does not depend only on one variable but on many. The population size of a species depends on various aspects, including the availability of food, the number of predators, and other environmental factors.\n",
+    "```"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "9ff452e5",
+   "metadata": {},
+   "source": []
+  },
+  {
+   "cell_type": "markdown",
+   "id": "f4795887",
+   "metadata": {},
+   "source": []
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 24,
+   "id": "2753df55",
+   "metadata": {
+    "tags": [
+     "remove-cell"
+    ]
+   },
+   "outputs": [
+    {
+     "data": {
+      "image/png": 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",
+      "text/plain": [
+       "<Figure size 360x360 with 1 Axes>"
+      ]
+     },
+     "metadata": {
+      "needs_background": "light"
+     },
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "import matplotlib.pyplot as plt\n",
+    "import numpy as np\n",
+    "import matplotlib.ticker as plticker\n",
+    "\n",
+    "# Create figure and add axes\n",
+    "#fig = plt.figure(figsize=(5, 5))\n",
+    "#ax = fig.add_subplot(111)\n",
+    "\n",
+    "fig, ax = plt.subplots(figsize=(5, 5)) \n",
+    "\n",
+    "# Move left y-axis and bottim x-axis to centre, passing through (0,0)\n",
+    "ax.spines['left'].set_position('zero')\n",
+    "ax.spines['bottom'].set_position('zero')\n",
+    "\n",
+    "# Eliminate upper and right axes\n",
+    "ax.spines['right'].set_color('none')\n",
+    "ax.spines['top'].set_color('none')\n",
+    "\n",
+    "# Show ticks in the left and lower axes only\n",
+    "ax.xaxis.set_ticks_position('bottom')\n",
+    "ax.yaxis.set_ticks_position('left')\n",
+    "\n",
+    "\n",
+    "#loc = plticker.MultipleLocator(base=3.0) # this locator puts ticks at regular intervals\n",
+    "#ax.xaxis.set_major_locator(loc)\n",
+    "#loc = plticker.MultipleLocator(base=2.0) # this locator puts ticks at regular intervals\n",
+    "#ax.yaxis.set_major_locator(loc)\n",
+    "\n",
+    "x = np.linspace(-3, 3, 100)\n",
+    "y = x**3\n",
+    "ax.plot(x, y, color='b', linewidth=2.5, label=('$f(x) = x^3$'))\n",
+    "\n",
+    "ylim = ax.get_ylim()\n",
+    "xlim = ax.get_xlim()\n",
+    "\n",
+    "ax.tick_params(axis='both', which='major', labelsize=13)\n",
+    "\n",
+    "\n",
+    "ax.legend(fontsize=13)\n",
+    "# Add legend\n",
+    "\n",
+    "#ax.legend(bbox_to_anchor=(1.05, 0.3),loc='lower left', \n",
+    "#          frameon=False, labelspacing=0.2, fontsize=13)\n",
+    "\n",
+    "plt.tight_layout()\n",
+    "plt.savefig('xcube.png')"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 25,
+   "id": "8401c1b7",
+   "metadata": {
+    "tags": [
+     "remove-cell"
+    ]
+   },
+   "outputs": [
+    {
+     "data": {
+      "image/png": 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4U4h8cIehpyKX7uXnadMJ2GQy8VrT1wC4EH+BpX8vNTgiIeybs7MziYmJRodRoCQmJuZqpY2b2XQCBuhVtxel3UsD8NVfX8m7txD3oHTp0pw/f56EhAT5X7pHSikSEhI4f/48pUuXztMxbKYaWk6KOhXlpYCXGL1lNHsj9rLtzDZaVbTs0BAhCiovLy8ALly4QEpKisHR2D9nZ2fKlClj/rnmls1NRc7O5euXqfB1BW6k3eCJGk/w81M/Gx2SsBEyFVnYCfuYipyd0u6leaaenou94vAKwq9abploIYSwFrtIwACvNn0VAIViwl8yMUMIYf/sJgHXKV2HDlU6ADA7dDZXE68aHJEQQtwbu0nAAMMfGg5AQkoC04KnGRyNEELcG7tKwO3ub0e9MroI87e7v+VG6g2DIxJCiLyzqwRsMpl446E3ALh47SILDyw0OCIhhMg7u0rAAD3r9MTX0xeA8TvHy2ByIYTdsrsEXMSxCMMaDwPg78i/WXd8ncERCSFE3thdAgZ4IeAFPIp4ADDuz3EGRyOEEHljlwnYu6g3zzd4HoBNJzcRciHE4IjEnbz55pvUrl0bLy8vfH19GThwIFevZh1KOG/ePKpUqYKbmxtNmjQhJER+r6Jgs8sEDPDaQ6/haHIEpBVsDxwdHZk/fz5RUVGEhYVx7tw5+vfvb358+/btvPjii0ydOpXo6GiCgoLo3LkzcXFxBkYthGXZRS2InPT5uQ/z98/HweTAsZePcX/x+40OSdyl3377jd69exMbGwtA3759SU9P54cffgB0pamKFSsyZswY+vbtm+NxpBaEsBP2WwsiJyOajQAgXaXz1c6vDI5G5MbGjRupV6+e+X5YWBiNGjUy3zeZTDRo0ICwsLBbnjt9+nQCAgIICAggMjLSKvEKYQl2nYDrlalHp6qdAJi9bzaR1+Wf0R4sW7aMGTNmMGFCZk2P+Ph4ihUrlmU/b2/vbLsgBg0aRHBwMMHBwZQqVcri8QphKXadgAFGNhsJQGJqIpN2TzI4GnEnS5cuZeDAgfz66680bNjQvN3T09PcHZEhJiYmz3VWhbAHdp+A21RqQ4BvAKDXjbuefN3giERO5syZwwsvvMDKlStp27Ztlsf8/f3Zu3ev+b5SitDQUPz9/a0dphBWY/cJ2GQy8WbzNwG4mniVGXtnGByRyM7EiRMZPnw469ato3nz5rc8PnDgQJYvX87GjRtJTk5m/PjxJCUlERgYaEC0QliHXY+CyJCWnkbNyTU5dvUY5b3Kc3zYcYo4FjE6LHETk8mEk5MTLi4uWbZfu3bNfHvevHmMHj2aiIgI6taty9SpU7NcmMuOjIIQdiLbURAFIgEDzNw7k4ErBwIw5/E59Kvfz9iAhFVIAhZ2ouANQ7tZn3p9zEV6Pt/xOekq3eCIhBDi9gpMAnZxcuH1pq8DcPjKYVYcXmFwREIIcXsFJgEDDGo0iOJFiwPw6fZPpVSlEMKmFagE7OniydDGQwHYc2EPG09uNDgiIYTIWYFKwADDmgzDzdkNgI+3fWxwNEIIkbMCl4BLupVkcKPBAGw+tZk/z/5pcERCCJG9ApeAAd5o9oZ5HLC0goUQtqpAJmBfT1+eq/8cAKuPrWZfxD6DIxJCiFsVyAQMMLL5SHPB9k+2f2JwNEIIcasCm4ArF6/M0/WeBmDZP8v4J/IfgyMSQoisCmwCBninxTuYMKFQ0hcshLA5BToBVy9ZnafqPAXA4oOLORp11OCIhBAiU4FOwACjWo4C9LJFn2yTvmAhhO0o8Am4dunaBNUMAmD+/vmciD5hcERCCKEV+AQMMKqVbgWnqTQ+3fapwdEIIYRWKBJw/bL1eaz6YwB8H/Y9p2JOGRuQEEJQSBIwwHut3gMgNT1V+oKFEDah0CTgAN8AulTrAsCc0DnSChZCGK7QJGCA0a1HA9IKFkLYhkKVgBv5NqJrta6AbgWfjD5pcERCiMKsUCVggA9afwBIK1gIYbxCl4Ab+TbKMiJCxgULIYxS6BIwZG0Fj9k6xuBohBCFVaFMwA3LNSSwRiAA88LmSY0IIYQhCmUCBviwzYeYMJGu0vlwy4dGhyOEKIQKbQKuW6auuVLaogOLOHj5oMERCSEKm0KbgEGPC3YwOaBQjN482uhwhBCFTKFOwNVLVueZes8AsOzQMlk7TghhVYU6AQO83+p9nBycABj1xyiDoxFCFCaFPgFX8anC8w2eB/QKytvPbDc4IiFEYVHoEzDoesFFnYoC8PbGt1FKGRyREKIwkAQM+Hn5MfTBoQBsP7OddcfXGRyREKIwkAT8r7davIVnEU8A3tn4Dukq3eCIhBAFnSTgf5VwK8HwZsMB2HdxHz/985PBEQkhCjpJwDd5relrlHQrCcCoTaNISUsxOCIhREEmCfgmni6e5mXsj109xqx9swyOSAhRkEkC/o/BAYOp5F0JgA+3fMj15OvGBiSEKLAkAf+Hi5MLY9rqEpUXr13km7++MTgiIURBJQk4G73r9qZemXoAfPHnF1xJuGJwREKIgkgScDYcTA589vBnAMTdiOPjrR8bHJEQoiCSBJyDTlU70aZSGwAm75nM8avHjQ1ICGEIS86MlQScA5PJxLj24wBISU/h3U3vGhyREMIIr659lX6/9ONc3Ll8P7Yk4NsI8A2gV51eAPz494/sPr/b4IiEENZ05MoRpgRPYW7YXF787cV8P74k4Dv45OFPKOJYBIDh64dLoR4hCpE3N7xJanoqJkyMbTs2348vCfgOKnlX4uXGLwOw7cw2fj3yq8ERCSGsYcupLaw4sgKAfvX74V/WP9/PIQn4LrzT8h28i3oDMHLDSJLTkg2OSAhhSekqnTfWvwGAm7ObeW5AfpMEfBd8XH14v9X7AByNOsq04GkGRySEsKSFBxYSEhECwPCHhuPn5WeR80gCvktDGg+hqk9VAEZvHs3VxKsGRySEsITElETe2fgOAGU9yjKi+QiLnUsS8F0q4ljEPCwtOimaMVss85FECGGs8TvHczbuLABj2o7Bo4iHxc4lCTgXHq/+OK0rtgZg0p5JHI06anBEQoj8dCH+Ap9u/xSAemXq0b9+f4ueTxJwLphMJr7q+BUmTKSmpzLy95FGhySEyEfvbHyHhJQEAL7p+A2ODo4WPZ8k4FxqWK4hz/o/C8CKIyvYcGKDwREJIfJD8IVg5obNBeCJGk/QtnJbi59TEnAefPrwp+Z+oVfWvkJqeqrBEdm+xYsX07JlS7y8vHBycrrl8bVr11K7dm1cXV2pU6cO69evNyBKUVgppXh17asAODs4m6/3WJok4Dwo51mOd1vq2hD/RP4jw9LuQvHixXnppZf45ptb6yufOHGCbt268fbbbxMbG8vbb79NYGAgp06dsn6golD68e8f2XF2BwCvNn3VPOLJ0kx3mFor825zkJSaRO0ptTkRfYLiRYtz7OVjlHArYXRYNm/z5s20a9eO1NTMTw0ffPABmzZtYtu2beZtLVu2pF27dnzwwQe3PV5AQADBwcEWi1cUfNeTr1N9UnXOx5+ntHtpjg49SrGixfL7NKbsNkoLOI+KOhVlfIfxgB6W9v4f7xsckf0KCwujUaNGWbY1bNiQsLCwbPefPn06AQEBBAQEEBkZaY0QRQH26fZPOR9/Xt9++FNLJN8cSQK+B49Xf5yHKz8MwLSQaYRdzD5hiNuLj4+nWLGsf/Te3t7ExcVlu/+gQYMIDg4mODiYUqVKWSNEUUAdv3qccX/q/t4A3wD61e9n1fNLAr4HJpOJCZ0m4GhyJF2l8/Kal6VaWh54enoSGxubZVtMTAxeXl4GRSQKizfWv2Gu7fLtI9/iYLJuSpQEfI9ql67NsCbDAF0tbdHBRQZHZH/8/f3Zu3dvlm379u3D3z//q08JkWFd+DpztbNn/Z+lafmmVo9BEnA++KD1B5RxLwPomsHxN+INjsj2pKWlkZSURHKybm0kJSWRlJSEUopnn32W4OBgFi1aREpKCosWLSIkJIS+ffsaHLUoqG6k3uDlNbrMrEcRD/MakNYmCTgfFCtajC/afwFAxLUIxmyVOhH/9cMPP+Dq6krHjh1JS0vD1dUVV1dXTp8+TZUqVVi+fDljx47Fy8uLsWPH8vPPP1OpUiWjwxYF1Pid4zl29RgAH7X5iHKe5QyJQ4ah5ZN0lU6L2S3YeW4nTg5OhA0Oo1apWkaHVeDJMDSRW6djTlNzck0SUxOpU7oOewftxdnR2dKnlWFoluRgcmBy58k4mBxITU/lpd9ekgtyQtig19e/TmJqIgCTO0+2RvLNkSTgfNSgXAOGPDgEgC2nt7DgwAKDIxJC3Gxt+FqWH1oOwDP1nqFVxVaGxiMJOJ+NaTvGfEHujfVvEJMUY3BEQgjQhdaHrNYNJM8innzR7guDI5IEnO+KFS3GVx2/AuDy9cuM2jTK4IiEEACfbPuEE9EnAPj4fx8bduHtZpKALaBXnV60raRL2U3ZM4U95/cYHJEQhduhyEN8vuNzABqVa8RLD75kcESaJGALMJlMTHl0Cs4OzigUg1YNkpKVQhhEKcWLv71ISnoKJkxM6zLN4oXW75YkYAupUbIGb7d4G4DQi6FM3DXR4IiEKJzmhc1jy+ktAAx5cAgBvgEGR5RJxgFbUFJqEvWm1uPY1WO4Obvxz0v/UNG7otFhFSgyDljcTuT1SGpOrklUYhRlPcpyeMhhq1Y7u4mMA7a2ok5FmdZFF2tPSElg6JqhMjZYCCt6ff3rRCVGAbrYjkHJN0eSgC3sf5X/R19/XdNg1dFV/PTPTwZHJEThsP74eubvnw9A12pdCaoZZHBEt5IEbAVfdviSkm4lARi6ZihXE68aHJEQBdv15OsMXjUY0MV2JneejMmUbS+AoSQBW0FJt5J801GvhXb5+mWGrx9ucERCFGyjN4/mZMxJAD753yfcV+w+gyPKniRgK+ldtzePVH0EgDmhc2Q5eyEsZM/5PXz1l54M1cSvic2M+c2OjIKwojOxZ6g9pTbXkq9R2bsyB148gHsRd4ucSymIiYELF/RXdDQkJemv9HQoWlR/ubtD2bJQrhyUKQPOxtUlyRMZBSFulpyWTKPpjTh4+SDODs7se2EftUvXNjosyGEUhJO1oyjMKhSrwKcPf8rLa17mZMxJ3t30Lt90unWZ9txKT4e//4Zt2yAsDA4c0PdzWFItR46OULky1KgBNWtCo0bQtClUqAA22H0mxC0+2fYJBy8fBOC9Vu/ZSvLNkbSArSxdpdNqTit2nN2BCRNb+m2hZcWWuT5OTAz8+iusWAFbtkBUlAWC/VeZMtCmDbRvr78qVLDcuXJLWsAiw4FLB2g4vSGp6anUK1OP4IHBhpaa/I9smzCSgA1wNOoo/tP8SUpNoqpPVcIGh+Hm7HbH5yUnw88/w9y5sGEDpKTcuk+xYlCnjv6qWhX8/PSXjw+4uuovkwlu3NDdEbGxcPEiRETAmTNw5AgcOgTHjkFqDrOn69aFoCD9Vbu2sa1jScACICUthYdmPURIRAiOJkd2Pb+LRr6NjA7rZtIFYSuqlajG2LZjGf77cMKvhvPepvcY33F8jvufPQuTJ8Ps2RAZmfUxHx/o0EG3UNu0gWrV8ichJiXBvn2waxfs2AEbN+p+ZNBdHAcOwOjRUKsW9O0LzzwDvr73fl4h8uLzHZ8TEhECwIhmI2wt+eZIWsAGSUtPo8WcFvx17i9MmNjWfxvNKzTPss+JE/DZZ/D991lbuz4+8OSTugXaurV1LpylpcHevbBmjW6Fh4ZmfdzBATp1gqFDoWNHfd8apAUswi6G8eCMB0lJT6F2qdqEDArBxcnF6LD+S7ogbM3hK4epP60+N9JuUKV4FcIGh+FexJ1Ll+C993SLNy0tc//mzWHwYOjeXY9gMNLx47B0Kcybp7ssblalik7EAwaAp6dl45AEXLglpyXTZGYTQi+G4mhy5K/n/7KpYjs3kVoQtqZGyRp8/L+PATgefZw31o3kiy/ggQdgxozM5Nu1K/z1F2zfrj/qG518QSfZt97Soy1279ZvDO7/jqg7fhxee01frBs1Ci5fNjZWUXB9vPVjQi/qj2Nvt3jbVpNvjqQFbLC09DTazm3LtjPb9IZ56+FEewAeeQQ++QTq13fhlA8AACAASURBVDcwwFyIjdUt4kmT4OjRzO2urvDSSzByJJQunb/nlBZw4bX7/G6azWpGmkqjXpl67Bm4hyKORYwOKyfSArZFSYmO3L//e0j+t/n4+HNUqxfDmjWwerX9JF/QIzBefll3SSxfDk2a6O2JiTB+vB5j/NZbmRfzhMirhJQE+vzchzSVhrODM3OfmGvLyTdHkoANtG+fTrBzv7kf1v07CqLYORq9P5ROnYyN7V44OEBgIOzcCZs2Qat/F55NSIDPP9fdF+PH65EWQuTFm7+/ydEo/THro7YfUb+sHbVUbiIJ2ABKwZQpepZZeLje1qzoIFqV07UiFh1cwI8HfzQwwvxhMkHbtrB5sx633KyZ3h4dDcOH6xl3S5bon4cQd2v98fVM2jMJgOb3NWdEsxEGR5R3koCt7Pp16NkThgzREyscHeHTT2HrFhM/9p5tLls5+LfBnIs7Z3C0+cNkgocf1hcRf/kFqlfX20+fhqee0kl6/35jYxT2ISohiv4r+gPg7uzO3Cfm2sz6bnkhCdiKzp+Hli11qw+gfHk9jfitt3QiLutRlhldZwAQkxRD31/6kq7SDYw4f5lM8PjjcPAgTJ0KJfV7DVu2QIMGMGxY7utXiMJDKcXAlQO5EH8BgG86fUMVnyoGR3VvJAFbSXAwPPig7vcFaNdO326ede4FT9R4gucbPA/AppOb+GrnV1aO1PKcnPSwtaNH4ZVX9JtPejp8+63ulli6VLolxK1m7p3Jz4d/BiCwRiADGgwwOKJ7JwnYCtas0ReiIiL0/Rdf1CMcMlqA//V1p6+pUly/s7+z8R2CLxTMYVbFi8M33+gKbq1b620REXqWX9euegq2EABHrhzh1XWvAuDr6cuMrjNscoWL3JIEbGFLlsBjj+mhWA4OMHGirutwu+nDHkU8WBi0ECcHJ1LSU+i1rBfxN+KtF7SV1a4Nf/yhp1xnvCn99pvePm2abh2LwutG6g16L+9NQkoCJkzMe2IeJdxKGB1WvpAEbEGzZkGvXrqqWJEi8NNPepzs3bxxN/ZrbJ4lF341nKFrhlo4WmOZTLqoz+HD+jtAfLz+tPDww3DypLHxCeO8teEt9kbsBWB4s+E8fP/DBkeUfyQBW8i0afD887r15u6uW3SBgbk7xvBmw2l3fzsA5oXNM6/wWpCVKKFbwmvXZtYd3rwZ6tWD6dOlb7iwWXlkJd/s0osWPOj7IGP/N9bgiPKXJGALmDNHt9wAvL31GNh27XJ/HAeTA/OemEcpt1IADF41mCNXjuRjpLarY0c9WmKwXtiWa9fghRegc+fMvnRRsJ2NPUu/Ff0A8HLxYnH3xXY52+12JAHns4ULdRUwAC8vnXybNs378cp5lmNe4DwArqdcp8fSHiSmJOZDpLbP01MPV1u3Tg/ZA90yrltXl8QUBVdqeiq9l/fmauJVAGZ0ncH9xe83OKr8Jwk4H/36Kzz7rP6Y7O6uk0WjfKgL3alqJ95u8TYABy4f4JW1r9z7Qe1Ihw66APzTT+v7UVHQrVtmF48oeEZtGsX2M9sBGNRwEE/WftLgiCxDqqHlk1279IyuxERd/WvNmsyhVfkhNT2VtnPbmv8o5wfO5+l6T+ffCezE4sW6eycmRt93cQngr7+C7apokbi9VUdX0XVRVwDqlanHXwP+wtXZ1eCo7plUQ7OU8HDo0iVzqNnSpfmbfAGcHJxYFLTIPFV50KpB/H357/w9iR3o2VNPW874+d64oauuTZwoF+gKglMxp3j252cB8CziyU89fioIyTdHkoDv0ZUrum7vlSv6/tSp8OijljlXea/yzA+cjwkTCSkJdFvSjbgbhW/u7n336TXqxozR95OT9Yy6bt2k1KU9u5F6gx5LexCdpH+Jsx+fzQMlHjA4KsuSBHwPUlL0umwZFc3efRcGDbLsOTtW7cjoNqMBvbrycyue4w7dSAWSo6NebaN69czhar/8Ag0b6hU6hP0ZtmaYedbnsMbD6F6ru8ERWZ4k4Hvw6quwdau+/fTTmS0ySxvVahSPVNWlK5cdWlYg60XcLQ8PXVPjscf0/VOnoEULXVeiEL4v2a2Ze2cyfe90AJqWb8q4DuMMjsg6JAHn0cyZuqYvQECAXsPNWlPTHUwO/BD4AxWLVQTgzQ1vsvHERuuc3Ab5+OjW71df6SneKSm6slrPnno2nbBtu8/vZsjqIQCUdi/NTz1+KnDjfXMiCTgPdu7Ua5yBXuNs+XI98sGaSriVYNmTyyjqVJQ0lcZTPz3FqZhT1g3ChphMeiHQbdt0HzHoOhwPPqgXDhW2KfJ6JN2XdCc5LRlHkyNLeyzFz8vP6LCsRhJwLkVF6WpdKSm6rOKyZZn/8NbWyLcR07voj21RiVEE/hhIQkqCMcHYiCZNYO9ePZMO4MgRaNxYD18TtiU5LZnuS7tzNk6XvRvfYTytKrYyOCrrkgScC0rpQjHn/l2o4ptvdH+jkfr49+HVJrpMX+jFUAb8OqBQXpS7WcmSutznhx/qlnFCgi6K9Oqr+o1T2IZha4ax9bS+iPJ03acZ1mSYwRFZnyTgXPj6a11UB6B798xuCKON6zCOtpXaArD44GI+2faJwREZz8EB3n9fJ+LixfW2CRN0TY5Ll4yNTcDUPVP5LuQ7QBfZKSj1fXNLZsLdpd279eoVqal6efW9e3WhHVtxJeEKjWc05mSMrtu47MlldKvZzeCoLC8gIIDg4NsXrD95Ug8XzFiNxM9Pdx01aWKFAMUt/jj5Bx3mdyA1PZVyHuXYM3BPYej3lZlweXXtWmZdX2dn+PFH20q+ACXdSrKy10o8i3gC0OfnPuyL2GdwVLahcmXYsUPX6QC9Nl+rVrpes7Cuo1FHCVoSRGp6KkUci7D8qeWFIfnmSBLwXRg+HE6c0Lc//VRfWbdFtUvXZnH3xTiYHEhISeCxxY+ZFzAs7FxddZ3hb7/VF0+Tk3Uxn4zVqYXlRSVE8ejCR80z3WY9Noum5e+hVGABIAn4Dtasge90VxWtW+uhTras8wOdGddeD2I/F3eOLgu7cC35msFR2QaTCYYOhU2b9PBB0GO5H35Y+oUtLTktmaAlQYRf1dNG32v1Hs/Ue8bgqIwnCfg2oqIya/t6euoWlIMd/MRea/oagxrqOdH7Lu6j17JepKWnGRyV7WjZUq9SHRCg72/frm+HhBgbV0GVsZz8ltNbAOhZpycftvnQ4Khsgx2kE+MMGZK5+sKECVCpkqHh3DWTycTkRyfTsYoeDLvq6CpeWftKoR+edrP77tPTyDP6hc+d00MKFy40Nq6C6P0/3mdemF5UoGn5psx5fE6hHPGQHUnAOfj1V32xDXSdgX79DA0n15wcnFjSYwl1S9cFYPKeyXz555cGR2VbMvqFv/5af7JJStI1PUaOhDT5wJAvpodMZ+w2vY5bVZ+q/NrzV4o6FTU4KtshCTgbcXGZY3y9vfUCm/b4hu3l4sXqp1fj6+kLwMgNI/kh7AeDo8pZWloaI0aMoFSpUnh6ehIUFMSVjDqfFmIy6Qka69ZljhceNw66ds0s+i7y5rejv/HSb/ofqaRbSdY8vYZS7qUMjsq2SALOxttv66FKoP8Zy5UzNp57Ud6rPGufXksxl2IAPPfrc6wLX2dwVNn77LPPWLFiBbt27eLcv9MN+/TpY5Vzt2sHe/ZA7dr6/po1epzw4cNWOX2B8+fZP+mxtAdpKg1XJ1dW9lpJVZ+qRodle5RSt/sqdHbsUMpkUgqUatNGqfR0oyPKH5tPblYuY1wUo1HuH7urXed2GR3SLSpUqKBmzpxpvh8eHq4AdfLkyRyf06hRo3yNIS5Oqccf179/UMrLS6nVq/P1FAXegUsHlPdn3orRKIcPHdQvh34xOiRbkG2Ove1MuE6dOql7+QgYGRlJqVL285FDKfjnH90XaDLp1pCLi37M3l5LdqKTojlxVQ9odnRwpHrJ6rg62cZyL2lpaYSGhlKzZk3c3NzM2/ft20flypXxvmnmS2RkpLlr4saNG9S3wIJwFy5kXoAFPXuubNl8P80t7P3vLDktmcNXDpOSpotuVPSuaF5Gy57d6+8lJCRknVKq0y0P5JSZVT60gPO7dWJp48Zltnw+/jjrY/b2WnLyXfB3itEoRqPKfVlOHb963OiQlFJKnTlzRgHqxIkTWbZXqFBB/fDDDzk+z83NzWIxLVmilJtb5t/EM88olZBgsdMppez77+xC3AVVdWJV89+X35N+RoeUb/Lh95JtjpU+4H9duKCrZ4Fe5mb4cGPjsZRBjQbh94+e+hlxLYL2P7S3idlynp56CnVsbGyW7TExMXh5eRkREj166CnMGUsezZ+vJ+NkXB8Qma4kXKHdD+3MEy2GPzScsset8JHBzkkC/tfIkbrmA+jpqkUKcEH+ssfL8lbztwA4EX2C/839H5euGTsVzNvbmwoVKrB3717zthMnThAXF0e9evUMi6t+fX1xrmVLfX/PHj0V/a+/DAvJ5sQkxdDhhw78E/kPAC80eoEv2n9hcFT2waIJeJClV6jMJ1u3woIF+nZQELRvf+s+9vJa7sagQYP45OFPeClADxE6EnWEh+c9TOT1SMPj+vzzzzl58iRxcXG8+eabdOzYkUq3mQFTsqTl+xdLl4YNG+CFF/T9iAjdEp47N//PZW9/Z3E34nhkwSPsu6gLPz1T7xmmPDoFk8lkd6/ldiz2WnLqm1CFZBRESopSdevqPj5XV6VOnzY6IutJS09Tz6943txnV29qPXXl+hXD4klNTVVvvPGGKlGihPLw8FCBgYEqMjLyts+xdp/plClKOTll9gu/+qr+GyqMYpNi1UMzHzL//QT9GKRS0grpD+POcj8KgkJQD3j69MyWzdixemn5wiRdpdN/RX/zVFH/Mv5seHaD3Vy5vpt6wPlt82bdP5wxQKhdOz1r0sfHqmEYKqPl++fZPwF4rPpjLO2xtNAsppkHUg/4v+Lj4b339O2KFeGNN4yNxwgOJgdmPzab3nV7AxB2KYy2c9ty+fplgyOzXW3a6L5gf399f8MG3S988KChYVlNbFJsoUq+N27oWuCWUKgT8BdfwOV/88ynn0LRQjpF3dHBkblPzKVPPT3r7ODlg7T5vg0R8RF3eGbhVamSHiHRo4e+f+IENG2qV8guyKISonh43sPm5Nu1WtcCnXwBRozQff4nT1rg4Dn1Tah86gPu37+/Kl++vPL09FRly5ZV/fv3V1evXs2PQ9+Ts2d1ny8o9eCDSqWl3fk5SUlJatCgQapq1arKw8ND3XfffWr48OEqMTHR8gFbwIQJE1Tjxo2Vq6urqlKlikpNS1X9f+lv7tOrMqGKOhl90ugws5WamqqGDx+unJyclIeHh+rWrdsd+4stIT1djxnPmD0JSr37rlKpqXd/jEWLFqkWLVooT09P5ejoaLlg71FEfISqM6WO+e8jcHGgSkpJyrLPyJEjVa1atZSnp6cqV66cev7551VUVJRBEd+7J5/83vx7dXbeqIKCgtTpvF0oMmYc8Ouvv87hw4eJi4vj0KFDJCQkMGTIEEuf9o5GjYLERH17/Pi7q/ObmppKyZIlWblyJTExMWzbto1Nmzbx5ptvWjZYC/H19WXkyJG8+2/Ht6ODIzMfm8kLjXSn+PHo47SY3YJDkYeMDDNbGXUjatSoYfW6ETczmeCdd2DlSsgYrvzxx7qC3t0W8ylevDgvvfQS33zzjeUCvUenY07T+vvWHLys+1l61enFj91/xMXJJct+jo6OzJ8/n6ioKMLCwjh37hz9+/c3IuR7dukSbNz4NKDrge/b9xAVKlSgZ8+e+XeSnDKzssAoiOjoaNWrVy/VsGHD/D50roSGZrZYAgPv7ViTJk1S9erVy5/ADDJnzhxVpUoV8/309HQ1fN1wc0un5BclVfD5YAMjvFVG3YiMURB3UzfC0g4fVqpGjcyWcNWqSh04cPfP/+OPP2yyBXzw0kHlN97P/PcwYMUAlZp2d038VatWKS8vLwtHmP/S05V65JHM3+W8eUpdu3ZNvfHGG8rHxycvhzRuJtxnn32Gp6cnxYsX55dffjG3uIwyapT+sTo6wuef39uxNm7caOhEAUswmUx80f4LPv7fx4Ce5dRmbhvWH19vcGRabGwsZ86coVGjRuZtVapUwcvLi/379xsWV/XqsGsXPP64vh8ervuFlywxLKR7tvPsTlrOacn5eD3979UmrzK963QcHRzv6vn2+v8xaZKuiAfw0EOnGDKkGB4eHkyYMIHRo0fn34lyyszqDi3gvn37KvQwtWy/3n333Vuec+LECTVq1CgVFhaWl3eQfLFjR+a72qBBmdvz8nq+/vprVapUqbz2CVlMbl/Lf1vAN5u8e7IyjTYpRqOcPnJS80LnWeMl3NbNdSNuHgd8p7oR1pKWptRHH2XtF37jjTuPF7a1FvCKwyuU61hXc8v3s22fqfRclAf86aeflIeHhwoJCbFglPkvJEQpFxf9e6tYUamYGL09IiJCjR07Vv3xxx95OWy2OTbPCTg+Pl5FRkbm+HX9+vVsn7dr1y7l5+en0u7mqlc+S0/XJSZB/4DPns18LLev56uvvlJlypRRB3LzGdNKcvtabpeAlVJq6d9LzaUsGY36ZOsnufpHzG/R0dEKUPv27cuSgL28vNSKFSsMi+u/Vq9Wyts7Mwm3bq1URETO+9tSAv5217fK4UMHc0nJmSEz7/ykmyxZskQVL15cbdq0yUIRWkZUlFKVKunfl6OjUtu2ZX380qVLyt3dPS8XFvM3AefVjh07FKDi4uIscfjb+v33rDOY8uqjjz5Sfn5+6vDhw/kXnIHulICVUmrLqS2q2KfFzEn4uV+eUzdSb1gpwltVqFBBzZo1y5yAjx8/bngfcHbCw5Xy98/8uytX7tZ/6gy2kIBT01LV62tfN/+e3T52UyuPrMzVMWbPnq2KFy+utm/fbqEoLSMtTalHH838XY0bd+s+58+fV0BeGl7WT8CXLl1Sc+fOVdHR0UoppY4cOaKaN2+uWrRoca+HzrX0dD3cDJRyd1fq0qW8HWf48OGqQoUKKjw8PH8DNEBKSopKTExU06dPV/fff79KTEy87ZC6A5cOqApfVzD/c7b5vo2KSjBmiNHYsWNVtWrVVJ06dVRsbKzq3r276tixoyGx3Mn160o9+2zmP7ajo1JffplZ7D81NVUlJiaqdevWKUdHR/PvwdqfMuKS4lTXhV3Nv9+yX5bN9cXXCRMmKB8fH7V7924LRWk5Y8dm/o66dVMqNTVNffvtt+rSv8ni7Nmz6oknnlCVKlVSKbmff279BHz58mXVtm1bVbx4ceXm5qYqVKigXnjhBXXx4sV7PXSurViRdZxmXpw6dUoBqkiRIsrd3d38VatWrfwN1ko++OCDbPuIbyciPkI1mdHE/E9adWJVdfDSQStFnCmjboSjo+Nd140wUnq6Ut99p1SRIpl/h48/rtTVq/oTSHa/B2u25o9fPa5qT65t/r3WmlxLnYo+levjAMrJySnL/4e7u7sFIs5fq1Zl9tk/8IDu901LS1OPPPKIKlWqlHJzc1O+vr6qd+/eeW18ZZtjC0UtCKX0VNGQEChWDE6d0ottirxJTEmk34p+LPlbX973KOLBD4E/8ESNJ6weixG1IO5FcLCePXfqlL5fubIeJREQYFxMm05u4smlTxKVGAVA5wc6syhoEV4uxtRhtrb9+6F5c12O1s1NlxqtWzffT1N4a0GsXauTL+gVcCX53htXZ1cWBS1iTNsxAFxLvkbgj4G8/8f7pKXLeu63ExAAe/fqiRqgp7c2awbffKMbCtaklOLz7Z/T/of25uQ7/KHh/Nrz10KTfC9d0itgX7umJ9UsWGCR5JuznJrGykIX4awtPV2ppk31RwtPT/2RT+SfXw//qjw/8TR/dG03r526dC2PHex5YK9L+KSn637gm0tbdumilLV6UWKTYlXg4kDz763o2KJqbuhc65zcRiQkZOYGUOrzzy16OuMmYhhp48bM1QtefhmKFzc2noKma/Wu7B64mxolawCw4cQGGnzXgG2ntxkcmW0zmXT1ve3bdWEfgFWrdIW1TZsse+7gC8E0+K4BPx/+GYDK3pXZOWAnz/o/a9kT25DUVHjqqczc0L+/LrpjbQU+AX/0kf7u7g6vvWZsLAVVjZI12DNwj7mk5YX4C7Sd25aPtnxEarqF6vgVEE2awL590L27vn/hgq4v/OabkJycv+dSSvH1zq9pNqsZJ6L16tidH+hMyKAQ6pfN/5WlbVV6Ojz/vK7fAbq86LRp+k3R2gp0At6yBbb92xB76SWwwuo1hZZHEQ/mB85n2qPTcHF0IU2l8cHmD2jzfRtOx5w2Ojyb5u2tL8R99x24uuoPxF98AQ89BP/8kz/niIiP4NGFj/L6+tdJSU/BycGJce3HsbLXSoq7Fp6PhUrpBXczlpNq2BBWrDBuDcgCPQrikUf0BThXV32xo0wZoyMqHMIuhtF7eW/zIo1eLl5M6DSBvv59MeVzM8PeRkHcyeHD0Lu3bhUDuLjAZ5/BsGF3V7EvO8v+WcYLq14wX2ir5F2JxUGLaVK+ST5FbR+Ugg8+gDH62jHVqukGWunSVjl94RoFsX+/Tr4AAwZI8rUm/7L+7Bm4hxcDXgT08jX9V/TnscWPcSH+gsHR2bYaNXS/5Jtv6o/EN27orrN27XJfEPxKwhWeXv403Zd2NyffZ+o9w74X9hXK5Pvuu5nJ188P1q+3WvLNWU5X55Sdj4J4+ml9ZdPBQakTJ4yOpvBaeWSlKvtlWfPV9uKfFVez9s7Kt1le9joK4m5s26ZU5cqZV+nd3ZWaPPnOiwekp6erxQcWq1JflDL/3H0+91FLDi6xTuA2Jj1dqZEjM3+O5csrdeyY1cMoPKMgTp+GxYv17Sef1IPdhTG6VOvC3y/9zdN1dWHr6KRoBvw6gDZz23D4ymGDo7NtLVpAWFjmorHXr8OQIfC//8GxY9k/52T0SR5b/Bg9l/UkMiES0MsGHXjxAD1q97BS5LYjLQ2GDtV96gAVKuhrQ1WrGhtXhgKZgL/6Sv/gAUaONDYWAT6uPszvNp+fn/oZX09fALae3or/NH/e2fgO15KvGRyh7fL01Ffof/9dLxwLOoHUrQuffJI5UiIpNYkxW8ZQa0otVh1dBUApt1IsDlrMip4rzD/3wiQxUY8umTJF369YUf/s7r/f2LhuVuAuwkVF6Xe5hARo31738wjbEXcjjnc3vsvkPZNR//55+Xr68kW7L+hdt3euL9IVtItwtxMfD2+9BVOnZs6aq11H8dTo5cw9/ybHo4+b9+1Xvx/j2o+jpFvhHPoTFaVnG/6p1w6lbl1dYN3Pz7CQCsdFuClTdPIFaf3aIi8XL77t/C1/Pf8Xjf0aA3rc8DM/P0PTWU3ZfGqzsQHaME9PmDxZT96oXRvw3cPfD7bm/YPdzcm3bum6bOu/jTmPzym0yTc0VE/5zki+bdvq0Q4GJt8cFagEfONG5seN+vXh4YeNjUfkrLFfY3YO2Mmcx+dQxl0PUdl9fjdt57al84LO7I3Ya3CEtsur6kGqvtsNBjWGiv8OdE8oQdFNE+kdv5cHy7QwNkADLVyoa2tkFDvq3Vu3fIsVMzSsHBWoBLxkCVy8qG+/9poxM1vE3XMwOdCvfj+OvnyUUS1H4ebsBsCa8DU0mt6IxxY9RvCFwtG9cDcOXj5I72W9qTe1HiuO6mnEzg7OVL38Bkw8RtLWl3n7TSdq1oSffrJ+cR8jJSToi21PP637fh0c9IW3+fP1WGpbVWD6gJXKrDRVpoweCWHLP3hxq4j4CD7a8hEz983MMoW5Y5WODG82nIcrP3xLH3Fh6APeeXYnn27/lJVHV5q3OZgc6Ovfl/dbv08l70qsXQuvvw6HDmU+r0kTfaHuf/8zIGgr2rtXJ97D/w6q8fHRo6Datzc2rv/IvjmY0/g0ZWfjgLdtyxznN3q00dGIe3Hi6gk18NeByukjJ/M4Vkaj/Kf6qzn75qiE5ATzvgV1HPCN1Btqfth81XhG4yw/A4cPHVTPn3qqQ5GHbnlOcrJSkyYpVaJE5v8CKNWunVJ//mnAi7CwpCS9+OnNFeWaN1fKxlalylCwC7J37w7Lluk53WfOyMy3guB0zGnG/TmOOaFzSEhJMG8vXrQ4ff37MqjRIPo80qdAtYCPRh1l9r7ZfB/6PZeuXzJvL+JYhL7+fRnZfCRVfW4/iDUmBr78Er7+OvOCNOhrIu+9B61bWyp66/njD3jxRThyRN93coLRo/UMQicnQ0PLSbYt4AKRgE+dgipVdJWjfv1gzhyjIxL5KSohimnB05i0ZxIXr13M8pjbbDc++fETetbpSRkP+3zXjbweybJDy1h4YCHbzmQt41nKrRSDAwYzOGBwrsfyXrwIH38M06dnrazWpInurujWzWaTVY7Cw/WbSMZEK4A6dWD2bL3qjQ0ruAl4xAj9jg+6iEn9wlNZr1BJSUthxZEVTAuexsaTG/XG74AXdJ9oiwotCKwRyBM1nqCSdyUjQ72jM7FnWHV0Fb8c/oVNJzeRprKuJNL8vuYMbDiQp+o8RVGnovd0rvPnYdw4XW0tKSlze4UKukpgv362/4nx/Hldx2HWLF3LF/TyQR9+CK+8As7OxsZ3FwpmAk5M1OP7oqOhVSs900UUfOFXw/kh7Ac+6/sZyQNuLZxbvUR1OlTpQIcqHWh2XzN8XH0MiDJT3I04tp3exh+n/uD3E7+z/9L+W/bx8/Sjd93ePNfgOXOB+/x06RJ8+62eWRcVlbndyQkef1wXJe/QwbaSWWio7kpZtAhSUjK39+wJn3+u30TsRMFMwN9/r/9wQA9D61H4prsXagEBAUz8eSLL/lnG8sPLORVzKtv9apWqRbPyzWhYriH+Zf2pV6YeHkU8LBLTteRrHIo8ROjFUPZc2MOeC3s4cOnALa1cgHIe5ehWsxtPzek3SgAAC+dJREFU1X6K5hWa42Cy/MjQhAQ9PGviRPj776yP+fjo6ylPPgktWxpTJzcuTl/P+f572Lo162OdO+tuFTv8lFswE3CTJrB7N5Qtqy++2dK7t7C8m4ehKaUIvRjKmvA1rD++nh1nd9x2RQ4/Tz+q+lSlqk9VynuVx9fTl3Ie5SjuWhwvFy88i3hSxLEIjg6OOJocSUlPISk1icSURGJvxHIl4QpXEq5wPu48p2NPczr2NMeijnE6NucC9I4mRwJ8A+j8QGe6VOtC/bL1rZJ0s6MU7NwJM2bAjz/qT5M38/KCjh2hUyd94e7++y03tv7SJV0+9rff9EoVN3eVODhAUJDut27a1DLnt4KCl4CDgzM73t97L3P5IVF43G4ccPyNeHae28mOMzvYcXYHu8/vJj453qrxeRTxIMA3gMa+jWldqTUtK7TE08XTqjHcjfh4vTLEwoW6fkpaNotb+/rqWWYNGugWaJ06uvvP0TF350pM1BfTQkJ042nnTt3V8F9lysAzz+gJFhnr5tmxgpeABwzQVz8dHfVIiPLljY5IWFtuJmKkq3ROxZwi9GIoBy4dIDw6nGNRxzgefZwrCVfuKY7S7qWpWKwilYtXplbJWtQuXZvapWpTrUQ1HB1ymaEMdvWqbo2uWqW/R0fnvK+zs64ydt99uvuieHHdcs5YvSMtTQ+Li46GK1d0Ufnz53M+nre37mbo00cXobe3URq3UbAScHS0fvdNTITAQFi+3OiIhBHyayZccloyl65dIuJaBLFJscTdiCPuRhwp6SmkpaeRptJwdnDG1dmVok5FKeZSjJJuJSnhVoIy7mVwdXbNh1dje9LS9OoyW7fqC9zBwXD2bP4d380NGjXS/c2PPKK7GApQ0r1ZtgnYbl/q3LmZfVYvvmhsLML+FXEswn3F7uO+YvcZHYpNcXTUXQ4NGujhXqBHUISF6UkQJ0/qT5/nz+tGUXS07s7IaNeZTLpVW7y4/qpYURdDr1IF6tWDWrUKbMK9K3bZAlYKatbUfwAPPKDngOd1wUJh3wpDLQhRIBScesDbt2dOQRw0SJKvEMI+2WXqmjlTf3d2hmefNTYWIYTIK7tLwDExsHSpvv344zawrLQQQuSR3SXgRYsyL74NHGhsLEIIcS/sLgHPmKG/V6yoxwkKIYS9sqsEvHevrnYG8NxzcvFNCGHf7CqFZVx8c3DILMAjhBD2ym4ScGKinqcOujjIfTJeXghh5+wmAa9YAbGx+ra0foUQBYHdJOC5c/X34sWha1djYxFCiPxgFwn4wgVdIg90JXxZbl4IURDYRQJesEAvuAnQt6+xsQghRH6x+QSsVGb3Q40a0LixsfEIIUR+sfkEvHdv5rpVfftabkkUYTkTJ06kSZMmuLm5UbVq1Wz3mTdvHlWqVMHNzY0mTZoQEhJi5SiFsD6bT8Dff6+/m0x6eRJhf3x9fRk5ciTvvvtuto9v376dF198kalTpxIdHU1QUBCdO3cmLi7OypEKYV02nYCTk3XtB9DTjmXJIfvUvXt3goKC8PPzy/bxGTNm0K1bNzp06ICLiwsjRozAxcWFn3/+2cqRCmFdNp2A16/X1fdByk4WZGFhYTRq1Mh832Qy0aBBA8LCwrLdf/r06QQEBBAQEEBkZKS1whQi39l0Al6wQH93c4MnnjA2FnGrfv36YTKZcvwaNWrUXR0nPj6eYsWKZdnm7e2dYxfEoEGDCA4OJjg4mFKlSt3z6xDCKDa7GtO1a3r2G+i6vx4exsYjbjVp0iS+/PLLHB93c3O7q+N4enoSmzHN8V8xMTFUqVLlnuITwtbZbAL+5ZfMur+9exsbi8ieh4cHHvnwzujv78/evXvN95VShIaG0q1bt3s+thC2zGa7IDIK75QoAR07GhuLuDepqakkJSWRkpKCUoqkpCSSkpLMjw8cOJDly5ezceNGkpOTGT9+PElJSQQGBhoYtRCWZ5Mt4MjIzKnHPXrotd+E/Ro7diwffvih+b6rqyugW7oALVq0YMqUKQwcOJCIiAjq1q3L6tWr8fLyMiReIazFJpelnzwZhg7Vt7dtgxYtjIhC2ANZll7YCftZlj6j+6FCBWjWzNhYhBDCUmwuAZ86BX/+qW/37i3LDgkhCi6bS28ZS86DLj0phBAFlc0l4B9/1N+rV4d69YyNRQghLMmmEvDx45BRBOvJJ6XymRCiYLOpBHxz98OTTxoXhxBCWINNJeAlS/T3WrWgTh1jYxFCCEuzmQR87Bjs26dvS+tXCFEY2EwCzmj9gp79JoQQBZ3NJeA6dXQXhBBCFHQ2kYCPHIH9+/Xtp54yNhYhhLAWm0jAy5Zl3u7e3bg4hBDCmmwiAS9frr/XqqWXnhdCiMLA8AR8+nTm5IugIGNjEUIIazI8Ad+88K0sgCCEKEwMT8AZ/b+VK4O/v7GxCCGENRmagC9ehB079O1u3aT2gxCicDE0Aa9YARkLckj/rxCisDE0AWeMfihXDpo0MTISIYSwPsMScHQ0bNqkbwcGysoXQojCx7C0t2oVpKbq2zL6QQhRGBmWgFes0N+9vaFVK6OiEEII4xiSgJOSYO1affvRR8HZ2YgohBDCWIYk4D/+gOvX9e3HHzciAiHE/9u7f5e21jiO459ga4sm6iAutlppRbBUB7PYimKRthAUoVNxcYp/QXDJFhzEgA46KK7+A4bW26KC6FKw2iBOgsXBdghI/AFq0eYOhyQ3t7H34u3znOT6foH4TXI838flw+HLyXngPlcCOD1+uH1bevnSjRUAgPusB/CPH1Is5tTPn0sVFbZXAACFwXoAf/okff3q1IwfANxk1gN4fj5b9/ba7g4AhcN6AKfnv21t0r17trsDQOGwGsBfvkhbW07d12ezMwAUHqsB/NfxA/NfADed1QB++9b5ff++1NJiszMAFB5rAXxyIq2sOHUgwLN/AcBaAC8uSt+/O3UgYKsrABQuawGcHj/cvet8AQMAbjorAZxKSe/eOXV3t1RWZqMrABQ2KwH8+XP222+MHwDAYSWA0+MHiQAGgDSrAfz4sfTggY2OAFD4jAdwIiF9/OjUXP0CQJbxAF5YyG49TwADQJaVAJakykrp6VPT3QCgeBgN4MtL6cMHp37xQrp1y2Q3ACguRgN4fV06OHDqV69MdgKA4mM0gN+/z9bs/QYAuYwGcHrr+SdPpNpak50AoPgYC+CDg+ztZ4wfAOBnxgJ4cdHZAVkigAEgH2MBnB4/lJdLz56Z6gIAxctIAKdS2QDu7pbu3DHRBQCKm5EA3tqSvn1zasYPAJCfkQBOX/1KBDAAXMVoAD96JD18aKIDisX5+bmGhobU2Ngon8+nuro6hUIhnZ2d5Rw3Njam2tpalZeXq6enR7u7uy6tGLDHSAC/eSP19kqvX5s4O4rJxcWFqqurFYvFlEwmtbq6quXlZQ0PD2eOmZub09jYmGKxmBKJhJqbm9XX16fLy0sXVw6Y50mlH1WW3y8/BK5jampKMzMzisfjkqSuri51dnYqEolIkk5OTlRTU6OFhQV1dXX98lx+v1/r6+vG1wz8R3n3gbe2KSeQtrS0pJaWlszreDyutra2zGuv16vGxsZMQAP/VwQwrm1wcFAej+fKn3A4/NPfTExMaG1tTSMjI5n3jo+PVVlZmXNcVVWVjo6O8vadmZmR3++X3+9XIpH4vf8UYBEPiMS1TU5OKhqNXvl52d+2vx4fH9fo6KiWl5dVV1eXed/n8+nw8DDn2GQyqYqKirznDQaDCgaDkpwRBFCsCGBcm9frldfr/VfHRiIRTU9Pa2VlRU1NTTmftba2amNjQ/39/ZKcGfDOzo5aW1t/+5qBQsIIAsaFQiHNzs7mDV/JuaKdnp7W5uamTk9PFQ6H1dDQoI6ODhdWC9jDFTCM2tvbUzQaVWlpac4VbX19vba3tyVJAwMD2t/fVyAQUDKZVHt7u+bn51VSUuLWsgEruA0NRY3b0FAkuA0NAAoJAQwALiGAAcAlBDAAuIQABgCXEMAA4JJ/ug0NKGgej+ePVCrFY/9RlAhgAHAJIwgAcAkBDAAuIYABwCUEMAC4hAAGAJf8CUkFhu4lo5cBAAAAAElFTkSuQmCC",
+      "text/plain": [
+       "<Figure size 360x360 with 1 Axes>"
+      ]
+     },
+     "metadata": {
+      "needs_background": "light"
+     },
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "import matplotlib.pyplot as plt\n",
+    "import numpy as np\n",
+    "import matplotlib.ticker as plticker\n",
+    "\n",
+    "# Create figure and add axes\n",
+    "#fig = plt.figure(figsize=(5, 5))\n",
+    "#ax = fig.add_subplot(111)\n",
+    "\n",
+    "fig, ax = plt.subplots(figsize=(5, 5)) \n",
+    "\n",
+    "# Move left y-axis and bottim x-axis to centre, passing through (0,0)\n",
+    "ax.spines['left'].set_position('zero')\n",
+    "ax.spines['bottom'].set_position('zero')\n",
+    "\n",
+    "# Eliminate upper and right axes\n",
+    "ax.spines['right'].set_color('none')\n",
+    "ax.spines['top'].set_color('none')\n",
+    "\n",
+    "# Show ticks in the left and lower axes only\n",
+    "ax.xaxis.set_ticks_position('bottom')\n",
+    "ax.yaxis.set_ticks_position('left')\n",
+    "\n",
+    "\n",
+    "#loc = plticker.MultipleLocator(base=3.0) # this locator puts ticks at regular intervals\n",
+    "#ax.xaxis.set_major_locator(loc)\n",
+    "#loc = plticker.MultipleLocator(base=2.0) # this locator puts ticks at regular intervals\n",
+    "#ax.yaxis.set_major_locator(loc)\n",
+    "\n",
+    "x = np.linspace(-3, 3, 100)\n",
+    "y = x**3-(3/2)*x**2-6*x+3\n",
+    "ax.plot(x, y, color='b', linewidth=2.5, label=('$f\\,(x)$'))\n",
+    "\n",
+    "x = np.linspace(-3, 3, 100)\n",
+    "y = 3*x**2-3*x-6\n",
+    "ax.plot(x, y, color='g', linewidth=2.5, label=('$f\\,\\'(x)$'))\n",
+    "\n",
+    "ylim = ax.get_ylim()\n",
+    "xlim = ax.get_xlim()\n",
+    "\n",
+    "ax.tick_params(axis='both', which='major', labelsize=13)\n",
+    "\n",
+    "\n",
+    "ax.legend(fontsize=15)\n",
+    "# Add legend\n",
+    "\n",
+    "#ax.legend(bbox_to_anchor=(1.05, 0.3),loc='lower left', \n",
+    "#          frameon=False, labelspacing=0.2, fontsize=13)\n",
+    "\n",
+    "plt.tight_layout()\n",
+    "plt.savefig('func_diff.png')"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 31,
+   "id": "cb32140f",
+   "metadata": {
+    "tags": [
+     "remove-cell"
+    ]
+   },
+   "outputs": [
+    {
+     "data": {
+      "image/png": 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",
+      "text/plain": [
+       "<Figure size 360x360 with 1 Axes>"
+      ]
+     },
+     "metadata": {
+      "needs_background": "light"
+     },
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "import matplotlib.pyplot as plt\n",
+    "import numpy as np\n",
+    "import matplotlib.ticker as plticker\n",
+    "\n",
+    "# Create figure and add axes\n",
+    "#fig = plt.figure(figsize=(5, 5))\n",
+    "#ax = fig.add_subplot(111)\n",
+    "\n",
+    "fig, ax = plt.subplots(figsize=(5, 5)) \n",
+    "\n",
+    "# Move left y-axis and bottim x-axis to centre, passing through (0,0)\n",
+    "ax.spines['left'].set_position('zero')\n",
+    "ax.spines['bottom'].set_position('zero')\n",
+    "\n",
+    "# Eliminate upper and right axes\n",
+    "ax.spines['right'].set_color('none')\n",
+    "ax.spines['top'].set_color('none')\n",
+    "\n",
+    "# Show ticks in the left and lower axes only\n",
+    "ax.xaxis.set_ticks_position('bottom')\n",
+    "ax.yaxis.set_ticks_position('left')\n",
+    "\n",
+    "\n",
+    "#loc = plticker.MultipleLocator(base=3.0) # this locator puts ticks at regular intervals\n",
+    "#ax.xaxis.set_major_locator(loc)\n",
+    "#loc = plticker.MultipleLocator(base=2.0) # this locator puts ticks at regular intervals\n",
+    "#ax.yaxis.set_major_locator(loc)\n",
+    "\n",
+    "x = np.linspace(-3, 3, 100)\n",
+    "y = x**3-(3/2)*x**2-6*x+3\n",
+    "ax.plot(x, y, color='b', linewidth=2.5, label=('$f\\,(x)$'))\n",
+    "\n",
+    "x = np.linspace(-3, 3, 100)\n",
+    "y = 6*x-3\n",
+    "ax.plot(x, y, color='orange', linewidth=2.5, label=('$f\\,\\'\\'(x)$'))\n",
+    "\n",
+    "ylim = ax.get_ylim()\n",
+    "xlim = ax.get_xlim()\n",
+    "\n",
+    "ax.tick_params(axis='both', which='major', labelsize=13)\n",
+    "\n",
+    "\n",
+    "ax.legend(fontsize=15)\n",
+    "# Add legend\n",
+    "\n",
+    "#ax.legend(bbox_to_anchor=(1.05, 0.3),loc='lower left', \n",
+    "#          frameon=False, labelspacing=0.2, fontsize=13)\n",
+    "\n",
+    "plt.tight_layout()\n",
+    "plt.savefig('plot_concav.png')"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 37,
+   "id": "26e54118",
+   "metadata": {
+    "tags": [
+     "remove-cell"
+    ]
+   },
+   "outputs": [
+    {
+     "data": {
+      "image/png": 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",
+      "text/plain": [
+       "<Figure size 360x360 with 1 Axes>"
+      ]
+     },
+     "metadata": {
+      "needs_background": "light"
+     },
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "import matplotlib.pyplot as plt\n",
+    "import numpy as np\n",
+    "import matplotlib.ticker as plticker\n",
+    "\n",
+    "# Create figure and add axes\n",
+    "#fig = plt.figure(figsize=(5, 5))\n",
+    "#ax = fig.add_subplot(111)\n",
+    "\n",
+    "fig, ax = plt.subplots(figsize=(5, 5)) \n",
+    "\n",
+    "# Move left y-axis and bottim x-axis to centre, passing through (0,0)\n",
+    "ax.spines['left'].set_position('zero')\n",
+    "ax.spines['bottom'].set_position('zero')\n",
+    "\n",
+    "# Eliminate upper and right axes\n",
+    "ax.spines['right'].set_color('none')\n",
+    "ax.spines['top'].set_color('none')\n",
+    "\n",
+    "# Show ticks in the left and lower axes only\n",
+    "ax.xaxis.set_ticks_position('bottom')\n",
+    "ax.yaxis.set_ticks_position('left')\n",
+    "\n",
+    "\n",
+    "#loc = plticker.MultipleLocator(base=3.0) # this locator puts ticks at regular intervals\n",
+    "#ax.xaxis.set_major_locator(loc)\n",
+    "#loc = plticker.MultipleLocator(base=2.0) # this locator puts ticks at regular intervals\n",
+    "#ax.yaxis.set_major_locator(loc)\n",
+    "\n",
+    "x = np.linspace(-3, 4, 100)\n",
+    "\n",
+    "y = (3/2)*x**4-2*x**3-6*x**2+2\n",
+    "ax.plot(x, y, color='b', linewidth=2.5, label=('$f\\,(x)$'))\n",
+    "\n",
+    "y = 6*x**3-6*x**2-12*x\n",
+    "ax.plot(x, y, color='g', linewidth=2.5, label=('$f\\,\\'(x)$'))\n",
+    "\n",
+    "y = 18*x**2-12*x-12\n",
+    "ax.plot(x, y, color='orange', linewidth=2.5, label=('$f\\,\\'\\'(x)$'))\n",
+    "\n",
+    "#ylim = ax.get_ylim()\n",
+    "#xlim = ax.get_xlim()\n",
+    "ax.set_ylim(-50,50)\n",
+    "\n",
+    "ax.tick_params(axis='both', which='major', labelsize=13)\n",
+    "\n",
+    "\n",
+    "ax.legend(fontsize=15)\n",
+    "# Add legend\n",
+    "\n",
+    "#ax.legend(bbox_to_anchor=(1.05, 0.3),loc='lower left', \n",
+    "#          frameon=False, labelspacing=0.2, fontsize=13)\n",
+    "\n",
+    "plt.tight_layout()\n",
+    "plt.savefig('plot_extrema.png')"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 51,
+   "id": "797a95f5",
+   "metadata": {
+    "tags": [
+     "remove-cell"
+    ]
+   },
+   "outputs": [
+    {
+     "data": {
+      "image/png": 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",
+      "text/plain": [
+       "<Figure size 432x288 with 1 Axes>"
+      ]
+     },
+     "metadata": {
+      "needs_background": "light"
+     },
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "import matplotlib.pyplot as plt\n",
+    "import numpy as np\n",
+    "    \n",
+    "dens = np.linspace(0,4)\n",
+    "\n",
+    "fig, ax = plt.subplots(figsize=(6, 4))\n",
+    "\n",
+    "\n",
+    "ax.plot(dens,2*dens**2/(1+dens**2))\n",
+    "    \n",
+    "ax.set_xlabel('$N$',fontsize = 16)\n",
+    "ax.set_ylabel('$f\\,(N)$',fontsize = 16)\n",
+    "\n",
+    "ax.set_yticks([])\n",
+    "ax.set_yticklabels([])\n",
+    "\n",
+    "ax.set_xticks([np.sqrt(1/3)])\n",
+    "ax.set_xticklabels([r'$\\sqrt{\\frac{1}{3ah}}$'],fontsize=16)\n",
+    "\n",
+    "ax.axvline(np.sqrt(1/3), ls='--', color='black', lw=0.5)\n",
+    "\n",
+    "ax.set_ylim(0, 2)\n",
+    "ax.set_xlim(0, 4)\n",
+    "\n",
+    "fig.tight_layout()\n",
+    "fig.savefig('hollingIII.png')"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 57,
+   "id": "1c07acfd",
+   "metadata": {
+    "tags": [
+     "remove-cell"
+    ]
+   },
+   "outputs": [
+    {
+     "data": {
+      "image/png": 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",
+      "text/plain": [
+       "<Figure size 360x360 with 1 Axes>"
+      ]
+     },
+     "metadata": {
+      "needs_background": "light"
+     },
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "import matplotlib.pyplot as plt\n",
+    "import numpy as np\n",
+    "import matplotlib.ticker as plticker\n",
+    "\n",
+    "# Create figure and add axes\n",
+    "#fig = plt.figure(figsize=(5, 5))\n",
+    "#ax = fig.add_subplot(111)\n",
+    "\n",
+    "fig, ax = plt.subplots(figsize=(5, 5)) \n",
+    "\n",
+    "# Move left y-axis and bottim x-axis to centre, passing through (0,0)\n",
+    "ax.spines['left'].set_position('zero')\n",
+    "ax.spines['bottom'].set_position('zero')\n",
+    "\n",
+    "# Eliminate upper and right axes\n",
+    "ax.spines['right'].set_color('none')\n",
+    "ax.spines['top'].set_color('none')\n",
+    "\n",
+    "# Show ticks in the left and lower axes only\n",
+    "ax.xaxis.set_ticks_position('bottom')\n",
+    "ax.yaxis.set_ticks_position('left')\n",
+    "\n",
+    "\n",
+    "#loc = plticker.MultipleLocator(base=3.0) # this locator puts ticks at regular intervals\n",
+    "#ax.xaxis.set_major_locator(loc)\n",
+    "#loc = plticker.MultipleLocator(base=2.0) # this locator puts ticks at regular intervals\n",
+    "#ax.yaxis.set_major_locator(loc)\n",
+    "\n",
+    "x = np.linspace(-3, 3, 100)\n",
+    "y = np.exp(x)\n",
+    "ax.plot(x, y, color='b', linewidth=2.5, label=('$e^x$'))\n",
+    "\n",
+    "y = 1+x\n",
+    "ax.plot(x, y, color='orange', linewidth=2.5, label=('$1+x$'))\n",
+    "\n",
+    "\n",
+    "ax.tick_params(axis='both', which='major', labelsize=13)\n",
+    "\n",
+    "\n",
+    "ax.legend(fontsize=15)\n",
+    "ax.set_ylim(-2,10)\n",
+    "# Add legend\n",
+    "\n",
+    "#ax.legend(bbox_to_anchor=(1.05, 0.3),loc='lower left', \n",
+    "#          frameon=False, labelspacing=0.2, fontsize=13)\n",
+    "\n",
+    "plt.tight_layout()\n",
+    "plt.savefig('exp_linapp.png')"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "f7510b82",
+   "metadata": {},
+   "source": [
+    "## Analysing change, continuously (Part 2)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "4d5884a5",
+   "metadata": {},
+   "source": [
+    "In many different contexts we may be interested in finding the maximum or minimum values a given function may assume. This process is referred to as *optimization*.  \n",
+    "  \n",
+    "In Ecology, some of the most common uses for optimization methods are:\n",
+    "\n",
+    "1 - *Statistical estimation and inference*: optimizing the likelihood of a model and its parameters being true, for a given set of observations.\n",
+    "\n",
+    "2 - *Resources management*: finding a balance between different wildlife and economic priorities (market, legal, natural constraints)\n",
+    "\n",
+    "3 - *Optimality theory*: how can the behavioural or life-history trade-offs result in optimal strategies for different organisms"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "1733c93f",
+   "metadata": {},
+   "source": [
+    "## Extreme values of functions  \n",
+    "  \n",
+    "For a given function $f(x)$ that is **continuous** in an interval $a \\le x \\le b$, there will always be a global maximum and a global minimum, points at which the function assumes its maximum and minimum values in the interval.  \n",
+    "```{image} extrema.png  \n",
+    "  \n",
+    "```\n",
+    "  \n",
+    "A function $f(x)$ presents a *local maximum* value if for a given vicinity of the point $x_0$, $f(x_0)$ is greater than all the other $f(x)$ for $x$ in this vicinity. The analogous definition can be made for a *local minimum* value when $f(x_0)$ is smaller than all the other $f(x)$ in the vicinity. Local minima and local maxima constitue the *local extrema* for $f(x)$, and with the set of all extrema we can find the global maximum and the global minimum.  \n",
+    "  \n",
+    "An important result is that, if the function $f$ has a local extremum at a given point $x=c$ in that interval ($a<c<b$) and the derivative of the function exists at that point (in other words, if $f'(c)$ is well defined), then we will have:  \n",
+    "  \n",
+    "$f'(c)=0$  \n",
+    "    \n",
+    "  \n",
+    "`````{warning}  \n",
+    "\n",
+    "````{panels}  \n",
+    "  \n",
+    "```{image} ./xcube.png  \n",
+    "\n",
+    "```  \n",
+    "\n",
+    "---  \n",
+    "  \n",
+    "Note, however, that the contrary is not always true. If $f'(c)=0$, not necessarily $f(c)$ will be an extremum point. On the left plot, $f(x)=x^3 \\implies f'(x)=2x^2$, thus $f'(0)=0$ but there is no extremum at $x=0$ .\n",
+    "      \n",
+    "```` \n",
+    "`````  \n",
+    "  \n",
+    "Thus the points $x=c$, for which $f'(c)$, will constitute **candidates** for local extrema, and will have to be evaluated under other sets of conditions. Also important to have in mind that point for which the derivative is not definided (kinks or discontinuities) may also constitute local extrema (see on the first figure).  \n",
+    "  \n",
+    "  \n",
+    "In general, when searching for local extrema, we have to adopt the following rules:  \n",
+    "\n",
+    "- Do not assume points $x$ for which $f'(x) = 0$ are extrema. They are candidates.\n",
+    "- Check points where derivative is not defined.\n",
+    "- Check endpoints of domain.\n",
+    "  \n",
+    "```{tip}\n",
+    "If possible, create a plot. Visuals are powerful tools and can help verify whether calculations have been done correctly.\n",
+    "```"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "fd6ac48c",
+   "metadata": {},
+   "source": [
+    "## Monotonicity and concavity  \n",
+    "  \n",
+    "\n",
+    "\n",
+    "Imagine a function $f(x)$ defined in a given interval $a \\le x \\le b$. For every $x_1$ and $x_2$ in this interval, if $x_{2} > x_{1}$ implies that $f(x_{2}) > f(x_{1})$ then the function is **increasing** in this interval. If on the other hand, $x_{2} > x_{1}$ implies that $f(x_{2}) < f(x_{1})$ then the function is **decreasing** in this interval.  This can be graphically shown as:  \n",
+    "  \n",
+    "```{image} incr_decr.png  \n",
+    "```  \n",
+    "  \n",
+    "\n",
+    "  \n",
+    "```{note}  \n",
+    "Since we are using the stric inequalities (\"$>$\" or \"$<$\") it can be added that the function is **monotonic**, or monotonically increasing or decreasing. If we had $x_{2} > x_{1}$ implying that $f(x_{2}) \\ge f(x_{1})$ (or $f(x_{2}) \\le f(x_{1})$), the function would be *non-decreasing* (or *non-increasing*).  \n",
+    "  \n",
+    "```\n",
+    "  \n",
+    "Now suppose that $f$ is well-behaved in a given interval $a \\le x \\le b$ (in mathematical terms, we say: $f$ is continuous and differentiable in $(a,b)$), then an important results is that\n",
+    "\n",
+    "$$\\mbox{If }\\ f'(x) > 0\\ \\mbox{ for}\\ a \\le x \\le b \\implies f(x)\\ \\mbox{is increasing for}\\ a \\le x \\le b$$\n",
+    "\n",
+    "$$\\mbox{If }\\ f'(x) < 0\\ \\mbox{ for}\\ a \\le x \\le b \\implies f(x)\\ \\mbox{is decreasing for}\\ a \\le x \\le b$$  \n",
+    "  \n",
+    "  \n",
+    "In other words, if the derivative of function $f$ is positive (negative) for every point $x$ in a given interval then the function is increasing (decreasing) in this interval.  \n",
+    "  \n",
+    "````{admonition} Example  \n",
+    "Let us calculate the intervals where the function $f(x) = x^{3} - \\displaystyle\\frac{3}{2}x^{2} - 6x + 3$ is increasing and decreasing. Since $f(x)$ is continuos and differentiable for all real values of $x$ (remember that all polynomial functions are well-behved), we can use the first derivative test:  \n",
+    "  \n",
+    "$$f'(x) = 3x^{2}-3x-6 = 3(x^{2}-x-2) = 3(x+1)(x-2)$$  \n",
+    "  \n",
+    "and we see that $f'(x)$ has roots in $x=-1$ and $x=2$. Since $f'(x)$ is a polynomial of degree 2 with the coefficient of the leading term positive, it is *concave up*. From this, we can conclude that:  \n",
+    "  \n",
+    "$$x<-1\\ \\mbox{or}\\ x>2 \\implies f'(x) > 0 \\implies f(x)\\ \\mbox{is}\\ increasing$$  \n",
+    "$$-1<x<2 \\implies f'(x) < 0 \\implies f(x)\\ \\mbox{is}\\ decreasing$$  \n",
+    "  \n",
+    "The following plot shows the function $f(x)$ and its derivative. Compare the intervals where the derivative is positive (negative) with the intervals where the function $f(x)$ is increasing (decreasing).  \n",
+    "  \n",
+    "```{image} func_diff 2.png  \n",
+    "```  \n",
+    "\n",
+    "````  \n",
+    "  \n",
+    "When we discussed polynomials of degree 2, depending on the sign of the coefficient of the leading term, we had the two cases:  \n",
+    "  \n",
+    "```{image}  concav.png  \n",
+    "```\n",
+    "  \n",
+    "Let us now generalise the notion of concavity for any arbitrary function. Suppose $f(x)$ is differentiable in a given interval $a \\le x \\le b$ (in other words, the derivative $f'(x)$ is well-defined for every point in this interval). Then we have the following results:  \n",
+    "  \n",
+    "$$\\mbox{If }\\ f'(x)\\ \\mbox{is increasing for}\\ a \\le x \\le b \\implies f(x)\\ \\mbox{is}\\ concave\\ up$$\n",
+    "\n",
+    "$$\\mbox{If }\\ f'(x)\\ \\mbox{is decreasing for}\\ a \\le x \\le b \\implies f(x)\\ \\mbox{is}\\ concave\\ down$$  \n",
+    "  \n",
+    "This means that if for a given function $f(x)$ the slopes of the tangent lines are increasing in a given interval of $x$, in this interval the function should resemble an \"upwards U-shape\". If on the other hand, the slopes of the tangent lines are decreasing in this interval, the function should resemble a \"downwards U-shape\". This can be visualised on the plots below.  \n",
+    "  \n",
+    "```{image}  slopes_inc_dec.png  \n",
+    "```\n",
+    "  \n",
+    "If the function $f$ is twice differentiable in $a \\le x \\le b$ ($f''(x)$ is well-define for $a \\le x \\le b$), we can use the last two results to find a general criterium for concavity. This will be given by:  \n",
+    "  \n",
+    "$$\\mbox{If }\\ f''(x) > 0\\ \\mbox{ for}\\ a \\le x \\le b \\implies f'(x)\\ \\mbox{is increasing for}\\ a \\le x \\le b \\implies f(x)\\ \\mbox{is}\\ concave\\ up$$\n",
+    "\n",
+    "$$\\mbox{If }\\ f''(x) < 0\\ \\mbox{ for}\\ a \\le x \\le b \\implies f'(x)\\ \\mbox{is decreasing for}\\ a \\le x \\le b \\implies f(x)\\ \\mbox{is}\\ concave\\ down$$  \n",
+    " \n",
+    "```{admonition} Example\n",
+    "If you have a population curve, these tools will help you analyze questions such as 'How fast is the population growing?' 'Is the growth accelerating?' or 'Are any of these factors changing over different periods?\n",
+    "```\n",
+    "  \n",
+    "````{admonition} Example  \n",
+    "Let us analyse the function $f(x) = x^{3}-\\displaystyle\\frac{3}{2}x^{2}-6x+3$ again. We have:  \n",
+    "  \n",
+    "$$\\begin{align} \n",
+    "f'(x) = 3x^{2}-3x-6 \\\\\n",
+    "f''(x) = 6x -3 \\end{align}$$  \n",
+    "  \n",
+    "The function f''(x) has a root in $x=\\displaystyle\\frac{1}{2}$. Since its slope is positive, we have:  \n",
+    "  \n",
+    "$$x<\\displaystyle\\frac{1}{2} \\implies f''(x) < 0 \\implies f(x)\\ \\mbox{is}\\ concave\\ down$$  \n",
+    "$$x>\\displaystyle\\frac{1}{2} \\implies f''(x) > 0 \\implies f(x)\\ \\mbox{is}\\ concave\\ up$$  \n",
+    "  \n",
+    "On the plots below, compare the intervals where the second derivative is positive (negative) with the intervals where the function $f(x)$ is concave up (concave down).  \n",
+    "  \n",
+    "```{image}  plot_concav.png  \n",
+    "```  \n",
+    "\n",
+    "````"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "8d0f05bc",
+   "metadata": {},
+   "source": [
+    "## Finding extrema  \n",
+    "  \n",
+    "  \n",
+    "Back to our strategy for finding maxima or minima of functions, we first determine the set:  \n",
+    "  \n",
+    "- find all points $c$ for which $f'(c) = 0$ or all point $c$ where $f'(c)$ does not exist. These are called **critical points**.\n",
+    "- find endpoints of domain of $f$.  \n",
+    "  \n",
+    "Additionally, if $f$ is twice differentiable (in some interval containing $c$):  \n",
+    "  \n",
+    "- If $f'(c) = 0$ and $f''(c) < 0 \\implies c$ is a local maximum (since the function is concave down)\n",
+    "- If $f'(c) = 0$ and $f''(c) > 0 \\implies c$ is a local minimum (since the function is concave up)  \n",
+    "  \n",
+    "````{admonition} Example  \n",
+    "Let us analyse the function $f(x) = \\displaystyle\\frac{3}{2}x^{4}-2x^{3}-6x^{2}+2$, and find its local extrema. We have:  \n",
+    "  \n",
+    "$$\\begin{align}\n",
+    "f'(x)&= 6x^{3}-6x^{2}-12x=6x(x-2)(x+1) \\\\\n",
+    "f''(x)&=18x^{2}-12x-12=6(3x^{2}-2x-2)\\end{align}$$  \n",
+    "  \n",
+    "Since this is a polynomial function, the derivative is well-defined for all points. Also, for the critical points (roots of $f'(x)$) we have $x=0$, $x=-1$, and $x=2$. Calculating the value of the second derivative at the critical points:  \n",
+    "  \n",
+    "$$\\begin{align}\n",
+    "&x=0 \\implies f''(0)=-12 \\text{ (local maximum)} \\\\\n",
+    "&x=-1 \\implies f''(-1)=18 \\text{ (local minimum)} \\\\\n",
+    "&x=2 \\implies f''(2)=36 \\text{ (local minimum)}\\end{align}$$  \n",
+    "  \n",
+    "Now we can check on the plot of $f(x)$ that we have successfully determined the local extrema.  \n",
+    "  \n",
+    "```{image} plot_extrema.png  \n",
+    "```\n",
+    "\n",
+    "````"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "15f830da",
+   "metadata": {},
+   "source": [
+    "## Inflection points  \n",
+    "  \n",
+    "  \n",
+    "As we saw before, given a function $f(x)$, for some values of $x$, there might be a change in the concavity of the function:  \n",
+    "  \n",
+    "```{image} inflection.png  \n",
+    "```\n",
+    "    \n",
+    "The points $x$ for where the function changes its concavity are called **inflection points**. If $f$ is twice differentiable and $c$ is an inflection point, then $f''(c)=0$.  \n",
+    "\n",
+    "Inflection points are the points where the function is (locally) steepest. It therefore helps us answer questions like when the growth/decline was strongest.\n",
+    "  \n",
+    "```{note}  \n",
+    "Note again that the points $x=c$ where $f''(c)=0$ are only candidates for inflection points. See that if $f(x)=x^{4}$, $f''(x)=12x^{2}$. Then, $f''(0)=0$, but $x=0$ is not an inflection point of $f$.  \n",
+    "  \n",
+    "After calculating the candidate points $x=c$, if the concavities of the function (given by the sign of the second derivative) to the left and to the right of $c$ change, then $c$ is an inflection point.  \n",
+    "\n",
+    "```  \n",
+    "\n",
+    "````{admonition} Example  \n",
+    "\n",
+    "Consider the Holling Type III functional response   \n",
+    "  \n",
+    "$$f(N)=\\frac{aN^{2}}{1+ahN^{2}},$$  \n",
+    "  \n",
+    "where $f(N)$ is prey consumption rate per predator, $N$ is the prey population density, $a$ is the attack rate, and $h$ is handling time. Let us check if this function presents any inflection point. We have:  \n",
+    "  \n",
+    "$\\begin{align}f'(N) \n",
+    "&= \\frac{2aN(1+ahN^{2})-aN^{2}(2ahN)}{(1+ahN^{2})^{2}} \\\\\n",
+    "&= \\frac{2aN}{(1+ahN^{2})^{2}}\\end{align}$  \n",
+    "  \n",
+    "<br />\n",
+    "  \n",
+    "$\\begin{align}f''(N)\n",
+    "&=\\frac{2a(1+ahN^{2})^{2}-2aN[2(1+ahN^{2})\\cdot2ahN]}{(1+ahN^{2})^{4}} \\\\\n",
+    "&=\\frac{2a(1+2ahN^{2}+a^{2}h^{2}N^{4})-2a[4ahN^{2}+4a^{2}h^{2}N^{4}]}{(1+ahN^{2})^{4}} \\\\\n",
+    "&=\\frac{2a(1-3ahN^{2})}{(1+ahN^{2})^{3}}\\end{align}$  \n",
+    "  \n",
+    "See that the function $f''(N)$ has two roots: $N=-\\sqrt{\\displaystyle\\frac{1}{3ah}}$ and $N=\\sqrt{\\displaystyle\\frac{1}{3ah}}$. Since $N$ is a population density, it only makes sense to analyse $N \\ge 0$. It is also possible to show that:\n",
+    "  \n",
+    "$$N<\\sqrt{\\displaystyle\\frac{1}{3ah}} \\implies f''(N) > 0$$  \n",
+    "$$N>\\sqrt{\\displaystyle\\frac{1}{3ah}} \\implies f''(N) < 0$$  \n",
+    "  \n",
+    "This means that we have a change of concavity at the point $N=\\sqrt{\\displaystyle\\frac{1}{3ah}}$ and thus this is and inflection point of this curve.  \n",
+    "  \n",
+    "Analyse how the function $f(N)$ changes with $N$.  \n",
+    "  \n",
+    "```{image} hollingIII.png  \n",
+    "```\n",
+    "  \n",
+    "How does this function compares with the Holling type II functional response?\n",
+    "\n",
+    "````  \n",
+    "  \n",
+    "```{tip}  \n",
+    "  \n",
+    "For the intervals where the function is concave up you can also think of it as representing \"accelerated change\" of that function (since the first derivative, the \"velocity\" is increasing), while the intervals where it is concave down represent \"deccelerated change\" of the function. Understanding this can be particularly important if you function has a specific biological meaning, as in the previous example.  \n",
+    "  \n",
+    "```"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "ed45757b",
+   "metadata": {},
+   "source": [
+    "## Series expansions \n",
+    "  \n",
+    "From the formal definition of a derivative  \n",
+    "  \n",
+    "$$f'(a)= \\lim_{x \\to a} \\frac{f(x)-f(a)}{(x-a)} \\xrightarrow{\\text{$x$ very close to $a$}} f'(a) \\approx \\frac{f(x)-f(a)}{(x-a)},$$  \n",
+    "  \n",
+    "which means that if $x$ is sufficiently close to $a$, the average rate of change is sufficiently close to the value of the derivative at that point. From the previous approximation, we have:  \n",
+    "  \n",
+    "$$f(x) = f(a)+f'(a)(x-a),$$  \n",
+    "  \n",
+    "which provides a linear approximation of $f(x)$ around $a$ (see that if $\\alpha= f(a)$ and $\\beta=f'(a)$, both constants, then $f(x) = \\alpha+\\beta(x-a)$, which can easily be arranged on the form $y=mx+b$).  \n",
+    "  \n",
+    "`````{admonition} Example  \n",
+    "  \n",
+    "````{panels}  \n",
+    "  \n",
+    "```{image} ./exp_linapp.png\n",
+    "\n",
+    "```  \n",
+    "\n",
+    "---  \n",
+    "  \n",
+    "Say we have $f(x)=e^{x}$ and we want to use this linear expression to approximate the function close to $a=0$: \n",
+    "\n",
+    "$$f(x)=f(0)+f'(0)(x-0)=1+x,$$  \n",
+    "  \n",
+    "since $f'(x) = e^{x}$ and thus $f(0)=f'(0)=1$.  \n",
+    "  \n",
+    "The approximation is reasonable if $x$ very close to $0$, but fails when $x$ increasingly departs from $0$. \n",
+    "  \n",
+    "````  \n",
+    "  \n",
+    "`````"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "dcb07889",
+   "metadata": {},
+   "source": [
+    "One way of trying to make the approximation better is to add terms of higher order of $x$. Suppose we want to approximate the function $f(x)$ at point $a$ by a given polynomial of degree $n$  \n",
+    "  \n",
+    "$$P(x)=a_{0}+a_{1}(x-a)+a_{2}(x-a)^{2}+\\cdots+a_n(x-a)^{n}$$  \n",
+    "  \n",
+    "so that the derivatives of $f(x)$ and $P(x)$ are the same at $x=a$:\n",
+    "\n",
+    "$$\\begin{align}f(a)&=P(a) \\\\\n",
+    "f'(a)&=P'(a)\\\\\n",
+    "&\\vdots \\\\\n",
+    "f^{(n)}(a)&=P^{(n)}(a)\\end{align}.$$  \n",
+    "  \n",
+    "But for the polynomial, we have:  \n",
+    "  \n",
+    "$$\\begin{align}  \n",
+    "P(a)&=a_{0} \\\\\n",
+    "P'(a)&=1 \\cdot a_{1} \\\\\n",
+    "P''(a)&=2 \\cdot 1 \\cdot a_{2} \\\\\n",
+    "P'''(a)&=3 \\cdot 2 \\cdot 1 \\cdot a_{3} \\\\  \n",
+    "P''''(a)&=4 \\cdot 3 \\cdot 2 \\cdot 1 \\cdot a_{4} \\\\\n",
+    "&\\vdots \\\\\n",
+    "P^{(n)}(a)&= \\underbrace{n(n-1)(n-2) \\cdots 3 \\cdot 2 \\cdot 1}_{\\displaystyle\\text{$$n!$$}}\\cdot a_{n} \\end{align}$$  \n",
+    "  \n",
+    "where $n! = n\\cdot(n-1)\\cdot(n-2) \\cdots 3 \\cdot 2 \\cdot 1$ is the factorial of $n$.  \n",
+    "  \n",
+    "Thus, by making derivatives of $f(x)$ and $P(x)$ equal at $x=a$, we obtain: \n",
+    "  \n",
+    "$$f^{(n)}(a) = P^{(n)}(a) \\implies f^{(n)}(a) = n!\\, a_n \\implies a_{n} = \\frac{f^{(n)}(a)}{n!}$$  \n",
+    "  \n",
+    "The polynomial with these coefficients is called **Taylor polynomial** of degree $n$ at $x = a$:  \n",
+    "  \n",
+    "$$P(x)=f(a)+f'(a)(x-a)+\\frac{f''(a)}{2!}(x-a)^{2}+\\cdots+\\frac{f^{(n)}(a)}{n!}(x-a)^{n}.$$  \n",
+    "  \n",
+    "````{admonition} Example  \n",
+    "  \n",
+    "To find approximation of degree 3 for $f(x)=e^{x}$, at $x=0$, we have:  \n",
+    "  \n",
+    "$$f(0)=1 \\\\\n",
+    "f'(0)=1 \\\\\n",
+    "f''(0)=1 \\\\\n",
+    "f'''(0)=1$$  \n",
+    "  \n",
+    "Thus, \n",
+    "  \n",
+    "$$P_{3}(x)=1+(x-0)+\\frac{1}{2!}(x-0)^2+\\frac{1}{3!}(x-0)^3=1+x+\\frac{x^2}{2}+\\frac{x^3}{6}$$  \n",
+    "  \n",
+    "See how the approximation to the function $f(x)=e^{x}$  at $x=0$ gets increasingly better when more terms are added to the polynomial.  \n",
+    "  \n",
+    "```{image} exp_series.gif  \n",
+    "```\n",
+    "  \n",
+    "````  \n",
+    "  \n",
+    "  \n",
+    "```{admonition} Example  \n",
+    "For the function $f(x)=\\displaystyle\\frac{1}{1-x}\\ (x\\neq1)$ at $x=0$, we have:  \n",
+    "  \n",
+    "$$f(x)=\\frac{1}{1-x} \\implies f(0)=1 \\\\\n",
+    "f'(x)=\\frac{1}{(1-x)^{2}} \\implies f'(0)=1 \\\\\n",
+    "f''(x)=\\frac{2}{(1-x)^{3}} \\implies f''(0)=2 \\\\\n",
+    "f'''(x)=\\frac{3\\cdot2}{(1-x)^{4}} \\implies f'''(0)=3! \\\\\n",
+    "f^{(n)}(x)=\\frac{n!}{(1-x)^{n+1}} \\implies f^{(n)}(0)= n!$$  \n",
+    "  \n",
+    "Thus the Taylor polynomial of degree $n$ will be:  \n",
+    "  \n",
+    "$$\\begin{align}P(x)\n",
+    "&=1+1(x-0)+\\frac{2}{2}(x-0)^{2}+\\frac{3!}{3!}(x-0)^{3}+\\cdots+\\frac{n!}{n!}(x-0)^{n} \\\\\n",
+    "&=1+x+x^{2}+x^{3}+\\cdots+x^{n}\\end{align}$$\n",
+    "\n",
+    "  \n",
+    "```  \n",
+    "  \n",
+    "For some functions, the approximation $f(x)=P_n(x)$ at a given reference point $x=a$ can be made increasingly better by adding more terms to $P(x)$, so that in the limit $n\\rightarrow \\infty$ the Taylor polynomials (called **Taylor series** in this limit) for the functions $e^{x}$, $\\mbox{sin}(x)$, $\\mbox{cos}(x)$ converge exactly for all values of $x$. For other functions, such as $\\mbox{ln}(1+x)$ and $\\displaystyle\\frac{1}{1-x}$, there is a restricted range of values of $x$ for which the approximation can be made better by adding terms.  \n",
+    "  \n",
+    "We have the following approximations by Taylor series, with ranges in $x$, at $x = 0$:\n",
+    "\n",
+    "$$e^x=\\sum_{n=0}^\\infty \\frac{x^n}{n!} =1+x+\\frac{x^2}{2}+\\frac{x^3}{3!}+\\cdots,\\ -\\infty<x<\\infty$$\n",
+    "\n",
+    "$$\\mbox{sin}(x)=\\sum_{n=0}^\\infty \\frac{(-1)^nx^{1+2n}}{(1+2n)!} = x-\\frac{x^3}{3}+\\frac{x^5}{5!}-\\frac{x^7}{7!}+\\cdots,\\ -\\infty<x<\\infty$$\n",
+    "\n",
+    "$$\\mbox{cos}(x)=\\sum_{n=0}^\\infty \\frac{(-1)^nx^{2n}}{(2n)!} = 1-\\frac{x^2}{2}+\\frac{x^4}{4!}-\\frac{x^6}{6!}+\\cdots,\\ -\\infty<x<\\infty$$\n",
+    "\n",
+    "$$\\mbox{ln}(1+x)=-\\sum_{n=1}^\\infty \\frac{(-1)^n x^n}{n} = \\frac{x^2}{2!}+\\frac{x^3}{3!}-\\frac{x^4}{4!}+\\frac{x^5}{5!}+\\cdots,\\ -1 < x \\leq 1$$\n",
+    "\n",
+    "$$\\frac{1}{1-x} = \\sum_{n=0}^\\infty x^n = 1+x+x^2+x^3+\\cdots,\\ -1 \\leq x < 1$$  \n",
+    "  \n",
+    "Knowing the Taylor series often gives insight into the properties of a function."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "b869064b",
+   "metadata": {},
+   "source": [
+    "## Calculating accumulated change"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "c400afba",
+   "metadata": {},
+   "source": [
+    "## Series  \n",
+    "  \n",
+    "For every sequence $\\{a_n\\}$ of real numbers, it is possible to define another sequence $\\{S_n\\}$ for which the elements are given by the sum of the first $n$ terms of the sequence $\\{a_n\\}$:\n",
+    "\n",
+    "$$\\begin{align}\n",
+    "S_0 &= a_0 \\\\\n",
+    "S_1 &= a_0 + a_1 \\\\\n",
+    "S_2 &= a_0 + a_1 + a_2 \\\\\n",
+    "&\\vdots \\\\\n",
+    "S_n &= a_0 + a_1 + \\cdots + a_{n-1} + a_n \\\\\n",
+    "S_n &= \\sum_{i = 0}^n a_i \\end{align}$$  \n",
+    "  \n",
+    "The sequence $\\{S_n\\}$ is called a *series* (or an infinite series, to emphasize the sum over increasing number of terms $\\{a_n\\}$).  \n",
+    "  \n",
+    "The same way as we asked about $\\lim a_n$, we can ask if $\\lim S_n$ exists. If it exists, or, in other words, if the sum of infinite terms $\\{a_n\\}$ is equal to a finite number, we say that the series $S_n$ is convergent.  \n",
+    "  \n",
+    "  \n",
+    "```{admonition} Examples  \n",
+    "  \n",
+    "The following expressions are example of convergent series:  \n",
+    "  \n",
+    "$$\\begin{align}&\\sum_{n=0}^\\infty \\frac{x^n}{n!} \\hspace{0.1cm}, \\hspace{0.5cm} x \\in \\mathbb{R} \\\\\n",
+    "&\\sum_{n=0}^\\infty x^n \\hspace{0.1cm}, \\hspace{0.5cm} -1<x<1 \\\\\n",
+    "&\\sum_{n=0}^\\infty (-1)^n \\frac{x^{2n+1}}{(2n+1)!} \\hspace{0.1cm}, \\hspace{0.5cm} x \\in \\mathbb{R}\\end{align}$$  \n",
+    "  \n",
+    "```  \n",
+    "  "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "99112c46",
+   "metadata": {},
+   "source": [
+    "## Calculating areas  \n",
+    "  \n",
+    "  \n",
+    "The idea of summing infinite quantities can be used to solve the problem of determining the area below a given curve. Let $f(x)$ be a function defined on the interval $a \\le x \\le b$ as:\n",
+    "\n",
+    "```{image}  sum_areas.png\n",
+    "```\n",
+    "\n",
+    "The area below the curve of $f(x)$ can be found dividing the interval $a \\le x \\le b$ into $n$ subintervals:\n",
+    "\n",
+    "$$a=x_0<x_1<x_2<\\cdots<x_{n-1}<x_n=b$$  \n",
+    "  \n",
+    "where each interval defines a rectangle of height $f(x_i)$ and width $(x_{i+1}-x_i) = \\Delta x_i = \\displaystyle\\frac{(b-a)}{n}$. The area below the curve will then be approximately the sum of the areas of the rectangles\n",
+    "\n",
+    "$$A \\approx \\sum_{i=0}^n f(x_i)\\Delta x_i.$$  \n",
+    "  \n",
+    "As we increasethe number of intervals $n$, while decreasing the widths of each rectangle, the sum of the areas of the rectangles gives a better approximation of the area below the curve. The exact result is given by the limit $n \\rightarrow \\infty$. In this limit, the sum is represented by the following symbol:\n",
+    "\n",
+    "$$\\lim_{n \\to \\infty} \\sum_{i=0}^n f(x_i) \\Delta x_i = \\int_a^b f(x)dx$$  \n",
+    "  \n",
+    "If this limit exists, we say that the function $f$ is **integrable** on the interval $a \\lt x \\lt b$ and denote the limit by the **definite integral** $\\int_a^b f(x)dx$.  \n",
+    "  \n",
+    "The interpretation of the integral as area can be generalized for when $f(x)$ assumes negative values  \n",
+    "  \n",
+    "```{image}  defin_int.png\n",
+    "``` \n",
+    "  \n",
+    "In the case of the previous figure, we will have:  \n",
+    "  \n",
+    "$$\\begin{align} \\int_a^b f(x)dx \n",
+    "&= A_1 - A_2 + A_3 \\hspace{0.1cm}, \\hspace{0.5cm} (A_i \\geq 0) \\\\\n",
+    "&= [\\text{area above x-axis}] - [\\text{area below x-axis}] \\end{align}$$  \n",
+    "  \n",
+    "Other intuitive results related to the calculation of the area are:\n",
+    "\n",
+    "$$\\int_a^a f(x)dx = 0\\quad \\mbox{(size of the interval equal to zero)}$$\n",
+    "\n",
+    "$$\\int_a^b f(x)dx = -\\int_b^a f(x)dx\\quad  \\mbox{(orientation of the integral)}$$   \n",
+    "  \n",
+    "Furthermore, some of the properties of the integral follows directly from the properties of limits:\n",
+    "\n",
+    "$$\\begin{align}\n",
+    "\\int_a^b kf(x)dx &= k \\int_a^b f(x)dx \\\\\n",
+    "\\int_a^b [f(x) + g(x)]dx &= \\int_a^b f(x)dx + \\int_a^b g(x)dx \\\\\n",
+    "\\int_a^b f(x)dx &= \\int_a^c f(x)dx + \\int_c^b f(x)dx ,\\ \\  (a \\le c \\le b) \\end{align}$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "9697057a",
+   "metadata": {},
+   "source": [
+    "## Fundamental theorem of Calculus  \n",
+    "  \n",
+    "  \n",
+    "The problem of calculating integrals as limits of sums requires specific methods for each function and becomes quite impractical. Fortunately, this problem can be solved with the very important result that the area problem is intrinsically related to the tangent problem.\n",
+    "\n",
+    "Suppose we define a function $F(x)$ that gives the area below the curve $f(u)$ from $a$ until a variable value $x$. By definition:  \n",
+    "  \n",
+    "$$F(x) = \\int_a^x f(u)du$$  \n",
+    "  \n",
+    "```{image} fund_calc.png  \n",
+    "```\n",
+    "  \n",
+    "If we consider a small increment $h$ to $x$, the new area below the curve shall be given by:  \n",
+    "  \n",
+    "$$F(x+h) = \\int_a^{x+h} f(u)du$$  \n",
+    "  \n",
+    "  \n",
+    "The difference between these two areas shall be approximately given by the area of the retangle with width $h$ and height $f(x)$:  \n",
+    "  \n",
+    "$$F(x+h) - F(x) \\approx hf(x) \\implies \\displaystyle\\frac{F(x+h) - F(x)}{h} \\approx f(x)$$  \n",
+    "  \n",
+    "But see that in the limit where the increment $h$ gets very small, the approximation gets exactly equal to the increment in area, and we obtain the definition of the derivative of $F(x)$:  \n",
+    "  \n",
+    "$$\\lim_{h\\rightarrow 0} \\displaystyle\\frac{F(x+h) - F(x)}{h} = \\displaystyle\\frac{dF}{dx} = f(x).$$  \n",
+    "  \n",
+    "This results is known as the *Fundamental theorem of Calculus*, and it say that the instantaneous rate of change of the area below a given curve $f$ at a point $x$ is given by the value of the function at that point. In other words, the area $F(x)$ is a function that, if differentiated, gives the function $f(x)$.  \n",
+    "  \n",
+    "The function $F(x)$ is then called an **antiderivative** of $f(x)$. In general, there is a *family* of antiderivates of $f(x)$, since $\\displaystyle\\frac{d(F(x) + C)}{dx}=\\displaystyle\\frac{dF}{dx}=f(x)$, independent of the value of the constant $C$. We therefore call **indefinite integral**:\n",
+    "\n",
+    "$$\\int f(x)dx = F(x) + C$$  \n",
+    "   \n",
+    "It also directly follows that\n",
+    "\n",
+    "$$\\int_b^c f(x)dx = F(c) - F(b)$$  \n",
+    "  \n",
+    "Thus the problem of calculating the area below the curve $f(x)$ from $b$ to $c$, which, by definition, is given by the infinite sum of terms $f(x_i)\\Delta x_i$ (with $\\Delta x_i \\rightarrow 0$), can be reduced to obtaining the antiderivative of $f$, $F(x)$, and subtracting the values of this function at points $c$ and $b$. The main problem then becomes how to calculate the antiderivates of a given function $f(x)$, for which different methods can be used."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "02bc5c04",
+   "metadata": {},
+   "source": [
+    "## Calculating integrals  \n",
+    "  \n",
+    "  \n",
+    "The simplest integrals can be calculated by inverting the differentiation of fundamental functions, as\n",
+    "\n",
+    "$$\\int x^n \\,dx = \\frac{x^{n+1}}{n+1} + C$$\n",
+    "\n",
+    "$$\\int e^x \\,dx = e^x + C$$\n",
+    "\n",
+    "$$\\int \\frac{1}{x}\\,dx = \\ln |x| + C$$\n",
+    "\n",
+    "$$\\int \\mbox{cos}(x)\\, dx = \\mbox{sin}(x) + C$$\n",
+    "\n",
+    "$$\\int \\mbox{sin}(x)\\, dx = -\\mbox{cos}(x) + C$$  \n",
+    "  \n",
+    "This are obtained simply because we know that the derivative of the functions on the righ-hand sides of the equations give exactly the function that is being integrated. When the integrals are not so simple, we must apply other methods to turn them into integrals we know. \n",
+    "  \n",
+    "  \n",
+    "### Integration by substitution  \n",
+    "  \n",
+    "  \n",
+    "This method consists in finding an appropriate substitution of the variable of integration so that we end up with a simpler integral to calculate.  \n",
+    "  \n",
+    "  \n",
+    "````{admonition} Examples  \n",
+    "  \n",
+    "- $\\int\\frac{1}{\\sqrt{x+4}}dx$  \n",
+    "    \n",
+    "    Note that if $u=x+4 \\implies \\displaystyle\\frac{du}{dx}=1 \\implies du=dx$. Thus, the integral can be given by:  \n",
+    "    \n",
+    "    $$\\int u^{-\\frac{1}{2}}du = \\frac{1}{\\left(-\\frac{1}{2}+1\\right)} \\cdot u^{\\frac{1}{2}} + C = 2\\sqrt{x+4} + C.$$  \n",
+    "      \n",
+    "      \n",
+    "    \n",
+    "    ```{note}  \n",
+    "    Note that three mathematicians die everytime we operate with $du$ and $dx$ as if they were numbers on a fraction. Formally, the correct substitution on the integral would be given by a function $u=u(x)$, so that $ \\int f(u(x)) u'(x)dx = \\int f(u)du$. But we can procede with the heresy above as long as we have this in mind.  \n",
+    "    ```  \n",
+    "    <br />\n",
+    "      \n",
+    "- $\\int xe^{(x^2-1)}dx$  \n",
+    "      \n",
+    "    See that $u=x^2-1 \\implies \\displaystyle\\frac{du}{dx}=2x \\implies  dx=\\displaystyle\\frac{du}{2x}$. Then:  \n",
+    "    \n",
+    "    $$\\int \\frac{1}{2}e^{u}du = \\frac{1}{2}e^u + C = \\frac{e^{(x^2-1)}}{2} + C $$  \n",
+    "      \n",
+    "````\n",
+    "\n",
+    "### Integration by parts  \n",
+    "  \n",
+    "When integrating by parts, we have to find an appropriate association $(u=u(x), v=v(x))$, so that:  \n",
+    "  \n",
+    "$$\\int uv'dx=uv-\\int u'vdx$$  \n",
+    "  \n",
+    "```{admonition} Example  \n",
+    "  \n",
+    "- $\\int x\\ln x\\, dx $  \n",
+    "      \n",
+    "    Note that  \n",
+    "      \n",
+    "    $$\\begin{align}\n",
+    "    u=\\ln x \\implies u' = \\displaystyle\\frac{1}{x} \\\\\n",
+    "    v'=x \\implies v=\\displaystyle\\frac{x^2}{2} \\end{align}$$  \n",
+    "      \n",
+    "    Thus:  \n",
+    "      \n",
+    "    $$\\begin{align}\\int x\\ln x\\, dx \n",
+    "&= \\frac{x^2}{2}\\ln x - \\int \\frac{x^2}{2} \\cdot \\frac{1}{x}dx \\\\\n",
+    "&= \\frac{x^2 \\ln x}{2} - \\frac{1}{2}\\int xdx \\\\\n",
+    "&= \\frac{x^2 \\ln x}{2} - \\frac{1}{4}x^2 + C \\\\\n",
+    "&= \\frac{1}{4} x^2 (2\\ln x - 1) + C \\end{align}$$  \n",
+    "  \n",
+    "- $\\int xe^x \\,dx$  \n",
+    "      \n",
+    "    Note that  \n",
+    "      \n",
+    "    $$\\begin{align}\n",
+    "    u=x \\implies u'=1 \\\\\n",
+    "    v'=e^x \\implies v=e^x \\end{align}$$  \n",
+    "      \n",
+    "    Thus:  \n",
+    "      \n",
+    "    $$\\begin{align}\\int xe^xdx  \n",
+    "    &= xe^x - \\int e^x dx \\hspace{0.1cm} \\\\\n",
+    "    &= xe^x-e^x + C \\\\\n",
+    "    &= e^x (x-1) + C \\end{align}$$  \n",
+    "      \n",
+    "```\n",
+    "\n",
+    "```{tip}\n",
+    "For complicated integrals, computer algebra programs like Mathematica or Sympy can be helpful. An especially user-friendly option is the website WolframAlpha.\n",
+    "```\n",
+    "  \n",
+    "### Integration by partial fractions  \n",
+    "  \n",
+    "If the integral has the form \n",
+    "\n",
+    "$$\\int \\frac{P(x)}{Q(x)}dx$$ \n",
+    "\n",
+    "where the degree of $P(x)$ is smaller than the degree of $Q(x)$, and $Q(x)=a(x-x_1)(x-x_2)\\cdots(x-x_n)$, we can write\n",
+    "\n",
+    "$$\\frac{P(x)}{Q(x)} = \\frac{1}{a}\\left[\\frac{A_1}{x-x_1}+\\frac{A_2}{x-x_2}+\\cdots+\\frac{A_n}{x-x_n}\\right]$$ \n",
+    "\n",
+    "with $\\{A_n\\}$ to be determined.  \n",
+    "  \n",
+    "```{admonition} Example  \n",
+    "  \n",
+    "- $\\int \\frac{1}{x(x-1)}\\, dx$  \n",
+    "      \n",
+    "    First, note that $\\displaystyle\\frac{1}{x(x-1)} =  \\displaystyle\\frac{A}{x} + \\displaystyle\\frac{B}{(x-1)} \\implies 1 = A(x-1) + Bx \\implies 1 = x(A+B) - A$. By comparing the coeffients of the terms in $x$ and the terms independent of $x$, we obtain:  \n",
+    "      \n",
+    "    $$\\begin{cases}\n",
+    "    A + B &= 0\\\\\n",
+    "       -A &= 1\n",
+    "    \\end{cases} \\implies  \n",
+    "    \\begin{cases}\n",
+    "    A &= -1 \\\\\n",
+    "    B &= 1\\end{cases}$$  \n",
+    "      \n",
+    "    Thus:  \n",
+    "      \n",
+    "    $$\\frac{1}{x(x-1)} = -\\frac{1}{x} + \\frac{1}{x-1}$$\n",
+    "      \n",
+    "    Then we will have for the integral:  \n",
+    "      \n",
+    "    $$\\begin{align} \\int\\frac{1}{x(x-1)}dx  \n",
+    "    &= \\int\\left(-\\frac{1}{x}+\\frac{1}{x-1}\\right)dx \\\\\n",
+    "    &= -\\int \\frac{1}{x}dx + \\int\\frac{1}{x-1}dx \\\\\n",
+    "    &= - \\ln|x| + \\ln|x-1| + C \\\\\n",
+    "    &= \\ln \\left|\\frac{x-1}{x}\\right| + C\\end{align}$$  \n",
+    "      \n",
+    "```\n",
+    "    "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "a24228ef",
+   "metadata": {
+    "tags": [
+     "remove-cell"
+    ]
+   },
+   "source": [
+    "## Fourier coefficients  \n",
+    "  \n",
+    "For a given periodic function $f(t)$ (period $P$), we have shown\n",
+    "\n",
+    "$$f(t)=a+\\sum_{n=1}^N[\\beta_ncos(\\omega nt) + \\gamma_n sin(\\omega nt)]$$  \n",
+    "  \n",
+    "For well-behaved functions $N \\rightarrow \\infty$ gives the exact correspondence between the function $f(t)$ and the series.  \n",
+    "  \n",
+    "If $$\\omega_n=\\frac{2\\pi}{P} \\cdot n$$\n",
+    "\n",
+    "then:\n",
+    "\n",
+    "$$f(t) = a + \\sum_{n=1}^\\infty \\beta_n cos\\left[\\frac{2\\pi nt}{P}\\right] + \\gamma_n sin\\left[\\frac{2\\pi nt}{P}\\right]$$  \n",
+    "  \n",
+    "Multiplying both sides by $cos\\left[\\frac{2\\pi nt}{P}\\right]$ and integrating over one period $P$:\n",
+    "\n",
+    "$$\\beta_n = \\frac{2}{P} \\int_{P} f(t) cos\\left[\\frac{2\\pi nt}{P}\\right]dt$$\n",
+    "\n",
+    "since \n",
+    "\n",
+    "$$\\begin{align} \\int_{-\\pi}^\\pi cos(nx)cos(mx)\n",
+    "&=\\pi\\gamma_{mn} \\\\\n",
+    "\\int_{-\\pi}^\\pi sin(mx)cos(nx)&=0\\end{align}$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Tutorial 01"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "1.  An autocatalytic reaction uses its resulting product for the formation of a new product, as in the reaction  \n",
+    "  \n",
+    "    $$A + X \\rightarrow X$$  \n",
+    "    \n",
+    "    If we assume that this reaction occurs in a closed vessel, then the reaction rate is given by\n",
+    "    \n",
+    "    $$\n",
+    "       R(x) = kx(a - x)\n",
+    "    $$\n",
+    "    \n",
+    "    for $0 \\le x \\le a$, where $a$ is the initial concentration of $A$ and $x$ is the concentration of $X$.\n",
+    "    \n",
+    "    <br/>\n",
+    "    \n",
+    "    (a) Show that $R(x)$ is a polynomial and determine its degree.\n",
+    "    <br/>\n",
+    "    (b) Without using any graphic visualiser, graph $R(x)$ for $k = 2$ and $a = 6$. Find the value of $x$ at which the reaction rate is maximal.  \n",
+    "    \n",
+    "    <br/>\n",
+    "      \n",
+    "2. Find the maximum/minimum value of the following polynomials  \n",
+    "  \n",
+    "    <br/>\n",
+    "    \n",
+    "    (a) $p(x) = x^2 - 2x - 3$\n",
+    "    \n",
+    "    (b) $p(x) = -x^2 - 6x + 8$\n",
+    "    \n",
+    "    (*Hint:* Calculate the roots and consider the symmetry of the parabola)  \n",
+    "    \n",
+    "    <br/>\n",
+    "      \n",
+    "3. There is a function that is frequently used to describe the per capita growth rate of organisms when the rate depends on the concentration of some nutrient and becomes saturated for large enough nutrient concentrations. If we denote the concentration of the nutrient by $N$, then the per capita growth rate $r(N)$ is given by the *Monod growth function*\n",
+    "\n",
+    "    $$\n",
+    "        r(N)=\\frac{aN}{k+N},\\ \\ \\ N \\ge 0\n",
+    "    $$\n",
+    "    \n",
+    "    where $a$ and $k$ are positive constants.\n",
+    "    \n",
+    "    <br/>\n",
+    "    \n",
+    "    (a) Use Python or R to investigate the Monod growth function. Graph $r(N)$ for (i) $a = 5$ and $k = 1$, (ii) $a = 5$ and $k = 3$, and (iii) $a = 8$ and k = 1. Place all three graphs in the same coordinate system.\n",
+    "    \n",
+    "    (b) On the basis of the graphs in (a), describe in words what happens when you change $a$.\n",
+    "    \n",
+    "    (c) On the basis of the graphs in (a), describe in words what happens when you change $k$.  \n",
+    "    \n",
+    "    <br/>\n",
+    "      \n",
+    "      \n",
+    "4. The function \n",
+    "\n",
+    "    $$\n",
+    "        f(x) = \\frac{x^n}{b^n + x^n},\\ \\ \\ x \\ge 0\n",
+    "    $$ \n",
+    "    \n",
+    "    where $n$ is a positive integer and $b$ is a positive real number, is used in biochemistry to model reaction rates as a function of the concentration of some reactants    \n",
+    "    \n",
+    "    <br/>\n",
+    "    \n",
+    "    (a) Use Python or R to graph $f(x)$ for $n = 1, 2,\\ \\mbox{and}\\ 3$ in the same coordinate system when $b = 2$.\n",
+    "    \n",
+    "    (b) Where do the three graphs in (a) intersect?\n",
+    "    \n",
+    "    (c) What happens to $f(x)$ as $x$ gets larger?\n",
+    "\n",
+    "    (d) For an arbitrary positive value of $b$, show that $f (b) = 1/2$. On the basis of this demonstration and your answer in C. ,  explain why $b$ is called the half-saturation constant.  \n",
+    "    \n",
+    "    <br/>\n",
+    "  \n",
+    "  \n",
+    "5. The file [words_moby_dick.csv](https://imperialcollegelondon.box.com/s/940y521dq1owarx8pt4d4mu2d0f2cz85) contains data on the cumulative distribution of the number of times specific words occur in the text of the novel *Moby Dick*, by Herman Melville. Assume that this distribution is a power law of the form \n",
+    "\n",
+    "    $$\n",
+    "        y = x^D\n",
+    "    $$\n",
+    "    \n",
+    "    (a) Using the provided data, and R/Python, estimate the exponent of this power law. \n",
+    "    (*Hint:* If you plot this data in log-log scale, what should correspond to the exponent $D$?)\n",
+    "    \n",
+    "    (b) Can you think of a better way to estimate D?  \n",
+    "    \n",
+    "    <br/>\n",
+    "  \n",
+    "  \n",
+    "6. Assume that a population size at time $t$ is $N(t)$ and that\n",
+    "\n",
+    "    $$\n",
+    "        N(t) = 40 \\cdot 2^t,\\ \\ \\ t \\ge 0\n",
+    "    $$    \n",
+    "\n",
+    "    (a) Find the population size at time t = 0.\n",
+    "\n",
+    "    (b) Show that\n",
+    "\n",
+    "    $$\n",
+    "        N(t) = 40\\, \\mbox{e}^{t\\, ln 2},\\ \\ \\ t \\ge 0\n",
+    "    $$    \n",
+    "\n",
+    "    (c) How long will it take until the population size reaches 1000?\n",
+    "    \n",
+    "   (*Hint*: Find $t$ so that $N(t) = 1000$.)  \n",
+    "   \n",
+    "   <br/>\n",
+    "     \n",
+    "7. (*Adapted from Moss, 1980*) Hall (1964) investigated the change in population size of the zooplankton species *Daphnia galeata mendota* in Base Line Lake, Michigan. The population size $N(t)$ at time $t$ was modeled by the equation\n",
+    "\n",
+    "    $$ \n",
+    "        N(t) = N_0\\mbox{e}^{rt}\n",
+    "    $$\n",
+    "    \n",
+    "    where $N_0$ denotes the population size at time $0$. The constant $r$ is called the **intrinsic rate of growth**.\n",
+    "    \n",
+    "    <br/>\n",
+    "    \n",
+    "    (a) Plot $N(t)$ as a function of $t$ if $N_0 = 100$ and $r = 2$. Compare your graph against the graph of $N(t)$ when $N_0 = 100$ and $r = 3$. Which population grows faster?\n",
+    "    \n",
+    "    (b) The constant $r$ is an important quantity because it describes how quickly the population changes. Suppose that you determine the size of the population at the beginning and at the end of a period of length $1$, and you find that at the beginning there were $200$ individuals and after one unit of time there were $250$ individuals. Determine $r$. (*Hint*: Consider the ratio $N(t+1)/N(t)$.)  \n",
+    "    \n",
+    "    <br/>\n",
+    "      \n",
+    "8. Fish are indeterminate growers; that is, they grow throughout their lifetime. The growth of fish can be described by the von Bertalanffy function\n",
+    "\n",
+    "    $$\n",
+    "        L(x) = L_{\\infty}(1-\\textrm{e}^{-kx} )\n",
+    "    $$\n",
+    "    \n",
+    "    for $x \\ge 0$, where $L(x)$ is the length of the fish at age $x$ and $k$ and $L_{\\infty}$ are positive\n",
+    "    \n",
+    "    <br/>\n",
+    "    \n",
+    "    (a) Using Python or R, graph $L(x)$ for $L_{\\infty} = 20$, for (i) $k = 1$ and (ii) $k = 0.1$.\n",
+    "    \n",
+    "    (b) For $k = 1$, find $x$ so that the length is 90\\% of $L_{\\infty}$. Repeat for 99\\% of $L_{\\infty}$. Can the fish ever attain length $L_{\\infty}$? Interpret the meaning of $L_{\\infty}$.\n",
+    "    \n",
+    "    (c) Compare the graphs obtained in A. Which growth curve reaches 90\\% of $L_{\\infty}$ faster? Can you explain what happens to the curve of $L(x)$ when you vary $k$ (for fixed $L_{\\infty}$)?  \n",
+    "    \n",
+    "    <br/>\n",
+    "    \n",
+    "9. For the following pairs of functions, plot the ratio (quotient) between the two using R or Python. Based on the behaviour of the ratio when $x \\rightarrow \\infty$ (very large values of $x$), how does each of these functions compare to the other in the velocity that they grow?\n",
+    "\n",
+    "    <br/>  \n",
+    "    \n",
+    "    (a) $ f(x) = e^x \\quad \\text{ and } \\quad g(x) = x^2 $\n",
+    "\n",
+    "    (b) $ f(x) = e^x \\quad \\text{ and } \\quad g(x) = x^{20} $\n",
+    "    \n",
+    "    (c) $ f(x) = x \\quad \\text{ and } \\quad g(x) = \\log(x) $  \n",
+    "    \n",
+    "    <br/>\n",
+    "      \n",
+    "10. A community measure that takes both species abundance and species richness into account is the Shannon diversity index $H$. To calculate $H$, the proportion $p_i$ of species $i$ in the community is used. Assume that the community consists of $S$ species. Then\n",
+    "\n",
+    "    $$\n",
+    "        H = -\\sum_{i=1}^S p_i\\, \\textrm{ln}\\, p_i = -( p_1 \\textrm{ln}\\, p_1 + p_2 \\textrm{ln}\\, p_2 + \\cdots + p_S \\textrm{ln}\\, p_S )\n",
+    "    $$\n",
+    "    \n",
+    "    <br/>\n",
+    "    \n",
+    "    (a) Assume that $S = 5$ and that all species are equally abundant; that is, $p_1 = p_2 = \\cdots = p_5$. Compute $H$.\n",
+    "    \n",
+    "    (b) Assume that $S = 10$ and that all species are equally abundant; that is, $p_1 = p_2 = \\cdots = p_{10}$. Compute $H$.\n",
+    "    \n",
+    "    (c) A measure of equitability (or evenness) of the species distribution can be measured by dividing the diversity index $H$ by $\\textrm{ln}\\, S$. Compute $H/\\textrm{ln}\\, S$ for $S = 5$ and $S = 10$."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Tutorial 05\n",
+    "\n",
+    "\n",
+    "1. Solve the following systems of linear equations\n",
+    "    <br/>\n",
+    "    \n",
+    "    $$\n",
+    "        \\mbox{(a)}  \\ \n",
+    "        \\begin{cases}\n",
+    "            5x - y + 2z = 6 \\\\\n",
+    "            x + 2y - z = -1 \\\\\n",
+    "            3x + 2y - 2z = 1\\\\\n",
+    "        \\end{cases}\n",
+    "    $$\n",
+    "    \n",
+    "    <br/>\n",
+    "    \n",
+    "    $$\n",
+    "        \\mbox{(b)}  \\ \n",
+    "        \\begin{cases}\n",
+    "            2x - y + 3z = 3 \\\\\n",
+    "            2x + y + 4z = 4 \\\\\n",
+    "            2x - 3y + 2z = 2\\\\\n",
+    "        \\end{cases}\n",
+    "    $$\n",
+    "    \n",
+    "    <br/>\n",
+    "    \n",
+    "2. Three different species of insects are reared together in a laboratory cage. They are supplied with two different types of food each day. Each individual of species 1 consumes 3 units of food $A$ and 5 units of food $B$, each individual of species 2 consumes 2 units of food $A$ and 3 units of food $B$, and individual of species 3 consumes 1 unit of food $A$ and 2 units of food $B$. Each day, 500 units of food $A$ and 900 units of food $B$ are supplied. How many individuals of each species can be reared together? Is there more than one solution? What happens if we add 550 units of a third type of food, called $C$, and each individual of species 1 consumes 2 units of food $C$, each individual of species 2 consumes 4 units of food $C$, and each individual of species 3 consumes 1 unit of food $C$?\n",
+    "\n",
+    "    <br/>\n",
+    "\n",
+    "3. Networks are abstractions of real systems used to identify and study patterns of connectivity between the individual units that compose those systems. Networks (often referred to as *graphs*) are mathematically defined as a set of *nodes*, or vertices, (representing the individual units) and a set of *links*, or connections, between these nodes, which account for the interaction patterns within the studied systems. In Ecology, graphs have been widely used in studies investigating food-webs, plant-pollinator interactions, and disease transmission in populations.  \n",
+    "    \n",
+    "    <br />\n",
+    "  \n",
+    "    If the nodes of a network of size $N$ are indexed with positive integers such as $i = 1, 2, \\ldots, N$, an useful method for representing the structure of a network is to construct a matrix $A$ for which the elements $a_{ij}$ are given by:  \n",
+    "      \n",
+    "    $$a_{ij} = \\begin{cases}1,\\ \\mbox{if nodes}\\ i \\ \\mbox{and}\\ j \\ \\mbox{are linked} \\\\ 0,\\ \\mbox{otherwise}  \\end{cases}$$  \n",
+    "    \n",
+    "    Matrix $A$ is called the *adjacency matrix* of the network. One example of a network with $6$ nodes is shown below.\n",
+    "\n",
+    "    <br/>  \n",
+    "    \n",
+    "    ```{image} graph_example.png  \n",
+    "    ```  \n",
+    "          \n",
+    "    (a) Based on the graphical representation of this network, write down its adjacency matrix.  \n",
+    "        \n",
+    "    <br/>\n",
+    "    \n",
+    "    (b) By the definition of the adjacency matrix, the sum of the matrix elements on row $i$, $k_i = \\sum_{j=1}^N a_{ij}$, corresponds to the number of links of node $i$. The quantity $k_i$ is called the *degree* of node $i$. Calculate the corresponding degrees for each node on the previous network.\n",
+    "        \n",
+    "    <br/>\n",
+    "    \n",
+    "    (c) The Laplacian matrix is a mathematical object that can be used to find many useful properties of a network. By definition, $L$ is constructed as  \n",
+    "    \n",
+    "    <br/>  \n",
+    "    \n",
+    "    $$ L = D - A $$  \n",
+    "    \n",
+    "    <br/>  \n",
+    "    \n",
+    "    Where $A$ is the adjacency matrix, and $D$ is the degree matrix. The degree matrix $D$ is a diagonal matrix which contains information about the degree of each node. With your answer for itens (a) and (b), calculate $L$.\n",
+    "        \n",
+    "    <br/>\n",
+    "    \n",
+    "    (d) Use R/Python to numerically calculate the eigenvalues of the Laplacian matrix.\n",
+    "    \n",
+    "    <br/>\n",
+    "   \n",
+    "4. Write each system in matrix form\n",
+    "   <br/>\n",
+    "    \n",
+    "    $$\n",
+    "        \\mbox{(a)}  \\ \n",
+    "        \\begin{cases}\n",
+    "            2x_2-x_1 = x_3 \\\\\n",
+    "            4x_1 + x_3 = 7x_2 \\\\\n",
+    "            x_2 - x_1 = x_3\\\\\n",
+    "        \\end{cases}\n",
+    "    $$\n",
+    "    \n",
+    "    <br/>\n",
+    "    \n",
+    "    $$\n",
+    "        \\mbox{(b)}  \\ \n",
+    "        \\begin{cases}\n",
+    "            2x_1 - 3x_2 = 4 \\\\\n",
+    "            -x_1 + x_2 = 3 \\\\\n",
+    "            3x_1 = 4\\\\\n",
+    "        \\end{cases}\n",
+    "    $$\n",
+    "    \n",
+    "    <br/>\n",
+    "    \n",
+    "5. Suppose that \n",
+    "\n",
+    "    <br/>\n",
+    "    \n",
+    "    $$A = \\begin{pmatrix}\n",
+    "            2 & 4 \\\\\n",
+    "            3 & 6 \n",
+    "            \\end{pmatrix}\n",
+    "        \\ \\ \\ \\text{,}\\ \\ \\\n",
+    "        X = \\begin{pmatrix}\n",
+    "            x \\\\\n",
+    "            y \n",
+    "            \\end{pmatrix} \\ \\ \\ \\text{and}\\ \\ \\\n",
+    "            B = \\begin{pmatrix}\n",
+    "            b_1 \\\\\n",
+    "            b_2 \\\\\n",
+    "            \\end{pmatrix}$$\n",
+    "\n",
+    "    (a) Write $AX = B$ as a system of linear equation.\n",
+    "        \n",
+    "    <br/>\n",
+    "    \n",
+    "    (b) Show that if \n",
+    "   \n",
+    "    $$\n",
+    "        B = \\begin{pmatrix}\n",
+    "            3 \\\\\n",
+    "            \\frac92 \\\\\n",
+    "            \\end{pmatrix}\n",
+    "    $$\n",
+    "    \n",
+    "    then $AX = B$ has infinitely many solutions. Graph the two straight lines associated with the corresponding system of linear equations, and explain why the system has infinitely many solutions.\n",
+    "        \n",
+    "    <br/>\n",
+    "    \n",
+    "    (c) Find a column vector\n",
+    "    \n",
+    "    <br/>\n",
+    "    \n",
+    "    $$B = \\begin{pmatrix}\n",
+    "            b_1 \\\\\n",
+    "            b_2 \\\\\n",
+    "            \\end{pmatrix}$$\n",
+    "\n",
+    "    so that $AX = B$ has no solutions.\n",
+    "    \n",
+    "    <br/>\n",
+    "    \n",
+    "6. Assume that a population is divided into three age classes and that 80\\% of the individuals age 0 and 10\\% of the individuals age 1 survive until the end of the next breeding season. Assume further that individuals age 1 have an average of 1.6 offspring and individuals age 2 have an average of 3.9 offspring. If, at time 0, the population consists of 1000 individuals age 0, 100 individuals age 1, and 20 individuals age 2, find the projection matrix and the age distribution at time 3.\n",
+    "\n",
+    "    <br/>\n",
+    "\n",
+    "7. Consider the following projection matrix representing a population with five age classes:\n",
+    "\n",
+    "    <br/>  \n",
+    "    \n",
+    "    $$\n",
+    "        P = \\begin{pmatrix}\n",
+    "        0 & 5 & 3 & 2 & 1 \\\\\n",
+    "        0.9 & 0 & 0 & 0 & 0 \\\\\n",
+    "        0 & 0.3 & 0 & 0 & 0 \\\\\n",
+    "        0 & 0 & 0.1 & 0 & 0 \\\\\n",
+    "        0 & 0 & 0 & 0.05 & 0 \\\\\n",
+    "    \\end{pmatrix}\n",
+    "    $$\n",
+    "    \n",
+    "    <br/>\n",
+    "    \n",
+    "    (a) Making use of Python or R code, begin with a population distributed as $(0,0,0,0,10)$ individuals at time $t=0$, project the population at time $t=20$, and plot the total number over time. Plot the logarithm of the total population over time.\n",
+    "        \n",
+    "    <br/>\n",
+    "    \n",
+    "    (b) Repeat the above but begin with $(80,16,5,1,1)$ individuals. How do these results compare to the results in (a) ?\n",
+    "    \n",
+    "    <br/>\n",
+    "    \n",
+    "8. In the problems below, find the eigenvalues $\\lambda_1$ and $\\lambda_2$ and corresponding eigenvectors $\\mathbf{v_1}$ and $\\mathbf{v_2}$ for each matrix $B$. Determine the equations of the lines through the origin in the direction of the eigenvectors $\\mathbf{v_1}$ and $\\mathbf{v_2}$, and graph the lines together with the eigenvectors $\\mathbf{v_1}$ and $\\mathbf{v_2}$ and the vectors $A\\mathbf{v_1}$ and $A\\mathbf{v_2}$.\n",
+    "\n",
+    "    <br/>\n",
+    "    \n",
+    "    $$\n",
+    "    \\text{(a)} \\ \\ B = \\begin{pmatrix}\n",
+    "    2 & 3\\\\\n",
+    "    0 & -1\n",
+    "    \\end{pmatrix}\n",
+    "    $$\n",
+    "\n",
+    "    <br/>\n",
+    "\n",
+    "    $$\n",
+    "    \\text{(b)} \\ \\ B = \\begin{pmatrix}\n",
+    "    3 & 6\\\\\n",
+    "    -1 & -4\n",
+    "    \\end{pmatrix}  \n",
+    "    $$\n",
+    "    \n",
+    "    <br/>\n",
+    "    \n",
+    "9. Suppse that \n",
+    "\n",
+    "    <br/>\n",
+    "    \n",
+    "    $$\n",
+    "    P = \\begin{pmatrix}\n",
+    "    0.5 & 2.3\\\\\n",
+    "    a & 0\n",
+    "    \\end{pmatrix}\n",
+    "    $$\n",
+    "    \n",
+    "    <br/>\n",
+    "    \n",
+    "    is the projection matrix of a population with two age classes. For which values of $a$ does this population grow?\n",
+    "    \n",
+    "    <br/>\n",
+    "    \n",
+    "10. Suppose that \n",
+    "\n",
+    "    <br/>\n",
+    "    \n",
+    "    $$\n",
+    "    P = \\begin{pmatrix}\n",
+    "    0.5 & 2.0\\\\\n",
+    "    0.1 & 0\n",
+    "    \\end{pmatrix}\n",
+    "    $$\n",
+    "    \n",
+    "    <br/>\n",
+    "    \n",
+    "    is the projection matrix of a population with two age classes.\n",
+    "        \n",
+    "    <br/>\n",
+    "    \n",
+    "    (a) If you were to manage this population, would you need to be concerned about its long-term survival?\n",
+    "    \n",
+    "    <br/>\n",
+    "    \n",
+    "    (b) Suppose that you can improve either the fecundity or the survival of the zero-year-olds, but due to physiological and environmental constraints, the fecundity of zero-year-olds will not exceed 1.5 and the survival of zero-year-olds will not exceed 0.4. Investigate how the growth rate of the population is affected by changing either the survival or the fecundity of zero-year-olds, or both. What would be the maximum achievable growth rate?\n",
+    "    \n",
+    "    <br/>\n",
+    "    \n",
+    "    (c) In real situations, what other factors might you need to consider when you decide on management strategies?"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Tutorial 06"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "1. **Island Model** Assume that a species lives in a habitat that consists of many islands close to a mainland. The species occupies both the mainland and the islands, but, although it is present on the mainland at all times, it frequently goes extinct on the islands. Islands can be recolonized by migrants from the mainland. The following model keeps track of the fraction of islands occupied: Denote the fraction of islands occupied at time $t$ by $p(t)$. Assume that each island experiences a constant risk of extinction and that vacant islands (the fraction $1-p$) are colonized from the mainland at a constant rate. Then\n",
+    "\n",
+    "    $$\n",
+    "        \\frac{dp}{dt}=c(1-p)-ep\n",
+    "    $$\n",
+    "\n",
+    "     where $c$ and $e$ are positive constants.\n",
+    "\n",
+    "    <br/>\n",
+    "    \n",
+    "    (a) The gain from colonization is $f (p) = c(1 - p)$ and the loss from extinction is $g(p) = ep$. Graph (without graph visualisers) $f(p)$ and $g(p)$ for $0 \\le p \\le 1$ in the same coordinate system. Explain why the two graphs intersect whenever $e$ and $c$ are both positive. Compute the point of intersection and interpret its biological meaning.\n",
+    "    \n",
+    "     <br/>\n",
+    "     \n",
+    "    (b) The parameter $c$ measures how quickly a vacant island becomes colonized from the mainland. The closer the islands, the larger is the value of $c$. Use your graph in (a) to explain what happens to the point of intersection of the two lines as c increases. Interpret your result in biological terms.  \n",
+    "      \n",
+    "    <br />  \n",
+    "      \n",
+    "2. **Biotic Diversity** (Adapted from Valentine, 1985.) Walker and Valentine (1984) suggested a model for species diversity which assumes that species extinction rates are independent of diversity but speciation rates are regulated by competition. Denoting the number of species at time $t$ by $S(t)$, the speciation rate by $b$, and the extinction rate by $a$, they used the model\n",
+    "\n",
+    "    $$\n",
+    "        \\frac{dS}{dt} = S\\left[b\\left(1-\\frac{S}{K}\\right)-a \\right]\n",
+    "    $$\n",
+    "      \n",
+    "    where $K$ denotes the number of \"niches\", or potential places for species in the ecosystem.\n",
+    "    \n",
+    "    <br/>\n",
+    "    \n",
+    "    (a) Find possible equilibria ($dS/dt = 0$) under the condition $a < b$.\n",
+    "    \n",
+    "    <br/>\n",
+    "    \n",
+    "    (b) Use your result in (a) to explain the following statement by Valentine (1985):\n",
+    "    \n",
+    "     ```{admonition} Quotation \n",
+    "     \"In this situation, ecosystems are never 'full', with all potential niches occupied by species so long as the extinction rate is above zero.\"\n",
+    "     ```\n",
+    "    \n",
+    "    <br/>\n",
+    "    \n",
+    "    (c) What happens when $a \\ge b$?  \n",
+    "      \n",
+    "    <br />  \n",
+    "      \n",
+    "3. Find a point on the curve\n",
+    "    \n",
+    "    $$y = 1 - 3x^3$$\n",
+    "    \n",
+    "     whose tangent line is parallel to the line $y = -x$. Is there more than one such point? If so, find all other points with this property.  \n",
+    "       \n",
+    "    <br />  \n",
+    "      \n",
+    "4. Find a point on the curve\n",
+    "\n",
+    "    $$y = 2x^3 - 4x + 1$$\n",
+    "    \n",
+    "    whose tangent line is parallel to the line $y-2x = 1$. Is there more than one such point? If so, find all other points with this property.  \n",
+    "      \n",
+    "    <br />  \n",
+    "      \n",
+    "5. In the following, use the product and quotient rules to find the derivative with respect to the independent variable.\n",
+    "\n",
+    "    <br/>\n",
+    "    \n",
+    "    (a) $f(x) = (2x^3 - 1)(3 + 2x^2)$\n",
+    "    \n",
+    "    <br/>\n",
+    "    \n",
+    "    (b) $h(s) = (4 - 3s^2 + 4s^3)^2$\n",
+    "    \n",
+    "    <br/>\n",
+    "    \n",
+    "    (c) $f(x) =\\sqrt{3x}(x^2 - 1)$\n",
+    "    \n",
+    "    <br/>\n",
+    "    \n",
+    "    (d) $g(t) = 4(3t^2 - 1)(2t + 1)$\n",
+    "    \n",
+    "    <br/>\n",
+    "    \n",
+    "    (e) $f(x) = 3(x^2 + 2)(4x^2 - 5x^4) - 3$\n",
+    "    \n",
+    "    <br/>\n",
+    "    \n",
+    "    (f) $g(x) = \\displaystyle\\frac{1 - 4x^3}{1 - x}$\n",
+    "    \n",
+    "    <br/>\n",
+    "    \n",
+    "    (g) $h(x) = \\displaystyle\\frac{1 + 2x^2 - 4x^4}{3x^3 - 5x^5}$\n",
+    "    \n",
+    "    <br/>\n",
+    "    \n",
+    "    (h) $f(t) =\\displaystyle\\frac{3 - t^2}{(t + 1)^2}$\n",
+    "    \n",
+    "    <br/>\n",
+    "    \n",
+    "    (i) $h(s) = \\displaystyle\\frac{2s^3 - 4s^2 + 5s - 7}{(s^2 - 3)^2}$\n",
+    "    \n",
+    "    <br/>\n",
+    "    \n",
+    "    (j) $g(s) = \\displaystyle\\frac{s^{1/7} - s^{2/7}}{s^{3/7} + s^{4/7}}$\n",
+    "      \n",
+    "    <br />  \n",
+    "      \n",
+    "6. **Logistic Growth**\n",
+    "\n",
+    "    <br/>\n",
+    "    \n",
+    "    (a) Find the derivative of the logistic growth curve  \n",
+    "      \n",
+    "    $$N(t)=\\frac{K}{1+\\left(\\frac{K}{N(0)}-1\\right)e^{-rt}}$$  \n",
+    "    \n",
+    "    where $r$ and $K$ are positive constants and $N(0)$ is the population size at time $0$.\n",
+    "        \n",
+    "    <br/>\n",
+    "            \n",
+    "    (b) Show that $N(t)$ satisfies the equation  \n",
+    "      \n",
+    "    $$\\frac{dN}{dt}=rN\\left(1-\\frac{N}{K}\\right)$$  \n",
+    "      \n",
+    "    [*Hint:* Use the function $N(t)$ given in (a) for the right-hand side, and simplify until you obtain the derivative of N(t) that you computed in A.]\n",
+    "        \n",
+    "    <br/>\n",
+    "        \n",
+    "    (c) What happens to the *per-capita* rate of growth $\\frac{1}{N}\\frac{dN}{dt}$ as you increase N? What biological principle and what type of interaction is your model describing?  \n",
+    "      \n",
+    "    <br />  \n",
+    "      \n",
+    "7. In the following, find the derivative with respect to the independent variable (*Hint*: use the chain rule).\n",
+    "    \n",
+    "    <br/>\n",
+    "    \n",
+    "    (a) $f(x) = \\mbox{sin}(e^{2x} + x)$\n",
+    "    \n",
+    "    <br/>\n",
+    "    \n",
+    "    (b) $h(x) = \\mbox{exp}[x^2 - 2 \\mbox{cos}\\, x]$\n",
+    "    \n",
+    "    <br/>\n",
+    "    \n",
+    "    (c) $f(x) =\\displaystyle\\frac{x-2^{−x}}\n",
+    "{1+x2^{−x}}$\n",
+    "    \n",
+    "    <br/>\n",
+    "    \n",
+    "    (d) $g(t) = \\mbox{log}\\left(\\displaystyle\\frac{1 - t}{1 + 2t}\\right)$\n",
+    "    \n",
+    "    <br/>\n",
+    "    \n",
+    "    (e) $f(t) = \\mbox{sin}(\\mbox{ln}(3t))$\n",
+    "    \n",
+    "    <br/>\n",
+    "    \n",
+    "    (f) $g(x) = \\mbox{ln}(\\mbox{cos}(1 - x))$\n",
+    "    \n",
+    "    <br/>\n",
+    "    \n",
+    "    (g) $h(x) = \\displaystyle\\frac{\\sqrt{x^2 - 1}}{2+\\sqrt{x^2+1}}$ \n",
+    "    \n",
+    "    <br/>\n",
+    "    \n",
+    "    (h) $f(t) =\\sqrt[3]{t^2+\\sqrt{1-t}}$\n",
+    "    \n",
+    "    <br/>\n",
+    "      \n",
+    "8. The functional responses of some predators are sigmoidal; that is, the number of prey attacked per predator as a function of prey density has a sigmoidal shape. If we denote the prey density by $N$, the predator density by $P$, the time available for searching for prey by $T$, and the handling time of each prey item per predator by $T_h$, then the number of prey encounters per predator as a function of $N$, $T$, and $T_h$ can be expressed as\n",
+    "\n",
+    "    $$\n",
+    "        f (N, T, T_h) = \\frac{b^2N^2T}{1 + cN + bT_hN^2}\n",
+    "    $$\n",
+    "    \n",
+    "    where $b$ and $c$ are positive constants.\n",
+    "    \n",
+    "    <br/>\n",
+    "    \n",
+    "    (a) Calculate the first-order partial derivative of $f(N, T, T_h)$ with respect to the prey density $N$. If $N$ increases, what happens to $f(N, T, T_h)$?\n",
+    "     \n",
+    "    <br/>\n",
+    "    \n",
+    "    (b) Calculate the first-order partial derivative of $f(N, T, T_h)$ with respect to the the time $T$ available for search. If $T$ increases, what happens to $f(N, T, T_h)$?\n",
+    "     \n",
+    "    <br/>\n",
+    "    \n",
+    "    (c) Calculate the first-order partial derivative of $f(N, T, T_h)$ with respect to the handling time $T_h$. If $T_h$ increases, what happens to $f(N, T, T_h)$?\n",
+    "    \n",
+    "    [*Hint:* The function $f(N, T, T_h)$ will be an increasing (decreasing) function of the independent variable if the first-order partial derivative of $f$ with respect to this variable is positive (negative).]"
+   ]
+  }
+ ],
+ "metadata": {
+  "celltoolbar": "Tags",
+  "kernelspec": {
+   "display_name": "Python 3 (ipykernel)",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.11.5"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 2
+}
diff --git a/_sources/notebooks/MetabolicBasis.ipynb b/_sources/notebooks/MetabolicBasis.ipynb
new file mode 100644
index 00000000..0faffbf5
--- /dev/null
+++ b/_sources/notebooks/MetabolicBasis.ipynb
@@ -0,0 +1,160 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# The Metabolic Basis of Everything"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "  * Scaling laws - space and time\n",
+    "  * Biochemical kinetics and thermodynamics\n",
+    "  * Implications for ecological and evolutionary dynamics (?)\n",
+    "  * Applications/Examples "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "\n",
+    "## Introduction\n",
+    "Metabolism is central to the understanding of ecological and evolutionary processes. As Ludwig Boltzmann noted in 1886:\n",
+    "\n",
+    "> \"The struggle for existence of living beings is not for the fundamental constituents of food, but for the possession of the free energy obtained, chiefly by means of the green plant, from the transfer of radiant energy from the hot sun to the cold earth.\"\n",
+    "\n",
+    "The metabolic theory of ecology (Brown et al., 2004) provides a framework for understanding how metabolic processes scale across levels of biological organization and influence ecological dynamics.\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "\n",
+    "## Energy and Metabolic Rate\n",
+    "Energy is a measurable property of an object that determines its ability to perform work. Life's purpose is to utilize energy for propagating itself through self-replication. Plants and other autotrophs harness photons via photosynthesis, converting light energy into chemical energy stored in glucose. Heterotrophs, such as animals, derive energy from breaking chemical bonds in organic molecules. ATP (adenosine triphosphate) is a critical energy carrier in cells, facilitating various biochemical processes.\n",
+    "\n",
+    "Metabolism, the set of chemical reactions within an organism, determines metabolic rate, which is the rate of energy use, often measured in J/s, kcal/day, or Watts. Organisms must balance energy consumption and expenditure:\n",
+    "\n",
+    "$$\n",
+    "\\text{Energy Consumed} = \\text{Energy Used (Maintenance + Growth + Movement)}.\n",
+    "$$\n",
+    "\n",
+    "Metabolic rates influence key life-history traits, including movement rates, development speed, lifespan, and reproductive output (Dell et al., 2011).\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "\n",
+    "## Importance of Size\n",
+    "Metabolic rate (\\(B\\)) increases with body size (\\(M\\)) according to the allometric scaling law (Kleiber, 1947):\n",
+    "\n",
+    "$$\n",
+    "\\log_{10}(B) = \\log_{10}(B_0) + b \\log_{10}(M),\n",
+    "$$\n",
+    "\n",
+    "or equivalently:\n",
+    "\n",
+    "$$\n",
+    "B = B_0 M^b,\n",
+    "$$\n",
+    "\n",
+    "where \\(b\\) is typically close to 3/4 for most organisms. This implies that larger organisms have higher absolute metabolic rates but lower mass-specific rates (\\(B/M\\)):\n",
+    "\n",
+    "$$\n",
+    "\\frac{B}{M} = B_0 M^{b-1}.\n",
+    "$$\n",
+    "\n",
+    "For example, a mouse (0.1 kg) needs approximately 12.3 kcal/day, while its mass-specific rate is 123 kcal/(kg\\(\\cdot\\)day). This reduction in per-mass energy use with size is hypothesized to reduce overheating risk and improve resource allocation efficiency (West et al., 2005).\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "\n",
+    "### Ecological Implications of Size\n",
+    "- Larger organisms have lower mass-specific metabolic rates, enabling longer lifespans and greater reproductive investment (Savage et al., 2004).\n",
+    "- Population growth rate scales negatively with body size:\n",
+    "\n",
+    "$$\n",
+    "    r_{\\text{max}} = r_0 M^{-0.25}.\n",
+    "$$\n",
+    "\n",
+    "\n",
+    "- Population density decreases with size (Damuth's law).\n",
+    "- Larger carnivores are rarer than herbivores due to energy loss across trophic levels (Colinvaux, 1978).\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "\n",
+    "## Importance of Temperature\n",
+    "Temperature profoundly impacts metabolism, as reaction rates increase with temperature. The Boltzmann-Arrhenius equation describes the exponential increase in reaction rates (Dell et al., 2011):\n",
+    "\n",
+    "$$\n",
+    "    k = A e^{-\\frac{E_a}{k_B T}},\n",
+    "$$\n",
+    "\n",
+    "where \\(k\\) is the reaction rate, \\(A\\) is a constant, \\(E_a\\) is the activation energy, \\(k_B\\) is the Boltzmann constant, and \\(T\\) is temperature in Kelvin. This relationship explains the characteristic thermal performance curve observed in biological processes, where rates increase with temperature to an optimum before declining due to enzyme denaturation or other stress factors (Johnson et al., 1974).\n",
+    "\n",
+    "- Ectotherms, such as reptiles and insects, rely on external temperature to regulate their metabolic processes.\n",
+    "- Endotherms, such as mammals and birds, generate internal heat to maintain a stable body temperature, but this comes at a significant energetic cost (Bar-On et al., 2018).\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "\n",
+    "## Summary\n",
+    "- Metabolism drives growth rates and influences population dynamics through its scaling with size and temperature (Brown et al., 2004).\n",
+    "- Larger organisms exhibit slower per-mass metabolic rates but greater absolute energy use (Savage et al., 2004).\n",
+    "- Temperature affects metabolic rates exponentially, shaping life-history strategies and population dynamics (Dell et al., 2011).\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "\n",
+    "## Discussion Questions\n",
+    "\n",
+    "1. Given the way resting metabolic rate scales with body size, what must consumption rate scale like?\n",
+    "2. Why did we focus on resting (or “basal”) metabolic rate?\n",
+    "3. How much energy would a resting rabbit (∼ 1 kg), a human (∼ 70 kg, and an Asian elephant (∼ 4000 kg) need for metabolic balance, at the individual level? What about per unit mass?\n",
+    "4. How does the value you calculated in the previous question compare with the recommended calorie intake for humans? Is it lower or higher? Why?\n",
+    "5. What are the optimal temperatures for the enzyme underlying bioluminescence in the two bacteria in the given example? Why might they have different thermal optima?\n"
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3 (ipykernel)",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.10.12"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 2
+}
diff --git a/_sources/notebooks/ModelFitting-NLLS.ipynb b/_sources/notebooks/ModelFitting-NLLS.ipynb
deleted file mode 100644
index 2477bb18..00000000
--- a/_sources/notebooks/ModelFitting-NLLS.ipynb
+++ /dev/null
@@ -1,4021 +0,0 @@
-{
- "cells": [
-  {
-   "cell_type": "code",
-   "execution_count": 1,
-   "metadata": {
-    "tags": [
-     "remove-cell"
-    ]
-   },
-   "outputs": [],
-   "source": [
-    "library(repr) ; options(repr.plot.width = 4, repr.plot.height = 4) # Change plot sizes (in cm)"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "# Model Fitting using Non-linear Least-squares"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "## Introduction\n",
-    "\n",
-    "In this Chapter, you will learn to fit non-linear mathematical models to data using Non-Linear Least Squares (NLLS).\n",
-    "\n",
-    "Specifically, you will learn to\n",
-    "\n",
-    "* Visualize the data and the mathematical model you want to fit to them\n",
-    "* Fit a non-linear model\n",
-    "* Assess the quality of the fit, and whether the model is appropriate for your data\n",
-    "* Compare and select between competing models\n",
-    "\n",
-    "We will work through various examples. These assume that you have at least a conceptual understanding of what Linear vs Non-linear models are, how they are fitted to data, and how the fits can be assessed statistically. You may want to see the [Linear Models lecture](https://github.com/mhasoba/TheMulQuaBio/tree/master/content/lectures/LinearModels) (you can also watch the [video](https://drive.google.com/drive/folders/12Sj56wHX6vcAnp9GE9qQ1gIXbn7QRHU2?usp=sharing)), and the [NLLS Lecture](https://github.com/mhasoba/TheMulQuaBio/blob/master/content/lectures/NLLS) lecture first (you can also watch the [video](https://web.microsoftstream.com/video/529b2560-4f45-4675-ae8b-28487d63ac41)). \n",
-    "\n",
-    "You may also (optionally) want to see the  [lecture on model fitting in Ecology and Evolution in general](https://github.com/mhasoba/TheMulQuaBio/tree/master/content/lectures/ModelFitting). "
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "We will use R. For starters, clear all variables and graphic devices and load necessary packages:"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 2,
-   "metadata": {},
-   "outputs": [],
-   "source": [
-    "rm(list = ls())\n",
-    "graphics.off()"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "## Traits data as an example \n",
-    "\n",
-    "Our first set of examples will focus on traits. \n",
-    "\n",
-    "A trait is any measurable feature of an individual organism. This includes physical traits (e.g., morphology, body mass, wing length), performance traits (e.g., biochemical kinetics, respiration rate, body velocity, fecundity), and behavioral traits (e.g., feeding preference, foraging strategy, mate choice). All natural populations show variation in traits across individuals. A trait is functional when it directly (e.g., mortality rate) or indirectly (e.g., somatic development or growth rate) determines individual fitness. Therefore, variation in (functional) traits can generate variation in the rate of increase and persistence of populations. When measured in the context of life cycles, without considering interactions with other organisms (e.g., predators or prey of the focal population), functional traits are typically called life history traits (such as mortality rate and fecundity). Other traits determine interactions both within the focal population (e.g., intra-specific interference or mating frequency) and between the focal population/species and others, including the species which may act as resources (prey, for example). Thus both life history and interaction traits determine population fitness and therefore abundance, which ultimately influences dynamics and functioning of the wider ecosystem, such as carbon fixation rate or disease transmission rate. "
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "## Biochemical Kinetics\n",
-    "\n",
-    "The properties of an organism's metabolic pathways, and the underlying (enzyme-mediated) biochemical reactions (kinetics) are arguably its most fundamental \"traits\", because these drive all \"performance\" traits, from photosynthesis and respiration, to movement and growth rate.\n",
-    "\n",
-    "The [Michaelis-Menten](https://en.wikipedia.org/wiki/Michaelis%E2%80%93Menten_kinetics) model is widely used to quantify reaction kinetics data and estimate key biochemical parameters. This model relates biochemical reaction rate ($V$) (rate of formation of the product of the reaction), to concentration of the substrate ($S$):\n",
-    "\n",
-    "$$ \n",
-    "V = \\frac{V_{\\max} S}{K_M + S} \n",
-    "$$(eq:M-M)\n",
-    "\n",
-    "Here,\n",
-    "\n",
-    "* $V_{\\max}$ is the maximum rate that can be achieved in the reaction system, which happens at saturating substrate concentration (as $S$ gets really large), and \n",
-    "* $K_M$ is the Michaelis or half-saturation constant, defined as the substrate concentration at which the reaction rate is half of $V_{\\max }$. This parameter controls the overall shape of the curve, i.e., whether $V$ approaches $V_{\\max}$ slowly or rapidly. In enzyme catalyzed reactions, it measures how loosely the substrate binds the enzyme: large $K_M$ indicates loose binding of enzyme to substrate, small $K_M$ indicates tight binding (it has units of the substrate concentration, $S$).\n",
-    "\n",
-    "Biochemical reactions involving a single substrate are often well fitted by the Michaelis-Menten kinetics.\n",
-    "\n",
-    "<img src=\"./graphics/MM.png\" alt=\"Michaelis-Menten model\" width=\"400px\">\n",
-    "\n",
-    "<small><center>The Michaelis-Menten model.</center></small>\n",
-    "\n",
-    "Let's fit the Michaelis-Menten model to some data. \n",
-    "\n",
-    "\n",
-    "### Generating data\n",
-    "\n",
-    "Instead of using real experimental data, we will actually *generate* some \"data\" because that way we know exactly what the errors in the data are. You can also import and use your own dataset for the fitting steps further below.\n",
-    "\n",
-    "We can generate some data as follows.\n",
-    "\n",
-    "First, generate a sequence of substrate concentrations from 1 to 50 in jumps of 5, using `seq()` (look up the documentation for `seq()`)."
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 3,
-   "metadata": {
-    "scrolled": true
-   },
-   "outputs": [
-    {
-     "data": {
-      "text/html": [
-       "<style>\n",
-       ".list-inline {list-style: none; margin:0; padding: 0}\n",
-       ".list-inline>li {display: inline-block}\n",
-       ".list-inline>li:not(:last-child)::after {content: \"\\00b7\"; padding: 0 .5ex}\n",
-       "</style>\n",
-       "<ol class=list-inline><li>1</li><li>6</li><li>11</li><li>16</li><li>21</li><li>26</li><li>31</li><li>36</li><li>41</li><li>46</li></ol>\n"
-      ],
-      "text/latex": [
-       "\\begin{enumerate*}\n",
-       "\\item 1\n",
-       "\\item 6\n",
-       "\\item 11\n",
-       "\\item 16\n",
-       "\\item 21\n",
-       "\\item 26\n",
-       "\\item 31\n",
-       "\\item 36\n",
-       "\\item 41\n",
-       "\\item 46\n",
-       "\\end{enumerate*}\n"
-      ],
-      "text/markdown": [
-       "1. 1\n",
-       "2. 6\n",
-       "3. 11\n",
-       "4. 16\n",
-       "5. 21\n",
-       "6. 26\n",
-       "7. 31\n",
-       "8. 36\n",
-       "9. 41\n",
-       "10. 46\n",
-       "\n",
-       "\n"
-      ],
-      "text/plain": [
-       " [1]  1  6 11 16 21 26 31 36 41 46"
-      ]
-     },
-     "metadata": {},
-     "output_type": "display_data"
-    }
-   ],
-   "source": [
-    "S_data <- seq(1,50,5)\n",
-    "S_data"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "Note that because we generated values only at intervals of, there will be 50/5 = 10 \"substrate\" values.\n",
-    "\n",
-    "Now generate a Michaelis-Menten reaction  velocity response with `V_max` = 12.5 and `K_M` = 7.1:"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 120,
-   "metadata": {
-    "scrolled": true
-   },
-   "outputs": [
-    {
-     "data": {
-      "image/png": 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-      "text/plain": [
-       "plot without title"
-      ]
-     },
-     "metadata": {
-      "image/png": {
-       "height": 240,
-       "width": 240
-      }
-     },
-     "output_type": "display_data"
-    }
-   ],
-   "source": [
-    "V_data <- ((12.5 * S_data)/(7.1 + S_data))\n",
-    "plot(S_data, V_data)"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "Note that our choice of $V_{\\max} = 12.5$ and $K_M = 7.1$ is completely arbitrary. As long as we make sure that $V_{\\max} > 0$, $K_H > 0$, and $K_M$ lies well within the lower half of the the range of substrate concentrations (0-50), these \"data\" will be physically biologically sensible.\n",
-    "\n",
-    "Now let's add some random (normally-distributed) fluctuations to the data to emulate experimental / measurement error:"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 121,
-   "metadata": {
-    "scrolled": true
-   },
-   "outputs": [
-    {
-     "data": {
-      "image/png": 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-      "text/plain": [
-       "plot without title"
-      ]
-     },
-     "metadata": {
-      "image/png": {
-       "height": 240,
-       "width": 240
-      }
-     },
-     "output_type": "display_data"
-    }
-   ],
-   "source": [
-    "set.seed(1456) # To get the same random fluctuations in the \"data\" every time\n",
-    "V_data <- V_data + rnorm(10,0,1) # Add 10 random fluctuations  with standard deviation of 1 to emulate error\n",
-    "plot(S_data, V_data)"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "That looks real!"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "## Fitting the model using NLLS\n",
-    "\n",
-    "Now, fit the model to the data:"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 122,
-   "metadata": {},
-   "outputs": [
-    {
-     "name": "stderr",
-     "output_type": "stream",
-     "text": [
-      "Warning message in nls(V_data ~ V_max * S_data/(K_M + S_data)):\n",
-      "“No starting values specified for some parameters.\n",
-      "Initializing ‘V_max’, ‘K_M’ to '1.'.\n",
-      "Consider specifying 'start' or using a selfStart model”\n"
-     ]
-    }
-   ],
-   "source": [
-    "MM_model <- nls(V_data ~ V_max * S_data / (K_M + S_data))"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "This warning arises because `nls` requires \"starting values\" for the parameters (two in this case: `V_max` and `K_M`) to start searching for optimal combinations of parameter values (ones that minimize the RSS). Indeed, all NLLS fitting functions / algorithms require this. If you do not provide starting values, `nls` gives you a warning (as above) and uses a starting value of 1 for every parameter by default. For simple models, despite the warning, this works well enough.\n",
-    "\n",
-    "```{tip}\n",
-    "Before proceeding further, have a look at what `nls()`'s arguments are using `?nls`, or looking at the documentation [online](https://www.rdocumentation.org/packages/stats/versions/3.6.2/topics/nls).\n",
-    "```\n",
-    "\n",
-    "We will address the issue of starting values soon enough, but first let's look at how good the fit that we obtained looks.\n",
-    "\n",
-    "### Visualizing the fit \n",
-    "\n",
-    "The first thing to do is to see how well the model fitted the data, for which plotting is the best  first option:"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 123,
-   "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "image/png": 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VQ6Jycn379/f/LkySXa1dXVx48ff5q5wggA/0JAQ5XdvEmtW9ODB0REbm50505l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\nJycnLi4uOzu7ffv2OPWseNu3ExE1bUr/PrkXAGq4Sj00Nj4+vl+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-      "text/plain": [
-       "plot without title"
-      ]
-     },
-     "metadata": {
-      "image/png": {
-       "height": 240,
-       "width": 240
-      }
-     },
-     "output_type": "display_data"
-    }
-   ],
-   "source": [
-    "plot(S_data,V_data, xlab = \"Substrate Concentration\", ylab = \"Reaction Rate\")  # first plot the data \n",
-    "lines(S_data,predict(MM_model),lty=1,col=\"blue\",lwd=2) # now overlay the fitted model "
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "This looks OK. \n",
-    "\n",
-    "```{note}\n",
-    "We used the `predict()` function here just as we did in any of the linear models chapters (e.g., [here](16-MulExp:Predicted-values)). In general, you can use most of the same commands/functions (e.g., `predict()` and `summary()`) on the output of a `nls()` model fitting object as you would on a `lm()` model fitting object. Please have a look at the [documentation](https://stat.ethz.ch/R-manual/R-devel/library/stats/html/predict.nls.html) of the predict function for `nls` before proceeding.   \n",
-    "```\n",
-    "\n",
-    "However, the above approach for plotting is not the best way to do it, because `predict()`, without further arguments (see its [documentation](https://stat.ethz.ch/R-manual/R-devel/library/stats/html/predict.nls.html)), by default only generates predicted values for the actual x-values (substrate) data used to fit the model. So if there are very few values in the original data, you will not get a smooth predicted curve (as you can see above). A better approach is to generate a sufficient number of x-axis values and then calculate the predicted line. Let's do it:"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 124,
-   "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "text/html": [
-       "<style>\n",
-       ".dl-inline {width: auto; margin:0; padding: 0}\n",
-       ".dl-inline>dt, .dl-inline>dd {float: none; width: auto; display: inline-block}\n",
-       ".dl-inline>dt::after {content: \":\\0020\"; padding-right: .5ex}\n",
-       ".dl-inline>dt:not(:first-of-type) {padding-left: .5ex}\n",
-       "</style><dl class=dl-inline><dt>V_max</dt><dd>12.9636891133139</dd><dt>K_M</dt><dd>10.6072253230542</dd></dl>\n"
-      ],
-      "text/latex": [
-       "\\begin{description*}\n",
-       "\\item[V\\textbackslash{}\\_max] 12.9636891133139\n",
-       "\\item[K\\textbackslash{}\\_M] 10.6072253230542\n",
-       "\\end{description*}\n"
-      ],
-      "text/markdown": [
-       "V_max\n",
-       ":   12.9636891133139K_M\n",
-       ":   10.6072253230542\n",
-       "\n"
-      ],
-      "text/plain": [
-       "   V_max      K_M \n",
-       "12.96369 10.60723 "
-      ]
-     },
-     "metadata": {},
-     "output_type": "display_data"
-    }
-   ],
-   "source": [
-    "coef(MM_model) # check the coefficients"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 125,
-   "metadata": {},
-   "outputs": [],
-   "source": [
-    "Substrate2Plot <- seq(min(S_data), max(S_data),len=200) # generate some new x-axis values just for plotting"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 126,
-   "metadata": {},
-   "outputs": [],
-   "source": [
-    "Predict2Plot <- coef(MM_model)[\"V_max\"] * Substrate2Plot / (coef(MM_model)[\"K_M\"] + Substrate2Plot) # calculate the predicted values by plugging the fitted coefficients into the model equation "
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 127,
-   "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "image/png": 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epl2zYMHw6pFBIJpk/Ht99Co2RfZ0yePDkwMDDvNbZq1qz566+/tmzZcsGCBTwN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\nAAAAAElFTkSuQmCC",
-      "text/plain": [
-       "plot without title"
-      ]
-     },
-     "metadata": {
-      "image/png": {
-       "height": 240,
-       "width": 240
-      }
-     },
-     "output_type": "display_data"
-    }
-   ],
-   "source": [
-    "plot(S_data,V_data, xlab = \"Substrate Concentration\", ylab = \"Reaction Rate\")  # first plot the data \n",
-    "lines(Substrate2Plot, Predict2Plot, lty=1,col=\"blue\",lwd=2) # now overlay the fitted model"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "That looks much better (smoother) than the plot above !"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "### Summary stats of the fit\n",
-    "\n",
-    "Now lets get some stats of this NLLS fit. Having obtained the fit object (`MM_model`), we can use `summary()` just like we would for a `lm()` fit object: "
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 128,
-   "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "text/plain": [
-       "\n",
-       "Formula: V_data ~ V_max * S_data/(K_M + S_data)\n",
-       "\n",
-       "Parameters:\n",
-       "      Estimate Std. Error t value Pr(>|t|)    \n",
-       "V_max   12.964      1.221  10.616 5.42e-06 ***\n",
-       "K_M     10.607      3.266   3.248   0.0117 *  \n",
-       "---\n",
-       "Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1\n",
-       "\n",
-       "Residual standard error: 0.8818 on 8 degrees of freedom\n",
-       "\n",
-       "Number of iterations to convergence: 5 \n",
-       "Achieved convergence tolerance: 9.503e-06\n"
-      ]
-     },
-     "metadata": {},
-     "output_type": "display_data"
-    }
-   ],
-   "source": [
-    "summary(MM_model)"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "This looks a lot like the output of a linear model, and to be specific, of a [Linear Regression](./14-regress.ipynb). For starters, compare the above output with the output of `summary(genomeSizeModelDragon)` in [this section](regress:perform) of the Linear Regression chapter. \n",
-    "\n",
-    "So here are the main things to note about the output of `summary()` of an `nls()` model object:\n",
-    "\n",
-    "* `Estimate`s are, as in the output of the `lm()` function for fitting linear models, the estimated values of the coefficients of the model that you fitted ($V_{\\max}$ and $K_M$). Note that although we generated our data using $V_{\\max} = 12.5$ and $K_M = 7.1$, the actual coefficients are quite different from what we are getting with the NLLS fitting ($\\hat{V}_{\\max} = 12.96$ and $\\hat{K}_M = 10.61$). This is because we added random (normally-distributed) errors when we generated the \"data\". This tells you something about how experimental and/or measurement errors can distort your image of the underlying mechanism or process.   \n",
-    "* `Std. Error`, `t value`, and `Pr(>|t|)` and `Residual standard error` have the same interpretation as in the output of `lm()` (please look back at the [Linear Regression Chapter](./14-regress.ipynb))\n",
-    "* `Number of iterations to convergence` tells you how many times the NLLS algorithm had to adjust the parameter values till it managed to find a solution that minimizes the Residual Sum of Squares (RSS)\n",
-    "* `Achieved convergence tolerance` tells you on what basis the algorithm decided that it was close enough to the a solution; basically if the RSS does not improve more than a certain threshold despite parameter adjustments, the algorithm stops searching. This may or may not be close to an optimal solution (but in this case it is).\n",
-    "\n",
-    "The last two items are specific to the output of an `nls()` fitting `summary()`, because unlike Ordinary Least Squares (OLS), which is what we used for Linear regression, NLLS is not an *exact* procedure, and the fitting requires computer simulations; revisit the [Lecture](https://github.com/mhasoba/TheMulQuaBio/tree/master/content/lectures/NLLS) for an explanation of this. This is all you need to know for now. As such, you do not need to report these last two items when presenting the results of an NLLS fit, but they are useful for problem solving in case the fitting does not work (more on this below). \n",
-    "\n",
-    "As noted above, you can use the same sort of commands on a `nls()` fitting result as you can on a `lm()` object.  \n",
-    "\n",
-    "Thus, much of the output of NLLS fitting using `nls()` is similar to the output of an `lm()`, and can be further analyzed or processed using analogous functions such as `coef()`, `residuals()`, and `confint()`. \n",
-    "\n",
-    "For example, you can get just the values of the estimated coefficients using `coef()` as you did above for the plotting.   "
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "## Statistical inference using NLLS fits\n",
-    "\n",
-    "So what do we do with the results of an NLLS fit? What statistical inferences can be made? We will address this issue here at a basic level, and then revisit it using model selection further below (in the Allometric growth example)."
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "### Goodness of fit\n",
-    "\n",
-    "Assessing whether an NLLS-fitted model is \"significant\" based on an F-value, by performing an Analysis of Variance (ANOVA) on the regression results as one can do for a Linear Model using the [`anova()` function on a linear model fit](regress:perform) is not advisable. Try `anova(MM_model)`, and see what happens:"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 156,
-   "metadata": {},
-   "outputs": [
-    {
-     "ename": "ERROR",
-     "evalue": "Error in anova.nls(MM_model): anova is only defined for sequences of \"nls\" objects\n",
-     "output_type": "error",
-     "traceback": [
-      "Error in anova.nls(MM_model): anova is only defined for sequences of \"nls\" objects\nTraceback:\n",
-      "1. anova(MM_model)",
-      "2. anova.nls(MM_model)",
-      "3. stop(\"anova is only defined for sequences of \\\"nls\\\" objects\")"
-     ]
-    }
-   ],
-   "source": [
-    "anova(MM_model)"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "So a ANOVA calculation is not implemented for a NLLS fit. The third item in the error message / traceback above says that you can however use `anova()` to compare two *nested* models. That is, you can do something like `anova(model1,model2)`, where model 1 is nested within model 1, as we did [previously](./14-regress.ipynb) to ask if a simpler model explains significantly more variation in the data than a more complicated one (with one or more addtional terms in the equation)."
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "To understand why you cannot use an ANOVA to assess the significance of a NLLS model fit, recall that the objective of an [`anova()` function applied to  a linear model fit](regress:perform) is o compare the fitted model to a *null* model, asking whether you can reject that null model (hypothesis). The problem with doing so for a nonlinear model is that unfortunately it is not clear what the null hypothesis should be when the model is not a linear combination of terms (the definition of a linear model). For an F-statistic to be meaningful, the null hypothesis model must be nested *within* the alternative model (in which case you can indeed use the `anova()` function using `anova(Model1,Model2)`). \n",
-    "\n",
-    "To make this clear, let's recall the equation for a simple linear regression:\n",
-    "\n",
-    "$$y = \\beta_0 + \\beta_1 x$$\n",
-    "\n",
-    "The null model for this is \n",
-    "\n",
-    "$$y = \\beta_0$$\n",
-    "\n",
-    "This then allows the TSS, RSS and ESS to be calculated in a meaninglful way, as we [did before](./14-regress.ipynb), leading to an ANOVA based assessment of significance of the regrssion moel fit to the data. \n",
-    "\n",
-    "Now consider a nonlinear model like the Michaelis-Menten eqn {eq}`eq:M-M`. *What would be an appropriate null model for this equation?*\n",
-    "\n",
-    "Thus, the best ways to assess a NLLS model's fit are:\n",
-    "\n",
-    "* Compare it's likelihood to those of other alternative models' fits (which may include your best guess for a null model).\n",
-    "\n",
-    "* Examine whether the fitted coefficients are reliable, i.e., are significant, based on their (low) standard errors, (high) t-values, and (low) p-values (see the example model's summary above)."
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "#### R-squared values\n",
-    "\n",
-    "To put it simply, unlike an R$^2$ value obtained by fitting a linear model, that obtained from NLLS fitting is not reliable, and should not be used. The reason for this is somewhat technical (e.g., see this [paper](https://dx.doi.org/10.1186%2F1471-2210-10-6)) and we won't go into it here. But basically, NLLS R$^2$ values do not always accurately reflect the quality of fit, and definitely cannot be used to select between competing models (Model selection, as you learned [previously](./18-ModelSimp.ipynb)). Indeed R$^2$ values obtained from NLLS fitting even be negative when the model fits very poorly! We will learn more about model selection with non-linear models later below."
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "### Confidence Intervals \n",
-    "\n",
-    "One particularly useful thing you can do after NLLS fitting is to calculate/construct the confidence intervals (CI's) around the estimated parameters in our fitted model. This is analogous to how we would in the OLS fitting used for Linear Models: "
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 129,
-   "metadata": {},
-   "outputs": [
-    {
-     "name": "stderr",
-     "output_type": "stream",
-     "text": [
-      "Waiting for profiling to be done...\n",
-      "\n"
-     ]
-    },
-    {
-     "data": {
-      "text/html": [
-       "<table class=\"dataframe\">\n",
-       "<caption>A matrix: 2 × 2 of type dbl</caption>\n",
-       "<thead>\n",
-       "\t<tr><th></th><th scope=col>2.5%</th><th scope=col>97.5%</th></tr>\n",
-       "</thead>\n",
-       "<tbody>\n",
-       "\t<tr><th scope=row>V_max</th><td>10.640478</td><td>17.00502</td></tr>\n",
-       "\t<tr><th scope=row>K_M</th><td> 4.924547</td><td>22.39247</td></tr>\n",
-       "</tbody>\n",
-       "</table>\n"
-      ],
-      "text/latex": [
-       "A matrix: 2 × 2 of type dbl\n",
-       "\\begin{tabular}{r|ll}\n",
-       "  & 2.5\\% & 97.5\\%\\\\\n",
-       "\\hline\n",
-       "\tV\\_max & 10.640478 & 17.00502\\\\\n",
-       "\tK\\_M &  4.924547 & 22.39247\\\\\n",
-       "\\end{tabular}\n"
-      ],
-      "text/markdown": [
-       "\n",
-       "A matrix: 2 × 2 of type dbl\n",
-       "\n",
-       "| <!--/--> | 2.5% | 97.5% |\n",
-       "|---|---|---|\n",
-       "| V_max | 10.640478 | 17.00502 |\n",
-       "| K_M |  4.924547 | 22.39247 |\n",
-       "\n"
-      ],
-      "text/plain": [
-       "      2.5%      97.5%   \n",
-       "V_max 10.640478 17.00502\n",
-       "K_M    4.924547 22.39247"
-      ]
-     },
-     "metadata": {},
-     "output_type": "display_data"
-    }
-   ],
-   "source": [
-    "confint(MM_model)"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "The `Waiting for profiling to be done...` message reflects the fact that calculating the standard errors from which the CI's are calculated requires a particular computational procedure (which we will not go into here) when it comes to NLLS fits. \n",
-    "\n",
-    "Calculating confidence intervals can be useful because, \n",
-    "\n",
-    "1. As you learned [here](13-t_F_tests:CI), [here](13-t_F_tests:CI2), and [here](15-anova:CI) (among other places) you can use a coefficient/parameter estimate's confidence intervals to test whether it is significantly different from some reference value. In our CI example above, the intervals for `K_M` do in fact include the original value of $K_M = 7.1$ that we used to generate the data.\n",
-    "\n",
-    "2. Confidence intervals can also be used to do a quick (but not robust - see warning below) test of whether coefficient estimates of the same model coefficient obtained from different populations (samples) are significantly different from each other (If their CI's don't overlap, they are significantly different).\n",
-    "\n",
-    "```{warning}\n",
-    "You can compare estimates of the same coefficient (parameter) from samples of different populations: if their CI's don't overlap, this indicates a statistically significant difference between the two populations as far as that coefficient is concerned (e.g., for 95% CI's this means at the 0.05 level of significance or p-value). However, the opposite is not necessarily true: when CI's overlap, there may *still* be a statistically significant difference between the coefficients (and therefore, populations).\n",
-    "```"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "## The starting values problem\n",
-    "\n",
-    "Now let's revisit the issue of starting values in NLLS fitting. Previously, we fitted the Michaelis-Menten Model without any starting values, and R gave us a warning but managed to fit the model to our synthetic \"data\" using default starting values. \n",
-    "\n",
-    "### Fitting the model with starting values\n",
-    "\n",
-    "Lets try the NLLS fitting again, but with some particular starting values:"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 130,
-   "metadata": {},
-   "outputs": [],
-   "source": [
-    "MM_model2 <- nls(V_data ~ V_max * S_data / (K_M + S_data), start = list(V_max = 12, K_M = 7))"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "Note that unlike before, we got no warning message about starting values.\n",
-    "\n",
-    "*As will become apparent below, using sensible starting values is important in NLLS fitting.*\n",
-    "\n",
-    "```{note}\n",
-    "*How do you find \"sensible\" starting values for NLLS fitting?* This very much depends on your understanding of the mathematical model that is being fitted to the data, the mechanistic interpretation of its parameters, and the specific dataset. For example, in the Michaelis-Menten Model example, we can know that $V_{\\max}$ is maximum reaction velocity and $K_M$ is the value of the substrate concentration at which half of $V_{\\max}$ is reached. So we can choose starting values by \"eye-balling\" a particular dataset and determining *approximately* what the `V_max`and `K_M` are be for that particular dataset. In this particular case, we chose `V_max` = 12 and `K_M` = 7 because looking at the data plot above, these values seem to be reasonable guesses for the two parameters.   \n",
-    "```\n",
-    "\n",
-    "\n",
-    "Let's compare the coefficient estimates from our two different model fits to the same dataset:"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 131,
-   "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "text/html": [
-       "<style>\n",
-       ".dl-inline {width: auto; margin:0; padding: 0}\n",
-       ".dl-inline>dt, .dl-inline>dd {float: none; width: auto; display: inline-block}\n",
-       ".dl-inline>dt::after {content: \":\\0020\"; padding-right: .5ex}\n",
-       ".dl-inline>dt:not(:first-of-type) {padding-left: .5ex}\n",
-       "</style><dl class=dl-inline><dt>V_max</dt><dd>12.9636891133139</dd><dt>K_M</dt><dd>10.6072253230542</dd></dl>\n"
-      ],
-      "text/latex": [
-       "\\begin{description*}\n",
-       "\\item[V\\textbackslash{}\\_max] 12.9636891133139\n",
-       "\\item[K\\textbackslash{}\\_M] 10.6072253230542\n",
-       "\\end{description*}\n"
-      ],
-      "text/markdown": [
-       "V_max\n",
-       ":   12.9636891133139K_M\n",
-       ":   10.6072253230542\n",
-       "\n"
-      ],
-      "text/plain": [
-       "   V_max      K_M \n",
-       "12.96369 10.60723 "
-      ]
-     },
-     "metadata": {},
-     "output_type": "display_data"
-    },
-    {
-     "data": {
-      "text/html": [
-       "<style>\n",
-       ".dl-inline {width: auto; margin:0; padding: 0}\n",
-       ".dl-inline>dt, .dl-inline>dd {float: none; width: auto; display: inline-block}\n",
-       ".dl-inline>dt::after {content: \":\\0020\"; padding-right: .5ex}\n",
-       ".dl-inline>dt:not(:first-of-type) {padding-left: .5ex}\n",
-       "</style><dl class=dl-inline><dt>V_max</dt><dd>12.9636297453629</dd><dt>K_M</dt><dd>10.6070555266004</dd></dl>\n"
-      ],
-      "text/latex": [
-       "\\begin{description*}\n",
-       "\\item[V\\textbackslash{}\\_max] 12.9636297453629\n",
-       "\\item[K\\textbackslash{}\\_M] 10.6070555266004\n",
-       "\\end{description*}\n"
-      ],
-      "text/markdown": [
-       "V_max\n",
-       ":   12.9636297453629K_M\n",
-       ":   10.6070555266004\n",
-       "\n"
-      ],
-      "text/plain": [
-       "   V_max      K_M \n",
-       "12.96363 10.60706 "
-      ]
-     },
-     "metadata": {},
-     "output_type": "display_data"
-    }
-   ],
-   "source": [
-    "coef(MM_model)\n",
-    "coef(MM_model2)"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "Not too different, but not exactly the same! \n",
-    "\n",
-    "In contrast, when you fit linear models you will get exactly the same coefficient estimates every single time, because OLS is an *exact* procedure."
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "Now, let's try even more different start values:"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 132,
-   "metadata": {},
-   "outputs": [],
-   "source": [
-    "MM_model3 <- nls(V_data ~ V_max * S_data / (K_M + S_data), start = list(V_max = .01, K_M = 20))"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "Compare the coefficients of this model fit to the two previous ones: "
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 133,
-   "metadata": {
-    "scrolled": true
-   },
-   "outputs": [
-    {
-     "data": {
-      "text/html": [
-       "<style>\n",
-       ".dl-inline {width: auto; margin:0; padding: 0}\n",
-       ".dl-inline>dt, .dl-inline>dd {float: none; width: auto; display: inline-block}\n",
-       ".dl-inline>dt::after {content: \":\\0020\"; padding-right: .5ex}\n",
-       ".dl-inline>dt:not(:first-of-type) {padding-left: .5ex}\n",
-       "</style><dl class=dl-inline><dt>V_max</dt><dd>12.9636891133139</dd><dt>K_M</dt><dd>10.6072253230542</dd></dl>\n"
-      ],
-      "text/latex": [
-       "\\begin{description*}\n",
-       "\\item[V\\textbackslash{}\\_max] 12.9636891133139\n",
-       "\\item[K\\textbackslash{}\\_M] 10.6072253230542\n",
-       "\\end{description*}\n"
-      ],
-      "text/markdown": [
-       "V_max\n",
-       ":   12.9636891133139K_M\n",
-       ":   10.6072253230542\n",
-       "\n"
-      ],
-      "text/plain": [
-       "   V_max      K_M \n",
-       "12.96369 10.60723 "
-      ]
-     },
-     "metadata": {},
-     "output_type": "display_data"
-    },
-    {
-     "data": {
-      "text/html": [
-       "<style>\n",
-       ".dl-inline {width: auto; margin:0; padding: 0}\n",
-       ".dl-inline>dt, .dl-inline>dd {float: none; width: auto; display: inline-block}\n",
-       ".dl-inline>dt::after {content: \":\\0020\"; padding-right: .5ex}\n",
-       ".dl-inline>dt:not(:first-of-type) {padding-left: .5ex}\n",
-       "</style><dl class=dl-inline><dt>V_max</dt><dd>12.9636297453629</dd><dt>K_M</dt><dd>10.6070555266004</dd></dl>\n"
-      ],
-      "text/latex": [
-       "\\begin{description*}\n",
-       "\\item[V\\textbackslash{}\\_max] 12.9636297453629\n",
-       "\\item[K\\textbackslash{}\\_M] 10.6070555266004\n",
-       "\\end{description*}\n"
-      ],
-      "text/markdown": [
-       "V_max\n",
-       ":   12.9636297453629K_M\n",
-       ":   10.6070555266004\n",
-       "\n"
-      ],
-      "text/plain": [
-       "   V_max      K_M \n",
-       "12.96363 10.60706 "
-      ]
-     },
-     "metadata": {},
-     "output_type": "display_data"
-    },
-    {
-     "data": {
-      "text/html": [
-       "<style>\n",
-       ".dl-inline {width: auto; margin:0; padding: 0}\n",
-       ".dl-inline>dt, .dl-inline>dd {float: none; width: auto; display: inline-block}\n",
-       ".dl-inline>dt::after {content: \":\\0020\"; padding-right: .5ex}\n",
-       ".dl-inline>dt:not(:first-of-type) {padding-left: .5ex}\n",
-       "</style><dl class=dl-inline><dt>V_max</dt><dd>0.816129930954726</dd><dt>K_M</dt><dd>-19.4993655809431</dd></dl>\n"
-      ],
-      "text/latex": [
-       "\\begin{description*}\n",
-       "\\item[V\\textbackslash{}\\_max] 0.816129930954726\n",
-       "\\item[K\\textbackslash{}\\_M] -19.4993655809431\n",
-       "\\end{description*}\n"
-      ],
-      "text/markdown": [
-       "V_max\n",
-       ":   0.816129930954726K_M\n",
-       ":   -19.4993655809431\n",
-       "\n"
-      ],
-      "text/plain": [
-       "      V_max         K_M \n",
-       "  0.8161299 -19.4993656 "
-      ]
-     },
-     "metadata": {},
-     "output_type": "display_data"
-    }
-   ],
-   "source": [
-    "coef(MM_model)\n",
-    "coef(MM_model2)\n",
-    "coef(MM_model3)"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "The estimates in our latest model fit are completely different (in fact, `K_M` is negative)! Let's plot this model's and the first model's fit together: "
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 134,
-   "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "image/png": 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D1bK7yoLg5BeLhcAvrpUxw6hB07EBKC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\nRK5CYII=",
-      "text/plain": [
-       "plot without title"
-      ]
-     },
-     "metadata": {
-      "image/png": {
-       "height": 240,
-       "width": 240
-      }
-     },
-     "output_type": "display_data"
-    }
-   ],
-   "source": [
-    "plot(S_data,V_data)  # first plot the data \n",
-    "lines(S_data,predict(MM_model),lty=1,col=\"blue\",lwd=2) # overlay the original model fit\n",
-    "lines(S_data,predict(MM_model3),lty=1,col=\"red\",lwd=2) # overlay the latest model fit"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "As you would have guessed from the really funky coefficient estimates that were obtained in `MM_model3`, this is a pretty poor model fit to the data, with the negative value of `K_M` causing the fitted version of the Michaelis-Menten model to behave strangely.\n",
-    "\n",
-    "*Also, you could have used the nicer plotting approach that was introduced before.*\n",
-    "\n",
-    "Let's try with even more different starting values.  "
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 135,
-   "metadata": {},
-   "outputs": [
-    {
-     "ename": "ERROR",
-     "evalue": "Error in nlsModel(formula, mf, start, wts, scaleOffset = scOff, nDcentral = nDcntr): singular gradient matrix at initial parameter estimates\n",
-     "output_type": "error",
-     "traceback": [
-      "Error in nlsModel(formula, mf, start, wts, scaleOffset = scOff, nDcentral = nDcntr): singular gradient matrix at initial parameter estimates\nTraceback:\n",
-      "1. nls(V_data ~ V_max * S_data/(K_M + S_data), start = list(V_max = 0, \n .     K_M = 0.1))",
-      "2. nlsModel(formula, mf, start, wts, scaleOffset = scOff, nDcentral = nDcntr)",
-      "3. stop(\"singular gradient matrix at initial parameter estimates\")"
-     ]
-    }
-   ],
-   "source": [
-    "nls(V_data ~ V_max * S_data / (K_M + S_data), start = list(V_max = 0, K_M = 0.1))"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "The `singular gradient matrix at initial parameter estimates` error arises from the fact that the starting values you provided were so far from the optimal solution, that the parameter searching in `nls()` failed at the very first step. The algorithm could not figure out where to go from those starting values. In fact, the starting value we gave it is biologically/ physically impossible, because `V_max` can't equal 0.  \n",
-    "\n",
-    "Let's look at another pair of starting values that causes the model fitting to fail:  "
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 136,
-   "metadata": {},
-   "outputs": [
-    {
-     "ename": "ERROR",
-     "evalue": "Error in nls(V_data ~ V_max * S_data/(K_M + S_data), start = list(V_max = -0.1, : singular gradient\n",
-     "output_type": "error",
-     "traceback": [
-      "Error in nls(V_data ~ V_max * S_data/(K_M + S_data), start = list(V_max = -0.1, : singular gradient\nTraceback:\n",
-      "1. nls(V_data ~ V_max * S_data/(K_M + S_data), start = list(V_max = -0.1, \n .     K_M = 100))"
-     ]
-    }
-   ],
-   "source": [
-    "nls(V_data ~ V_max * S_data / (K_M + S_data), start = list(V_max = -0.1, K_M = 100))"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "In this case, the model fitting did get started, but eventually failed, again because the starting values were too far from the (approximately) optimal values ($V_{\\max} \\approx 12.96, K_M \\approx 10.61$).\n",
-    "\n",
-    "```{note}\n",
-    "There are other types of errors (other than the \"singular gadient matrix\" one) that the nlls fitting can run into because of poor starting parameter values.\n",
-    "```"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "And what happens if we start really close to the optimal values? Let's try: "
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 137,
-   "metadata": {},
-   "outputs": [],
-   "source": [
-    "MM_model4 <- nls(V_data ~ V_max * S_data / (K_M + S_data), start = list(V_max = 12.96, K_M = 10.61))"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 138,
-   "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "text/html": [
-       "<style>\n",
-       ".dl-inline {width: auto; margin:0; padding: 0}\n",
-       ".dl-inline>dt, .dl-inline>dd {float: none; width: auto; display: inline-block}\n",
-       ".dl-inline>dt::after {content: \":\\0020\"; padding-right: .5ex}\n",
-       ".dl-inline>dt:not(:first-of-type) {padding-left: .5ex}\n",
-       "</style><dl class=dl-inline><dt>V_max</dt><dd>12.9636891133139</dd><dt>K_M</dt><dd>10.6072253230542</dd></dl>\n"
-      ],
-      "text/latex": [
-       "\\begin{description*}\n",
-       "\\item[V\\textbackslash{}\\_max] 12.9636891133139\n",
-       "\\item[K\\textbackslash{}\\_M] 10.6072253230542\n",
-       "\\end{description*}\n"
-      ],
-      "text/markdown": [
-       "V_max\n",
-       ":   12.9636891133139K_M\n",
-       ":   10.6072253230542\n",
-       "\n"
-      ],
-      "text/plain": [
-       "   V_max      K_M \n",
-       "12.96369 10.60723 "
-      ]
-     },
-     "metadata": {},
-     "output_type": "display_data"
-    },
-    {
-     "data": {
-      "text/html": [
-       "<style>\n",
-       ".dl-inline {width: auto; margin:0; padding: 0}\n",
-       ".dl-inline>dt, .dl-inline>dd {float: none; width: auto; display: inline-block}\n",
-       ".dl-inline>dt::after {content: \":\\0020\"; padding-right: .5ex}\n",
-       ".dl-inline>dt:not(:first-of-type) {padding-left: .5ex}\n",
-       "</style><dl class=dl-inline><dt>V_max</dt><dd>12.9636623970243</dd><dt>K_M</dt><dd>10.6071489188677</dd></dl>\n"
-      ],
-      "text/latex": [
-       "\\begin{description*}\n",
-       "\\item[V\\textbackslash{}\\_max] 12.9636623970243\n",
-       "\\item[K\\textbackslash{}\\_M] 10.6071489188677\n",
-       "\\end{description*}\n"
-      ],
-      "text/markdown": [
-       "V_max\n",
-       ":   12.9636623970243K_M\n",
-       ":   10.6071489188677\n",
-       "\n"
-      ],
-      "text/plain": [
-       "   V_max      K_M \n",
-       "12.96366 10.60715 "
-      ]
-     },
-     "metadata": {},
-     "output_type": "display_data"
-    }
-   ],
-   "source": [
-    "coef(MM_model)\n",
-    "coef(MM_model4)"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "The results of the first model fit and this last one are still not exactly the same! This drives home the point that NLLS is not an \"exact\" procedure. However, the differences between these two solutions are minuscule, so the main thing to take away is that if the starting values are reasonable, NLLS is *exact enough*.       "
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "Note that and even if you started the NLLS fitting with the exact parameter values with which you generated the data before introducing errors (so use `start = list(V_max = 12.5, K_M = 7.1)` above instead), you would still get the same result for the coefficients (try it). This is because the NLLS fitting will converge back to the parameter estimates based on the actual data, errors and all."
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "## A more robust NLLS algorithm\n",
-    "\n",
-    "The standard NLLS function in R, `nls`, which we have been using so far, does the NLLS fitting by implementing an algorithm called the Gauss-Newton algorithm. While the Gauss-Newton algorithm works well for most simple non-linear models, it has a tendency to \"get lost\" or \"stuck\" while searching for optimal parameter estimates (that minimize the residual sum of squares, or RSS). Therefore, `nls` will often fail to fit your model to the data if you start off at starting values for the parameters that are too far from what the optimal values would be, as you saw above (e.g., when you got the `singular gradient matrix` error).    \n",
-    "\n",
-    "Some nonlinear models are especially difficult for nls to fit to data because such model have a mathematical form that makes it hard to find parameter combinations that minimize the residual sum of squared (RSS). If this does not makes sense, don't worry about it.\n",
-    "\n",
-    "One solution to this is to use a different algorithm than Gauss-Newton. `nls()` has one other algorithm that can be more robust in some situations, called the \"port\" algorithm. However, there is a better solution still: the Levenberg-Marqualdt algorithm, which is less likely to get stuck (is more robust than) than Gauss-Newton (or port). If you want to learn more about the technicalities of this, [here](https://en.wikipedia.org/wiki/Gauss%E2%80%93Newton_algorithm) are [here](https://en.wikipedia.org/wiki/Levenberg%E2%80%93Marquardt_algorithm) are good places to start (also see the Readings list at the end of this chapter).\n",
-    "\n",
-    "\n",
-    "To be able to use nlsLM, we will need to switch to a different NLLS function called `nlsLM`. In order to be able to use `nlsLM`, you will need the `nls.lm` R package, which you can install using the method appropriate for your operating system  (e.g., linux users will launch R in `sudo` mode first) and then use:\n",
-    "\n",
-    "```r\n",
-    "> install.packages(\"minpack.lm\") \n",
-    "```\n",
-    "\n",
-    "Now load the `minpack.lm` package:"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 139,
-   "metadata": {
-    "run_control": {
-     "marked": true
-    },
-    "scrolled": true
-   },
-   "outputs": [],
-   "source": [
-    "require(\"minpack.lm\")"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "Now let's try it (using the same starting values as `MM_model2` above):"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 140,
-   "metadata": {},
-   "outputs": [],
-   "source": [
-    "MM_model5 <- nlsLM(V_data ~ V_max * S_data / (K_M + S_data), start = list(V_max = 12, K_M = 7))"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "Now compare the `nls` and `nlsLM` fitted coefficients:"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 141,
-   "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "text/html": [
-       "<style>\n",
-       ".dl-inline {width: auto; margin:0; padding: 0}\n",
-       ".dl-inline>dt, .dl-inline>dd {float: none; width: auto; display: inline-block}\n",
-       ".dl-inline>dt::after {content: \":\\0020\"; padding-right: .5ex}\n",
-       ".dl-inline>dt:not(:first-of-type) {padding-left: .5ex}\n",
-       "</style><dl class=dl-inline><dt>V_max</dt><dd>12.9636297453629</dd><dt>K_M</dt><dd>10.6070555266004</dd></dl>\n"
-      ],
-      "text/latex": [
-       "\\begin{description*}\n",
-       "\\item[V\\textbackslash{}\\_max] 12.9636297453629\n",
-       "\\item[K\\textbackslash{}\\_M] 10.6070555266004\n",
-       "\\end{description*}\n"
-      ],
-      "text/markdown": [
-       "V_max\n",
-       ":   12.9636297453629K_M\n",
-       ":   10.6070555266004\n",
-       "\n"
-      ],
-      "text/plain": [
-       "   V_max      K_M \n",
-       "12.96363 10.60706 "
-      ]
-     },
-     "metadata": {},
-     "output_type": "display_data"
-    },
-    {
-     "data": {
-      "text/html": [
-       "<style>\n",
-       ".dl-inline {width: auto; margin:0; padding: 0}\n",
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-       ".dl-inline>dt::after {content: \":\\0020\"; padding-right: .5ex}\n",
-       ".dl-inline>dt:not(:first-of-type) {padding-left: .5ex}\n",
-       "</style><dl class=dl-inline><dt>V_max</dt><dd>12.9636298178077</dd><dt>K_M</dt><dd>10.6070557350073</dd></dl>\n"
-      ],
-      "text/latex": [
-       "\\begin{description*}\n",
-       "\\item[V\\textbackslash{}\\_max] 12.9636298178077\n",
-       "\\item[K\\textbackslash{}\\_M] 10.6070557350073\n",
-       "\\end{description*}\n"
-      ],
-      "text/markdown": [
-       "V_max\n",
-       ":   12.9636298178077K_M\n",
-       ":   10.6070557350073\n",
-       "\n"
-      ],
-      "text/plain": [
-       "   V_max      K_M \n",
-       "12.96363 10.60706 "
-      ]
-     },
-     "metadata": {},
-     "output_type": "display_data"
-    }
-   ],
-   "source": [
-    "coef(MM_model2)\n",
-    "coef(MM_model5)"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "Close enough. \n",
-    "\n",
-    "Now, let's try fitting the model using all those starting parameter combinations that failed previously:  "
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 142,
-   "metadata": {},
-   "outputs": [],
-   "source": [
-    "MM_model6 <- nlsLM(V_data ~ V_max * S_data / (K_M + S_data), start = list(V_max = .01, K_M = 20))"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 143,
-   "metadata": {},
-   "outputs": [],
-   "source": [
-    "MM_model7 <- nlsLM(V_data ~ V_max * S_data / (K_M + S_data), start = list(V_max = 0, K_M = 0.1))"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 144,
-   "metadata": {
-    "scrolled": true
-   },
-   "outputs": [],
-   "source": [
-    "MM_model8 <- nlsLM(V_data ~ V_max * S_data / (K_M + S_data), start = list(V_max = -0.1, K_M = 100))"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 145,
-   "metadata": {
-    "scrolled": true
-   },
-   "outputs": [
-    {
-     "data": {
-      "text/html": [
-       "<style>\n",
-       ".dl-inline {width: auto; margin:0; padding: 0}\n",
-       ".dl-inline>dt, .dl-inline>dd {float: none; width: auto; display: inline-block}\n",
-       ".dl-inline>dt::after {content: \":\\0020\"; padding-right: .5ex}\n",
-       ".dl-inline>dt:not(:first-of-type) {padding-left: .5ex}\n",
-       "</style><dl class=dl-inline><dt>V_max</dt><dd>12.9636487209143</dd><dt>K_M</dt><dd>10.6071097943627</dd></dl>\n"
-      ],
-      "text/latex": [
-       "\\begin{description*}\n",
-       "\\item[V\\textbackslash{}\\_max] 12.9636487209143\n",
-       "\\item[K\\textbackslash{}\\_M] 10.6071097943627\n",
-       "\\end{description*}\n"
-      ],
-      "text/markdown": [
-       "V_max\n",
-       ":   12.9636487209143K_M\n",
-       ":   10.6071097943627\n",
-       "\n"
-      ],
-      "text/plain": [
-       "   V_max      K_M \n",
-       "12.96365 10.60711 "
-      ]
-     },
-     "metadata": {},
-     "output_type": "display_data"
-    },
-    {
-     "data": {
-      "text/html": [
-       "<style>\n",
-       ".dl-inline {width: auto; margin:0; padding: 0}\n",
-       ".dl-inline>dt, .dl-inline>dd {float: none; width: auto; display: inline-block}\n",
-       ".dl-inline>dt::after {content: \":\\0020\"; padding-right: .5ex}\n",
-       ".dl-inline>dt:not(:first-of-type) {padding-left: .5ex}\n",
-       "</style><dl class=dl-inline><dt>V_max</dt><dd>12.9636360068123</dd><dt>K_M</dt><dd>10.6070734344093</dd></dl>\n"
-      ],
-      "text/latex": [
-       "\\begin{description*}\n",
-       "\\item[V\\textbackslash{}\\_max] 12.9636360068123\n",
-       "\\item[K\\textbackslash{}\\_M] 10.6070734344093\n",
-       "\\end{description*}\n"
-      ],
-      "text/markdown": [
-       "V_max\n",
-       ":   12.9636360068123K_M\n",
-       ":   10.6070734344093\n",
-       "\n"
-      ],
-      "text/plain": [
-       "   V_max      K_M \n",
-       "12.96364 10.60707 "
-      ]
-     },
-     "metadata": {},
-     "output_type": "display_data"
-    },
-    {
-     "data": {
-      "text/html": [
-       "<style>\n",
-       ".dl-inline {width: auto; margin:0; padding: 0}\n",
-       ".dl-inline>dt, .dl-inline>dd {float: none; width: auto; display: inline-block}\n",
-       ".dl-inline>dt::after {content: \":\\0020\"; padding-right: .5ex}\n",
-       ".dl-inline>dt:not(:first-of-type) {padding-left: .5ex}\n",
-       "</style><dl class=dl-inline><dt>V_max</dt><dd>12.9636401038953</dd><dt>K_M</dt><dd>10.6070851518134</dd></dl>\n"
-      ],
-      "text/latex": [
-       "\\begin{description*}\n",
-       "\\item[V\\textbackslash{}\\_max] 12.9636401038953\n",
-       "\\item[K\\textbackslash{}\\_M] 10.6070851518134\n",
-       "\\end{description*}\n"
-      ],
-      "text/markdown": [
-       "V_max\n",
-       ":   12.9636401038953K_M\n",
-       ":   10.6070851518134\n",
-       "\n"
-      ],
-      "text/plain": [
-       "   V_max      K_M \n",
-       "12.96364 10.60709 "
-      ]
-     },
-     "metadata": {},
-     "output_type": "display_data"
-    }
-   ],
-   "source": [
-    "coef(MM_model6)\n",
-    "coef(MM_model7)\n",
-    "coef(MM_model8)"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "Nice, these all worked with `nlsLM` even though they had failed with `nls`! \n",
-    "\n",
-    "But `nlsLM` also has its limits. Let's try more absurd starting values:"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 146,
-   "metadata": {
-    "scrolled": true
-   },
-   "outputs": [
-    {
-     "ename": "ERROR",
-     "evalue": "Error in nlsModel(formula, mf, start, wts): singular gradient matrix at initial parameter estimates\n",
-     "output_type": "error",
-     "traceback": [
-      "Error in nlsModel(formula, mf, start, wts): singular gradient matrix at initial parameter estimates\nTraceback:\n",
-      "1. nlsLM(V_data ~ V_max * S_data/(K_M + S_data), start = list(V_max = -10, \n .     K_M = -100))",
-      "2. nlsModel(formula, mf, start, wts)",
-      "3. stop(\"singular gradient matrix at initial parameter estimates\")"
-     ]
-    }
-   ],
-   "source": [
-    "nlsLM(V_data ~ V_max * S_data / (K_M + S_data), start = list(V_max = -10, K_M = -100))"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "Here again, NLLS fitting fails because these starting values are just too far from the best solution.  "
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "(20-ModelFitting-NLLS:Bounding)=\n",
-    "## Bounding parameter values"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "You can also bound the starting values, i.e., prevent them from exceeding some minimum and maximum value *during* the NLLS fitting process.\n",
-    "\n",
-    "For example let's first re-run the fitting without bounding the parameters (and some relatively-far-from-optimal starting values):"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 148,
-   "metadata": {
-    "scrolled": true
-   },
-   "outputs": [
-    {
-     "data": {
-      "text/plain": [
-       "Nonlinear regression model\n",
-       "  model: V_data ~ V_max * S_data/(K_M + S_data)\n",
-       "   data: parent.frame()\n",
-       "V_max   K_M \n",
-       "12.96 10.61 \n",
-       " residual sum-of-squares: 6.22\n",
-       "\n",
-       "Number of iterations to convergence: 12 \n",
-       "Achieved convergence tolerance: 1.49e-08"
-      ]
-     },
-     "metadata": {},
-     "output_type": "display_data"
-    }
-   ],
-   "source": [
-    "nlsLM(V_data ~ V_max * S_data / (K_M + S_data), start = list(V_max = 0.1, K_M = 0.1))"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "Now, the same with parameter bounds:"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 149,
-   "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "text/plain": [
-       "Nonlinear regression model\n",
-       "  model: V_data ~ V_max * S_data/(K_M + S_data)\n",
-       "   data: parent.frame()\n",
-       "V_max   K_M \n",
-       "12.96 10.61 \n",
-       " residual sum-of-squares: 6.22\n",
-       "\n",
-       "Number of iterations to convergence: 10 \n",
-       "Achieved convergence tolerance: 1.49e-08"
-      ]
-     },
-     "metadata": {},
-     "output_type": "display_data"
-    }
-   ],
-   "source": [
-    "nlsLM(V_data ~ V_max * S_data / (K_M + S_data), start = list(V_max = 0.1, K_M = 0.1), lower=c(0.4,0.4), upper=c(100,100))"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "So when you added bounds, the solution was found in two fewer iterations (not a spectacular improvement, but an improvement nevertheless).\n",
-    "\n",
-    "```{note}\n",
-    "The `nls()` function too has an option to provide `lower` and `upper` parameter bounds, but that is only in effect available when using `algorithm = \"port\"` (only available for a particular algorithm). \n",
-    "```\n",
-    "However, if you bound the parameters too much (to excessively narrow ranges), the algorithm cannot search sufficient parameter space (combinations of parameters), and will fail to converge on a good solution. For example:   "
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 150,
-   "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "text/plain": [
-       "Nonlinear regression model\n",
-       "  model: V_data ~ V_max * S_data/(K_M + S_data)\n",
-       "   data: parent.frame()\n",
-       "V_max   K_M \n",
-       "16.09 20.00 \n",
-       " residual sum-of-squares: 9.227\n",
-       "\n",
-       "Number of iterations to convergence: 3 \n",
-       "Achieved convergence tolerance: 1.49e-08"
-      ]
-     },
-     "metadata": {},
-     "output_type": "display_data"
-    }
-   ],
-   "source": [
-    "nlsLM(V_data ~ V_max * S_data / (K_M + S_data), start =  list(V_max = 0.5, K_M = 0.5), lower=c(0.4,0.4), upper=c(20,20))"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "Here the algorithm converged on a poor solution, and in fact took fewer iterations (3) than before to do so. This is because it could not explore sufficient parameter combinations of `V_max` and `K_M` as we have narrowed the range that both these parameters could be allowed to take during the optimization too much."
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "## Diagnostics of an NLLS fit\n",
-    "\n",
-    "NLLS regression carries the same three key assumptions as Linear models:\n",
-    "\n",
-    "* No (in practice, minimal) measurement error in explanatory/independent/predictor variable ($x$-axis variable)\n",
-    "\n",
-    "* Data have constant normal variance &mdash; errors in the $y$-axis are homogeneously distributed over the $x$-axis range\n",
-    "\n",
-    "* The measurement/observation errors are Normally distributed (Gaussian)\n",
-    "\n",
-    "At the very least, it is a good idea to plot the residuals of a fitted NLLS model. Let's do that for our Michaelis-Menten Model fit:"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 151,
-   "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "image/png": 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O8z8d80/sEff0kr2aF63cVn/UDe0ZbwKkUR8xvqnFN02bCMhv8WheObpXqaHKkl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\nyVUAAAAASUVORK5CYII=",
-      "text/plain": [
-       "Plot with title “Histogram of residuals(MM_model6)”"
-      ]
-     },
-     "metadata": {
-      "image/png": {
-       "height": 240,
-       "width": 240
-      }
-     },
-     "output_type": "display_data"
-    }
-   ],
-   "source": [
-    "hist(residuals(MM_model6))"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "The residuals look OK. But this should not come as a surprise because we generated these \"data\" ourselves using normally-distributed errors!\n",
-    "\n",
-    "You may also want to look at further diagnostics, as we did [previously](14-regress:Diagnostics) in the case of Linear models. The most convenient way to do this is to use the `nlstools` package. We will not go into it here, but you can have a look at its [documentation](https://rdrr.io/rforge/nlstools/). Note that you will need to install this package as it is not one of the core (base) R packages. `nlstools` has some other handy utilities as well; for example, its `preview` command allows you to visualise how good the starting values are (by evaluating your non-linear function at those values) in advance of the actual fitting.    "
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "```{note}\n",
-    "For the remaining examples, we will switch to using `nlsLM` instead of `nls`.\n",
-    "```"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "## Allometric scaling of traits\n",
-    "\n",
-    "Now let's move on to a very common class of traits in biology: physical traits like body weight, wing span, body length, limb length, eye size, ear width, etc. \n",
-    "\n",
-    "We will look at a very common phenomenon called [allometric scaling](https://en.wikipedia.org/wiki/Allometry). Allometric relationships between linear measurements such as body length, limb length, wing span, and thorax width are a good way to obtain estimates of body weights of individual organisms. We will look at allometric scaling of body weight vs. total body length in dragonflies and damselfiles. \n",
-    "\n",
-    "Allometric relationships take the form:"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "$$\n",
-    "y = a x^b\n",
-    "$$(eq:allom)"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "where $x$ and $y$ are morphological measures (body length and body weight respectively, in our current example), the constant is the value of $y$ at body length $x = 1$ unit, and $b$ is the scaling \"exponent\". This is also called a power-law, because $y$ relates to $x$ through a simple power. "
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "Let's fit a power low to a typical allometric relationship: The change in body weight vs change in body length. In general, this relationship is a allometry; that is, body weight does not increase proportionally with some measure of body length.    \n",
-    "\n",
-    "First, let's look at the data. You can get the data [here](https://raw.githubusercontent.com/mhasoba/TheMulQuaBio/master/content/data/GenomeSize.csv) (first click on link and use \"Save as\" or `Ctrl+S` to download it as a csv). \n",
-    "\n",
-    "$\\star$ Save the `GenomeSize.csv` data file to your `data` directory, and import it into your R workspace:"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 56,
-   "metadata": {
-    "scrolled": true
-   },
-   "outputs": [
-    {
-     "data": {
-      "text/html": [
-       "<table class=\"dataframe\">\n",
-       "<caption>A data.frame: 6 × 16</caption>\n",
-       "<thead>\n",
-       "\t<tr><th></th><th scope=col>Suborder</th><th scope=col>Family</th><th scope=col>Species</th><th scope=col>GenomeSize</th><th scope=col>GenomeSE</th><th scope=col>GenomeN</th><th scope=col>BodyWeight</th><th scope=col>TotalLength</th><th scope=col>HeadLength</th><th scope=col>ThoraxLength</th><th scope=col>AdbdomenLength</th><th scope=col>ForewingLength</th><th scope=col>HindwingLength</th><th scope=col>ForewingArea</th><th scope=col>HindwingArea</th><th scope=col>MorphologyN</th></tr>\n",
-       "\t<tr><th></th><th scope=col>&lt;chr&gt;</th><th scope=col>&lt;chr&gt;</th><th scope=col>&lt;chr&gt;</th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;int&gt;</th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;int&gt;</th></tr>\n",
-       "</thead>\n",
-       "<tbody>\n",
-       "\t<tr><th scope=row>1</th><td>Anisoptera</td><td>Aeshnidae</td><td>Aeshna canadensis   </td><td>2.20</td><td>  NA</td><td>1</td><td>0.159</td><td>67.58</td><td>6.83</td><td>11.81</td><td>48.94</td><td>45.47</td><td>45.40</td><td>369.57</td><td>483.61</td><td>2</td></tr>\n",
-       "\t<tr><th scope=row>2</th><td>Anisoptera</td><td>Aeshnidae</td><td>Aeshna constricta   </td><td>1.76</td><td>0.06</td><td>4</td><td>0.228</td><td>71.97</td><td>6.84</td><td>10.72</td><td>54.41</td><td>46.00</td><td>45.48</td><td>411.15</td><td>517.38</td><td>3</td></tr>\n",
-       "\t<tr><th scope=row>3</th><td>Anisoptera</td><td>Aeshnidae</td><td>Aeshna eremita      </td><td>1.85</td><td>  NA</td><td>1</td><td>0.312</td><td>78.80</td><td>6.27</td><td>16.19</td><td>56.33</td><td>51.24</td><td>49.47</td><td>460.72</td><td>574.33</td><td>1</td></tr>\n",
-       "\t<tr><th scope=row>4</th><td>Anisoptera</td><td>Aeshnidae</td><td>Aeshna tuberculifera</td><td>1.78</td><td>0.10</td><td>2</td><td>0.218</td><td>72.44</td><td>6.62</td><td>12.53</td><td>53.29</td><td>49.84</td><td>48.82</td><td>468.74</td><td>591.42</td><td>2</td></tr>\n",
-       "\t<tr><th scope=row>5</th><td>Anisoptera</td><td>Aeshnidae</td><td>Aeshna umbrosa      </td><td>2.00</td><td>  NA</td><td>1</td><td>0.207</td><td>73.05</td><td>4.92</td><td>11.11</td><td>57.03</td><td>46.51</td><td>45.97</td><td>382.48</td><td>481.44</td><td>1</td></tr>\n",
-       "\t<tr><th scope=row>6</th><td>Anisoptera</td><td>Aeshnidae</td><td>Aeshna verticalis   </td><td>1.59</td><td>  NA</td><td>1</td><td>0.220</td><td>66.25</td><td>6.48</td><td>11.64</td><td>48.13</td><td>45.91</td><td>44.91</td><td>400.40</td><td>486.97</td><td>1</td></tr>\n",
-       "</tbody>\n",
-       "</table>\n"
-      ],
-      "text/latex": [
-       "A data.frame: 6 × 16\n",
-       "\\begin{tabular}{r|llllllllllllllll}\n",
-       "  & Suborder & Family & Species & GenomeSize & GenomeSE & GenomeN & BodyWeight & TotalLength & HeadLength & ThoraxLength & AdbdomenLength & ForewingLength & HindwingLength & ForewingArea & HindwingArea & MorphologyN\\\\\n",
-       "  & <chr> & <chr> & <chr> & <dbl> & <dbl> & <int> & <dbl> & <dbl> & <dbl> & <dbl> & <dbl> & <dbl> & <dbl> & <dbl> & <dbl> & <int>\\\\\n",
-       "\\hline\n",
-       "\t1 & Anisoptera & Aeshnidae & Aeshna canadensis    & 2.20 &   NA & 1 & 0.159 & 67.58 & 6.83 & 11.81 & 48.94 & 45.47 & 45.40 & 369.57 & 483.61 & 2\\\\\n",
-       "\t2 & Anisoptera & Aeshnidae & Aeshna constricta    & 1.76 & 0.06 & 4 & 0.228 & 71.97 & 6.84 & 10.72 & 54.41 & 46.00 & 45.48 & 411.15 & 517.38 & 3\\\\\n",
-       "\t3 & Anisoptera & Aeshnidae & Aeshna eremita       & 1.85 &   NA & 1 & 0.312 & 78.80 & 6.27 & 16.19 & 56.33 & 51.24 & 49.47 & 460.72 & 574.33 & 1\\\\\n",
-       "\t4 & Anisoptera & Aeshnidae & Aeshna tuberculifera & 1.78 & 0.10 & 2 & 0.218 & 72.44 & 6.62 & 12.53 & 53.29 & 49.84 & 48.82 & 468.74 & 591.42 & 2\\\\\n",
-       "\t5 & Anisoptera & Aeshnidae & Aeshna umbrosa       & 2.00 &   NA & 1 & 0.207 & 73.05 & 4.92 & 11.11 & 57.03 & 46.51 & 45.97 & 382.48 & 481.44 & 1\\\\\n",
-       "\t6 & Anisoptera & Aeshnidae & Aeshna verticalis    & 1.59 &   NA & 1 & 0.220 & 66.25 & 6.48 & 11.64 & 48.13 & 45.91 & 44.91 & 400.40 & 486.97 & 1\\\\\n",
-       "\\end{tabular}\n"
-      ],
-      "text/markdown": [
-       "\n",
-       "A data.frame: 6 × 16\n",
-       "\n",
-       "| <!--/--> | Suborder &lt;chr&gt; | Family &lt;chr&gt; | Species &lt;chr&gt; | GenomeSize &lt;dbl&gt; | GenomeSE &lt;dbl&gt; | GenomeN &lt;int&gt; | BodyWeight &lt;dbl&gt; | TotalLength &lt;dbl&gt; | HeadLength &lt;dbl&gt; | ThoraxLength &lt;dbl&gt; | AdbdomenLength &lt;dbl&gt; | ForewingLength &lt;dbl&gt; | HindwingLength &lt;dbl&gt; | ForewingArea &lt;dbl&gt; | HindwingArea &lt;dbl&gt; | MorphologyN &lt;int&gt; |\n",
-       "|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|\n",
-       "| 1 | Anisoptera | Aeshnidae | Aeshna canadensis    | 2.20 |   NA | 1 | 0.159 | 67.58 | 6.83 | 11.81 | 48.94 | 45.47 | 45.40 | 369.57 | 483.61 | 2 |\n",
-       "| 2 | Anisoptera | Aeshnidae | Aeshna constricta    | 1.76 | 0.06 | 4 | 0.228 | 71.97 | 6.84 | 10.72 | 54.41 | 46.00 | 45.48 | 411.15 | 517.38 | 3 |\n",
-       "| 3 | Anisoptera | Aeshnidae | Aeshna eremita       | 1.85 |   NA | 1 | 0.312 | 78.80 | 6.27 | 16.19 | 56.33 | 51.24 | 49.47 | 460.72 | 574.33 | 1 |\n",
-       "| 4 | Anisoptera | Aeshnidae | Aeshna tuberculifera | 1.78 | 0.10 | 2 | 0.218 | 72.44 | 6.62 | 12.53 | 53.29 | 49.84 | 48.82 | 468.74 | 591.42 | 2 |\n",
-       "| 5 | Anisoptera | Aeshnidae | Aeshna umbrosa       | 2.00 |   NA | 1 | 0.207 | 73.05 | 4.92 | 11.11 | 57.03 | 46.51 | 45.97 | 382.48 | 481.44 | 1 |\n",
-       "| 6 | Anisoptera | Aeshnidae | Aeshna verticalis    | 1.59 |   NA | 1 | 0.220 | 66.25 | 6.48 | 11.64 | 48.13 | 45.91 | 44.91 | 400.40 | 486.97 | 1 |\n",
-       "\n"
-      ],
-      "text/plain": [
-       "  Suborder   Family    Species              GenomeSize GenomeSE GenomeN\n",
-       "1 Anisoptera Aeshnidae Aeshna canadensis    2.20         NA     1      \n",
-       "2 Anisoptera Aeshnidae Aeshna constricta    1.76       0.06     4      \n",
-       "3 Anisoptera Aeshnidae Aeshna eremita       1.85         NA     1      \n",
-       "4 Anisoptera Aeshnidae Aeshna tuberculifera 1.78       0.10     2      \n",
-       "5 Anisoptera Aeshnidae Aeshna umbrosa       2.00         NA     1      \n",
-       "6 Anisoptera Aeshnidae Aeshna verticalis    1.59         NA     1      \n",
-       "  BodyWeight TotalLength HeadLength ThoraxLength AdbdomenLength ForewingLength\n",
-       "1 0.159      67.58       6.83       11.81        48.94          45.47         \n",
-       "2 0.228      71.97       6.84       10.72        54.41          46.00         \n",
-       "3 0.312      78.80       6.27       16.19        56.33          51.24         \n",
-       "4 0.218      72.44       6.62       12.53        53.29          49.84         \n",
-       "5 0.207      73.05       4.92       11.11        57.03          46.51         \n",
-       "6 0.220      66.25       6.48       11.64        48.13          45.91         \n",
-       "  HindwingLength ForewingArea HindwingArea MorphologyN\n",
-       "1 45.40          369.57       483.61       2          \n",
-       "2 45.48          411.15       517.38       3          \n",
-       "3 49.47          460.72       574.33       1          \n",
-       "4 48.82          468.74       591.42       2          \n",
-       "5 45.97          382.48       481.44       1          \n",
-       "6 44.91          400.40       486.97       1          "
-      ]
-     },
-     "metadata": {},
-     "output_type": "display_data"
-    }
-   ],
-   "source": [
-    "MyData <- read.csv(\"../data/GenomeSize.csv\") # using relative path assuming that your working directory is \"code\"\n",
-    "\n",
-    "head(MyData)"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "[Anisoptera](https://en.wikipedia.org/wiki/Dragonfly) are dragonflies, and [Zygoptera](https://en.wikipedia.org/wiki/Damselfly) are Damselflies. The variables of interest are `BodyWeight` and `TotalLength`. Let's use the dragonflies data subset. "
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "So subset the data accordingly and remove NAs:"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 57,
-   "metadata": {},
-   "outputs": [],
-   "source": [
-    "Data2Fit <- subset(MyData,Suborder == \"Anisoptera\")\n",
-    "\n",
-    "Data2Fit <- Data2Fit[!is.na(Data2Fit$TotalLength),] # remove NA's"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "Plot the data:"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 58,
-   "metadata": {
-    "scrolled": true
-   },
-   "outputs": [
-    {
-     "data": {
-      "image/png": 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AJvwUm2PzHbVP/8re46zK5Mr3lVvkKo4acR7HQx376h9qrgqSkoSb2L2ZFZ1eCY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\nc4hgzCGCMYcIxhwiGHP+Dy4zSGvwT6wKAAAAAElFTkSuQmCC",
-      "text/plain": [
-       "plot without title"
-      ]
-     },
-     "metadata": {
-      "image/png": {
-       "height": 240,
-       "width": 240
-      }
-     },
-     "output_type": "display_data"
-    }
-   ],
-   "source": [
-    "plot(Data2Fit$TotalLength, Data2Fit$BodyWeight, xlab = \"Body Length\", ylab = \"Body Weight\")"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "Or, using `ggplot`:"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 59,
-   "metadata": {
-    "scrolled": true
-   },
-   "outputs": [
-    {
-     "data": {
-      "image/png": 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zZthg4dKiowngaQy+Vbt26Njo5OS0urVavWiBEjfHx89PrHEGL1RCIcPIjRo7F7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\nmKARQoih/gf/MmD7hk3h5AAAAABJRU5ErkJggg==",
-      "text/plain": [
-       "plot without title"
-      ]
-     },
-     "metadata": {
-      "image/png": {
-       "height": 240,
-       "width": 240
-      }
-     },
-     "output_type": "display_data"
-    }
-   ],
-   "source": [
-    "library(\"ggplot2\")\n",
-    "\n",
-    "ggplot(Data2Fit, aes(x = TotalLength, y = BodyWeight)) + \n",
-    "geom_point(size = (3),color=\"red\") + theme_bw() + \n",
-    "labs(y=\"Body mass (mg)\", x = \"Wing length (mm)\")"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "You can see these body weights of dragonflies does not increase proportionally with body length &ndash; they curve upwards w.r.t. wing length (so the allometric constant $b$ in eqn {eq}`eq:allom` mustbe greater than 1), instead of increasing as a straight line (in which case $b = 1$ (isometry, instead of allometry). "
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "Now fit the model to the data using NLLS:"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 60,
-   "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "text/html": [
-       "60"
-      ],
-      "text/latex": [
-       "60"
-      ],
-      "text/markdown": [
-       "60"
-      ],
-      "text/plain": [
-       "[1] 60"
-      ]
-     },
-     "metadata": {},
-     "output_type": "display_data"
-    }
-   ],
-   "source": [
-    "nrow(Data2Fit)"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 61,
-   "metadata": {},
-   "outputs": [],
-   "source": [
-    "PowFit <- nlsLM(BodyWeight ~ a * TotalLength^b, data = Data2Fit, start = list(a = .1, b = .1))"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "### NLLS fitting using a model object \n",
-    "\n",
-    "Another way to tell `nlsLM` which model to fit, is to first create a function object for the power law model:"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 62,
-   "metadata": {},
-   "outputs": [],
-   "source": [
-    "powMod <- function(x, a, b) {\n",
-    " return(a * x^b)\n",
-    "}"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "Now fit the model to the data using NLLS by calling the model:"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 63,
-   "metadata": {},
-   "outputs": [],
-   "source": [
-    "PowFit <- nlsLM(BodyWeight ~ powMod(TotalLength,a,b), data = Data2Fit, start = list(a = .1, b = .1))"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "Which gives the same result as before (you can check it)."
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "### Visualizing the fit\n",
-    "\n",
-    "The first thing to do is to see how well the model fitted the data, for which plotting is the best first option. So let's visualize the fit. For this, first we need to generate a vector of body lengths (the x-axis variable) for plotting: "
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 64,
-   "metadata": {},
-   "outputs": [],
-   "source": [
-    "Lengths <- seq(min(Data2Fit$TotalLength),max(Data2Fit$TotalLength),len=200)"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 65,
-   "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "text/html": [
-       "<strong>a:</strong> 3.94068495559397e-06"
-      ],
-      "text/latex": [
-       "\\textbf{a:} 3.94068495559397e-06"
-      ],
-      "text/markdown": [
-       "**a:** 3.94068495559397e-06"
-      ],
-      "text/plain": [
-       "           a \n",
-       "3.940685e-06 "
-      ]
-     },
-     "metadata": {},
-     "output_type": "display_data"
-    },
-    {
-     "data": {
-      "text/html": [
-       "<strong>b:</strong> 2.58504796499038"
-      ],
-      "text/latex": [
-       "\\textbf{b:} 2.58504796499038"
-      ],
-      "text/markdown": [
-       "**b:** 2.58504796499038"
-      ],
-      "text/plain": [
-       "       b \n",
-       "2.585048 "
-      ]
-     },
-     "metadata": {},
-     "output_type": "display_data"
-    }
-   ],
-   "source": [
-    "coef(PowFit)[\"a\"]\n",
-    "coef(PowFit)[\"b\"]"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 66,
-   "metadata": {},
-   "outputs": [],
-   "source": [
-    "Predic2PlotPow <- powMod(Lengths,coef(PowFit)[\"a\"],coef(PowFit)[\"b\"])"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "Next, calculate the predicted line. For this, we will need to extract the coefficient from the model fit object using the `coef()`command. "
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "Now plot the data and the fitted model line:"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 67,
-   "metadata": {
-    "scrolled": true
-   },
-   "outputs": [
-    {
-     "data": {
-      "image/png": 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INfuPgnaF1d3c5/xJs3bwQUDEKIuo4dA+4cECYmEB0NcnJiDegbwD9B9+zZk3f3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\nChM0QghRFCZohBCiqP8DDduZBQhASCIAAAAASUVORK5CYII=",
-      "text/plain": [
-       "plot without title"
-      ]
-     },
-     "metadata": {
-      "image/png": {
-       "height": 240,
-       "width": 240
-      }
-     },
-     "output_type": "display_data"
-    }
-   ],
-   "source": [
-    "plot(Data2Fit$TotalLength, Data2Fit$BodyWeight)\n",
-    "lines(Lengths, Predic2PlotPow, col = 'blue', lwd = 2.5)"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "### Summary of the fit\n",
-    "\n",
-    "Now lets get some stats of this NLLS fit. Having obtained the fit object (`PowMod`), we can use `summary()` just like we would for a `lm()` fit object: "
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 68,
-   "metadata": {
-    "scrolled": true
-   },
-   "outputs": [
-    {
-     "data": {
-      "text/plain": [
-       "\n",
-       "Formula: BodyWeight ~ powMod(TotalLength, a, b)\n",
-       "\n",
-       "Parameters:\n",
-       "   Estimate Std. Error t value Pr(>|t|)    \n",
-       "a 3.941e-06  2.234e-06   1.764    0.083 .  \n",
-       "b 2.585e+00  1.348e-01  19.174   <2e-16 ***\n",
-       "---\n",
-       "Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1\n",
-       "\n",
-       "Residual standard error: 0.02807 on 58 degrees of freedom\n",
-       "\n",
-       "Number of iterations to convergence: 39 \n",
-       "Achieved convergence tolerance: 1.49e-08\n"
-      ]
-     },
-     "metadata": {},
-     "output_type": "display_data"
-    }
-   ],
-   "source": [
-    "summary(PowFit)"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 69,
-   "metadata": {},
-   "outputs": [
-    {
-     "name": "stderr",
-     "output_type": "stream",
-     "text": [
-      "Waiting for profiling to be done...\n",
-      "\n"
-     ]
-    },
-    {
-     "name": "stdout",
-     "output_type": "stream",
-     "text": [
-      "          2.5%        97.5%\n",
-      "a 1.171935e-06 1.205273e-05\n",
-      "b 2.318292e+00 2.872287e+00\n"
-     ]
-    }
-   ],
-   "source": [
-    "print(confint(PowFit))"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "### Examine the residuals \n",
-    "\n",
-    "\n",
-    "As we did before, let's plot the residuals around the fitted model:"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 70,
-   "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "image/png": 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IIdAsFSsSv497VMRMgJwVKxK3jvLzKo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\nIFhx/h/S9XOlYQeW+QAAAABJRU5ErkJggg==",
-      "text/plain": [
-       "Plot with title “Histogram of residuals(PowFit)”"
-      ]
-     },
-     "metadata": {
-      "image/png": {
-       "height": 240,
-       "width": 240
-      }
-     },
-     "output_type": "display_data"
-    }
-   ],
-   "source": [
-    "hist(residuals(PowFit))"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "The residuals look OK."
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "```{tip}\n",
-    "Remember, when you write this analysis into a stand-alone R script, you should put all commands for loading packages (`library()`, `require()`) at the start of the script. *  \n",
-    "```"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "##### Exercises <a id='Allom_Exercises'></a>\n",
-    "\n",
-    "(a) Make the same plot as above, fitted line and all, in `ggplot`, and add (display) the equation you estimated to your new (ggplot) plot. The equation is: $\\text{Weight} = 3.94 \\times 10^{-06} \\times \\text{Length}^{2.59}$\n",
-    "\n",
-    "(b) Try playing with the starting values, and see if you can \"break\" the model fitting -- that is, change the starting values till the NLLS fitting does not converge on a solution.\n",
-    "\n",
-    "(c) Repeat the model fitting (including a-b above) using the Zygoptera data subset.\n",
-    "\n",
-    " \n",
-    "(d) There is an alternative (and in fact, more commonly-used) approach for fitting the allometric model to data: using Ordinary Least Squares on bi-logarithamically transformed data. That is, if you take a log of both sides of the [allometric equation](#eq:allom) we get,\n",
-    "\n",
-    "$$\n",
-    "\\log(y) = \\log(a) + b \\log(x)\n",
-    "$$\n",
-    "\n",
-    "This is a straight line equation of the form $c = d + b z $, where $c = \\log(c)$, $d = \\log(a)$, $z = \\log(x)$, and $b$ is now the slope parameter. So you can use Ordinary Least Squares and the linear models framework (with `lm()`) in R to estimate the parameters of the allometric equation. \n",
-    "\n",
-    "In this exercise, try comparing the NLLS vs OLS methods to see how much difference you get in the parameter estimates between them. For example, see the methods used in this paper by [Cohen et al 2012](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3465447/).\n",
-    "\n",
-    "(e) The allometry between Body weight and Length is not the end of the story. You have a number of other linear morphological measurements (`HeadLength`, `ThoraxLength`, `AdbdomenLength`, `ForewingLength`, `HindwingLength`, `ForewingArea`, and `HindwingArea`) that can also be investigated. In this exercise, try two lines of investigation (again, repeated separately for Dragonflies and Damselfiles): \n",
-    "\n",
-    " (i) How do each of these measures allometrically scale with Body length (obtain estimates of scaling constant and exponent)? (Hint: you may want to use the `pairs()` command in R to get an overview of all the pairs of potential scaling relationships. \n",
-    "\n",
-    " (ii) Do any of the linear morphological measurements other than body length better predict Body weight? That is, does body weight scale more tightly with a linear morphological measurement other than total body length? You would use model selection here, which we will learn next. But for now, you can just look at and compare the $R^2$ values of the models."
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "(Model-Fitting-R-Comparing-Models)=\n",
-    "## Comparing models\n",
-    "\n",
-    " *How do we know that there isn't a better or alternative model that adequately explains the pattern in your dataset?* \n",
-    "\n",
-    "This is important consideration in all data analyses (and more generally, the scientific method!), so you must aim to compare your NLLS model with an one or more alternatives for a more extensive and reliable investigation of the problem. \n",
-    "\n",
-    "Let's use model comparison to investigate whether the relationship between body weight and length we found above is indeed allometric. For this, we need an alternative model that can be fitted to the same data. Let's try a quadratic curve, which is of the form:\n",
-    "\n",
-    "$$\n",
-    "y = a + b x + c x^2\n",
-    "$$\n",
-    "\n",
-    "This can also capture curvature in data, and is an alternative model to the [allometric equation](#eq:allom). Note that this mode is linear in its parameters (a linear model), which you can fit to the simply data using your familiar `lm()` function: "
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 71,
-   "metadata": {},
-   "outputs": [],
-   "source": [
-    "QuaFit <- lm(BodyWeight ~ poly(TotalLength,2), data = Data2Fit)"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "And like before, we obtain the predicted values (but this time using the `predict.lm` function):"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 72,
-   "metadata": {},
-   "outputs": [],
-   "source": [
-    "Predic2PlotQua <- predict.lm(QuaFit, data.frame(TotalLength = Lengths))"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "Now let's plot the two fitted models together:"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 73,
-   "metadata": {
-    "scrolled": true
-   },
-   "outputs": [
-    {
-     "data": {
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66Fh4ffunWroqICIaSrq7t8+fLRo0fb2Niw/4QxNzefMmWKubn5tWvXIEED8HNj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\nYII=",
-      "text/plain": [
-       "plot without title"
-      ]
-     },
-     "metadata": {
-      "image/png": {
-       "height": 240,
-       "width": 240
-      }
-     },
-     "output_type": "display_data"
-    }
-   ],
-   "source": [
-    "plot(Data2Fit$TotalLength, Data2Fit$BodyWeight)\n",
-    "lines(Lengths, Predic2PlotPow, col = 'blue', lwd = 2.5)\n",
-    "lines(Lengths, Predic2PlotQua, col = 'red', lwd = 2.5)"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "Very similar fits, except that the quadratic model seems to deviate a bit from the data at the lower end of the data range. Let's do a proper, formal model comparison now to check which model better-fits the data.\n",
-    "\n",
-    "First calculate the R$^2$ values of the two fitted models:"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 74,
-   "metadata": {
-    "scrolled": true
-   },
-   "outputs": [
-    {
-     "data": {
-      "text/html": [
-       "0.90054752976309"
-      ],
-      "text/latex": [
-       "0.90054752976309"
-      ],
-      "text/markdown": [
-       "0.90054752976309"
-      ],
-      "text/plain": [
-       "[1] 0.9005475"
-      ]
-     },
-     "metadata": {},
-     "output_type": "display_data"
-    },
-    {
-     "data": {
-      "text/html": [
-       "0.900302864503218"
-      ],
-      "text/latex": [
-       "0.900302864503218"
-      ],
-      "text/markdown": [
-       "0.900302864503218"
-      ],
-      "text/plain": [
-       "[1] 0.9003029"
-      ]
-     },
-     "metadata": {},
-     "output_type": "display_data"
-    }
-   ],
-   "source": [
-    "RSS_Pow <- sum(residuals(PowFit)^2) # Residual sum of squares\n",
-    "TSS_Pow <- sum((Data2Fit$BodyWeight - mean(Data2Fit$BodyWeight))^2) # Total sum of squares\n",
-    "RSq_Pow <- 1 - (RSS_Pow/TSS_Pow) # R-squared value\n",
-    "\n",
-    "RSS_Qua <- sum(residuals(QuaFit)^2) # Residual sum of squares\n",
-    "TSS_Qua <- sum((Data2Fit$BodyWeight - mean(Data2Fit$BodyWeight))^2) # Total sum of squares\n",
-    "RSq_Qua <- 1 - (RSS_Qua/TSS_Qua) # R-squared value\n",
-    "\n",
-    "RSq_Pow \n",
-    "RSq_Qua"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "Not very useful. In general, R$^2$ is a good measure of model fit, but cannot be used for model selection &ndash; especially not here, given the tiny difference in R$^2$'s.\n",
-    "\n",
-    "Instead, as explained in the [lecture](https://github.com/mhasoba/TheMulQuaBio/blob/master/lectures/ModelFitting), we can use the Akaike Information Criterion (AIC):"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 75,
-   "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "text/html": [
-       "-2.14742608125084"
-      ],
-      "text/latex": [
-       "-2.14742608125084"
-      ],
-      "text/markdown": [
-       "-2.14742608125084"
-      ],
-      "text/plain": [
-       "[1] -2.147426"
-      ]
-     },
-     "metadata": {},
-     "output_type": "display_data"
-    }
-   ],
-   "source": [
-    "n <- nrow(Data2Fit) #set sample size\n",
-    "pPow <- length(coef(PowFit)) # get number of parameters in power law model\n",
-    "pQua <- length(coef(QuaFit)) # get number of parameters in quadratic model\n",
-    "\n",
-    "AIC_Pow <- n + 2 + n * log((2 * pi) / n) + n * log(RSS_Pow) + 2 * pPow\n",
-    "AIC_Qua <- n + 2 + n * log((2 * pi) / n) + n * log(RSS_Qua) + 2 * pQua\n",
-    "AIC_Pow - AIC_Qua"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "Of course, as you might have suspected, we can do this using an in-built function in R! "
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 76,
-   "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "text/html": [
-       "-2.1474260812509"
-      ],
-      "text/latex": [
-       "-2.1474260812509"
-      ],
-      "text/markdown": [
-       "-2.1474260812509"
-      ],
-      "text/plain": [
-       "[1] -2.147426"
-      ]
-     },
-     "metadata": {},
-     "output_type": "display_data"
-    }
-   ],
-   "source": [
-    "AIC(PowFit) - AIC(QuaFit)"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    " * So which model wins? * As we had dicussed in the NLLS lecture, a rule of thumb is that a AIC value difference (typically denoted as $\\Delta$AIC) >-2 (more negative is better) is a acceptable cutoff for calling a winner. So the power law (allometric model) is a better fit here. Read the [Johnson & Omland paper](https://github.com/mhasoba/TheMulQuaBio/blob/master/readings/Modelling/JohnsonOmland2004.pdf) for more on model selection in Ecology and Evolution. "
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "### Exercises <a id='ModelSelection_Exercises'></a>\n",
-    "\n",
-    "(a) Calculate the Bayesian Information Criterion (BIC), also know as the Schwarz Criterion (see your Lecture notes and the [Johnson & Omland paper](https://github.com/mhasoba/TheMulQuaBio/blob/master/readings/Modelling/JohnsonOmland2004.pdf), and use $\\Delta$BIC to select the better fitting model. \n",
-    "\n",
-    "(b) Fit a straight line to the same data and compare with the allometric and quadratic models.\n",
-    "\n",
-    "(c) Repeat the model comparison (incuding 1-2 above) using the Damselflies (Zygoptera) data subset -- does the allometric model still win?\n",
-    "\n",
-    "(d) Repeat exercise (e)(i) and (ii) from the [above set](#Allom_Exercises), but with model comparison (e.g., again using a quadratic as an alternative model) to establish that the relationships are indeed allometric.\n",
-    "\n",
-    "(e) Repeat exercise (e)(ii) from the [above set](#Allom_Exercises), but with model comparison to establish which linear measurement is the best predictor of Body weight."
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "## Albatross chick growth\n",
-    "\n",
-    "Now let's look at a different trait example: the growth of an individual albatross chick (you can find similar data for vector and non-vector arthropods in [VecTraits](https://vectorbyte.org/)). First load and plot the data:"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 77,
-   "metadata": {
-    "scrolled": true
-   },
-   "outputs": [
-    {
-     "data": {
-      "image/png": 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EymF9x+EBWoFpDVv+rm+BCpCH0FWwxmBA/VtEKirhhbcOLULn0Wqq2WU5Ve+mq8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",
-      "text/plain": [
-       "plot without title"
-      ]
-     },
-     "metadata": {
-      "image/png": {
-       "height": 240,
-       "width": 240
-      }
-     },
-     "output_type": "display_data"
-    }
-   ],
-   "source": [
-    "alb <- read.csv(file=\"../data/albatross_grow.csv\")\n",
-    "alb <- subset(x=alb, !is.na(alb$wt))\n",
-    "plot(alb$age, alb$wt, xlab=\"age (days)\", ylab=\"weight (g)\", xlim=c(0, 100))"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "##### Fitting the three models using NLLS\n",
-    "\n",
-    "Let's fit multiple models to this dataset.\n",
-    "\n",
-    "The Von Bertalanffy model is commonly used for modelling the growth of an individual. It's formulation is:\n",
-    "\n",
-    "$$\n",
-    "W(t) = \\rho (L_{\\infty}(1-e^{-Kt})+L_0 e^{-Kt})^3\n",
-    "$$\n",
-    "\n",
-    "If we pull out $L_{\\infty}$ and define $c=L_0/L_{\\infty}$ and $W_{\\infty}=\\rho L_{\\infty}^3$ this equation becomes:\n",
-    "\n",
-    "$$\n",
-    "W(t) = W_{\\infty}(1-e^{-Kt}+ c e^{-Kt})^3.\n",
-    "$$\n",
-    "\n",
-    "$W_{\\infty}$ is interpreted as the mean asymptotic weight, and $c$ the ratio between the initial and final lengths. This second equation is the one we will fit.\n",
-    "\n",
-    "We will compare this model against the classical Logistic growth equation and a straight line.\n",
-    "\n",
-    "The logistic equation is:\n",
-    "\n",
-    "$$\n",
-    "N_t = \\frac{N_0 K e^{r t}}{K + N_0 (e^{r t} - 1)}\n",
-    "$$\n",
-    "\n",
-    "Here $N_t$ is population size at time $t$, $N_0$ is initial population size, $r$ is maximum growth rate (AKA $r_\\text{max}$), and $K$ is carrying capacity.\n",
-    "\n",
-    "\n",
-    " First, as we did before, let's define the R functions for the two models:"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 78,
-   "metadata": {},
-   "outputs": [],
-   "source": [
-    "logistic1 <- function(t, r, K, N0){\n",
-    " N0 * K * exp(r * t)/(K+N0 * (exp(r * t)-1))\n",
-    "}\n",
-    "\n",
-    "vonbert.w <- function(t, Winf, c, K){\n",
-    " Winf * (1 - exp(-K * t) + c * exp(-K * t))^3\n",
-    "}"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "For the straight line, we use simply use R's `lm()` function, as that is a linear least squares problem. Using NLLS will give (approximately) the same answer, of course. Now fit all 3 models using least squares. \n",
-    "\n",
-    "We will scale the data before fitting to improve the stability of the estimates:"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 79,
-   "metadata": {},
-   "outputs": [],
-   "source": [
-    "scale <- 4000\n",
-    "\n",
-    "alb.lin <- lm(wt/scale ~ age, data = alb)"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 80,
-   "metadata": {},
-   "outputs": [],
-   "source": [
-    "alb.log <- nlsLM(wt/scale~logistic1(age, r, K, N0), start=list(K=1, r=0.1, N0=0.1), data=alb)\n",
-    "\n",
-    "alb.vb <- nlsLM(wt/scale~vonbert.w(age, Winf, c, K), start=list(Winf=0.75, c=0.01, K=0.01), data=alb)"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "Next let's calculate predictions for each of the models across a range of ages."
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 81,
-   "metadata": {},
-   "outputs": [],
-   "source": [
-    "ages <- seq(0, 100, length=1000)\n",
-    "\n",
-    "pred.lin <- predict(alb.lin, newdata = list(age=ages)) * scale\n",
-    "\n",
-    "pred.log <- predict(alb.log, newdata = list(age=ages)) * scale\n",
-    "\n",
-    "pred.vb <- predict(alb.vb, newdata = list(age=ages)) * scale"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "And finally plot the data with the fits:"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 82,
-   "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "image/png": 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ds67/u2srJauHBhTEzM4cOH/fz8YED3LTab3ZV9kDEYDAaD4fP5AAAsFosIbTON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KjQUAFg9f9kdDI4uPAWh+vPEu97rtf/oD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\nYII=",
-      "text/plain": [
-       "plot without title"
-      ]
-     },
-     "metadata": {
-      "image/png": {
-       "height": 240,
-       "width": 240
-      }
-     },
-     "output_type": "display_data"
-    }
-   ],
-   "source": [
-    "plot(alb$age, alb$wt, xlab=\"age (days)\", ylab=\"weight (g)\", xlim=c(0,100))\n",
-    "lines(ages, pred.lin, col=2, lwd=2)\n",
-    "lines(ages, pred.log, col=3, lwd=2)\n",
-    "lines(ages, pred.vb, col=4, lwd=2)\n",
-    "\n",
-    "legend(\"topleft\", legend = c(\"linear\", \"logistic\", \"Von Bert\"), lwd=2, lty=1, col=2:4)"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "Next examine the residuals between the 3 models:"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 83,
-   "metadata": {
-    "scrolled": true
-   },
-   "outputs": [
-    {
-     "data": {
-      "image/png": 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FoDYaGw3NcMuxqEd0B9X363V9EA9OmD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\nCyYdZgAAAABJRU5ErkJggg==",
-      "text/plain": [
-       "Plot with title “VB resids”"
-      ]
-     },
-     "metadata": {
-      "image/png": {
-       "height": 240,
-       "width": 240
-      }
-     },
-     "output_type": "display_data"
-    }
-   ],
-   "source": [
-    "par(mfrow=c(3,1), bty=\"n\")\n",
-    "plot(alb$age, resid(alb.lin), main=\"LM resids\", xlim=c(0,100))\n",
-    "plot(alb$age, resid(alb.log), main=\"Logisitic resids\", xlim=c(0,100))\n",
-    "plot(alb$age, resid(alb.vb), main=\"VB resids\", xlim=c(0,100))"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "The residuals for all 3 models still exhibit some patterns. In particular, the data seems to go down near the end of the observation period, but none of these models can capture that behavior. \n",
-    "\n",
-    "Finally, let's compare the 3 models using a simpler approach than the AIC/BIC one that we used [above](#Allom_Exercises) by calculating adjusted Sums of Squared Errors (SSE's):"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 84,
-   "metadata": {
-    "scrolled": true
-   },
-   "outputs": [
-    {
-     "data": {
-      "text/html": [
-       "<dl>\n",
-       "\t<dt>$lin</dt>\n",
-       "\t\t<dd>0.00958</dd>\n",
-       "\t<dt>$log</dt>\n",
-       "\t\t<dd>0.0056</dd>\n",
-       "\t<dt>$vb</dt>\n",
-       "\t\t<dd>0.00628</dd>\n",
-       "</dl>\n"
-      ],
-      "text/latex": [
-       "\\begin{description}\n",
-       "\\item[\\$lin] 0.00958\n",
-       "\\item[\\$log] 0.0056\n",
-       "\\item[\\$vb] 0.00628\n",
-       "\\end{description}\n"
-      ],
-      "text/markdown": [
-       "$lin\n",
-       ":   0.00958\n",
-       "$log\n",
-       ":   0.0056\n",
-       "$vb\n",
-       ":   0.00628\n",
-       "\n",
-       "\n"
-      ],
-      "text/plain": [
-       "$lin\n",
-       "[1] 0.00958\n",
-       "\n",
-       "$log\n",
-       "[1] 0.0056\n",
-       "\n",
-       "$vb\n",
-       "[1] 0.00628\n"
-      ]
-     },
-     "metadata": {},
-     "output_type": "display_data"
-    }
-   ],
-   "source": [
-    "n <- length(alb$wt)\n",
-    "list(lin=signif(sum(resid(alb.lin)^2)/(n-2 * 2), 3), \n",
-    " log= signif(sum(resid(alb.log)^2)/(n-2 * 3), 3), \n",
-    " vb= signif(sum(resid(alb.vb)^2)/(n-2 * 3), 3))   "
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "The adjusted SSE accounts for sample size and number of parameters by dividing the RSS by the residual degrees of freedom. Adjusted SSE can also be used for model selection like AIC/BIC (but is less robust than AIC/BIC). The residual degrees of freedom is calculated as the number of response values (sample size, $n$) minus 2, times the number of fitted coefficients $m$ (= 2 or 3 in this case) estimated.\n",
-    "\n",
-    "The logistic model has the lowest adjusted SSE, so it's the best by this measure. It is also, visually, a better fit. "
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "### Exercises <a id='Albatross_Exercises'></a>\n",
-    "\n",
-    "(a) Use AIC/BIC to perform model selection on the Albatross data as we did for the trait allometry example.\n",
-    "\n",
-    "(b) Write this example as a self-sufficient R script, with ggplot istead of base plotting "
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "## Aedes aegypti fecundity\n",
-    "\n",
-    "Now let's look at a disease vector example. These data measure the reponse of * Aedes aegypti * fecundity to temperature. \n",
-    "\n",
-    "First load and visualize the data:"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 85,
-   "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "image/png": 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KF0CFYRlA0hkEukmgEvKf8UlOP43+DGvB7/mlwRHKQ4wltwcPU/Pzwps3MkfxPn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\nUDDloGDKQcGUg4IpBwVTDgqmHBRMOf8PIxfXks2/Y1gAAAAASUVORK5CYII=",
-      "text/plain": [
-       "plot without title"
-      ]
-     },
-     "metadata": {
-      "image/png": {
-       "height": 240,
-       "width": 240
-      }
-     },
-     "output_type": "display_data"
-    }
-   ],
-   "source": [
-    "aedes <- read.csv(file=\"../data/aedes_fecund.csv\")\n",
-    "\n",
-    "plot(aedes$T, aedes$EFD, xlab=\"temperature (C)\", ylab=\"Eggs/day\")"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "(Model-Fitting-NLLS-TPCs)=\n",
-    "### The Thermal Performance Curve models\n",
-    "\n",
-    "Let's define some models for Thermal Performance Curves:"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 86,
-   "metadata": {},
-   "outputs": [],
-   "source": [
-    "quad1 <- function(T, T0, Tm, c){\n",
-    " c * (T-T0) * (T-Tm) * as.numeric(T<Tm) * as.numeric(T>T0)\n",
-    "}"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "Instead of using the inbuilt quadratic function in R, we we defined our own to make it easier to choose starting values, and so that we can force the function to be equal to zero above and below the minimum and maximum temperature thresholds (more on this below)."
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 87,
-   "metadata": {},
-   "outputs": [],
-   "source": [
-    "briere <- function(T, T0, Tm, c){\n",
-    " c * T * (T-T0) * (abs(Tm-T)^(1/2)) * as.numeric(T<Tm) * as.numeric(T>T0)\n",
-    "}"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "The Briere function is a commonly used model for temperature dependence of insect traits. See here [section](Miniproj-TPCs-Models) for more info. Unlike the original [model definition](Miniproj-TPCs-Models), we have used `abs()` to allow the NLLS algorithm to explore the full parameter space of $T_m$; if we did not do this, the NLLS could fail as soon as a value of $T_m < T$ was reached during the optimization, because the [square root of a negative number is complex](https://en.wikipedia.org/wiki/Square_root). Another way to deal with this issue is to set parameter bounds on $T_m$ so that it is can never be less than T. However, this is a more technical approach that we will not go into here.   \n",
-    " \n",
-    "As in the case of the albatross growth data, we will also compare the above two models with a * straight line * (again, its a linear model, so we can just use `lm()` without needing to define a function for it). \n",
-    "\n",
-    "Now fit all 3 models using least squares. Although it's not as necessary here (as the data don't have as large values as the albatross example), lets again scale the data first: "
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 88,
-   "metadata": {},
-   "outputs": [],
-   "source": [
-    "scale <- 20\n",
-    "\n",
-    "aed.lin <- lm(EFD/scale ~ T, data=aedes)\n",
-    "\n",
-    "aed.quad <- nlsLM(EFD/scale~quad1(T, T0, Tm, c), start=list(T0=10, Tm=40, c=0.01), data=aedes)\n",
-    "\n",
-    "aed.br <- nlsLM(EFD/scale~briere(T, T0, Tm, c), start=list(T0=10, Tm=40, c=0.1), data=aedes)"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "### Exercises <a id='Aedes_Exercises'></a>\n",
-    "\n",
-    "(a) Complete the * Aedes * data analysis by fitting the models, calculating predictions and then comparing models. Write a single, self-standing script for it. Which model fits best? By what measure?\n",
-    "\n",
-    "(b) In this script, use ggplot instead of base plotting."
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "## Abundance data as an example\n",
-    "\n",
-    "Fluctuations in the abundance (density) of single populations may play a crucial role in ecosystem dynamics and emergent functional characteristics, such as rates of carbon fixation or disease transmission. For example, if vector population densities or their traits change at the same or shorter timescales than the rate of disease transmission, then (vector) abundance fluctuations can cause significant fluctuations in disease transmission rates. Indeed, most disease vectors are small ectotherms with short generation times and greater sensitivity to environmental conditions than their (invariably larger, longer-lived, and often, endothermic) hosts. So understanding how populations vary over time, space, and with respect to environmental variables such as temperature and precipitation is key. We will look at fitting models to the growth of a single population here. \n",
-    "\n",
-    "(Model-Fitting-R-Population-Growth)=\n",
-    "\n",
-    "## Population growth rates\n",
-    "\n",
-    "A population grows exponentially while its abundance is low and resources are not limiting (the Malthusian principle). This growth then slows and eventually stops as resources become limiting. There may also be a time lag before the population growth really takes off at the start. We will focus on microbial (specifically, bacterial) growth rates. Bacterial growth in batch culture follows a distinct set of phases; lag phase, exponential phase and stationary phase. During the lag phase a suite of transcriptional machinery is activated, including genes involved in nutrient uptake and metabolic changes, as bacteria prepare for growth. During the exponential growth phase, bacteria divide at a constant rate, the population doubling with each generation. When the carrying capacity of the media is reached, growth slows and the number of cells in the culture stabilizes, beginning the stationary phase. \n",
-    "\n",
-    "Traditionally, microbial growth rates were measured by plotting cell numbers or culture density against time on a semi-log graph and fitting a straight line through the exponential growth phase &ndash; the slope of the line gives the maximum growth rate ($r_{max}$). Models have since been developed which we can use to describe the whole sigmoidal bacterial growth curve (e.g., using NLLS). Here we will take a look at these different approaches, from applying linear models to the exponential phase, through to fitting non-linear models to the full growth curve.\n",
-    "\n",
-    "Let's first generate some \"data\" on the number of bacterial cells as a function of time that we can play with:"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 89,
-   "metadata": {
-    "scrolled": true
-   },
-   "outputs": [
-    {
-     "data": {
-      "text/html": [
-       "<table class=\"dataframe\">\n",
-       "<caption>A data.frame: 6 × 2</caption>\n",
-       "<thead>\n",
-       "\t<tr><th></th><th scope=col>Time</th><th scope=col>N</th></tr>\n",
-       "\t<tr><th></th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th></tr>\n",
-       "</thead>\n",
-       "<tbody>\n",
-       "\t<tr><th scope=row>1</th><td> 0</td><td>  28577.04</td></tr>\n",
-       "\t<tr><th scope=row>2</th><td> 2</td><td>  33915.52</td></tr>\n",
-       "\t<tr><th scope=row>3</th><td> 4</td><td>  42120.88</td></tr>\n",
-       "\t<tr><th scope=row>4</th><td> 6</td><td>  80370.17</td></tr>\n",
-       "\t<tr><th scope=row>5</th><td> 8</td><td> 464096.05</td></tr>\n",
-       "\t<tr><th scope=row>6</th><td>10</td><td>1502365.99</td></tr>\n",
-       "</tbody>\n",
-       "</table>\n"
-      ],
-      "text/latex": [
-       "A data.frame: 6 × 2\n",
-       "\\begin{tabular}{r|ll}\n",
-       "  & Time & N\\\\\n",
-       "  & <dbl> & <dbl>\\\\\n",
-       "\\hline\n",
-       "\t1 &  0 &   28577.04\\\\\n",
-       "\t2 &  2 &   33915.52\\\\\n",
-       "\t3 &  4 &   42120.88\\\\\n",
-       "\t4 &  6 &   80370.17\\\\\n",
-       "\t5 &  8 &  464096.05\\\\\n",
-       "\t6 & 10 & 1502365.99\\\\\n",
-       "\\end{tabular}\n"
-      ],
-      "text/markdown": [
-       "\n",
-       "A data.frame: 6 × 2\n",
-       "\n",
-       "| <!--/--> | Time &lt;dbl&gt; | N &lt;dbl&gt; |\n",
-       "|---|---|---|\n",
-       "| 1 |  0 |   28577.04 |\n",
-       "| 2 |  2 |   33915.52 |\n",
-       "| 3 |  4 |   42120.88 |\n",
-       "| 4 |  6 |   80370.17 |\n",
-       "| 5 |  8 |  464096.05 |\n",
-       "| 6 | 10 | 1502365.99 |\n",
-       "\n"
-      ],
-      "text/plain": [
-       "  Time N         \n",
-       "1  0     28577.04\n",
-       "2  2     33915.52\n",
-       "3  4     42120.88\n",
-       "4  6     80370.17\n",
-       "5  8    464096.05\n",
-       "6 10   1502365.99"
-      ]
-     },
-     "metadata": {},
-     "output_type": "display_data"
-    }
-   ],
-   "source": [
-    "t <- seq(0, 22, 2)\n",
-    "N <- c(32500, 33000, 38000, 105000, 445000, 1430000, 3020000, 4720000, 5670000, 5870000, 5930000, 5940000)\n",
-    "\n",
-    "set.seed(1234) # To ensure we always get the same random sequence in this example \"dataset\" \n",
-    "\n",
-    "data <- data.frame(t, N * (1 + rnorm(length(t), sd = 0.1))) # add some random error\n",
-    "\n",
-    "names(data) <- c(\"Time\", \"N\")\n",
-    "\n",
-    "head(data)"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "Note how we added some random \"sampling\" error using `N * (1 + rnorm(length(t), sd = .1))`. \n",
-    "\n",
-    "This means that we are adding an error at each time point $t$ (let's call this fluctuation $\\epsilon_t$) as a *percentage* of the population ($N_t$) at that time point in a vectorized way. That is, we are performing the operation $N_t \\times (1 + \\epsilon_t)$ at all time points at one go. This is important to note because this is often the way that errors appear &ndash; proportional to the value being measured."
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "Now let's plot these data:"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 90,
-   "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "image/png": 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LaIkDEdAFzZ5bcfFISwRxbmWs0AnNTg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",
-      "text/plain": [
-       "plot without title"
-      ]
-     },
-     "metadata": {
-      "image/png": {
-       "height": 240,
-       "width": 240
-      }
-     },
-     "output_type": "display_data"
-    }
-   ],
-   "source": [
-    "ggplot(data, aes(x = Time, y = N)) + \n",
-    " geom_point(size = 3) +\n",
-    " labs(x = \"Time (Hours)\", y = \"Population size (cells)\")"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "#### Basic approach\n",
-    "\n",
-    "The size of an exponentially growing population ($N$) at any given time ($t$) is given by:\n",
-    "\n",
-    "$\n",
-    "N(t) = N_0 e^{rt} ,\n",
-    "$\n",
-    "\n",
-    "where $N_0$ is the initial population size and $r$ is the growth rate. We can re-arrange this to give:\n",
-    "\n",
-    "$\n",
-    "r = \\frac{\\log(N(t)) - \\log(N_0)}{t} ,\n",
-    "$\n",
-    "\n",
-    "That is, in exponential growth at a constant rate, the growth rate can be simply calculated as the difference in the log of two population sizes, over time. We will log-transform the data and estimate by eye where growth looks exponential."
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 91,
-   "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "image/png": 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-      "text/plain": [
-       "plot without title"
-      ]
-     },
-     "metadata": {
-      "image/png": {
-       "height": 240,
-       "width": 240
-      }
-     },
-     "output_type": "display_data"
-    }
-   ],
-   "source": [
-    "data$LogN <- log(data$N)\n",
-    "\n",
-    "# visualise\n",
-    "ggplot(data, aes(x = t, y = LogN)) + \n",
-    " geom_point(size = 3) +\n",
-    " labs(x = \"Time (Hours)\", y = \"log(cell number)\")"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "By eye the logged data looks fairly linear (beyond the initial \"lag phase\" of growth; see below) between hours 5 and 10, so we'll use that time-period to calculate the growth rate. "
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 92,
-   "metadata": {
-    "scrolled": true
-   },
-   "outputs": [
-    {
-     "data": {
-      "text/html": [
-       "0.732038333517017"
-      ],
-      "text/latex": [
-       "0.732038333517017"
-      ],
-      "text/markdown": [
-       "0.732038333517017"
-      ],
-      "text/plain": [
-       "[1] 0.7320383"
-      ]
-     },
-     "metadata": {},
-     "output_type": "display_data"
-    }
-   ],
-   "source": [
-    "(data[data$Time == 10,]$LogN - data[data$Time == 6,]$LogN)/(10-6)"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "This is our first, most basic estimate of $r$.\n",
-    "\n",
-    "Or, we can decide not to eyeball the data, but just pick the maximum observed gradient of the curve. For this, we can use the the `diff()` function:"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 93,
-   "metadata": {
-    "scrolled": true
-   },
-   "outputs": [
-    {
-     "data": {
-      "text/html": [
-       "<style>\n",
-       ".list-inline {list-style: none; margin:0; padding: 0}\n",
-       ".list-inline>li {display: inline-block}\n",
-       ".list-inline>li:not(:last-child)::after {content: \"\\00b7\"; padding: 0 .5ex}\n",
-       "</style>\n",
-       "<ol class=list-inline><li>0.171269154259665</li><li>0.216670872460636</li><li>0.646099642770272</li><li>1.75344839347772</li><li>1.17470494059035</li><li>0.639023867964838</li><li>0.44952974020198</li><li>0.181493481601755</li><li>-0.000450183952025895</li><li>0.0544907101941003</li><li>-0.0546009242768832</li></ol>\n"
-      ],
-      "text/latex": [
-       "\\begin{enumerate*}\n",
-       "\\item 0.171269154259665\n",
-       "\\item 0.216670872460636\n",
-       "\\item 0.646099642770272\n",
-       "\\item 1.75344839347772\n",
-       "\\item 1.17470494059035\n",
-       "\\item 0.639023867964838\n",
-       "\\item 0.44952974020198\n",
-       "\\item 0.181493481601755\n",
-       "\\item -0.000450183952025895\n",
-       "\\item 0.0544907101941003\n",
-       "\\item -0.0546009242768832\n",
-       "\\end{enumerate*}\n"
-      ],
-      "text/markdown": [
-       "1. 0.171269154259665\n",
-       "2. 0.216670872460636\n",
-       "3. 0.646099642770272\n",
-       "4. 1.75344839347772\n",
-       "5. 1.17470494059035\n",
-       "6. 0.639023867964838\n",
-       "7. 0.44952974020198\n",
-       "8. 0.181493481601755\n",
-       "9. -0.000450183952025895\n",
-       "10. 0.0544907101941003\n",
-       "11. -0.0546009242768832\n",
-       "\n",
-       "\n"
-      ],
-      "text/plain": [
-       " [1]  0.171269154  0.216670872  0.646099643  1.753448393  1.174704941\n",
-       " [6]  0.639023868  0.449529740  0.181493482 -0.000450184  0.054490710\n",
-       "[11] -0.054600924"
-      ]
-     },
-     "metadata": {},
-     "output_type": "display_data"
-    }
-   ],
-   "source": [
-    "diff(data$LogN)"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "This gives all the (log) population size differences between successive timepoint pairs. The max of this is what we want, divided by the time-step."
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 94,
-   "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "text/html": [
-       "0.87672419673886"
-      ],
-      "text/latex": [
-       "0.87672419673886"
-      ],
-      "text/markdown": [
-       "0.87672419673886"
-      ],
-      "text/plain": [
-       "[1] 0.8767242"
-      ]
-     },
-     "metadata": {},
-     "output_type": "display_data"
-    }
-   ],
-   "source": [
-    "max(diff(data$LogN))/2 # 2 is the difference in any successive pair of timepoints"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "### Using OLS\n",
-    "\n",
-    "But we can do better than this. To account for some error in measurement, we shouldn't really take the data points directly, but fit a linear model through them instead, where the slope gives our growth rate. This is pretty much the \"traditional\" way of calculating microbial growth rates &ndash; draw a straight line through the linear part of the log-transformed data."
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 95,
-   "metadata": {
-    "scrolled": true
-   },
-   "outputs": [
-    {
-     "data": {
-      "text/plain": [
-       "\n",
-       "Call:\n",
-       "lm(formula = LogN ~ Time, data = data[data$Time > 2 & data$Time < \n",
-       "    12, ])\n",
-       "\n",
-       "Residuals:\n",
-       "       3        4        5        6 \n",
-       " 0.21646 -0.38507  0.12076  0.04785 \n",
-       "\n",
-       "Coefficients:\n",
-       "            Estimate Std. Error t value Pr(>|t|)   \n",
-       "(Intercept)   7.9366     0.5350  14.835  0.00451 **\n",
-       "Time          0.6238     0.0728   8.569  0.01335 * \n",
-       "---\n",
-       "Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1\n",
-       "\n",
-       "Residual standard error: 0.3256 on 2 degrees of freedom\n",
-       "Multiple R-squared:  0.9735,\tAdjusted R-squared:  0.9602 \n",
-       "F-statistic: 73.42 on 1 and 2 DF,  p-value: 0.01335\n"
-      ]
-     },
-     "metadata": {},
-     "output_type": "display_data"
-    }
-   ],
-   "source": [
-    "lm_growth <- lm(LogN ~ Time, data = data[data$Time > 2 & data$Time < 12,])\n",
-    "summary(lm_growth)"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "Npw we get $r \\approx 0.62$, which is probably closer to the \"truth\". \n",
-    "\n",
-    "But this is still not ideal because we only guessed the exponential phase by eye. We could do it better by iterating through different windows of points, comparing the slopes and finding which the highest is to give the maximum growth rate, $r_{max}$. This is called a \"rolling regression\". \n",
-    "\n",
-    "Or better still, we can fit a more appropriate mathematical model using NLLS!"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "### Using NLLS\n",
-    "\n",
-    "For starters, a classical, (somewhat) mechanistic model is the logistic equation:\n",
-    "\n",
-    "$$\n",
-    "N_t = \\frac{N_0 K e^{r t}}{K + N_0 (e^{r t} - 1)}\n",
-    "$$(eq:logist_growth_sol)\n",
-    "\n",
-    "Here $N_t$ is population size at time $t$, $N_0$ is initial population size, $r$ is maximum growth rate (AKA $r_\\text{max}$), and $K$ is carrying capacity (maximum possible abundance of the population). Note that this model is actually the solution to the differential equation that defines the classic logistic population growth model (eqn. {eq}`eq:logist_growth`). \n",
-    "\n",
-    "```{note}\n",
-    "The derivation of eqn. {eq}`eq:logist_growth_sol` is covered [here](Logistic-Population-Growth). But you don't need to know the derivation to fit eqn. {eq}`eq:logist_growth_sol` to data. \n",
-    "```\n",
-    "\n",
-    "Let's fit it to the data. First, we need to define it as a function object:"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 96,
-   "metadata": {},
-   "outputs": [],
-   "source": [
-    "logistic_model <- function(t, r_max, K, N_0){ # The classic logistic equation\n",
-    " return(N_0 * K * exp(r_max * t)/(K + N_0 * (exp(r_max * t) - 1)))\n",
-    "}"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "Now fit it:"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 97,
-   "metadata": {
-    "scrolled": true
-   },
-   "outputs": [
-    {
-     "data": {
-      "text/plain": [
-       "\n",
-       "Formula: N ~ logistic_model(t = Time, r_max, K, N_0)\n",
-       "\n",
-       "Parameters:\n",
-       "       Estimate Std. Error t value Pr(>|t|)    \n",
-       "r_max 6.309e-01  3.791e-02  16.641 4.56e-08 ***\n",
-       "N_0   3.317e+03  1.451e+03   2.286   0.0481 *  \n",
-       "K     5.538e+06  7.192e+04  76.995 5.32e-14 ***\n",
-       "---\n",
-       "Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1\n",
-       "\n",
-       "Residual standard error: 119200 on 9 degrees of freedom\n",
-       "\n",
-       "Number of iterations to convergence: 12 \n",
-       "Achieved convergence tolerance: 1.49e-08\n"
-      ]
-     },
-     "metadata": {},
-     "output_type": "display_data"
-    }
-   ],
-   "source": [
-    "# first we need some starting parameters for the model\n",
-    "N_0_start <- min(data$N) # lowest population size\n",
-    "K_start <- max(data$N) # highest population size\n",
-    "r_max_start <- 0.62 # use our estimate from the OLS fitting from above\n",
-    "\n",
-    "fit_logistic <- nlsLM(N ~ logistic_model(t = Time, r_max, K, N_0), data,\n",
-    "      list(r_max=r_max_start, N_0 = N_0_start, K = K_start))\n",
-    "\n",
-    "summary(fit_logistic)"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "We did not pay much attention to what starting values we used in the simpler example of fitting the allometric model because the power-law model is easy to fit using NLLS, and starting far from the optimal parameters does not matter too much. Here, we used the actual data to generate more realistic start values for each of the three parameters (`r_max`, `N_0`, `K`) of the Logistic equation."
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "Now, plot the fit:"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 98,
-   "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "image/png": 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0aCEQCEaMGCGXy3v37m3J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\nBDQAAEchoAEAOAoBDQDAUf8PHH0c6gu0+0gAAAAASUVORK5CYII=",
-      "text/plain": [
-       "plot without title"
-      ]
-     },
-     "metadata": {
-      "image/png": {
-       "height": 240,
-       "width": 240
-      }
-     },
-     "output_type": "display_data"
-    }
-   ],
-   "source": [
-    "timepoints <- seq(0, 22, 0.1)\n",
-    "\n",
-    "logistic_points <- logistic_model(t = timepoints, \n",
-    "         r_max = coef(fit_logistic)[\"r_max\"], \n",
-    "         K = coef(fit_logistic)[\"K\"], \n",
-    "         N_0 = coef(fit_logistic)[\"N_0\"])\n",
-    "df1 <- data.frame(timepoints, logistic_points)\n",
-    "df1$model <- \"Logistic equation\"\n",
-    "names(df1) <- c(\"Time\", \"N\", \"model\")\n",
-    "\n",
-    "ggplot(data, aes(x = Time, y = N)) +\n",
-    " geom_point(size = 3) +\n",
-    " geom_line(data = df1, aes(x = Time, y = N, col = model), size = 1) +\n",
-    " theme(aspect.ratio=1)+ # make the plot square \n",
-    " labs(x = \"Time\", y = \"Cell number\")"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "That looks nice, and the $r_{max}$ estimate we get (0.64) is fairly close to what we got above with OLS fitting. \n",
-    "\n",
-    "Note that we've done this fitting to the original non transformed data, whilst the linear regressions earlier were on log transformed data. What would this function look like on a log-transformed axis?"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 99,
-   "metadata": {
-    "scrolled": true
-   },
-   "outputs": [
-    {
-     "data": {
-      "image/png": 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FNwcLCDg0OPHj02b95c4b3u3r07ceJET09PBweHkJCQgwcP1scnhHLmnkEfPny4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\nRtgAAAAASUVORK5CYII=",
-      "text/plain": [
-       "plot without title"
-      ]
-     },
-     "metadata": {
-      "image/png": {
-       "height": 240,
-       "width": 240
-      }
-     },
-     "output_type": "display_data"
-    }
-   ],
-   "source": [
-    "ggplot(data, aes(x = Time, y = LogN)) +\n",
-    " geom_point(size = 3) +\n",
-    " geom_line(data = df1, aes(x = Time, y = log(N), col = model), size = 1) +\n",
-    " theme(aspect.ratio=1)+ \n",
-    " labs(x = \"Time\", y = \"log(Cell number)\")"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "The model actually diverges from the data at the lower end! This was not visible in the previous plot where you examined the model in linear scale (without taking a log) because the deviation of the model is small, and only becomes clear in the log scale. This is because of the way logarithms work. Let's have a look at this in our Cell counts \"data\":"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 100,
-   "metadata": {
-    "scrolled": true
-   },
-   "outputs": [
-    {
-     "data": {
-      "image/png": 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-      "text/plain": [
-       "plot without title"
-      ]
-     },
-     "metadata": {
-      "image/png": {
-       "height": 240,
-       "width": 240
-      }
-     },
-     "output_type": "display_data"
-    }
-   ],
-   "source": [
-    "ggplot(data, aes(x = N, y = LogN)) +\n",
-    " geom_point(size = 3) +\n",
-    " theme(aspect.ratio = 1)+ \n",
-    " labs(x = \"N\", y = \"log(N)\")"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "As you can see the logarithm is a strongly nonlinear transformation of any sequence of real numbers, with small numbers close to zero yielding disproportionately large deviations."
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "```{note} \n",
-    "You may play with increasing the error (by increasing the value of `sd` in synthetic data generation step above) and re-evaluating all the subsequent model fitting steps above. However, note that above some values of `sd`, you will start to get negative values of populations, especially at early time points, which will raise issues with taking a logarithm.\n",
-    "```"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "The above seen deviation of the Logistic model from the data is because this model assumes that the population is growing right from the start (Time = 0), while in \"reality\" (in our synthetic \"data\"), this is not what's happening; the population takes a while to grow truly exponentially (i.e., there is a time lag in the population growth). This time lag is seen frequently in the lab, and is also expected in nature, because when bacteria encounter fresh growth media (in the lab) or a new resource/environment (in the field), they take some time to acclimate, activating genes involved in nutrient uptake and metabolic processes, before beginning exponential growth. This is called the lag phase and can be seen in our example data where exponential growth doesn't properly begin until around the 4th hour.\n",
-    "\n",
-    "To capture the lag phase, more complicated bacterial growth models have been designed. \n",
-    "\n",
-    "One of these is the modified Gompertz model (Zwietering et. al., 1990), which has been used frequently in the literature to model bacterial growth:\n",
-    "\n",
-    "$$ \n",
-    "  \\log(N_t) = N_0 + (N_{max} - N_0) e^{-e^{r_{max} \\exp(1) \\frac{t_{lag} - t}{(N_{max} - N_0) \\log(10)} + 1}}\n",
-    "$$(eq:Gompertz)\n",
-    "\n",
-    "Here maximum growth rate ($r_{max}$) is the tangent to the inflection point, $t_{lag}$ is the x-axis intercept to this tangent (duration of the delay before the population starts growing exponentially) and $\\log\\left(\\frac{N_{max}}{N_0}\\right)$ is the asymptote of the log-transformed population growth trajectory, i.e., the log ratio of maximum population density $N_{max}$ (aka \"carrying capacity\") and initial cell (Population) $N_0$ density."
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "```{note}\n",
-    "Note that unlike the Logistic growth model above, the Gompertz model is in the log scale. This is because the model is not derived from a differential equation, but was designed * specifically * to be fitted to log-transformed data.\n",
-    "```\n",
-    "\n",
-    "Now let's fit and compare the two alternative nonlinear growth models: Logistic and Gompertz. \n",
-    "\n",
-    "First, specify the function object for the Gompertz model (we already defined the function for the Logistic model above):"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 101,
-   "metadata": {},
-   "outputs": [],
-   "source": [
-    "gompertz_model <- function(t, r_max, K, N_0, t_lag){ # Modified gompertz growth model (Zwietering 1990)\n",
-    " return(N_0 + (K - N_0) * exp(-exp(r_max * exp(1) * (t_lag - t)/((K - N_0) * log(10)) + 1)))\n",
-    "}   "
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "Again, note that unlike the Logistic growth function above, this function has been written in the log scale. \n",
-    "\n",
-    "Now let's generate some starting values for the NLLS fitting of the Gompertz model. \n",
-    "\n",
-    "As we did above for the logistic equation, let's derive the starting values by using the actual data: "
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 102,
-   "metadata": {},
-   "outputs": [],
-   "source": [
-    "N_0_start <- min(data$LogN) # lowest population size, note log scale\n",
-    "K_start <- max(data$LogN) # highest population size, note log scale\n",
-    "r_max_start <- 0.62 # use our previous estimate from the OLS fitting from above\n",
-    "t_lag_start <- data$Time[which.max(diff(diff(data$LogN)))] # find last timepoint of lag phase"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    " * So how did we find a reasonable time lag from the data? * \n",
-    "\n",
-    "Let's break the last command down:"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 103,
-   "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "text/html": [
-       "<style>\n",
-       ".list-inline {list-style: none; margin:0; padding: 0}\n",
-       ".list-inline>li {display: inline-block}\n",
-       ".list-inline>li:not(:last-child)::after {content: \"\\00b7\"; padding: 0 .5ex}\n",
-       "</style>\n",
-       "<ol class=list-inline><li>0.171269154259665</li><li>0.216670872460636</li><li>0.646099642770272</li><li>1.75344839347772</li><li>1.17470494059035</li><li>0.639023867964838</li><li>0.44952974020198</li><li>0.181493481601755</li><li>-0.000450183952025895</li><li>0.0544907101941003</li><li>-0.0546009242768832</li></ol>\n"
-      ],
-      "text/latex": [
-       "\\begin{enumerate*}\n",
-       "\\item 0.171269154259665\n",
-       "\\item 0.216670872460636\n",
-       "\\item 0.646099642770272\n",
-       "\\item 1.75344839347772\n",
-       "\\item 1.17470494059035\n",
-       "\\item 0.639023867964838\n",
-       "\\item 0.44952974020198\n",
-       "\\item 0.181493481601755\n",
-       "\\item -0.000450183952025895\n",
-       "\\item 0.0544907101941003\n",
-       "\\item -0.0546009242768832\n",
-       "\\end{enumerate*}\n"
-      ],
-      "text/markdown": [
-       "1. 0.171269154259665\n",
-       "2. 0.216670872460636\n",
-       "3. 0.646099642770272\n",
-       "4. 1.75344839347772\n",
-       "5. 1.17470494059035\n",
-       "6. 0.639023867964838\n",
-       "7. 0.44952974020198\n",
-       "8. 0.181493481601755\n",
-       "9. -0.000450183952025895\n",
-       "10. 0.0544907101941003\n",
-       "11. -0.0546009242768832\n",
-       "\n",
-       "\n"
-      ],
-      "text/plain": [
-       " [1]  0.171269154  0.216670872  0.646099643  1.753448393  1.174704941\n",
-       " [6]  0.639023868  0.449529740  0.181493482 -0.000450184  0.054490710\n",
-       "[11] -0.054600924"
-      ]
-     },
-     "metadata": {},
-     "output_type": "display_data"
-    }
-   ],
-   "source": [
-    "diff(data$LogN) # same as what we did above - get differentials"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 104,
-   "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "text/html": [
-       "<style>\n",
-       ".list-inline {list-style: none; margin:0; padding: 0}\n",
-       ".list-inline>li {display: inline-block}\n",
-       ".list-inline>li:not(:last-child)::after {content: \"\\00b7\"; padding: 0 .5ex}\n",
-       "</style>\n",
-       "<ol class=list-inline><li>0.0454017182009707</li><li>0.429428770309636</li><li>1.10734875070745</li><li>-0.578743452887371</li><li>-0.535681072625511</li><li>-0.189494127762858</li><li>-0.268036258600224</li><li>-0.181943665553781</li><li>0.0549408941461262</li><li>-0.109091634470984</li></ol>\n"
-      ],
-      "text/latex": [
-       "\\begin{enumerate*}\n",
-       "\\item 0.0454017182009707\n",
-       "\\item 0.429428770309636\n",
-       "\\item 1.10734875070745\n",
-       "\\item -0.578743452887371\n",
-       "\\item -0.535681072625511\n",
-       "\\item -0.189494127762858\n",
-       "\\item -0.268036258600224\n",
-       "\\item -0.181943665553781\n",
-       "\\item 0.0549408941461262\n",
-       "\\item -0.109091634470984\n",
-       "\\end{enumerate*}\n"
-      ],
-      "text/markdown": [
-       "1. 0.0454017182009707\n",
-       "2. 0.429428770309636\n",
-       "3. 1.10734875070745\n",
-       "4. -0.578743452887371\n",
-       "5. -0.535681072625511\n",
-       "6. -0.189494127762858\n",
-       "7. -0.268036258600224\n",
-       "8. -0.181943665553781\n",
-       "9. 0.0549408941461262\n",
-       "10. -0.109091634470984\n",
-       "\n",
-       "\n"
-      ],
-      "text/plain": [
-       " [1]  0.04540172  0.42942877  1.10734875 -0.57874345 -0.53568107 -0.18949413\n",
-       " [7] -0.26803626 -0.18194367  0.05494089 -0.10909163"
-      ]
-     },
-     "metadata": {},
-     "output_type": "display_data"
-    }
-   ],
-   "source": [
-    "diff(diff(data$LogN)) # get the differentials of the differentials (approx 2nd order derivatives)"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 105,
-   "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "text/html": [
-       "3"
-      ],
-      "text/latex": [
-       "3"
-      ],
-      "text/markdown": [
-       "3"
-      ],
-      "text/plain": [
-       "[1] 3"
-      ]
-     },
-     "metadata": {},
-     "output_type": "display_data"
-    }
-   ],
-   "source": [
-    "which.max(diff(diff(data$LogN))) # find the timepoint where this 2nd order derivative really takes off "
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 106,
-   "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "text/html": [
-       "4"
-      ],
-      "text/latex": [
-       "4"
-      ],
-      "text/markdown": [
-       "4"
-      ],
-      "text/plain": [
-       "[1] 4"
-      ]
-     },
-     "metadata": {},
-     "output_type": "display_data"
-    }
-   ],
-   "source": [
-    "data$Time[which.max(diff(diff(data$LogN)))] # This then is a good guess for the last timepoint of the lag phase"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "Now fit the model using these start values:"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 107,
-   "metadata": {
-    "scrolled": true
-   },
-   "outputs": [],
-   "source": [
-    "fit_gompertz <- nlsLM(LogN ~ gompertz_model(t = Time, r_max, K, N_0, t_lag), data,\n",
-    "      list(t_lag=t_lag_start, r_max=r_max_start, N_0 = N_0_start, K = K_start))"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "You might one or more warning(s) that the model fitting iterations generated NaNs during the fitting procedure for these data (because at some point the NLLS fitting algorithm \"wandered\" to a combination of K and N_0 values that yields a NaN for log(K/N_0)). \n",
-    "\n",
-    "You can ignore these warning in this case. But not always &ndash; sometimes these NaNs mean that the equation is wrongly written, or that it generates NaNs across the whole range of the x-values, in which case the model is inappropriate for these data.   "
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "Get the model summary:"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 108,
-   "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "text/plain": [
-       "\n",
-       "Formula: LogN ~ gompertz_model(t = Time, r_max, K, N_0, t_lag)\n",
-       "\n",
-       "Parameters:\n",
-       "      Estimate Std. Error t value Pr(>|t|)    \n",
-       "t_lag  4.80680    0.18433   26.08 5.02e-09 ***\n",
-       "r_max  1.86616    0.08749   21.33 2.45e-08 ***\n",
-       "N_0   10.39142    0.05998  173.24 1.38e-15 ***\n",
-       "K     15.54956    0.05056  307.57  < 2e-16 ***\n",
-       "---\n",
-       "Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1\n",
-       "\n",
-       "Residual standard error: 0.09418 on 8 degrees of freedom\n",
-       "\n",
-       "Number of iterations to convergence: 10 \n",
-       "Achieved convergence tolerance: 1.49e-08\n"
-      ]
-     },
-     "metadata": {},
-     "output_type": "display_data"
-    }
-   ],
-   "source": [
-    "summary(fit_gompertz)"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "And see how the fits of the two nonlinear models compare:"
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 109,
-   "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "image/png": 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BTlrMe/PKTvpsAOX7IzRaFf9lJnv8y9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-      "text/plain": [
-       "plot without title"
-      ]
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-      "image/png": {
-       "height": 240,
-       "width": 240
-      }
-     },
-     "output_type": "display_data"
-    }
-   ],
-   "source": [
-    "timepoints <- seq(0, 24, 0.1)\n",
-    "\n",
-    "logistic_points <- log(logistic_model(t = timepoints, \n",
-    "          r_max = coef(fit_logistic)[\"r_max\"], \n",
-    "          K = coef(fit_logistic)[\"K\"], \n",
-    "          N_0 = coef(fit_logistic)[\"N_0\"]))\n",
-    "\n",
-    "gompertz_points <- gompertz_model(t = timepoints, \n",
-    "         r_max = coef(fit_gompertz)[\"r_max\"], \n",
-    "         K = coef(fit_gompertz)[\"K\"], \n",
-    "         N_0 = coef(fit_gompertz)[\"N_0\"], \n",
-    "         t_lag = coef(fit_gompertz)[\"t_lag\"])\n",
-    "\n",
-    "df1 <- data.frame(timepoints, logistic_points)\n",
-    "df1$model <- \"Logistic model\"\n",
-    "names(df1) <- c(\"Time\", \"LogN\", \"model\")\n",
-    "\n",
-    "df2 <- data.frame(timepoints, gompertz_points)\n",
-    "df2$model <- \"Gompertz model\"\n",
-    "names(df2) <- c(\"Time\", \"LogN\", \"model\")\n",
-    "\n",
-    "model_frame <- rbind(df1, df2)\n",
-    "\n",
-    "ggplot(data, aes(x = Time, y = LogN)) +\n",
-    " geom_point(size = 3) +\n",
-    " geom_line(data = model_frame, aes(x = Time, y = LogN, col = model), size = 1) +\n",
-    " theme_bw() + # make the background white\n",
-    " theme(aspect.ratio=1)+ # make the plot square \n",
-    " labs(x = \"Time\", y = \"log(Abundance)\")"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "Clearly, the Gompertz model fits way better than the logistic growth equation in this case! Note also that there is a big difference in the fitted value of $r_{max}$ from the two models; the value is much lower from the Logistic model because it ignores the lag phase, including it into the exponential growth phase. \n",
-    "\n",
-    "You can now perform model selection like you did above in the allometric scaling example. "
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "### Exercises\n",
-    "\n",
-    "(a) Calculate the confidence intervals on the parameters of each of the two fitted models, and use model selection (using AIC and/or BIC) as you did before to see if you can determine the best-fitting model among the three.\n",
-    "\n",
-    "(b) Alternatively, for a different random sequence of fluctuations, one or more of the models may fail to fit (a `singular gradiant matrix` error). Try repeating the above fitting with a different random seed (change the integers given to the `random.seed( )` function), or increase the sampling error by increasing the standard deviation and see if it happens. If/when the NLLS optimization does fail to converge (the RSS minimum was not found), you can try to fix it by changing the starting values. \n",
-    "\n",
-    "(c) Repeat the model comparison exercise 1000 times (You will have to write a loop), and determine if/whether one model generally wins more often than the others. Note that each run will generate a slightly different dataset, because we are adding a vector of random errors every time the \"data\" are generated. This may result in failure of the NLLS fitting to converge, in which case you will need to use the [`try()` or `tryCatch` functions](https://nbviewer.jupyter.org/github/mhasoba/TheMulQuaBio/blob/master/notebooks/07-R.ipynb).\n",
-    "\n",
-    "(d) Repeat (b), but increase the error by increasing the standard deviation of the normal error distribution, and see if there are differences in the robustness of the models to sampling/experimental errors. You may also want to try changing the distribution of the errors to some non-normal distribution and see what happens."
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "## Some tips and tricks for NLLS fitting \n",
-    "\n",
-    "(Model-Fitting-NLLS-Starting-Values)=\n",
-    "### Starting values \n",
-    "\n",
-    "The main challenge for NLLS fitting is finding starting (initial) values for the parameters, which the algorithm needs to proceed with the fitting/optimization. Inappropriate starting values can result in the algorithm finding parameter combinations represent convergence to a local optimum rather than the (globally) optimal solution. Starting parameter estimates can also result in or complete \"divergence\", i.e., the search results in a combination of parameters that cause mathematical \"singularity\" (e.g., log(0) or division by zero).\n",
-    "\n",
-    "#### Obtaining them \n",
-    "\n",
-    "Finding the starting values is a [bit of an art](https://en.wikipedia.org/wiki/Non-linear_least_squares#Initial_parameter_estimates). There is no method for finding starting values that works universally (across different types of models). \n",
-    "\n",
-    "The one universal rule though, is that finding starting values requires you to understand the meaning of each of the parameters in your model. So, for example, in the population [growth rate example](Model-Fitting-R-Population-Growth), the parameters in both the nonlinear models that [we covered](Model-Fitting-R-Population-Growth) (Logistic growth, eqn. {eq}`eq:logist_growth_sol` , Gompertz model; eqn. {eq}`eq:Gompertz`) have a clear meaning.\n",
-    "\n",
-    "Furthermore, you will typically need to determine starting values *specific* to each model *and* each dataset that that you are wanting to fit that model to (e.g., every distinct functional response dataset to be [fitted to the Holling Type II model](Miniproject-FR-Models)). To do so, understanding how each parameter in the model corresponds to features of the actual data is really necessary. \n",
-    "\n",
-    "For example, in the Gompertz population growth rate model (eqn. {eq}`eq:Gompertz`), your starting values generator function would, for each dataset,   \n",
-    "*  Calculate a starting value for $r_{max}$ by searching for the steepest slope of the growth curve (e.g., with a rolling OLS regression)\n",
-    "* Calculate a starting value of $t_{lag}$ by intersecting the fitted line with the x (time)-axis \n",
-    "* Calculate a starting value for the asymptote $K$ as the highest data (abundance) value in the dataset. \n",
-    "\n",
-    "```{tip}\n",
-    "Ideally, you should write a separate a function that calculates starting values for the model parameters.\n",
-    "```\n",
-    "\n",
-    "#### Sampling them\n",
-    "\n",
-    "Once you have worked out how to generate starting values for each non-linear model and dataset, a good next step for optimizing the fitting across multiple datasets (and thus maximize how many datasets are successfully fitted to the model) is to *rerun fitting attempts multiple times, sampling each of the starting values (simultaneously) randomly* (that is, randomly vary the set of starting values a bit each time). This sampling of starting values will increase the likelihood of the NLLS optimization algorithm finding a solution (optimal combination of parameters), and not getting stuck in a combination of parameters somewhere far away from that optimal solution. \n",
-    "\n",
-    "In particular, \n",
-    "* You can choose a Gaussian/Normal distribution if you have high confidence in mean value of parameter, or\n",
-    "* You can uniform distribution if you have low confidence in the mean, but higher confidence in the range of values that the parameter can take.\n",
-    "In both cases, the *mean* of the sampling distribution will be the starting value you inferred from the model and the data (previous section).\n",
-    "\n",
-    "Furthermore,\n",
-    "* Whichever distribution you choose (gaussian vs uniform), you will need to determine what range of values to restrict each parameter's samples to. In the case of the normal distribution, this is determined by what standard deviation parameter (you choose), and in the case of the uniform distribution, this is determined by what lower and upper bound (you choose). Generally, a good approach is to set the bound to be some *percent* (say 5-10%) of the parameter's (mean) starting value. In both cases the chosen range to restrict the sampling to would typically be some subset of the model's *parameter bounds* (next section).  \n",
-    "* *How many times to re-run* the fitting for a single dataset and model?* &ndash; this depends on how \"difficult\" the model is, and how much computational power you have. \n",
-    "\n",
-    "```{tip}\n",
-    "For the sampling of starting values, recall that you learned to generate random numbers from probability distributions in both the [R](R-random-numbers) and [Python](Python-scipy-stats) chapters). \n",
-    "```\n",
-    "You may also try and use a more sophisticated approach such as [grid searching](https://en.wikipedia.org/wiki/Hyperparameter_optimization#Approaches) for varying your starting values randomly. An example is in the [MLE chapter](ModelFitting-MLE-LikelihoodSurface).\n",
-    "\n",
-    "### Bounding parameters revisited\n",
-    "\n",
-    "At the start, we looked at an [example](20-ModelFitting-NLLS:Bounding) of NLLS fitting where we bounded the parameters. It can be a good idea to restrict the range of values that the each of the model's parameters can take *during any one fitting/optimization run*. To \"bound\" a parameter in this way means to give it upper and lower limits. By doing so, during one optimization/fitting (e.g., one call to `nlsLM`, to fit one model, to one dataset), the fitting algorithm does not allow a parameter to go outside some limits. This reduces the chances of the optimization getting stuck too far from the solution, or failing completely due to some mathematical singularity (e.g., log(0)).\n",
-    "\n",
-    "The bounds are typically fixed for each parameter of a model *at the level of the model* (e.g., they do not change based on each dataset). For example, in the Gompertz model for growth rates (eqn. {eq}`eq:Gompertz`), you can limit the growth rate parameter to never be negative (the bounds would be $[0,\\infty]$), or restrict it further to be some value between zero and an upper limit (say, 10) that you know organismal growth rates cannot exceed (the bounds would in this case would be $[0,10]$). \n",
-    "\n",
-    "However, as we saw in the Michaelis-Menten model fitting [example](20-ModelFitting-NLLS:Bounding), bounding the parameters too much (excessively narrow ranges) can result in poor solutions because the algorithm cannot explore sufficient parameter space.   \n",
-    "\n",
-    "```{tip}\n",
-    "The values of the parameter bounds you choose, of course, may depend on the *units of measurement* of the data. For example, in [SI](https://en.wikipedia.org/wiki/International_System_of_Units), growth rates in the Logistic or Gompertz models would be in units of s$^{-1}$).\n",
-    "```\n",
-    "Irrespective of which computer language the NLLS fitting algorithm is implemented in (`nlsLM`  in R or `lmfit` in Python), the fitting command/method will have options for setting the parameter bounds. In particular,\n",
-    "\n",
-    "* For `nlsLM` in R, look up https://www.rdocumentation.org/packages/minpack.lm/versions/1.2-1/topics/nlsLM (the `lower` and `upper` arguments to the function). \n",
-    "\n",
-    "* For `lmfit` in Python, look up https://lmfit.github.io/lmfit-py/parameters.html (and in particular, https://lmfit.github.io/lmfit-py/parameters.html#lmfit.parameter.Parameter) (the `min` and `max` arguments).\n",
-    "\n",
-    "*Bounding the parameter values has nothing to do, per se, with sampling the starting values of each parameter, though if you choose to sample starting values (explained in previous section), you need to make sure that the samples don't exceed the pre-set bounds (explained in this section).*\n",
-    "\n",
-    "```{note}\n",
-    "Python's `lmfit` has an option to also internally vary the parameter. So by using a sampling approach as described in the previous section, *and* allowing the parameter to vary (note that `vary=True` is the default) within `lmfit`, you will be in essence be imposing sampling twice. This may or may not improve fitting performance &ndash; try it out both ways.\n",
-    "```"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "## Readings and Resources\n",
-    " \n",
-    "* Motulsky, Harvey, and Arthur Christopoulos. Fitting models to biological data using linear and nonlinear regression: a practical guide to curve fitting. OUP USA, 2004: <https://www.facm.ucl.ac.be/cooperation/Vietnam/WBI-Vietnam-October-2011/Modelling/RegressionBook.pdf>\n",
-    "\n",
-    "* These are a pretty good series of notes on NLLS (even if you are using R instead of Python): <https://lmfit.github.io/lmfit-py/intro.html>\n",
-    "\n",
-    "* Another technical description of NLLS  algorithms: <https://www.gnu.org/software/gsl/doc/html/nls.html>\n",
-    "\n",
-    "* Johnson, J. B. & Omland, K. S. 2004 Model selection in ecology and evolution. Trends Ecol. Evol. 19, 101–108.\n",
-    "\n",
-    "* The *nlstools* package for NLLS fit diagnostics: <https://rdrr.io/rforge/nlstools>\n",
-    "    * The original paper: <http://dx.doi.org/10.18637/jss.v066.i05>"
-   ]
-  }
- ],
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-   "display_name": "R",
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diff --git a/_sources/notebooks/Populations.ipynb b/_sources/notebooks/Populations.ipynb
new file mode 100644
index 00000000..26535ea3
--- /dev/null
+++ b/_sources/notebooks/Populations.ipynb
@@ -0,0 +1,56 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Single Populations"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "* Single populations\n",
+    "  * Are there *any* single populations? \n",
+    "  * Exponential growth\n",
+    "    * Metabolic basis\n",
+    "  * From populations to alleles\n",
+    "    * Basic pop gen\n",
+    "  * Logistic growth\n",
+    "    * Density dependence\n",
+    "    * Logistic growth as an \"effective\" model\n",
+    "      * SIS model (VJ)\n",
+    "    * Bifurcations (+Intro to theory - VJ) \n",
+    "  * Evolutionary dynamics\n",
+    "  * Stochasticity\n",
+    "  * Applications/Examples\n",
+    "  * Stage and age structured populations\n",
+    "    * Stage- vs Age-structured models\n",
+    "    * IPMs\n",
+    "  * Applications/Examples"
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3 (ipykernel)",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.10.12"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 2
+}
diff --git a/_sources/notebooks/Space.ipynb b/_sources/notebooks/Space.ipynb
new file mode 100644
index 00000000..f084dde9
--- /dev/null
+++ b/_sources/notebooks/Space.ipynb
@@ -0,0 +1,44 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Spatial Ecology and Evolution"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "* Spatial structure\n",
+    "  * Island biogeography and neutral theory\n",
+    "  * Metapopulations (VJ)\n",
+    "  * Evolutionary dynamics in space\n",
+    "    * Kin selection?\n",
+    "  * Applications/Examples"
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3 (ipykernel)",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.10.12"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 2
+}
diff --git a/_sources/notebooks/TurbulentMarriage.ipynb b/_sources/notebooks/TurbulentMarriage.ipynb
new file mode 100644
index 00000000..2ccf345e
--- /dev/null
+++ b/_sources/notebooks/TurbulentMarriage.ipynb
@@ -0,0 +1,631 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# The Turbulent Marriage of Ecology and Evolutionary Biology"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "```{epigraph}\n",
+    "'If there's no meaning in it,' said the King, 'that saves a world of trouble, you know, as we needn't try to find any…' \n",
+    "\n",
+    "-- Alice in Wonderland\n",
+    "\n",
+    "```"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Introduction\n",
+    "\n",
+    "Few would hesitate in identifying ecology and evolutionary biology as distinct field of research with somwhat disparate goals. In this  chapter, we will examine the alliance between these two, which have been distinct programs of research for more than a century now. In particular, we will focus on the most theoretically and empirically tangible part of this alliance: the unification of population ecology and population genetics. \n",
+    "\n",
+    "We are generally taught Ecological and Evoluonary theory and practice separately; one need only flip though an introductory text in biology to see how well-established this separation is. Indeed, a majority of the practitioners of either discipline continue to conduct research with limited perspectives from the other, perhaps, albeit subliminally, taking the stance of the King of Hearts in Alice quoted above! In this introductory chapter we will examine whether this lack of collaboration hinders each discipline's progress, what the roots for discord are, and the current status of the collaboration (the \"Marriage\"). \n",
+    "\n",
+    "This is an appropriate time to make this evaluation, as the last two decades decade or so in particular have seen a resurgence of theoretical and empirical work that combines ecological and evolutionary concepts, theory and empirical data (and apporaches, driven by advances in molecular bilogy and muti-omics). Another, more philosophical motivation for this chapter is that ecology and evolution have been noticeably underrepresented in accounts of the philosophy of biology {cite}`Hull1974,Sober1993,grene2004philosophy`. The reasons for this ostensible disinterest are worth exploring. Both cultural and conceptual differences between these disciplines have hindered their synthesis, and that this discord, and the nature of the current synthesis themselves are fertile grounds for philosophical analyses. Moreover, by highlighting the interdependence of ecological and evolutionary theory, and the importance of a synthesis to the overall progress of biology, we will also reexamine the view that ecology appears dull to philosophers because in its classical mold it is theoretically and causally redundant within biology {cite}`Haila2001`. \n",
+    "\n",
+    "Ecology today comprises a heterogeneous array of disciplines, transcending levels of complexity from individuals and populations at one end to ecosystems at the other. An attempt to examine the role of evolutionary biology at each of these and vice versa is beyond the scope of this Chapter. We will therefore focus mainly on two fields that form the traditional foundation of ecology and evolutionary biology, and which have a long history of exchange and attempted union: population biology and population genetics. Nevertheless, instances where such a union has obvious implications for higher levels of complexity will be dwelt upon. \n",
+    "\n",
+    "In the following sections we will address: \n",
+    "\n",
+    "* Goals of contemporary ecology and evolutionary biology.  \n",
+    "* A brief review of historical relationship with focus on perceptible roots of discord as well as attempts for collaboration between them[^2].   \n",
+    "* Current sources of discord between the two disciplines.  \n",
+    "* Based upon the approach taken towards resolving these differences, a classification of current research themes in evolutionary ecology.\n",
+    "* The current status of this synthesis in light of recent theoretical and empirical advances.\n",
+    "* And finally, a discussion in light of the preceding results."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## The Roots of Disharmony\n",
+    "\n",
+    "### Contemporary ecology vis-à-vis evolutionary biology\n",
+    "\n",
+    "Ecology today covers a wide range of topics ranging from the level of the individual, to ecosystems. This naturally implies a varied set of objectives, but a common underlying theme can still be identified as being along the lines of what the inventor of the term Haeckel (see below) originally defined it: the relationship between organisms and their environment. These two elements also comprise distinct ways in which things \"ecological\" are taken to mean: to some, ecology is synonymous with the characteristics of individuals, populations, communities or ecosystems of organisms, while to others, ecology is the environment that affects the organisms. This environment of course, can be biotic as well as abiotic. Disciplines in contemporary ecology are largely devoted to analyses of patterns observed in nature, their origins, and their changes with respect to the environment. Compared to evolutionary biology, ecology tends to dwell more in the present, its practitioners being most concerned with the relationship between organisms and their biotic as well as abiotic environment in \"ecological time scales\" (temporal scales in the order of one of few generations of the studied organisms). The common approach used to study these aspects is a mechanistic, physical one, with a tacit assumption of purely non-evolutionary response, and without consideration of the genetic variation (Mayr 1997)(pp. 210-211). Relative to evolutionary biology, the hope for general principles that govern these relationships have spurred the development of very detailed analyses of organism-environment interactions. \n",
+    "\n",
+    "Evolutionary biology on the other hand, consists of both prospective and retrospective theoretical frameworks that are concerned with the origins of, and changes in living entities over varying time scales. At one end lies the mathematical theory of genetics that forms a basis for microevolutionary (evolution on relatively short temporal scales) research. At the same time, the fecund collaboration of molecular biology and classical evolutionary theory (Huxley 1942) now also contributes to macroevolutionary (evolution over longer, generally geological time scales) research through an increasing collaboration between phylogenetics, systematics, and paleontology. I now present a brief historical background with the objective of exploring the roots of discord as well as collaboration between these two disciplines.\n",
+    "\n",
+    "### An early association and subsequent divergence: 1850-1950\n",
+    "\n",
+    "{cite:t}`Darwin1859`, in the process of laying the foundations of evolutionary biology did so with explicit and deep consideration of the relationship between the organism and the environment. Arguably, this prompted Haeckel to coin the term *ökologie* in 1866 {cite}`Haeckel1866`[^3], in whose words ecology was \n",
+    "\n",
+    "```{epigraph} \"*…the investigation of the total relations of the animal both to it's inorganic and to its organic environment ….in other words, ecology is the study of all those complex interrelationships referred to by Darwin as the conditions for the struggle for existence…*\" \n",
+    "```\n",
+    "(quoted in {cite:t}`Stauffer1957`). \n",
+    "\n",
+    "Naturally, the first step towards achieving this goal was to study the effect of the environment on individuals and populations.\n",
+    "\n",
+    "The period from Haeckel's definition to the early 19th century was a crucial period for the discipline, although the term \"ecology\" was not used much. This was when the tradition of detailed botanical and zoological study of morphology, anatomy and physiology of the individual in relation to its adaptive environment was established. This was also the period when Herbert Spencer published his ideas about the emergence of order in nature {cite}`Spencer1864`. Spencer's writings strongly influenced the approach of future generations by embedding the notion of stability and the \"balance of nature\" in the ecological psyche. By the beginning of the 20th century, when the units of study in ecology had already been defined at multiple levels beyond the individual (population, communities, ecosystems, etc), and population as well as community ecology we coming to the fore as distinct disciplines, a strong belief in the equilibrium state of nature and its units had already taken hold. This was followed by the adoption of a physical view by many ecologists, wherein living units were transformers of energy enmeshed in larger systems with considerable resilience to perturbation, and where details of the evolutionary process could be glossed in interest of a more systemic approach {cite}`Kingsland1985`(\"The beginnings of \"systems ecology\"\"; pp. 34-49)    \n",
+    "\n",
+    "Apart from the systems approach to ecology, this was also the period when faith in the pre-adaptation of species and the slowness of evolution took firm hold. These beliefs freed the ecologist of evolutionary concerns, and allowed a single-minded commitment to the study of the nature of the adaptations in relation to the environment. The roots of the viewpoint that ecology and evolution operate at vastly different time scales are probably to be found in fact that including Darwin's work, early evolutionary ideas were based on the fossil record, and tended to be \"gradualist\" {cite}`stanley1989`. In Darwin's {cite}`Darwin1859`(pp. 126-127) words, \n",
+    "```{epigraph}\n",
+    "I do believe that natural selection will generally act very slowly, only at long intervals of time, and only on a few of the inhabitants of the same region. I further believe that these slow, intermittent results accord well with what geology tells us of the rate and manner at which the inhabitants of the world have changed.\n",
+    "```\n",
+    "This view was reinforced by the fact that examples of natural life tangible to early observers mostly happened to be relatively long-lived vertebrates. In contrast, modern biologists we can see evolution in action due to advances in molecular and phenotyping technologies, especially in microbial populations (something we will revisit later in this chapter). Thus, in its development through the first part of the 20th century, the rapidly proliferating branches of ecology had largely diverged from evolutionary biology with only sporadic exchange between the two disciplines. Ecologists during time focused increasingly on the effect of the abiotic environment and biotic interactions on populations to the exclusion of any considerations of genetics or evolution, trying to deal with the onerous task of defining the characteristics and emergent properties of the various levels from individuals to ecosystems.\n",
+    "\n",
+    "The \"The golden age of theoretical ecology\" from 1920 to 1940 {cite}`Scudo1978`, was marked by the rediscovery of the Verhulst logistic equation by Raymond Pearl in the 1920's and its extension to interacting populations, as well as experimental evaluation {cite}`Gause1934` (e.g., {cite}`Kingsland1985`). The approach of theoretical ecology then was to define state variables representing properties of the population as a whole with a tacit assumption of genetic homogeneity. For example, the coupled differential equations of the classic Lotka-Volterra equations of competition between two species with population sizes $N_1$, $N_2$,\n",
+    "\n",
+    "$$\\begin{align}\n",
+    "    \\frac{dN_1}{dt} &= r_1 N_1 \\left( 1 - \\frac{N_1 + \\alpha_{12} N_2}{K_1} \\right) \\\\\n",
+    "    \\frac{dN_2}{dt} &= r_2 N_2 \\left( 1 - \\frac{N_2 + \\alpha_{21} N_1}{K_2} \\right),\n",
+    "\\end{align}$$(eq:turbulent:LV_competition)\n",
+    "\n",
+    "include the following immutable parameters: $r_1$, $r_2$ (intrinsic growth rates of species' 1 & 2), $K_1$, $K_2$ (carrying capacities), and $α_{12}$, $α_{21}$ (coefficients of competition). \n",
+    "\n",
+    "It is clear that many of the pioneers of theoretical population biology during this period were, to various degrees, motivated by Darwin's theory of natural selection and the then prevalent schools of evolutionary thought[^4]. This period also coincided with the mathematical fusion of Mendelian genetics and Darwin's theory by Fisher, Haldane and Wright {cite}`Huxley1942`(the \"Modern Synthesis\" of evolutionary biology)."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "image/png": 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",
+      "text/plain": [
+       "<Figure size 1000x600 with 1 Axes>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "import numpy as np\n",
+    "import matplotlib.pyplot as plt\n",
+    "from scipy.integrate import odeint\n",
+    "\n",
+    "# Define Lotka-Volterra competition model\n",
+    "def lotka_volterra(y, t, r1, r2, K1, K2, alpha12, alpha21):\n",
+    "    N1, N2 = y\n",
+    "    dN1_dt = r1 * N1 * (1 - (N1 + alpha12 * N2) / K1)\n",
+    "    dN2_dt = r2 * N2 * (1 - (N2 + alpha21 * N1) / K2)\n",
+    "    return [dN1_dt, dN2_dt]\n",
+    "\n",
+    "# Parameters\n",
+    "r1, r2 = 0.5, 0.3  # Growth rates\n",
+    "K1, K2 = 50, 30    # Carrying capacities\n",
+    "alpha12, alpha21 = 0.6, 0.8  # Competition coefficients\n",
+    "y0 = [10, 5]  # Initial populations\n",
+    "t = np.linspace(0, 50, 500)  # Time range\n",
+    "\n",
+    "# Solve ODEs\n",
+    "result = odeint(lotka_volterra, y0, t, args=(r1, r2, K1, K2, alpha12, alpha21))\n",
+    "\n",
+    "# Plot results\n",
+    "plt.figure(figsize=(10, 6))\n",
+    "plt.plot(t, result[:, 0], label=\"Species 1 (N1)\", lw=2)\n",
+    "plt.plot(t, result[:, 1], label=\"Species 2 (N2)\", lw=2)\n",
+    "plt.title(\"Lotka-Volterra Competition Dynamics\")\n",
+    "plt.xlabel(\"Time\")\n",
+    "plt.ylabel(\"Population Size\")\n",
+    "plt.legend()\n",
+    "plt.grid()\n",
+    "plt.show()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Both population ecology and population genetics had, and still do, the same mathematical foundations at their core. For example, consider the case of an asexual haploid population with discrete, non-overlapping generations. Let time be measured in generations, such that population size at time $t$, $N(t)$, is the result of the previous generation's population size, $N(t-1)$ multiplied by a growth factor which combines survival and reproduction, say $w$:\n",
+    "\n",
+    "$$\n",
+    "\\begin{align}\n",
+    "    N(t) &= w N(t-1)\\\\\n",
+    "        &= w^2 N(t-2)\\\\\n",
+    "        &= ~...\\\\\n",
+    "        &= w^t N(0),\n",
+    "\\end{align}\n",
+    "$$(eq:turbulent:fitness)\n",
+    "\n",
+    "yielding the classical model of population growth in discrete time. It is straightforward to extend equation {eq}`(eq:turbulent:fitness)` in population genetics terms. The growth factor $w$ is the absolute fitness of the population, and can be written as $w = 1+s$, such that the population increases if $s>0$ and decreases if $s<0$. The change in population size in one generation then is,\n",
+    "\n",
+    "$$\n",
+    "\\begin{align}\n",
+    "\\Delta N(t) \n",
+    "  &= N(t+1) - N(t)\\\\ \n",
+    "  &= (1 + s)N(t) - N(t)\\\\ \n",
+    "  &= sN(t).\n",
+    "\\end{align}\n",
+    "$$(eq:turbulent:fitness_2)\n",
+    "\n",
+    "```{note}\n",
+    "You will learn about this fitness equation and its usage in ecology as well as evolutionary biology in the next Chapter. \n",
+    "```\n",
+    "\n",
+    "Now, suppose the population contains $k$ genotypes (alleles), each with its own growth rate (fitness), $w_i, \\mid i = 1,2,...,k$. Let $n_i(t)$ denote the population size of genotype $i$ at time (=generation) $t$. Then, total population size at time $t$ is $N(t) = \\sum_in_i(t)$, and the mean fitness of the population is\n",
+    "\n",
+    "$$\n",
+    "    \\overline{W}(t) = \\sum_{i=1}^{k} W_i \\frac{n_i(t)}{N(t)}.\n",
+    "$$(eq:turbulent:fitness_mean)\n",
+    "\n",
+    "$\\overline{W}$ will will change over generations with changes in $n_i$. A continuous time extension of the population growth model shows the relationship between the population biologist's Malthusian parameter $r$ (also known as the intrinsic growth rate) and fitness $W$. Consider the population growth in continuous time $\\lim_{\\mathrm{d}t \\to 0} \\frac{\\mathrm{d}N}{\\mathrm{d}t} = rN.\n",
+    "$, which has the solution (next Chapter),\n",
+    "\n",
+    "$$\n",
+    "    N(t) = N(0)e^{rt}.\n",
+    "$$(eq:turbulent:exponential_growth)\n",
+    "\n",
+    "This is the familiar equation of exponential population growth. Then by comparing eqns. {eq}`eq:turbulent:fitness` and {eq}`eq:turbulent:exponential_growth`, $w = e^{r t}$, or $r = \\ln(w)$. If $w$ is close to 1 (population size is not changing), i.e., $s<<1$, $\\ln{w} = \\ln(1+s) \\approx s$ i.e., $r \\approx s$, thus allowing the Malthusian parameter to be used as a measure of fitness. This illustrates the interchangeability of the core concepts in population genetics and ecology, under some assumptions. \n",
+    "\n",
+    "During this coincident period of the development of population genetics and population biology however, little effort was made to combine the theories, although there were voices such as {cite}`Elton1927` and {cite}`Gause1934` that espoused the importance of exchange between the two disciplines. In part, this was because the extensions of models such as eqns. {eq}`eq:turbulent:LV_competition` by themselves posed mathematical challenges without introducing the added complexity of genetics. Another reason is that these early voices of evolutionary ecology were drowned in the clamor created by the rapid proliferation of ecological subdisciplines {cite}`Kingsland1985`. For example, Gause, who was clearly motivated by Darwin's theory (see introductory remarks in {cite:t}`Gause1934`) expressed the desire to eventually combine the population ecology of Volterra with the mathematical genetics of Haldane and others, but apparently never attempted this {cite}`Kingsland1985`(p. 160). A rare example of such work appears to be that of V. A. Kostinzin, whose efforts however passed largely unnoticed, and were cut short by the Second World War {cite}`Kingsland1985`(p. 144)[^5].  \n",
+    "\n",
+    "On the other hand, the direction taken by evolutionary theorists during the early 20th century also resulted in disassociation from many aspects of ecology. The foundations of Mendelian population genetics were built by a bare-bones approach of achieving mathematical tractability by judicious approximation and selective omission of ecological reality. For instance, as shown above (and you will learn more about later), approximation of the Malthusian parameter $r$ by the selection coefficient $s$ involved (and still does in classical population genetics theory) the assumption of a fixed population size. The Hardy-Weinberg theorem (next Chapter) served as a useful model by allowing an estimation of expected allele frequencies in diploid populations under assumption of panmictic, random mating, infinitely sized populations that experience no selection, mutation, emigration or immigration. A relaxation of different assumptions then allowed a modeling of the various agents of evolution, an approach that is still used in classical population genetics.\n",
+    "\n",
+    "For example, a traditional approach to modeling the role of natural selection in a diploid sexual population is to track the changes in allele frequencies for a single locus and two alleles. Consider two alleles $1$ & $2$, with frequencies $p$ & $q$, respectively. Assume an infinite population size, which assures that there is no random loss of alleles (no genetic drift), and that the condition of fixed population size derived in eqns. {eq}`eq:turbulent:fitness` - {eq}`eq:turbulent:exponential_growth` is satisfied. Furthermore, assume completely random mating followed by selective mortality, such that zygotes in every generation are in Hardy-Weinberg (HW) proportions of $p^2$, $2pq$ and $q^2$ before selection. Denoting population sizes of the three genotypes as $N_{11}$, $N_{12}$, $N_{22}$, and assigning fitnesses $w_{11}$, $w_{12}$ and $w_{22}$ (loss of zygotes from mortality followed by reproductive gain) respectively, the number of individuals of each genotype in the next generation are, $N_{11}' = w_{11} N_{11}$, $N_{12}' = w_{12} N_{12}$, and $N_{22}' = w_{22} N_{22}$. Then the frequency of allele *1* in the next generation is, \n",
+    "\n",
+    "$$\\begin{align}\n",
+    "    p' &= \\frac{N_{11}' + \\tfrac12 N_{12}'}{N_{11}' + N_{12}' + N_{22}'}\\\\\n",
+    "       &= \\frac{w_{11}N_{11} + w_{12}\\tfrac12N_{12}}\n",
+    "       {w_{11}N_{11} + w_{12}N_{12} + w_{22}N_{22}}\\\\\n",
+    "       &= \\frac{w_{11}p^{2} + w_{12}pq}\n",
+    "         {w_{11}p^{2} + w_{12}\\bigl(2pq\\bigr) + w_{22}q^{2}}.\n",
+    "\\end{align}$$\n",
+    "\n",
+    "Now, dividing through by $w_{11}$ adds the convenience of reducing the fitnesses to dimensionless quantities by redefining them *relative* to fitness of the $w11$ genotype. The percentage advantage or disadvantage can then be denoted by the coefficient of selection, $s$ which appeared in an *absolute* context in eqn. {eq}`eq:turbulent:fitness_2` above. For example, assuming a multiplicative fitness scheme,  \n",
+    "$$\\begin{array}\n",
+    "    W_{11} &= \\frac{w_{11}}{w_{11}} = 1,\\\\\n",
+    "    W_{12} &= \\frac{w_{12}}{w_{11}} = 1 - s,\\\\\n",
+    "    W_{22} &= \\frac{w_{22}}{w_{11}} = (1 - s)^{2}.\n",
+    "\\end{array} $$\n",
+    "\n",
+    "Plugging these fitness values into eqn. {eq}`eq:turbulent:fitness` and remembering that $q = 1 - p$, yields after some algebraic manipulation, \n",
+    "\n",
+    "$$\n",
+    "\\Delta p = p' - p \n",
+    "  = p(1 - p) \\frac{s}{ 1 - s + s p }.\n",
+    "$$\n",
+    "\n",
+    "Thus by introducing relative fitnesses, and retaining all other, albeit ecologically unrealistic assumptions for HW equilibrium, a simple two parameter model of natural selection can be analyzed. The issue of population size and the manner in which fitness is measured, are important issues in the marriage of population biology and population genetics, and will be considered in the next section. Furthermore, population geneticists during this period increasingly channeled their efforts towards extending one locus two allele cases towards more genetically realistic multi-locus models involving epistasis and genetic recombination {cite}`Provine2001`. For mathematical tractability, this had to be at the cost of ecological reality such as demographic fluctuations and environmental heterogeneity, topics very much at the heart of ecological research.\n",
+    "\n",
+    "Evolutionary theory was not entirely oblivious to ecological reality though, and two distinct topics in particular impinged on evolutionary thought. One was demography, the evolutionary importance of which was championed by {cite}`Elton1930`, and the other was ecological heterogeneity and niche partitioning, on which Gause {cite}`Gause1934` did seminal ecological work, and which really came to the fore in evolutionary ecology with Lack's work on the Galapagos finches in the 1940's {cite}`Lack1947`. In the \"Causes of Evolution\" Haldane paid some attention to conjectures by {cite}`Elton1930` and other evolutionary-minded ecologists about the role of finite population size and random extinction in evolution, but concluded that these factors had a minor role to play {cite}`Haldane1932`(pp. 201-204). By the 1930's, groundwork had been laid for modeling the effects of finite population sizes and spatial heterogeneity {cite}`Haldane1932,Fisher1930`, which was developed further by the succeeding generation of population geneticists using stochastic theory. Wright's \"Shifting Balance Theory\" for evolution, which invokes both finite population sizes and spatial heterogeneity is perhaps the most ecologically realistic population genetics theory coming from an evolutionary theorist of the period {cite}`Wright1931`.   \n",
+    "It is worth noting that by the early 20th century, the two aspects of ecology mentioned above, i.e., the study of populations, and the study of the environment, were clearly being given different weightage by different scientists in ecology and evolutionary biology. This is discussed further below, and it will be seen in the next section that this fact has had a strong impact on how evolutionary ecology or the synthesis of population biology and genetics is pursued today.\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Marriage and the quest for harmony: 1950-1985\n",
+    "\n",
+    "{cite:t}(Kingsland1985)(pp. 143-144) observes that there was a widespread disinterest apparent in the activities and views of ecologists and evolutionary biologists towards the other field from the 1920's into the 1950's. From closely entwined origins, ecology and evolutionary biology had diverged in many respects, resulting in deep-rooted cultural differences[^6]. The historical background above indicates various factors that contributed to the divergence of the two fields. Most of these persist today, and will be considered in more detail in the next section.  \n",
+    "\n",
+    "Around 1950, a renewed awareness of the chasm that had developed between evolutionary biology and ecology arose, with work such as David Lack's research on the Galapagos finches reiterating the importance of ecological heterogeneity, biotic interactions, and niche partitioning {cite}`Lack1947`. A shift away from the essentially equilibrium \"balance of nature\" view also played an important part in this. While a large body of ecologists continued to pursue the program of physical ecology involving detailed studies of the interface between the environment and ecological units ranging from individual to whole ecosystems {cite}`Odum1953,Weiling1965,lotka1956`, the view had become distinctly more dynamic for many others. In part this was due to the publication of the \"The Theory of Island Biogeography\" by MacArthur and Wilson {cite}`MacArthur1967` which simultaneously introduced two ideas atypical to the ecological thought of that time; a neutral theory of maintenance of communities immigration-extinction dynamics, and a strong synthesis of ecological and evolutionary thought. MacArthur's other work was also very influential, and within a short period of less than two decades, he initiated considerable shifts in the ecological community's viewpoints about theory as well as empirical work {cite}`Kingsland1985`. He made many contributions to the theoretical fusion of population genetics and population biology, and had a big impact in drawing the attention of the ecologists towards evolutionary thinking. As Rosenzweig {cite}`Rosenzweig1987` says, \n",
+    "> *MacArthur's greatest contribution may have been that he seems to have got through to ecologists that they ignore evolution at the peril of missing the most predictive aspects and beguiling issues in ecology.*\n",
+    ">\n",
+    "\n",
+    "Additionally, the interface between genetics and ecology was also beginning to be seen as an exciting and important area of theoretical and empirical work as the discovery of the genetic code, and advances in bio-molecular techniques allowed hitherto impossible quantification of genetic variation in nature {cite}`Ford1964,Lewontin1974`. This spurred on the efforts to combine models of population genetics and ecology. The period from 1960 onwards saw frequent appearances of papers, seminars, and books that displayed the intention to bring the theories and practices of ecology and evolution together. These included the works of both, ecological minded evolutionary biologists and vice versa {cite}`Ford1964,Lewontin1974,Levins1968,Roughgarden1979,Solbrig1980,Connell1970,Drake1968,Brussard1978,Dawson1971,Karlin1976,Bradshaw1984,Strong1984,Wohrmann1984,Lewontin1968,Pianka1970,Emlen1973,Cody1975,Stonehouse1977,Solbrig1979,Sammeta1970`. The first textbooks explicitly attempting to present an evolutionary ecology perspective also appeared during this period {cite}`MacArthur1966,Hedrick1984,Pianka1970`).\n",
+    "\n",
+    "Naturally, these forays into evolutionary ecology reflected the difference in approach of scientists. Introducing the aims and scope of the journal Evolutionary Ecology {cite}`Rosenzweig1987` says, \n",
+    "\n",
+    "> *...some of our articles will examine the evolution of natural ecosystems. Others will stress how the external environment, biotic or abiotic, influences the course of evolution. Finally, there will be some that study how the environment actually molds the evolutionary mechanisms themselves.*\n",
+    "\n",
+    "Among these, the most common theme was the role of the abiotic and biotic milieu in guiding evolution, well expressed in Hutchinson's {cite}`Hutchinson1965`metaphor of \"*The ecological theater and the evolutionary play\".* This focus on ecology as a background for evolutionary processes continues to be a dominant theme in evolutionary ecology (see below)[^7]. \n",
+    "\n",
+    "As always, there was a difference in approach between ecologists and evolutionary biologists towards the synthesis. Whereas ecologists focused on evolution with respect to population dynamics, interspecific interactions and environmental aspects external to focal populations (e.g., heterogeneity of the biotic and abiotic habitat), ecologically oriented evolutionary geneticists such as Lewontin and Ford made detailed investigations of genetic variation in nature with a more population genetics approach. The term \"Ecological Genetics\" was coined by Ford in 1964, a field which in his words, *\"...deals with the adjustments and adaptations of wild populations to their environment\"* {cite}`Ford1964`. This program was bolstered by further development of population genetics theory on evolution in spatially heterogeneous environments, that had been initially formalized by Wright, Fisher and Haldane {cite}`Provine2001`[^8]. \n",
+    "\n",
+    "A significant development during this period, which had a huge impact on the evolutionary ecology synthesis, was the development of quantitative genetics theory {cite}`Falconer1960`. Based on Fisher's {cite}`Fisher1930` initial work and the ideas of the biometrician school of genetics {cite}`Provine2001`, this involved modeling evolution of quantitative, polygenic traits without explicit description of the underlying genetics, by using statistical descriptors (e.g., mean & variance) of trait values. This approach was extremely useful from the viewpoint of empirical work, as it generated predictions about expected changes in continuously distributed trait values and indirect measurements of the strength of selection and fitness components from field data. Quantitative genetics therefore quickly became an integral component of ecological genetics. In a way, quantitative genetics was the ideal middle ground between population genetics and population ecology, as it involved a simplification of certain genetic as well as ecological complexities, yet addressing realistic scenarios. For example, consider a trait value $\\theta$, which is continuously distributed in a population as a Gaussian function `p(θ)`(i.e, the trait values are assumed to be normally distributed). Assume that there is no gene flow or mutation affecting the character, nor linkage disequilibrium {cite}`Slatkin1980`. As before, Let $w(\\theta)$ be the absolute fitness of an individual with trait $\\theta$. Various selection models can be used to generate values of $w(θ)$ (see below). Assuming that the $w(\\theta)$ functions are given, the change in mean and variance of the trait values are respectively, \n",
+    "\n",
+    "$$\n",
+    "\\overline{z}(t+1) \n",
+    "  = \\frac{\\sigma^{2}}{\\sigma^{2} + \\sigma_{e}^{2}}\n",
+    "    \\int \\theta  p(\\theta) w(\\theta) \\frac{d\\theta}{ \\overline{W} }\n",
+    "$$\n",
+    "\n",
+    "and\n",
+    "\n",
+    "$$\n",
+    "\\sigma^{2}(t+1)\n",
+    "  = h^{4} \\int \\bigl[\\theta - \\overline{\\theta}(t)\\bigr]^{2}  \n",
+    "    p(\\theta) w(\\theta) \\frac{d\\theta}{\\overline{W}}\n",
+    "    \\;+\\;\n",
+    "    \\Bigl(1 \\;-\\; \\frac{\\sigma^{2}}{\\sigma^{2} + \\sigma_{e}^{2}}\\Bigr) \n",
+    "    \\sigma^{2}(t).\n",
+    "$$\n",
+    "\n",
+    "where $\\sigma_e$ is the environmental component of the variance, and $\\overline{W} = \\int p(\\theta) w(\\theta) d \\theta$. The term $\\frac{\\sigma^{2}}{\\sigma^{2} + \\sigma_{e}^{2}}$ is traditionally called the heritability ($h^2$). Such models can be used to examine the effect of a wide range of selection mechanisms, depending upon what functions are used to define the $w(\\theta)$'s. Quantitative genetics circumvented some of the problems faced by previous attempts to model selection under frequency and density dependent population growth {cite}`Lewontin1974,Macarthur1962`[^9] by allowing tractable modeling of continuously distributed traits with definable fitness measures (see below). Its application to modeling niche width, species interactions, character displacement, and niche partitioning had a huge impact on the progress of the evolutionary ecology synthesis in the 1970's {cite}`Slatkin1970,Roughgarden1972,Slatkin1979,Slatkin1980,Slatkin1979b` by creating the first realistic evolutionary versions of hitherto purely demographic models {cite}`MacArthur1967`.   \n",
+    "\n",
+    "Another important step towards a demographically realistic evolutionary theory was the extension of models of selection to continuously breeding, age-structured populations {cite}`Charlesworth1980`. The temporal aspect of environmental heterogeneity and its effect on evolution also received attention during this period from both ecologists and evolutionary biologists. The approach again, was rather different, with population biologists tending to use demographically explicit models {cite}`Levins1968,Slatkin1976`, whereas population geneticists, following Kimura's neutral theory of molecular evolution {cite}`Kimura1983`, focused on genetic polymorphism, using stochastic theory to address questions of evolution under fluctuating selection by building on preliminary work by Haldane and Jayakar {cite}`Haldane1963,Gillespie1991`. The idea of temporally fluctuating environments was also central to the development of theory on the evolution of sex {cite}`Maynard_Smith1978`.   \n",
+    "\n",
+    "However, along with this upsurge of exchange between ecology and evolutionary biology from the 1950's onwards, other issues within the two disciplines increasingly claimed attention, contributing further to the overall divergence of the two. Evolutionary biologists on one hand, enthused by advances in molecular biology were increasingly caught up in the gene-centered view of life, which was essentially reductionist `Sarkar1998`. Apart from the different ways, as mentioned above, in which ecological realism was brought in by population geneticists into their models, a large body of biologists delved deeper into genetic models of selection and evolution, ignoring many aspects of genotype environment relationships. This reductionism has been lamented as being one of the causes of disunity between population biology and population genetics {cite}`Ayala1974,Levins1968,Levins2004`.  \n",
+    "\n",
+    "In ecology on the other hand, debates raged over the nature of community assembly, measures of stability in biological systems, the relative role of extrinsic vs. intrinsic factors driving ecological dynamics, the relationship between species diversity and community stability, the role of history in ecological patterns, and importance of processes at different spatial scales in determining diversity. Along the way, ecological sub-disciplines multiplied, with the addition of fields such as metapopulation ecology, metacommunity ecology, spatial ecology and landscape ecology. Unlike in evolutionary biology, a large rift also developed between theory and empirical work, with field biologists finding it tough going to glean testable predictions from complex, mechanistic models often inspired by physics, raising deep philosophical concerns about the role of theory in ecology and the place of ecology in biology {cite}`Kareiva1989,Kingsland1985, Sterelny1999`. Field ecology became increasingly preoccupied with conducting detailed characterizations of ecological patterns at various levels of complexity ranging from individuals to ecosystems. The hope was that cataloging patterns and processes would somehow reveal general principles. This was the period when global patterns such as \"Rapoport's rule\" {cite}`Gaston1998` and \"Bergmann's rule\" {cite}`Freckleton2003` were described. A formidable array of statistical techniques were also developed during this period to allow a mechanistic characterization of organism environment interrelationships. Evolutionary ecology took somewhat of a back-seat in these pursuits; especially because apart from a few exceptions, most of the focal biological systems for empirical work were vertebrates, where the role of evolution was considered insignificant at ecological time scales. Instead, complexity of the systems being studied commanded almost complete attention.   \n",
+    "\n",
+    "Nevertheless, in the meantime the unification of population genetics and population biology continued by a small group of biologists. From the 1980's onwards in particular, the effort came closer to achieving a more seamless union due to the use of increasingly sophisticated mathematical treatment and careful empirical testing with new model systems by a new breed of biologists who dwelt comfortably in both ecological and evolutionary hemispheres. In particular, the convergence of multilocus population genetic and quantitative genetic models (Burger 2000)(https://paperpile.com/c/lwDPbc/fycI) allowed greater ecological flexibility as well as genetic reality to evolutionary ecology modeling (Michael Doebeli 1996b, 1996a)(https://paperpile.com/c/lwDPbc/Abha+dbVj). In the process, these efforts diversified into various distinct approaches, which will be considered in the following section. \n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## The marriage of ecology and evolutionary biology: the ups, the downs, and a taxonomy\n",
+    "\n",
+    "We can now attempt a classification of contemporary approaches in evolutionary ecology, based on three themes that appear to have traditionally been at the core of dissent between ecology and evolutionary biology: the issue of time-scale, the role of demography, and the effect of environmental heterogeneity. We will discuss these before presenting the classification. \n",
+    "\n",
+    "Of the three issues, \"time-scale\" is mainly a cultural issue, referring to the importance given to history in population biology. Although there is no definable boundary between micro- vs. macro- evolutionary, or ecological vs. evolutionary time scales, the question of relative importance of processes at different time scales have generated distinct disciplines and this issue has apparently been a key factor affecting the unification of ecology and evolutionary biology.   \n",
+    "\n",
+    "The other two criteria are conceptual. The issue of demography is closely tied to the concept of fitness, and the two can be discussed together. Richard Levins gave a succinct account of these conceptual issues more than forty years ago, during an early phase of the evolutionary ecology synthesis {cite}`Levins1968`(p. 5): \n",
+    "\n",
+    "> *Darwinian fitness (Wright's $\\overbar{W}$) has to be interpreted in terms of its ecological components, such as the intrinsic capacity for increase (Andrewartha and Birch's $r_0$) and the carrying capacity of the environment for a given genotype ($K$). ...\n",
+    "> While the population geneticist's models generally assume stable age distributions, fixed populations sizes, and constant environments in order to study the pattern of genetic heterogeneity in single species, the population ecologist considers genetically uniform populations in multispecies systems in a heterogeneous environment*. \n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### The issue of ecological vs. evolutionary time scales\n",
+    "\n",
+    "As mentioned above, the early school of evolutionary thought was essentially gradualist, with a firm belief in the slow and steady progress of evolution over the geological time scale, a viewpoint that was prevalent till relatively recently {cite}`Stanley1989`. A competing view of punctuated equilibrium too exists, but this does not alter the argument that is being made here. The relevance of micro-evolutionary processes to observed macroevolutionary patterns, is still a contentious issue in evolutionary biology because of the observed opposition of trends at different time scales {cite}`Bennett1997,Stanley1989`. As such, this issue does not affect the evolutionary ecology synthesis much because evolutionary biologists naturally emphasize lineage-dependent processes and incorporate them into microevolutionary studies in an ecological context. However, a related view in ecology, that ecological and evolutionary time scales are independent of each other, has had considerable influence on the importance given to evolutionary processes in ecology. Tied to this is the strong faith among many ecologists of the existence of adaptation and co-adaptation in natural populations and their persistence over long, geologically significant periods of time {cite}`Hubbell2011`. For example Pimm {cite}`Pimm1991` discusses the stability and persistence of populations with respect to time scales ranging from centuries to millennia or more, invoking evolution only as a purely historical process. Similarly, Tokeshi {cite}`Tokeshi1999` discusses evolution in community structure on purely geological time scales. These perspectives naturally cause omission of evolutionary thought from ecological research. \n",
+    "\n",
+    "We have already mentioned above that a view of disjunct temporal processes is rooted in the tradition in biology to study a small set of vertebrate model systems {cite}`Bonnet2002,Pawar2003`. There is no denying that evolutionary history is an important factor to consider in ecology, and has been given due consideration. However, the role of microevolution in ecological patterns and processes has not been well evaluated till the two decade or so. In the last two decades or so, ecological geneticists and molecular ecologists have shown ample evidence for the existence of functional polymorphism and gene frequency shifts in nature. Not surprisingly, there is now an increasing awareness of the need to integrate macro- and microevolutionary processes in ecology at different scales of organization {cite}`Cavender-Bares2003`. In addition, the use of microbial model systems in ecology has also allowed many ecological questions to be answered in an evolutionary context, often adding important insights relevant to other systems where such methods are impossible {cite}`Jessup2004`. Perhaps ironically, the empirical insights from contemporary microbial work have come more than 60 years after Gause's {cite}`Gause1934` ecological experiments with microorganisms, which too had a strong evolutionary context. No doubt advances in molecular biology techniques have a large part to play in this, but this also reflects the reticence of ecologists to deviate from established model biological systems.  \n",
+    "\n",
+    "The argument here of course, is not that research on elephants and tigers be completely abandoned in favor of laboratory cultures; it is a plea for a pluralism of empirical approach that will provide insights for finding avenues of escape from the theoretical cul-de-sac that ecologists often find themselves due to the inability to test crucial aspects of their models {cite}`Kareiva1989`. Indeed, in the subsequent discussion on the empirical support for evolutionary ecology theory, there will be a striking preponderance of work on rapidly evolving model organisms such as bacteria and *Drosophila* that have provided new insights into questions that have traditionally challenged ecologists.\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Demographic considerations and measures of fitness\n",
+    "\n",
+    "From a population biology perspective, many questions are raised by the treatment of demography and the closely tied concept of fitness in evolutionary biology. The issue of fitness in particular, is a controversial one even within evolutionary biology {cite}`Ariew2004,Lewontin2004`. For our purposes however, we will focus on three aspects of population genetics that have particularly troubled population ecologists. The first two of these are visible in the illustration of the traditional population genetics approach in the equations above. The problems are, \n",
+    "\n",
+    "(i) population sizes are assumed to be infinite, which circumvents the complications of random birth-death processes in finite populations (called \"ecological drift\" or \"demographic stochasticity\") {cite}`Abrams2001`.\n",
+    "(ii) The Malthusian parameter $r$ is converted into a fitness measure $s$ by assuming fixed population size, which allows the convenience of reducing the parameters to just genotype proportions and relative fitnesses, and \n",
+    "(iii) When considering inter-specific interactions, the mean fitness (eqn. {eq}` `) of species' population becomes an absolute measure, creating the need to develop a general algorithm for translating absolute fitnesses between populations. As will be emphasized below, the use of quantitative genetic models helped resolve the problem theoretically, but the practicability of measuring fitness components in the field remains a difficult issue.\n",
+    "\n",
+    "Population geneticists dealt with the issue of infinite population size by introducing population size as a statistical parameter that affected the rate of random genetic drift– effective population size, $N_e$ {cite}`Kimura1983`. Although it was not an entirely satisfactory solution from a population biology perspective as it is only a phenomenological model of demographic stochasticity, it was acceptable in ecology because it could still be used to consider the evolutionary consequences of finite population size in terms of extinction probabilities {cite}`Caughley1994`. The issue of fixed population sizes on the other hand, raised some crucial questions for population biologists. What happens when populations change in size as they inevitably do? Or when generations are not discrete, but overlapping? Or when population change is density dependent? Extensions of population genetics theory to address questions of population change and overlapping generations had been accomplished by the 1970's, albeit with some issues still unresolved {cite}`Lewontin2004,Charlesworth1980`.   \n",
+    "\n",
+    "Evolution under density dependent population growth was first modeled by Kostitzin between 1935-40 {cite}`Christiansen2004`, and Fisher {cite}`Fisher1930` by assuming weak selection, and fitnesses dependent upon density dependent death rates. Subsequently, MacArthur {cite}`Macarthur1962` used a similar approach, but used the then familiar $r$ & $K$ parameterization of the Pearl-Verhulst logistic growth equation of density dependent growth,\n",
+    "\n",
+    "$$\n",
+    "    \\dfrac{dN}{dt} = rN\\left( 1-\\frac{N}{K}\\right)\n",
+    "$$\n",
+    "\n",
+    "This general approach, which was developed further by others {cite}`Christiansen2004` will be briefly illustrated here. Consider a haploid, asexual population with only two alleles 1 & 2 at a single locus. Assuming that the population is under density dependent growth, the fitnesses can then be defined as (cf eqns 8-10),\n",
+    "\n",
+    "$$\n",
+    "    w_{1} = 1 + r_{1}\\Bigl(1 - \\frac{N}{K}\\Bigr)\n",
+    "    \\quad \\text{and} \\quad\n",
+    "    w_{2} = 1 + r_{2}\\Bigl(1 - \\frac{N}{K}\\Bigr)\n",
+    "$$\n",
+    "Frequency dependent selection and genotype-specific carrying capacities can now be included by extending these eqns to,\n",
+    "\n",
+    "$$\n",
+    "    w_{1} = 1 + r_{1} \\Bigl(1 - \\frac{N_{1} + N_{2}}{K_{1}}\\Bigr)\n",
+    "    \\quad \\text{and} \\quad\n",
+    "    w_{2} = 1 + r_{2} \\Bigl(1 - \\frac{N_{1} + N_{2}}{K_{2}}\\Bigr)\n",
+    "$$\n",
+    "\n",
+    "which amounts to rephrasing the Lotka-Volterra equations (eqns. {eq}` `) for competition of two species to competition between two genotypes of one species, assuming the intra-specific competition parameters to be $\\alpha_{12} = α_{21} = 1$. Extensions of such models to overlapping generations under different single-population scenarios, and to coevolution of two or more species have provided many interesting insights, and contributed to the unification of population genetics and ecology {cite}`Christiansen2004`. The first major result of this research, of course, was the development in the 1970's of the ideas $r$\\-$K$ selection and the role of the environment in shaping life history strategies {cite}`MacArthur1967,Christiansen2004,Roughgarden1979`.  \n",
+    "\n",
+    "However, modeling fitnesses under frequency as well as density dependent growth in multiple populations does create the third problem mentioned above: that of absolute fitnesses. The interpretation of density dependent fitnesses becomes difficult in multi-species systems because mapping the effect of each populations absolute mean fitness on another population becomes context dependent because fitnesses are a function of the environment under this scheme (because they are tied to $K$), instead of being an inherent property of individuals' genotype {cite}`Ariew2004,Lewontin2004`. This makes it difficult to develop a framework for making general predictions about the evolutionary outcomes of species interactions without taking the situation-specific environment-phenotype-genotype map into consideration. For example, if we want to extend eqns. {eq}` ` to two haploid asexual species $A$ & $B$, each experiencing intra- and inter-specific density as well as frequency dependent selection, we could begin with, assuming one locus and two alleles, and using the Lotka-Volterra scheme,\n",
+    "\n",
+    "$$\n",
+    "\\begin{aligned}\n",
+    "w_{1A}\n",
+    "  &= 1 + r_{1A}\n",
+    "     \\;-\\;\n",
+    "     \\frac{r_{1A}}{K_{1A}}\\;N_{A}\n",
+    "     \\;-\\;\n",
+    "     \\frac{r_{1A}}{K_{1A}}\n",
+    "     \\Bigl(\\alpha_{1A1B}\\,N_{1B}\n",
+    "            \\;+\\;\n",
+    "            \\alpha_{1A2B}\\,N_{2B}\n",
+    "     \\Bigr)\n",
+    "  \\\\[6pt]\n",
+    "  &= 1 + r_{1A}\n",
+    "     \\;-\\;\n",
+    "     \\frac{r_{1A}}{K_{1A}}\n",
+    "     \\Bigl[\n",
+    "        N_{A}\n",
+    "        \\;-\\;\n",
+    "        \\bigl(\\alpha_{1A1B}\\,p_{B}\n",
+    "               \\;+\\;\n",
+    "               \\alpha_{1A2B}\\,\\bigl(1 - p_{B}\\bigr)\n",
+    "        \\bigr)\\,N_{B}\n",
+    "     \\Bigr]\n",
+    "\\end{aligned}\n",
+    "$$\n",
+    "\n",
+    "and similarly, \n",
+    "\n",
+    "$$\n",
+    "w_{2A}\n",
+    "  = 1 + r_{2A}\n",
+    "    \\;-\\;\n",
+    "    \\frac{r_{2A}}{K_{2A}}\n",
+    "    \\Bigl[\n",
+    "      N_{A}\n",
+    "      \\;-\\;\n",
+    "      \\bigl(\\alpha_{2A1B}\\,p_{B}\n",
+    "             \\;+\\;\n",
+    "             \\alpha_{2A2B}\\,\\bigl(1 - p_{B}\\bigr)\n",
+    "      \\bigr)\\,N_{B}\n",
+    "    \\Bigr]\n",
+    "$$\n",
+    "\n",
+    "where $w_{1A}$ and $w_{2A}$ are as before, the fitnesses of the two genotypes, while $r_{1A}$, $r_{2A}$ & $K_{1A}$, $K_{2A}$ are the genotype-specific intrinsic rate of increase and carrying capacity respectively. This model assumes that a genotype in a species $A$ interacts with all available genotypes of species $B$ in the community in proportion to their frequency. Thus, $\\alpha_{1A 1B}$ and $\\alpha_{1A 2B}$ represent the competition coefficients of genotype 1 in species $A$ with genotype 1 and 2 in species $B$, respectively. These are multiplied by the frequency of the respective genotype ($p$, $1-p$). The competition coefficient introduces the effect of interspecific frequency dependence relative to intraspecific density dependence {cite}`Laska1998,MacArthur1972,MacArthur1972b`. If $\\lapha$ were zero, the interacting populations would experience purely intra-specific density dependent growth and selection. Rewriting eqn. (4) eq in terms of eqns. (12 & 13) gives the mean fitness of the population:\n",
+    "\n",
+    "$$\n",
+    "\\overline{W}_{A}\n",
+    "  = 1 + \\overline{r}_{A}\n",
+    "    \\;-\\;\n",
+    "    \\frac{\\overline{r}_{A}}{K_{A}}\n",
+    "    \\Bigl(\n",
+    "      N_{A}\n",
+    "      \\;-\\;\n",
+    "      \\overline{\\alpha}_{AB}\\,N_{B}\n",
+    "    \\Bigr),\n",
+    "$$\n",
+    "\n",
+    "where\n",
+    "\n",
+    "![][image40],\t\t\t\t\t\t\t\t      (17)\n",
+    "\n",
+    "![][image41],\t\t\t\t\t\t\t      (18)\n",
+    "\n",
+    "and \n",
+    "\n",
+    "![][image42]\t\t\t\t      (19)\n",
+    "\n",
+    "In each generation, the population growth is a function of the mean fitness![][image43]of the population in the previous generation, made up of the relative contributions of the two genotypes. For example, if allele A is fixed in the species *A*, its population will grow at a density dependent growth determined by the *r* and *K* values of the genotype 1*A* and its interaction with genotypes of species *B*. Eqns.(14-19) illustrate the difficulty with measuring fitness in such situations. In this model, one would have to keep track of no less than 10 parameters to evaluate the evolutionary outcome of interaction between just two species\\! And this is assuming that *α* is a reasonable representation of both direct and indirect interactions in nature. Computer simulations and mathematical analyses (with some reparameterization) of these models do yield insights, but measuring all the components of fitness in such a situation to be an almost impossible task in reality, let alone defining a more general scheme for translating absolute fitnesses between species.  \n",
+    "\n",
+    "The problem is considerably simplified by using a quantitative genetics framework (see eqns. (9-10) above). For example, for the two competing species in the above example, the fitness functions from eqns (14 & 16) could be easily modified to feed into eqns. (9 & 10), thus creating a quantitative map between fitnesses of the two populations generated by frequency- and density-dependent interactions within and between species. However, the empirical problem of measuring all components of fitness remains a difficult problem. On the whole then, the manner in which fitnesses are dealt with will be an important criterion for the classification presented below. It will become apparent that certain model systems that have been used in recent empirical work are relatively amenable to estimation of fitness components. This has gone a long way in testing some theoretical predictions.   \n",
+    "\n",
+    "It should also be noted that in addition to examining the role of demography in evolution, it is possible to flip the question over, and focus on the effects of evolution on population dynamics. Such work has also emerged in recent years and is an area of active theoretical development and empirical research, adding yet another facet to evolutionary ecology. \n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### The role of environmental heterogeneity\n",
+    "\n",
+    "The 'problem' of complex population behavior and the role of \"external\" vs. \"internal\" factors play an important role in ecology (Hastings et al. 1993)(https://paperpile.com/c/lwDPbc/GV43). In search for answers to such questions, ecologists have expended considerable effort over the last century for a detailed characterization of heterogeneity in nature. Areas such as landscape and spatial ecology have steadily built a realistic characterization of heterogeneity in nature, and to many biologists, ecology is synonymous with environmental characteristics; the ecology of a species is at least as likely to connote the organism's environment as its inherent properties, such as life history characteristics. As was pointed out above, the field of ecological genetics has developed around this view of ecology.   \n",
+    "The incorporation of environmental heterogeneity into a unified evolutionary ecology framework is far from complete though. This is one area where the flow of information must largely be from the direction of ecology (in its environmental avatar) towards evolution. As Levins (Levins 2004)(https://paperpile.com/c/lwDPbc/Uyrg) points out, \" '*There has always been a much finer sophistication of our understanding of genetic variation than of the environment,*' '*…a successful study of evolution requires the recognition of the complexity, not just of the genotype, but also of the environment and of the whole organism in its development and its physiological flux'* \". This is of course, a reiteration of the argument made above about the effect of genetic reductionism on the conceptual unification of population genetics and population biology. Practitioners of ecological genetics do indeed often pay just lip service to environmental heterogeneity, using a set approach of employing statistical tools to fit observed genetic variation to predictions of traditional population genetics models, most of which have tenuous connections with ecological reality. Statistically reasonable fits may be observed, but the results do not reveal whether the model being tested describes the real mechanisms for the maintenance of genetic variation. Although this is a general concern in all scientific disciplines, it is also a function of how much of the universe of possibilities has been explored while formulating theories.   \n",
+    "In the quest for a formal description of all the nuances of environment complexity, ecologists have been no laggards; the mathematical structure of many aspects of the environment in which evolution can potentially operate has been well characterized in both time (Vasseur and Yodzis 2004)(https://paperpile.com/c/lwDPbc/dKQx) and space (Keitt et al. 2002)(https://paperpile.com/c/lwDPbc/3b7Q). For example, consider Figure 1, which shows an empirical observation of temporal environmental pattern: seasonal variation in temperature across latitudes. It is well known that the amplitude of seasonal cycles as well as random environmental noise increases away from the latitudes (Vasseur and Yodzis 2004)(https://paperpile.com/c/lwDPbc/dKQx). \n",
+    "\n",
+    "Fig. 1\\. Variation in temperature across months over 11 years for four sites spanning 400 of latitude, from New Zealand (46.70S), to Papua New Guinea (6.730S). All sites are from a similar altitude (0–20 m above sea level (S. S. Pawar 2005)(https://paperpile.com/c/lwDPbc/BCIv). Note that the fluctuations increase away from the equator. \n",
+    "\n",
+    "![][image44]\n",
+    "\n",
+    "This observation is easily translated into a model of fluctuating selection (Bürger and Gimelfarb 2002)(https://paperpile.com/c/lwDPbc/3fia). Consider a selectively optimum trait value *θ*, such as the one introduced in eqn.(9). Assume that the position of the optimum fluctuates cyclically about the midpoint of the range of trait values (*θo*), with a scaled random perturbation around it. This can be modeled by assuming that at time *t* the optimum *θt* is drawn from a normal distribution with mean\n",
+    "\n",
+    "![][image45]\t\t\t\t\t\t\t\t\t\t      (20)\n",
+    "\n",
+    "where *A* is the amplitude and *L* the length of the selection cycle relative to the organism's generation time. The perturbations have a standard deviation![][image46], where *d* is a measure of the magnitude of stochasticity. If *d*\\=0, there is purely cyclical selection; if, in addition, *A*\\=0, there is pure Gaussian stabilizing selection. Figure 2 shows the temporal pattern of cyclical selection generated from this model with arbitrary parameter values. This matches the empirical pattern in Figure 1 quite well.\n",
+    "\n",
+    "Fig. 2\\. Cyclical selection with random fluctuations over 100 generations, based on the model in eqn. (20). \n",
+    "\n",
+    "![untitled][image47]\n",
+    "\n",
+    "This idea can then be further developed to consider the effect of such environmental grain on evolution as well as persistence of populations. After the initial beginning in the 1960's of work on the effect of fluctuating environments on adaptation and persistence of populations, the role of periodic instead of completely random fluctuations, partly motivated by empirical patterns like the one in Figure 1, increasingly drew attention over the last decade or so, yielding new insights (Kirzhner, Korol, and Nevo 1998; Wichmann et al. 2003; Bürger and Gimelfarb 2002; Lande and Shannon 1996)(https://paperpile.com/c/lwDPbc/032V+9RCS+3fia+fzdV). Detailed examination of such empirical patterns of environmental heterogeneity can also help resolve questions about the likelihood of seeing certain evolutionary processes in nature. For example, Gillespie (J. H. Gillespie, Singh, and Uyenoyama 2004)(https://paperpile.com/c/lwDPbc/Nb5C) long argued that widely accepted population genetic models of adaptive evolution in DNA sequences need to be reevaluated in light of the fact that certain patterns of fluctuating selection can generate patterns of polymorphism akin to those from models of neutral evolution, a claim that remains largely untested. The evolutionary ecology synthesis will definitely benefit from the use of environmental information; the extent to which this has been achieved is considered in the classification that follows.\n"
+   ]
+  },
+  {
+   "attachments": {},
+   "cell_type": "markdown",
+   "metadata": {
+    "jp-MarkdownHeadingCollapsed": true
+   },
+   "source": [
+    "### Taxonomy of evolutionary ecology**\n",
+    " classification of contemporary approaches that combine ecological and evolutionary perspectives from population genetics and population biology are presented in Table 1\\. Keeping the tradition initiated by stalwarts of the evolutionary ecology synthesis, I drop the term \"evolutionary ecology\" in favor of \"population biology\" for this classification (Levins 2004; Lewontin et al. 2004; Travis et al. 1989)(https://paperpile.com/c/lwDPbc/Uyrg+fPH1+bZbA).\n",
+    "\n",
+    "**Table 1**. A taxonomy of research that unifies ecological and evolutionary perspectives in biology. The issues considered for classification are: approach towards time-scale, demographic considerations, fitness measures, and treatment of ecological heterogeneity.\n",
+    "\n",
+    "| Category | Scientific approach   |\n",
+    "| :---- | :---- |\n",
+    "| **I. Paleo-population biology** | *Retrospective studies considering historical effects on properties of current populations.*  |\n",
+    "| **(a) *Non-demographic schema*** | Reconstruction of evolutionary relationships without paleo-demographic inferences, with focus on extant populations of two or more species. Fitness concepts used are absolute and often qualitative. Structural or spatial aspects of habitat interpretable as niches play an important part.  |\n",
+    "| **(b) *Demographic schema*** | Reconstruction of phylogenies and gene genealogies with paleo-demographic and paleo-ecological inferences with focus on one or more extant species' populations. Fitness measures and ecological heterogeneity can be considered only in extant populations.  |\n",
+    "| **II. Population biology**  | *Essentially prospective studies restricted to \"microevolutionary\" or \"ecological\" time scales.*  |\n",
+    "| **(a) *Non-demographic schema*** | Study of populations without explicit consideration of demography, using traditional population genetic tools. Relative fitnesses are used, assuming fixed population sizes. Ecological heterogeneity, if considered, can be spatial and/or temporal.   |\n",
+    "| **(b) *Indirect-demographic schema*** | Study of populations with inference of demographic characteristics (in general, population size), indirectly, by using population genetics theory to estimate N*e*. Fitness measures are absolute, and generally qualitative. Ecological heterogeneity is generally spatial. |\n",
+    "| **(c) *Demographic schema*** | Entail explicit consideration of the interplay between evolutionary changes and demography of populations. Fitness measures vary, often involving density as well as frequency dependence, or are absolute. Ecological heterogeneity, if considered, can be spatial and/or temporal.  |\n",
+    "\n",
+    "Each of these research programs is now discussed with reference to the nature of scientific approach, the conceptual issues it addresses, its theory and supporting empirical data. Together, these disciplines comprise a number of research topics and a huge body of literature. This discussion below does not claim to be an exhaustive survey of all topics. Within each research program however, poorly explored avenues of interesting research are also mentioned.\n",
+    "\n",
+    "**I. Paleo-population biology**  \n",
+    "One solution towards fusing evolution and ecology is to incorporate historical processes by reconstructing the evolutionary history and paleo-ecology of populations and drawing inferences about contemporary properties of organisms. Two distinct approaches are currently being used:\n",
+    "\n",
+    "**(a) *Non-demographic schema***\n",
+    "\n",
+    "**Approach**.This program of research is an outcome of the above-mentioned 'historical' viewpoint mentioned that has been so prevalent in ecology. This program was invigorated in the 1990's by the introduction of phylogenetic techniques, thus allowing a more reliable reconstruction of evolutionary history (Eggleton and Vane-Wright 1994; Losos 1996)(https://paperpile.com/c/lwDPbc/6vwA+rB1C). This method is non-demographic in the sense that population size considerations rarely enter the analyses, either in the historical reconstructions, or in the study of contemporary adaptations. The concepts of fitness used here are absolute and qualitative, with the outcome of interactions between populations described in terms of the apparent level of fitness in populations.\n",
+    "\n",
+    "**Research themes:** *theory and empirical results*. This approach is strongly retrospective, generally involving verbal extensions of prospective evolutionary ecology models such as those of character displacement and the mapping of traits onto phylogenies (Eggleton and Vane-Wright 1994; Losos 1996)(https://paperpile.com/c/lwDPbc/6vwA+rB1C). Some prominent themes for research are adaptive radiation into ecological niches (Schluter 2000; Losos and Miles 2002)(https://paperpile.com/c/lwDPbc/H1J6+uLfC), Outcome of species interactions (e.g., evolutionary character displacement; (Radtkey, Fallon, and Case 1997)(https://paperpile.com/c/lwDPbc/Qrll), and the manner of community assembly (R. Gillespie 2004)(https://paperpile.com/c/lwDPbc/GUZ0). The insights from these lines of work have served to greatly clarify the role of evolution in structuring communities and molding species interactions. For example, Gillespie (R. Gillespie 2004)(https://paperpile.com/c/lwDPbc/GUZ0) has shown that convergent evolution into similar niches has an important role in community assembly, shedding light on the long unresolved debate in community ecology about the manner of community assembly, and the existence of \"assembly rules\" (Weiher and Keddy 1995)(https://paperpile.com/c/lwDPbc/3lUy). On the flipside, such studies contribute a lot towards clarifying the role of ecology in speciation, a burgeoning topic in evolutionary biology (Coyne and Orr 2004)(https://paperpile.com/c/lwDPbc/fVZt).\n",
+    "\n",
+    "**(b) *Demographic schema***\n",
+    "\n",
+    "**Approach***.*This relatively new area involves the inference of \"historical demography\" combined with analyses of properties of extant populations. This has been made possible by recent attempts to bring the coalescence theory (Kingman 2000)(https://paperpile.com/c/lwDPbc/PMBy) of population genetics and phylogenetics closer together to evaluate the interrelationships between micro- and macroevolution (Wakeley, Singh, and Uyenoyama 2004)(https://paperpile.com/c/lwDPbc/yxzS). This approach also opens up the possibility of explicitly including historical population structure in analyses of current populations, a topic that has hitherto been difficult to deal with except by verbal arguments in ecology as well as evolutionary biology. As paleo-population structure is inferred purely from nucleotide polymorphism data, this approach obviously cannot infer past environmental heterogeneity explicitly unless paleo-ecological events can be dated and implicated using molecular clock approaches. Fitnesses obviously do not even enter the picture as far as the historical component is concerned. The analyses of extant populations can of course include both, fitness measures and consideration of environmental heterogeneity.\n",
+    "\n",
+    "**Research themes:** *theory and empirical results.* Obviously, the past can impinge upon almost every feature of extant organisms. The themes that will be developed in this area will only be bound by the limits of the collaboration between paleobiology, phylogenetics, coalescent theory, and ecology. These are widely separated disciplines, and a coherent theory will probably take a while to develop. Research in this area could include, but need not be restricted to, all of the themes of the approaches under the non-demographic schema mentioned above. For example, recently, Flanagan et al (Flanagan et al. 2004)(https://paperpile.com/c/lwDPbc/GW6y) used such an approach to clarify the role of historical demography in the coevolution of two Müllerian comimetic butterfly species. The potential of this area of research is huge; all the questions asked in the non-demographic schema above can be better addressed by such analyses.\n",
+    "\n",
+    "**II. Microevolutionary ecology** \n",
+    "\n",
+    "This is a much larger area of research than paleo-population biology, because prospective study of extant populations is the predominant approach in population biology. This area involves the consideration of microevolutionary changes in generating ecological patterns and processes, as well as the effect of the environment on observable evolution. Two broad categories of approach can be identified here:\n",
+    "\n",
+    "**(a) *Non-demographic schema***\n",
+    "\n",
+    "**Approach***.*This program involves the extension of population genetics (including quantitative genetics) to realistic ecological (in the sense of the adaptive environment) scenarios, for a detailed examination of populations' characteristics in relation to the environment. In keeping with the population genetic framework, population size only appears if at all, as *Ne*, and fitnesses are relative, or if multiple species are considered, may be juxtaposed by analyses of mean population fitnesses (generally using quantitative genetics). The most common aspect of ecological heterogeneity is spatial, while temporal variation in the abiotic environment is increasingly being studied, with rare examples of a combination of the two (Frank and Slatkin 1990)(https://paperpile.com/c/lwDPbc/YH5u). \n",
+    "\n",
+    "**Research themes:** *theory and empirical results.* Most studies in ecological genetics and molecular ecology use this approach to establish the relationships between genetic variation in nature and ecological heterogeneity (Brian Charlesworth, Charlesworth, and Barton 2003)(https://paperpile.com/c/lwDPbc/H4zB), wherein the focal population's environs can be both biotic (e.g., herbivore-plant interactions), as well as abiotic (DeSalle and Schierwater 1998; Beebee and Rowe 2004; Conner and Hartl 2004)(https://paperpile.com/c/lwDPbc/S0h5+1SH7+fazZ). Other themes are evolution in temporally fluctuating environments (A. Sasaki and Ellner 1997; Ellner 1996; Akira Sasaki and Ellner 1995; J. H. Gillespie 1991)(https://paperpile.com/c/lwDPbc/gSfW+mNWb+aTAq+xa9v), evolution in spatio-temporally fluctuating environments (Frank and Slatkin 1990)(https://paperpile.com/c/lwDPbc/YH5u), evolutionary outcome of species interactions with such as competition and predation with or without interactions with the environment (P. a. Abrams 2000; M. Slatkin and Maynard Smith 1979; Montogemery Slatkin 1980; Agrawal 2003)(https://paperpile.com/c/lwDPbc/uGXo+o46W+O2ah+DBMh). A large empirical area corresponding to the last is the whole area of evolutionary ecology of plant-animal interactions, which involves the study of coevolution between populations, which does not generally consider demography explicitly, defines fitnesses largely qualitatively in terms of apparent adaptations of populations, with or without consideration of abiotic environment (Wöhrmann and Loeschcke 1984; Price and Schluter 1991; Emlen 1973; Beattie 1985; Drake 1968; Karlin and Nevo 1976; Poulin 1998)(https://paperpile.com/c/lwDPbc/VruJ+4cMl+E8U4+kCz4+zzc1+xIOL+j3Gh). An important area that has recently come to the fore is the study of the role of the environment in determining evolutionary rates at the molecular level (S. S. Pawar 2005)(https://paperpile.com/c/lwDPbc/BCIv). This has raised some interesting new questions about the evolutionary mechanisms behind latitudinal diversity gradients, an issue that has traditionally been an ecological topic. In addition, work with microorganisms has also provided valuable data for evolution in temporally fluctuating (Bell and Reboud 1997; Reboud and Bell 1997; Scheiner and Yampolsky 1998; Kassen and Bell 1998)(https://paperpile.com/c/lwDPbc/0eyC+P2GH+HqQK+XFKD) as well as spatially variable (Travisano, Vasi, and Lenski 1995)(https://paperpile.com/c/lwDPbc/Sc6a) environments.\n",
+    "\n",
+    "**(b) Indirect-demographic schema**\n",
+    "\n",
+    "**Approach***.* This area involves the study of populations in nature using ecological genetic techniques, and inferring demographic characteristics (generally, effective population size, *Ne*) indirectly, by using population genetics theory. Fitness measures are absolute, and generally qualitative. Ecological heterogeneity, if considered, is generally spatial.\n",
+    "\n",
+    "**Research themes:** *theory and empirical results–*   \n",
+    "The theoretical basis for this line of research is traditional population genetics theory, where genetic variation is used to estimate demographic characteristics such as N*e* and population structure (Hedrick 1984)(https://paperpile.com/c/lwDPbc/HqLP). As most inferences about population characteristics are indirectly inferred, this area has a restricted range of themes, and will probably be absorbed into one or more of the other categories. Most results in this area have yielded insights into the impact of natural or anthropogenic environment variation on characteristics of single populations  (Crawford, Macleod, and Mitchell 2003; Woolfit 2003)(https://paperpile.com/c/lwDPbc/tPSr+D6TV).\n",
+    "\n",
+    "**(b) Direct-Demographic schema**\n",
+    "\n",
+    "**Approach**. This area of research is a rapidly growing area, and includes a range of distinct research themes that have their origins in ecology or evolutionary biology. All these involve explicit consideration of demography. The most complex topics involve a three-way interaction between the environment, demographic dynamics, and evolution, perhaps capturing the essence of what has been visualized to be a unified population biology (Travis et al. 1989; Levins 2004; Lewontin et al. 2004)(https://paperpile.com/c/lwDPbc/bZbA+Uyrg+fPH1). The characterization of fitness in this diverse research area is naturally variable. To a great extent, the problems that have been mentioned above with the measurement of fitness components is perhaps the most relevant to demographically-oriented objectives of these studies.\n",
+    "\n",
+    "**Research themes:** *theory and empirical results.* The theory that underlies this area has been covered in parts in the previous sections, and involves demographically explicit models that combine population genetics (including quantitative genetics) with different levels of ecological heterogeneity. The themes of research are wide ranging (Brussard and Allard 1978)(https://paperpile.com/c/lwDPbc/okQu) include the effects of evolutionary change on demographic characteristics and vice versa (Michael Doebeli and Koella 1994, 1995; Michael Doebeli 1996b; M. Doebeli 1996; Michael Doebeli 1996a)(https://paperpile.com/c/lwDPbc/2IfO+n8sU+Abha+L8fS+dbVj), evolutionary changes in communities (Norberg et al. 2001)(https://paperpile.com/c/lwDPbc/t6g5), evolution of population stability, evolution of life history traits (B. Charlesworth 1993; Brian Charlesworth 1980; Reznick, Bryant, and Bashey 2002; Mueller and Joshi 2000; Levins 1968; A. D. Bradshaw and Shorrocks 1984; Norberg et al. 2001; M. Doebeli and de Jong 1999; Michael Doebeli 1996b; Michael Doebeli and Koella 1995)(https://paperpile.com/c/lwDPbc/a4Tw+VakU+09ni+69mL+WlSX+QWRf+t6g5+h1PE+Abha+n8sU), and evolution in fluctuating environments (Reznick, Bryant, and Bashey 2002; Mueller and Joshi 2000)(https://paperpile.com/c/lwDPbc/09ni+69mL).   \n",
+    "This area has had a long history of disjunction between theory and the empirical work, partly because many of the predictions of microevolutionary ecology models cannot be tested easily either because fitness components are difficult to measure, or the rate of evolution in the model organisms is too slow to be observed in reasonable time scales. However, the inclusion of new model systems, which was mentioned in section 3.1 above, has had a big impact on this situation. For example, work with *Drosophila* and *Poecilia (Reznick, Bryant, and Bashey 2002; Mueller and Joshi 2000)(https://paperpile.com/c/lwDPbc/09ni+69mL)* has produced the first concrete support for some of the predictions of theory on evolution of life history traits. Work with microorganisms too has provided strong evidence for the role of evolution in molding population dynamics (Yoshida et al. 2003; Fussmann, Ellner, and Hairston 2003)(https://paperpile.com/c/lwDPbc/UP82+gqas). This has important implications for the one of the longest-lived problems in ecology of pinpointing the factors driving population cycles in microtine rodents and their predators, for which evolutionary hypotheses had been proposed before, but had received little attention by ecologists (Peter Turchin 2003; Chitty 1996)(https://paperpile.com/c/lwDPbc/VtwM+P2dw). A number of equally valuable insights on other ecological questions from work with other microorganisms (Jessup et al. 2004)(https://paperpile.com/c/lwDPbc/8fxM). \n",
+    "\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Discussion\n",
+    "\n",
+    "### Has The Marriage Come Of Age?\n",
+    "\n",
+    "So what has been the outcome of this intricate branching of evolutionary ecology? In a recent paper, Lewontin and Levins, two key figures in the program for a unified population biology, provide an assessment, which ranges from being cautiously optimistic to rather dyspeptic [(Lewontin et al. 2004\\)](https://paperpile.com/c/lwDPbc/fPH1). After discussing a range of topics that are at the heart of the synthesis, Levins laments that \"This program for an integrated population biology remains an aspiration that has not been carried out in practice. Instead, we see a gross imbalance among these components and a continued separation of the disciplines\". This is definitely true to the extent that the mainstream of ecology and evolutionary biology do indeed proceed happily along a divergent (or parallel) path, as was observed at the beginning of this paper. However, I am inclined to be more optimistic that Levins is.   \n",
+    "The taxonomy of a unified population biology that has been presented here is no doubt temporary, but serves the purpose of demonstrating the impressive range of topics that studies undertaking an evolutionary approach have managed to cover, both in theory, and empirical work. One factor that Lewontin and Levins do not seem to have considered, is the crucial role that the quantitative genetics models have played in providing at the very least, heuristic insights into the role of ecology in evolution, and vice versa. Moreover, there is ample evidence for the bi-directional flow of contribution, which indicates a greater balance, perhaps not yet fully appreciated, in the nature of the unification. There is no doubt that the marriage has not yet come of age. However, its rapid progress in the last two decades or so, bode good tidings for the not too distant near future.\n",
+    "\n",
+    "### On the Dullness of Ecology\n",
+    "\n",
+    "It was pointed out at the beginning that the philosophy of biology appears to suffer from a considerable disinterest in ecology, especially in comparison to evolutionary biology [(Grene and Depew 2004; Sober 1993a; Hull and Ruse 1998; Hull 1974\\)](https://paperpile.com/c/lwDPbc/PICF+MDgF+Tpjb+JR1s). To some extent, this is understandable, as no other work in history has shaken the teleological foundations of human culture, as Darwin's radical theory [(Hull 1974; Sober 1993a)](https://paperpile.com/c/lwDPbc/JR1s+MDgF). Considering the fact that the role of teleology in evolutionary biology have been an important topic of discussion [(Sterelny and Griffiths 1999)](https://paperpile.com/c/lwDPbc/rVc1)[^10], this disinterest in the philosophy of ecology is surprising because there has been a widespread belief in ecology of the existence of optimal strategies, stability and persistence, the roots and implications of which are worth contemplating about. In this context, Hull [(Hull 1974\\)](https://paperpile.com/c/lwDPbc/JR1s) has a definition of teleology in the biological context that applies well to ecology: \"*prevalence of preferred states, closed feedback loops and programs, and the origin of such systems by means of selection processes*.\" On the other hand, apart from the issue of teleology, evolutionary biology also offers a much more coherent theoretical structure compared to ecology, and has thrown up its share of interesting topics ranging from the role of biological determinism, to issues such as the meaning of species, units of evolution and selection, and so on [(Grene and Depew 2004; Sterelny and Griffiths 1999\\)](https://paperpile.com/c/lwDPbc/PICF+rVc1).   \n",
+    "This paper offers another, perhaps more fundamental reason for a reconsideration of the dullness of ecology. The question of the existence of laws in biology of has been a topic that has troubled biologists and philosophers for a while, and still provides ample fodder for rumination [(Bergandi and Blandin 1998; Kingsland 1985; Ginzburg and Colyvan 2004; Yrjö Haila and Taylor 2001; Cooper 2003\\)](https://paperpile.com/c/lwDPbc/FE1J+WSwP+tvrS+KQ8r+wyg0). And if philosophers are interested in the laws of biology, where better to look at the rather discordant structure of ecology? Opinions about the validity of the theoretical foundations of ecology and the presence of laws vary widely, ranging from agreement about the presence of general principles at one end [(Berryman and Uni 2003; P. Turchin 2002\\)](https://paperpile.com/c/lwDPbc/SoZk+ugaQ), to the declaration of the theoretical redundancy of ecology to the advancement of biology on the other [(Yrjö Haila and Taylor 2001\\)](https://paperpile.com/c/lwDPbc/KQ8r). Writing about the relationship between ecology and evolution, Sterelny and Griffiths [(Sterelny and Griffiths 1999\\)](https://paperpile.com/c/lwDPbc/rVc1) invoke Sober's [(Sober 1993b)](https://paperpile.com/c/lwDPbc/08Ck) distinction between \"source laws\" and \"consequence laws\". Source laws explain the origins of fitness differences, whereas consequence laws explain the actual evolutionary outcome of these differences. Sterelny and Griffiths judge that the role of ecology is to define the source laws in evolutionary biology.   \n",
+    "However, I claim that the results of this paper indicate that this conclusion is incorrect, given the balanced nature of the current collaboration between ecology and evolutionary biology. Ecology and evolutionary biology have clearly made reciprocal contributions to each other's conceptual advancement. Thus While some may consider the future of ecology as a whole to lie in a stronger integration into the human social context [(Kingsland 1985\\)](https://paperpile.com/c/lwDPbc/WSwP)(pp 65-66) [(G. A. Bradshaw and Bekoff 2001; Yrjö Haila and Taylor 2001; Yrjšo Haila and Levins 1992\\)](https://paperpile.com/c/lwDPbc/mDgv+KQ8r+A4SO), the progress of the evolutionary ecology synthesis suggests that while parts of ecology might indeed have a important social role, a large part of ecology might actually come to rest in an synthetic theory that comprises a more sturdy theoretical foundation than either discipline alone. Neither discipline is theoretically redundant either to each other, or to biology as a whole. While the existence of laws in biology is an important issue, it is perhaps time to consider the theoretical and philosophical implications of the ongoing reunification of the two disciplines that were born together with Darwin's theory, which marks the birth of modern biology itself.\n",
+    "\n",
+    "## Conclusion\n",
+    "\n",
+    "In this Chapter we have examined the alliance between ecology and evolutionary biology, which has been a distinct program of research for more than a century now. In particular swe will focus on the most theoretically and empirically tangible part of this alliance: the unification of population ecology and population genetics. The potential benefits of this unification are wide ranging, and it has been sought for more than half a century now. We began by highlighting the disparities between ecological and evolutionary viewpoints and reviewed aspects of their history that point at the roots of discord as well as efforts for synthesis. There are three important differences in viewpoints that population genetics and population ecology differ at: the role of processes at different time-scales, demographic considerations (including measures of fitness), and environmental heterogeneity. Based on these, a taxonomy of theoretical approaches that merge these two disciplines was presented, with focus on their conceptual approach, themes of research, and empirical advances. This allowed an evaluation of the current status of the synthesis - it is clear that the unification of population biology through an evolutionary ecology synthesis has made steady progress, but remains a work in progress. The nature of this unification also calls for a re-evaluation of the reasons for which ecology has been considered a sterile subject by philosophers. The current structure of the synthesis clearly indicates a balanced role for the two disciplines in the conceptual advancement of biology, and philosophers should perhaps pay more attention to the progress of this synthesis. This will also perhaps be a refreshing diversion from the rather bedraggled issue of the existence of laws in ecology and in biology."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## References\n",
+    "\n",
+    "```{bibliography}\n",
+    "```"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Footnotes\n",
+    "\n",
+    "[^1]:  In doing so, for reasons of brevity many significant figures and interesting details will necessarily have to be glossed over, and the focus will, perhaps a tad unfairly, be polarized towards developments in the United States of America.\n",
+    "\n",
+    "[^2]:  The area of niche construction, which involves the study of evolutionary dynamics in situations where populations modify their own environment [(Odling-Smee, Laland, and Feldman 2003\\)](https://paperpile.com/c/lwDPbc/1C1p), although an interesting interface between ecology and evolution, poses a distinct set of challenges, and will not covered here.\n",
+    "\n",
+    "[^3]:  Ideas of population growth, which formed the basis for Darwin's theory of natural selection predate the theory of natural selection in Malthus' treatise on populations in 1798, and the mathematical formulation of the first model of regulated (\"logistical\") population growth, by Verhulst [(Kingsland 1985\\)](https://paperpile.com/c/lwDPbc/WSwP) (pp. 64-66).\n",
+    "\n",
+    "[^4]:  I am referring to the Biometric and the Mendelian schools [(Provine 2001\\)](https://paperpile.com/c/lwDPbc/L0vt); for examples, see the preface to Gause's book [(Gause 1934\\)](https://paperpile.com/c/lwDPbc/2qwm), and Kingsland's analysis [(Kingsland 1985\\)](https://paperpile.com/c/lwDPbc/WSwP).\n",
+    "\n",
+    "[^5]:  Efforts to fuse genetics and ecology were also made in the Russian school of ecology by people such as V.V. Stanchinskii [(Kingsland 1985\\)](https://paperpile.com/c/lwDPbc/WSwP)(pp. 150, 157-160) [(Mirzoyan 1996\\)](https://paperpile.com/c/lwDPbc/ROCY), the details of which I am unable to access. \n",
+    "\n",
+    "[^6]:  \"Cultural differences\" has been invoked as a reason for lack of communication between the two disciplines by proponents of the population genetics – population ecology synthesis. For example, Roughgarden [(J. Roughgarden 1979\\)](https://paperpile.com/c/lwDPbc/Ccks) says,*\"The fusion of population genetics with population ecology can be compared to a prearranged marriage between partners who speak different languages. Although both families agree that the marriage is advantageous, it is somewhat difficult to achieve because of cultural differences.\"*\n",
+    "\n",
+    "[^7]:  In Russia as well, there appears to have been developments of a similar nature [(Shvarts 1977\\)](https://paperpile.com/c/lwDPbc/IuTu).\n",
+    "\n",
+    "[^8]:  In its objectives and methods, this field is now barely distinguishable from a recent off-shoot of ecology called \"Molecular Ecology\", which originated in the 1990's as a program to use molecular genetic methods to address problems in conservation biology, but then ceased to be a purely applied field, expanding and converging in its objectives and methods on Ecological Genetics. This is an interesting example of an inadvertent convergence of ecological and evolutionary fields driven by a common motivation. \n",
+    "\n",
+    "[^9]:  The first model was actually developed much earlier, by Kostitzin in 1936 [(Christiansen, Singh, and Uyenoyama 2004\\)](https://paperpile.com/c/lwDPbc/VMHP).\n",
+    "\n",
+    "[^10]:  Due to two main reasons: Adaptationism (a belief in the determinism of evolution) and theology (belief in a creator as the determining factor in biology). "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "<!-- Some key recommendations and areas for revision / updating:\n",
+    "\n",
+    "### **1\\. New Theoretical Insights and Themes**\n",
+    "\n",
+    "* **Eco-Evolutionary Dynamics**: Emphasize how eco-evolutionary feedbacks integrate both ecology and evolutionary biology, showcasing research on how evolutionary changes can affect ecological interactions and vice versa (e.g., Palkovacs et al., 2018).  \n",
+    "* **Adaptive Dynamics**: Expand on adaptive dynamics as a formal framework for studying evolutionary outcomes in ecological contexts, highlighting recent advancements and applications (e.g., Kisdi, 2010).  \n",
+    "* **Niche Construction**: Discuss how organisms can shape their own selective environments, integrating this with ecological theory to highlight mutual impacts (Laland et al., 2016).\n",
+    "\n",
+    "### **2\\. Updated Empirical Examples**\n",
+    "\n",
+    "* **Rapid Evolution in Natural Populations**: Provide examples from recent studies showing real-time evolution, such as guppy life-history evolution in response to predation (Reznick et al., 1997; Travis et al., 2014).  \n",
+    "* **Microbial Evolution Experiments**: Highlight long-term experiments with *E. coli* (e.g., Lenski's experiments) that reveal evolutionary and ecological principles in action (Lenski et al., 2015).  \n",
+    "* **Climate Change Impacts**: Use contemporary examples to discuss how climate change drives rapid evolutionary responses and shifts in population dynamics (e.g., Franks et al., 2018).\n",
+    "\n",
+    "### **3\\. Contemporary Challenges and Critiques**\n",
+    "\n",
+    "* **Philosophical Implications**: Expand the philosophical exploration of whether ecology and evolutionary biology can be fully unified, considering recent critiques of reductionism and the debate on predictive models (e.g., Sterelny and Griffiths, 2019).  \n",
+    "* **Data Integration Challenges**: Discuss the difficulties in integrating large-scale ecological data with evolutionary models due to complexity and variability in environmental interactions.\n",
+    "\n",
+    "### **4\\. Methodological Advances**\n",
+    "\n",
+    "* **Genomic Tools and Phylogenetics**: Highlight the role of whole-genome sequencing in connecting phylogenetic history with ecological traits and evolutionary processes.  \n",
+    "* **Computational Ecology**: Explore how computational tools, such as agent-based modeling and machine learning, have revolutionized the analysis of eco-evolutionary systems (e.g., Schreiber et al., 2021).\n",
+    "\n",
+    "### **5\\. Additional Citations and References**\n",
+    "\n",
+    "* Integrate more recent references, such as:  \n",
+    "  * Laland, K. N., Matthews, B., & Feldman, M. W. (2016). \"An update on niche construction theory.\"  \n",
+    "  * Palkovacs, E. P., Kinnison, M. T., Correa, C., et al. (2018). \"Eco-evolutionary feedbacks in community and ecosystem ecology.\"  \n",
+    "  * Franks, S. J., Weber, J. J., & Aitken, S. N. (2018). \"Evolutionary and plastic responses to climate change in terrestrial plant populations.\"\n",
+    "\n",
+    "### **6\\. Emerging Topics for Discussion**\n",
+    "\n",
+    "* **Evolutionary Rescue**: Introduce how populations facing environmental changes may evolve rapidly to avoid extinction, drawing on cases of evolutionary rescue.  \n",
+    "* **Anthropocene Impacts**: Discuss how human-induced changes create novel selective pressures and ecological shifts, which merge evolutionary and ecological responses.  \n",
+    "* **Trait-Based Approaches**: Expand on how trait-based frameworks can elucidate the intersection of evolutionary adaptation and ecological dynamics.\n",
+    "\n",
+    "**Include adaptive dynamics \\- (Kisdi, 2010\\)**\n",
+    "\n",
+    " -->"
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3 (ipykernel)",
+   "language": "python",
+   "name": "python3"
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+   "mimetype": "text/x-python",
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+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.10.12"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 4
+}
diff --git a/genindex.html b/genindex.html
index 19ee1f47..6d48619f 100644
--- a/genindex.html
+++ b/genindex.html
@@ -7,7 +7,7 @@
   <head>
     <meta charset="utf-8" />
     <meta name="viewport" content="width=device-width, initial-scale=1.0" />
-    <title>Index &#8212; TheMulQuaBio</title>
+    <title>Index &#8212; MulQuaBio</title>
   
   
   
@@ -150,8 +150,8 @@
       
     
     
-    <img src="_static/logo.png" class="logo__image only-light" alt="TheMulQuaBio - Home"/>
-    <script>document.write(`<img src="_static/logo.png" class="logo__image only-dark" alt="TheMulQuaBio - Home"/>`);</script>
+    <img src="_static/logo.png" class="logo__image only-light" alt="MulQuaBio - Home"/>
+    <script>document.write(`<img src="_static/logo.png" class="logo__image only-dark" alt="MulQuaBio - Home"/>`);</script>
   
   
 </a></div>
@@ -185,23 +185,9 @@
 <li class="toctree-l1"><a class="reference internal" href="notebooks/Python.html">Biological Computing in Python</a></li>
 <li class="toctree-l1"><a class="reference internal" href="notebooks/R.html">Biological Computing in R</a></li>
 </ul>
-<p aria-level="2" class="caption" role="heading"><span class="caption-text">Maths for Biologists</span></p>
-<ul class="nav bd-sidenav">
-<li class="toctree-l1"><a class="reference internal" href="mathscourse/lecture01/01_Making_mathematical_statements.html">01 - Making mathematical statements</a></li>
-<li class="toctree-l1"><a class="reference internal" href="mathscourse/lecture02/02_Describing_shapes_and_patterns.html">02   - Describing shapes and patterns</a></li>
-<li class="toctree-l1"><a class="reference internal" href="mathscourse/lecture03/03_Analysing_change%2C_one_step_at_a_time.html">03   - Analysing change, one step at a time</a></li>
-<li class="toctree-l1"><a class="reference internal" href="mathscourse/lecture04/04_From_table_arrangements_to_powerful_tools.html">04 - From table arrangements to powerful tools</a></li>
-<li class="toctree-l1"><a class="reference internal" href="mathscourse/lecture05/05_From_table_arrangements_to_powerful_tools_part2.html">05 - From table arrangements to powerful tools - part 2</a></li>
-<li class="toctree-l1"><a class="reference internal" href="mathscourse/lecture06/06_Analysing_change_continuously.html">06 - Analysing change, continuously</a></li>
-<li class="toctree-l1"><a class="reference internal" href="mathscourse/lecture07/07_Analysing_change_continuously_part2.html">07 - Analysing change, continuously (Part 2)</a></li>
-<li class="toctree-l1"><a class="reference internal" href="mathscourse/lecture08/08_Calculating_accumulated_change.html">08 - Calculating accumulated change</a></li>
-<li class="toctree-l1"><a class="reference internal" href="mathscourse/tutorials/lecture01/Tutorial_01.html">Tutorial 01</a></li>
-<li class="toctree-l1"><a class="reference internal" href="mathscourse/tutorials/lecture05/Tutorial_05.html">Tutorial 05</a></li>
-<li class="toctree-l1"><a class="reference internal" href="mathscourse/tutorials/lecture06/Tutorial_06.html">Tutorial 06</a></li>
-</ul>
 <p aria-level="2" class="caption" role="heading"><span class="caption-text">Basic Data Analyses and Statistics</span></p>
 <ul class="nav bd-sidenav">
-<li class="toctree-l1"><a class="reference internal" href="Stats-Intro.html">Basic Data Science and Statistics: Introduction</a></li>
+<li class="toctree-l1"><a class="reference internal" href="intro-Stats.html">Introduction</a></li>
 <li class="toctree-l1"><a class="reference internal" href="notebooks/Data_R.html">Data Management and Visualization</a></li>
 <li class="toctree-l1"><a class="reference internal" href="notebooks/ExpDesign.html">Experimental design and Data exploration</a></li>
 <li class="toctree-l1"><a class="reference internal" href="notebooks/t_F_tests.html">Basic hypothesis testing: <span class="math notranslate nohighlight">\(t\)</span> and <span class="math notranslate nohighlight">\(F\)</span> tests</a></li>
@@ -221,6 +207,7 @@
 <li class="toctree-l1"><a class="reference internal" href="notebooks/Appendix-JupyIntro.html">Introduction to Jupyter</a></li>
 <li class="toctree-l1"><a class="reference internal" href="notebooks/Appendix-Data-Python.html">Data analyses  with Python &amp; Jupyter</a></li>
 <li class="toctree-l1"><a class="reference internal" href="notebooks/Appendix-Maths.html">Mathematical models in Jupyter</a></li>
+<li class="toctree-l1"><a class="reference internal" href="notebooks/Appendix-MathsForBiologists/MathsForBiologists.html">Maths for Biologists</a></li>
 <li class="toctree-l1"><a class="reference internal" href="notebooks/Appendix-Databases.html">Databases <span class="tocSkip"></span></a></li>
 <li class="toctree-l1"><a class="reference internal" href="notebooks/Appendix-NLLS-Python.html">Model fitting in Python</a></li>
 <li class="toctree-l1"><a class="reference internal" href="notebooks/Appendix-MiniProj.html">The Computing Miniproject</a></li>
@@ -280,7 +267,7 @@
       
       
       
-      <li><a href="https://github.com/mhasoba/TheMulQuaBio" target="_blank"
+      <li><a href="https://github.com/MulQuaBio/MQB" target="_blank"
    class="btn btn-sm btn-source-repository-button dropdown-item"
    title="Source repository"
    data-bs-placement="left" data-bs-toggle="tooltip"
@@ -297,7 +284,7 @@
       
       
       
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+      <li><a href="https://github.com/MulQuaBio/MQB/issues/new?title=Issue%20on%20page%20%2Fgenindex.html&body=Your%20issue%20content%20here." target="_blank"
    class="btn btn-sm btn-source-issues-button dropdown-item"
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@@ -409,7 +396,7 @@ <h1 id="index">Index</h1>
   <div class="footer-item">
     
 <p class="component-author">
-By TheMulQuaBio collective!
+By MulQuaBio collective!
 </p>
 
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diff --git a/intro-EnE_Modelling.html b/intro-EnE_Modelling.html
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+
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+
+
+<html lang="en" data-content_root="./" >
+
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+    <meta name="viewport" content="width=device-width, initial-scale=1.0" /><meta name="viewport" content="width=device-width, initial-scale=1" />
+
+    <title>Introduction &#8212; MulQuaBio</title>
+  
+  
+  
+  <script data-cfasync="false">
+    document.documentElement.dataset.mode = localStorage.getItem("mode") || "";
+    document.documentElement.dataset.theme = localStorage.getItem("theme") || "";
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+  <section class="tex2jax_ignore mathjax_ignore" id="introduction">
+<h1>Introduction<a class="headerlink" href="#introduction" title="Link to this heading">#</a></h1>
+<p>This MQB section is aimed at teaching you core concepts, theories, mathematical models, and modelling methods in Ecology and Evolution.</p>
+<section id="overview">
+<h2>Overview<a class="headerlink" href="#overview" title="Link to this heading">#</a></h2>
+<p>We will cover key ecological and evolutionary concepts, models, understand how they behave, and learn to interpret and classify their behaviour using dynamical systems theory and bifurcation analysis. Dynamical systems theory plays a major role in modern theoretical approaches to ecological concepts and phenomena such as competition, predation, metapopulation dynamics and disease spread. This section will introduce some of the key basics of dynamical systems theory in application to these topics. We will look at ordinary differential equations and difference equation models and will use stability analysis and bifurcation analysis as tools to understand the qualitative behaviour of ecological models.</p>
+<section id="structure-of-this-section">
+<h3>Structure of this section<a class="headerlink" href="#structure-of-this-section" title="Link to this heading">#</a></h3>
+<p>We will go from individual (single cells, organisms) to ecosystem levels, integratng ecology and evolution (as seamlessly as possible) at each level. To lay the foundation for this, we will start with a philosphically- and historically-flavored chapter on the (brief) history of Ecology and Evolutionary biology and theory, the interlinkages between the two fields, and the status of their unification into a single discipline. There you will learn that its basically a matter of time scales, and whether and when ecology and evolutionary dynamics play out at the same timescales, and when they can be separated.</p>
+<ul class="simple">
+<li><p>Real world applications in</p>
+<ul>
+<li><p>Microbial ecology and engineering</p></li>
+<li><p>Fisheries</p></li>
+<li><p>Global change</p></li>
+<li><p>Agriculture</p></li>
+<li><p>Disease</p></li>
+<li><p>Etc</p></li>
+</ul>
+</li>
+</ul>
+</section>
+</section>
+<section id="pre-requisites">
+<h2>Pre-requisites<a class="headerlink" href="#pre-requisites" title="Link to this heading">#</a></h2>
+<ul class="simple">
+<li><p>Maths</p>
+<ul>
+<li><p>Each chapter linked to specific maths appendix sections for pre-requisites (Move current maths materials to a single Appendix)</p></li>
+<li><p>Bifurcation analyses / theory</p></li>
+</ul>
+</li>
+<li><p>Computing</p>
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+<li><p>Python</p></li>
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+<p>This MQB section builds on the computing component, introducing you to basic
+Data Science including visualization and statistical analyses of data.</p>
+<p>You will learn basic to somewhat advanced
+<a class="reference external" href="https://en.wikipedia.org/wiki/Frequentist_inference"><em>frequentist</em></a> statistical
+methods ad inference in a hands-on on way, interspersed with lectures on
+underlying concepts.</p>
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+<p>The chapters in this Section assume that you have already worked through at least the basic sections of the <a class="reference internal" href="notebooks/R.html"><span class="std std-doc">R Chapter</span></a> of this book.</p>
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+<p>Look up the Readings directory on <a class="reference external" href="https://github.com/MulQuaBio/MQB">MulQuaBio</a>.</p>
+<ul class="simple">
+<li><p>Bolker, B. M.: Ecological Models and Data in R (eBook and Hardcover available).</p></li>
+<li><p>Beckerman, A. P. &amp; Petchey, O. L. (2012) Getting started with R: an introduction for biologists. Oxford, Oxford University Press. (<em>Good, short, general introduction</em>)</p></li>
+<li><p>Crawley, R. (2013) The R book. 2nd edition. Chichester, Wiley.Excellent but enormous reference book, with code and data <a class="reference external" href="http://www.bio.ic.ac.uk/research/mjcraw/therbook/index.htm">available online</a></p></li>
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diff --git a/intro.html b/intro.html
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+    <img src="_static/logo.png" class="logo__image only-light" alt="MulQuaBio - Home"/>
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 <ul class="nav bd-sidenav">
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 <li class="toctree-l1"><a class="reference internal" href="notebooks/Data_R.html">Data Management and Visualization</a></li>
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 <li class="toctree-l1"><a class="reference internal" href="notebooks/t_F_tests.html">Basic hypothesis testing: <span class="math notranslate nohighlight">\(t\)</span> and <span class="math notranslate nohighlight">\(F\)</span> tests</a></li>
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 <li class="toctree-l1"><a class="reference internal" href="notebooks/Appendix-Databases.html">Databases <span class="tocSkip"></span></a></li>
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-    <h1>01 - Making mathematical statements</h1>
-    <!-- Table of contents -->
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-                <h2> Contents </h2>
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-            <nav aria-label="Page">
-                <ul class="visible nav section-nav flex-column">
-<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#sets-and-relations">Sets and relations</a></li>
-<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#polynomial-functions">Polynomial functions</a><ul class="nav section-nav flex-column">
-<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#degree-0">Degree 0</a></li>
-<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#degree-1">Degree 1</a></li>
-<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#degree-2">Degree 2</a><ul class="nav section-nav flex-column">
-<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#concavity">Concavity</a></li>
-<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#intercept-roots">Intercept &amp; Roots</a></li>
-<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#minimum-maximum">Minimum/ maximum</a></li>
-</ul>
-</li>
-</ul>
-</li>
-<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#rational-functions">Rational functions</a></li>
-<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#power-functions">Power functions</a></li>
-<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#exponential-functions">Exponential functions</a></li>
-<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#logarithmic-function">Logarithmic function</a></li>
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-                  
-  <section class="tex2jax_ignore mathjax_ignore" id="making-mathematical-statements">
-<h1>01 - Making mathematical statements<a class="headerlink" href="#making-mathematical-statements" title="Link to this heading">#</a></h1>
-<section id="sets-and-relations">
-<h2>Sets and relations<a class="headerlink" href="#sets-and-relations" title="Link to this heading">#</a></h2>
-<p>Start by watching the <a class="reference external" href="https://web.microsoftstream.com/video/a9ad1a8c-3d80-4910-b9f2-fab98c01b40b">video</a>.</p>
-</section>
-<section id="polynomial-functions">
-<h2>Polynomial functions<a class="headerlink" href="#polynomial-functions" title="Link to this heading">#</a></h2>
-<p>One of the simplest elemntary functions is given by:</p>
-<div class="math notranslate nohighlight">
-\[ f(x) = a_n x^n + a_{n-1} x^{n-1}+ \cdots + a_1 x^1 + a_0, \]</div>
-<p>where <span class="math notranslate nohighlight">\(n\)</span> is a <em>non-negative integer</em>, and the coefficients <span class="math notranslate nohighlight">\(a_n\)</span>, <span class="math notranslate nohighlight">\(a_{n-1}\)</span>, …, <span class="math notranslate nohighlight">\(a_0\)</span>. The highest value of <span class="math notranslate nohighlight">\(n\)</span> (for which the coefficient <span class="math notranslate nohighlight">\(a_n\)</span> is different from <span class="math notranslate nohighlight">\(0\)</span>) defines the <em>degree</em> of the polynomial.</p>
-<div class="admonition-examples admonition">
-<p class="admonition-title">Examples</p>
-<p><span class="math notranslate nohighlight">\(f(x) = 3x^5 + 12x^4 - 6x^3 - x^2 + x - 16\)</span> (Polynomial function of degree 5)<br />
-<span class="math notranslate nohighlight">\(g(x) = \frac{3}{4}x^2 - \frac{5}{2} \)</span> (Polynomial function of degree 2)<br />
-<span class="math notranslate nohighlight">\(h(x) = 1.6x^8 - 0.576x^4 + 1.89x^2 - 0.7\)</span> (Polynomial function of degree 8)</p>
-</div>
-<div class="admonition tip">
-<p class="admonition-title">Tip</p>
-<p>When the domain of the polynomial function is not explicitly given, it is frequently assumed to be all real numbers, <span class="math notranslate nohighlight">\({\rm I\!R}\)</span>.</p>
-</div>
-<div class="admonition tip">
-<p class="admonition-title">Tip</p>
-<p>When no application is specified, the function name ‘<span class="math notranslate nohighlight">\(f\)</span>’ and variable name ‘<span class="math notranslate nohighlight">\(x\)</span>’ are usually chosen. For application purposes, the function and variable are often named for what they represent; for example, <span class="math notranslate nohighlight">\(A(r)=\pi r^2\)</span> for the area of a circle dependent on its radius.</p>
-</div>
-<p>Let us discuss three of the most common forms of polynomial functions with degrees <span class="math notranslate nohighlight">\(0\)</span>, <span class="math notranslate nohighlight">\(1\)</span>, and <span class="math notranslate nohighlight">\(2\)</span>.</p>
-<section id="degree-0">
-<h3>Degree 0<a class="headerlink" href="#degree-0" title="Link to this heading">#</a></h3>
-<p>Polynomial functions of degree 0 assume the form:</p>
-<div class="math notranslate nohighlight">
-\[ f(x) = c, \]</div>
-<p>where <span class="math notranslate nohighlight">\(c\)</span> is a real number. This function is graphically represented by:</p>
-<img alt="../../_images/poly_deg0.png" src="../../_images/poly_deg0.png" />
-<div class="admonition-example admonition">
-<p class="admonition-title">Example</p>
-<p>For most mobile phone contracts nowadays a fixed fee is paid for the month with unlimited usage. So the cost as a function of used minutes would be a polynomial of degree zero, since the cost does not depend on the usage.</p>
-</div>
-</section>
-<section id="degree-1">
-<h3>Degree 1<a class="headerlink" href="#degree-1" title="Link to this heading">#</a></h3>
-<p>Polynomial functions of degree 1 are also called <em>linear functions</em> and are given by:</p>
-<div class="math notranslate nohighlight">
-\[ f(x) = mx + b,\]</div>
-<p>where again <span class="math notranslate nohighlight">\(m\)</span> and <span class="math notranslate nohighlight">\(b\)</span> are real numbers (with <span class="math notranslate nohighlight">\(m \ne 0\)</span>).</p>
-<p>The constants <span class="math notranslate nohighlight">\(m\)</span> and <span class="math notranslate nohighlight">\(b\)</span> are frequently referred to as <em>slope</em> and <em>intercept</em>. The intercept <span class="math notranslate nohighlight">\(b\)</span> shows where the function crosses the <span class="math notranslate nohighlight">\(y\)</span>-axis, while the slope <span class="math notranslate nohighlight">\(m\)</span> is related to the inclination of the function.</p>
-<p>The sign of the slope <span class="math notranslate nohighlight">\(m\)</span> also determines if the function increases or decreases as a function of <span class="math notranslate nohighlight">\(x\)</span>. We have:</p>
-<div class="math notranslate nohighlight">
-\[\begin{split} \begin{align}
-  m&gt;0 &amp;\rightarrow \mbox{Increasing function of } x \\
-  m&lt;0 &amp;\rightarrow \mbox{Decreasing function of } x
-  \end{align} \end{split}\]</div>
-<p>One can also ask what is the value of <span class="math notranslate nohighlight">\(x\)</span> where the function <span class="math notranslate nohighlight">\(f(x)\)</span> crosses de <span class="math notranslate nohighlight">\(x\)</span>-axis, or, in mathematical notation, what is <span class="math notranslate nohighlight">\(x\ \mbox{so that}\ f(x) = 0\)</span>. This value of x is called a <strong>root of <span class="math notranslate nohighlight">\(f\)</span></strong> and it can be obtained by:</p>
-<div class="math notranslate nohighlight">
-\[\begin{split}\begin{align}
-    f(x) &amp;= 0 \\
-    mx + b &amp;= 0 \\
-    mx &amp;= -b \\
-    x &amp;= -\frac{b}{m}
-    \end{align} \end{split}\]</div>
-<div class="admonition-example admonition">
-<p class="admonition-title">Example</p>
-</div>
-<div class="admonition-examples admonition">
-<p class="admonition-title">Examples</p>
-<p>The metabolic rate of some organisms depends (approximately) linearly on the body’s temperature, with higher rates at higher temperatures.</p>
-</div>
-</section>
-<section id="degree-2">
-<h3>Degree 2<a class="headerlink" href="#degree-2" title="Link to this heading">#</a></h3>
-<p>Now we look at polynomial functions of degree 2, given by the general form:</p>
-<div class="math notranslate nohighlight">
-\[f(x) = ax^2 + bx +c,\]</div>
-<p>the coefficients <span class="math notranslate nohighlight">\(a\)</span>, <span class="math notranslate nohighlight">\(b\)</span>, and <span class="math notranslate nohighlight">\(c\)</span> are real numbers (with <span class="math notranslate nohighlight">\(a \ne 0\)</span>). The graph of a polynomial function of degree 2 is a <em>parabola</em> and many of its properties can be readily known by analysing the coefficients.</p>
-<section id="concavity">
-<h4>Concavity<a class="headerlink" href="#concavity" title="Link to this heading">#</a></h4>
-</section>
-<section id="intercept-roots">
-<h4>Intercept &amp; Roots<a class="headerlink" href="#intercept-roots" title="Link to this heading">#</a></h4>
-<p>The expression inside the square root distinguish between different properties of this function and therefore receives the name of <em>discriminant</em>:</p>
-<div class="math notranslate nohighlight">
-\[ \Delta = b^2 -4ac.\]</div>
-<p>Depending on the value of the discriminant a quick conclusion can be reached regarding the roots of the function.</p>
-<div class="admonition-example admonition">
-<p class="admonition-title">Example</p>
-<p>If we throw a ball that is heavy enough so that air resistance is negligible (alternatively we can travel to the moon) the height of the ball as a function of time follows a quadratic equation. Knowing the roots we can calculate when and where the ball will land.</p>
-</div>
-</section>
-<section id="minimum-maximum">
-<h4>Minimum/ maximum<a class="headerlink" href="#minimum-maximum" title="Link to this heading">#</a></h4>
-<p>Depending on the concavity of the parabola (up or down), there will be a point where the function assumes a minimum or a maximum value. This point is called the <em>vertex</em> of the parabola. This extreme point can be easily determined by the symmetry properties of the function curve.</p>
-<div class="admonition-example admonition">
-<p class="admonition-title">Example</p>
-<p>Imagine the population of a specific region having a negative quadratic relationship - it undergoes initial growth, reaches its peak, and then declines. To determine both the maximum population size and the time when it peaked, we need to identify the extrema of this parabolic curve.</p>
-</div>
-<p>Now watch this <a class="reference external" href="https://web.microsoftstream.com/video/f64abd8a-9997-4b5e-b11f-0ca09b7603fe">video</a> for a short introduction to complex numbers.</p>
-</section>
-</section>
-</section>
-<section id="rational-functions">
-<h2>Rational functions<a class="headerlink" href="#rational-functions" title="Link to this heading">#</a></h2>
-<p>Rational functions are defined as quotients of polynomial functions:</p>
-<div class="math notranslate nohighlight">
-\[f(x) = \frac{p(x)}{q(x)},\ \ \ q(x) \ne 0\]</div>
-<div class="admonition-example admonition">
-<p class="admonition-title">Example</p>
-<p>Rational functions are often used to express ratios, for example of chemicals. Lets assume chemical A increases quadratically <span class="math notranslate nohighlight">\(A(t) = 2 t^2\)</span> and chemical B increases linearly <span class="math notranslate nohighlight">\(B(t)=t+5\)</span>, the ratio will be:</p>
-<div class="math notranslate nohighlight">
-\[r(t) = \frac{A(t)}{B(t)} = 2\frac{t^2}{t+5}\]</div>
-</div>
-</section>
-<section id="power-functions">
-<h2>Power functions<a class="headerlink" href="#power-functions" title="Link to this heading">#</a></h2>
-<p>Another type of function used in many applications are power functions (sometimes also called power laws). These functions have the functional form:</p>
-<div class="math notranslate nohighlight">
-\[f(x) = Cx^{\alpha}\]</div>
-<p>where <span class="math notranslate nohighlight">\(C\)</span> and <span class="math notranslate nohighlight">\(\alpha\)</span> are real constants (with <span class="math notranslate nohighlight">\(C \ne 0\)</span>).</p>
-<p>Many commonly used functions are examples of power functions, e.g. the square root <span class="math notranslate nohighlight">\(\sqrt{x}=x^{\frac 12}\)</span> or the inverse function <span class="math notranslate nohighlight">\(x^{-1}\)</span>.</p>
-<p>Watch this <a class="reference external" href="https://web.microsoftstream.com/video/bf7415e1-823f-449e-ace1-e379fac8fe0a">video</a> about additional properties of power law functions.</p>
-</section>
-<section id="exponential-functions">
-<h2>Exponential functions<a class="headerlink" href="#exponential-functions" title="Link to this heading">#</a></h2>
-<p>Similar to power functions, but with very different properties are the exponential functions:</p>
-<div class="math notranslate nohighlight">
-\[f(x) = a^x, \]</div>
-<p>where <span class="math notranslate nohighlight">\(a\)</span> is a positive constant. The parameter <span class="math notranslate nohighlight">\(a\)</span> is called <em>base</em> and the variable <span class="math notranslate nohighlight">\(x\)</span> is the exponent (remember that for power functions, the variable <span class="math notranslate nohighlight">\(x\)</span> is the base, raised to the power of the parameter <span class="math notranslate nohighlight">\(\alpha\)</span>).</p>
-<p>Exponential functions are used to model many different real-world applications, from bacterial growth to radioactive decay.</p>
-<div class="admonition-example admonition">
-<p class="admonition-title">Example</p>
-<p>Suppose we start a bacterial colony with a single individual on a Petri dish. At each time step, counted as an average reproduction interval, each individual bacterium duplicates, originating two other individuals. The number of bacteria in the colony as a function of time, <span class="math notranslate nohighlight">\(N(t)\)</span>, can be counted as:</p>
-<div class="pst-scrollable-table-container"><table class="table">
-<thead>
-<tr class="row-odd"><th class="head"><p><span class="math notranslate nohighlight">\(t\)</span></p></th>
-<th class="head text-center"><p><span class="math notranslate nohighlight">\(N(t)\)</span></p></th>
-</tr>
-</thead>
-<tbody>
-<tr class="row-even"><td><p><span class="math notranslate nohighlight">\(0\)</span></p></td>
-<td class="text-center"><p><span class="math notranslate nohighlight">\(1\ (= 2^0)\)</span></p></td>
-</tr>
-<tr class="row-odd"><td><p><span class="math notranslate nohighlight">\(1\)</span></p></td>
-<td class="text-center"><p><span class="math notranslate nohighlight">\(2\ (= 2^1)\)</span></p></td>
-</tr>
-<tr class="row-even"><td><p><span class="math notranslate nohighlight">\(2\)</span></p></td>
-<td class="text-center"><p><span class="math notranslate nohighlight">\(4\ (= 2^2)\)</span></p></td>
-</tr>
-<tr class="row-odd"><td><p><span class="math notranslate nohighlight">\(3\)</span></p></td>
-<td class="text-center"><p><span class="math notranslate nohighlight">\(8\ (= 2^3)\)</span></p></td>
-</tr>
-<tr class="row-even"><td><p><span class="math notranslate nohighlight">\(4\)</span></p></td>
-<td class="text-center"><p><span class="math notranslate nohighlight">\(16\ (= 2^4)\)</span></p></td>
-</tr>
-<tr class="row-odd"><td><p><span class="math notranslate nohighlight">\(5\)</span></p></td>
-<td class="text-center"><p><span class="math notranslate nohighlight">\(32\ (= 2^5)\)</span></p></td>
-</tr>
-</tbody>
-</table>
-</div>
-<p>If no limitation (on space or resources) is imposed to the colony, the number of bacteria at each time step <span class="math notranslate nohighlight">\(t\)</span> can be well modelled by an exponential function of the form:</p>
-<div class="math notranslate nohighlight">
-\[ N(t) = 2^t. \]</div>
-</div>
-<p>Exponential functions of the form <span class="math notranslate nohighlight">\(f(x) = a^x\)</span> (with <span class="math notranslate nohighlight">\(a&gt;0\)</span>) can never assume negative values and their behaviour depends on the value of <span class="math notranslate nohighlight">\(a\)</span>.</p>
-<img alt="../../_images/exponential_01.png" class="align-center" src="../../_images/exponential_01.png" />
-<div class="admonition tip">
-<p class="admonition-title">Tip</p>
-<p>Note that the function <em>increases</em> with <span class="math notranslate nohighlight">\(x\)</span> if <span class="math notranslate nohighlight">\(a&gt;1\)</span> (growth) and <em>decreases</em> with <span class="math notranslate nohighlight">\(x\)</span> if <span class="math notranslate nohighlight">\(0&lt;a&lt;1\)</span> (decay). To remember this, one can directly apply the properties of exponentiation. If <span class="math notranslate nohighlight">\(f(x) = (1/3)^x\)</span> (thus <span class="math notranslate nohighlight">\(a = (1/3) &lt; 1\)</span>), we have:</p>
-<div class="math notranslate nohighlight">
-\[f(x) = \left(\frac{1}{3}\right)^x = \frac{1}{3^x},\]</div>
-<p>and since <span class="math notranslate nohighlight">\(3^x\)</span> is an increasing function of <span class="math notranslate nohighlight">\(x\)</span>, then <span class="math notranslate nohighlight">\(1/3^x\)</span> should be a decreasing function of <span class="math notranslate nohighlight">\(x\)</span> (as the denominator increases, the fraction decreases).</p>
-</div>
-<div class="admonition-example admonition">
-<p class="admonition-title">Example</p>
-<p>In the early stages of a new desease, we can approximate the spread as exponential. In this case <span class="math notranslate nohighlight">\(a\)</span> is the number of people every infected person infects themselves. For <span class="math notranslate nohighlight">\(a&lt;1\)</span> every person infects less than one other person on average and the desease will slow down, for <span class="math notranslate nohighlight">\(a&gt;1\)</span> the desease will spread. At the beginning of the COVID-19 pandemic, an estimate for <span class="math notranslate nohighlight">\(a\)</span> (or in this case <span class="math notranslate nohighlight">\(R\)</span> for reproductive value) was <span class="math notranslate nohighlight">\(a=4\)</span>. If we start with 1 infected person, how many do we have after 10 infection cycles?</p>
-</div>
-<p>Two of the most used bases are <span class="math notranslate nohighlight">\(a=10\)</span> and <span class="math notranslate nohighlight">\(a=e\)</span>, where <span class="math notranslate nohighlight">\(e=2.71828\)</span> (called <em>Euler’s number</em> or <em>Napier’s constant</em>). Exponential functions with <span class="math notranslate nohighlight">\(a=e\)</span> are commonly written as:</p>
-<div class="math notranslate nohighlight">
-\[f(x) = e^x = \mbox{exp}(x) \]</div>
-<p>The unique charachteristic of the exponential of base <span class="math notranslate nohighlight">\(e\)</span> is, that it is its own derivative</p>
-<div class="math notranslate nohighlight">
-\[\frac{\text{d}}{\text{d}x} e^x = e^x \]</div>
-<p>The both formulations can be changed into one another via</p>
-<div class="math notranslate nohighlight">
-\[f(x) = e^{\lambda x} = ({\underbrace{e^\lambda}_a})^x.\]</div>
-<p>We have exponential growth when <span class="math notranslate nohighlight">\(a&gt;1\)</span> and therefore when <span class="math notranslate nohighlight">\(\lambda&gt;0\)</span> and decay when <span class="math notranslate nohighlight">\(0&lt;a&lt;1\)</span> and therefore <span class="math notranslate nohighlight">\(\lambda&lt;0\)</span>.</p>
-</section>
-<section id="logarithmic-function">
-<h2>Logarithmic function<a class="headerlink" href="#logarithmic-function" title="Link to this heading">#</a></h2>
-<p>Let us start this section by understanding what the expression</p>
-<div class="math notranslate nohighlight">
-\[ \mbox{log}_a b = x \]</div>
-<p>actually means. One possible way to read it is: “<span class="math notranslate nohighlight">\(\mbox{log}_a b\)</span> is the number with which I have to exponentiate the base <span class="math notranslate nohighlight">\(a\)</span> so that I find the number <span class="math notranslate nohighlight">\(b\)</span>. Since “<span class="math notranslate nohighlight">\(\mbox{log}_a b = x\)</span>, the above expression is equivalent to:</p>
-<div class="math notranslate nohighlight">
-\[ a^x = b.\]</div>
-<p>The expression <span class="math notranslate nohighlight">\(\mbox{log}_a b\)</span> is called the <em>logarithm of b in the base a</em> and it is also possible to define a logarithmic function given by:</p>
-<div class="math notranslate nohighlight">
-\[f(x) = \mbox{log}_a x,\]</div>
-<p>where <span class="math notranslate nohighlight">\(a\)</span> is the base of the logarithm and it is assumed <span class="math notranslate nohighlight">\(a&gt;0\)</span> and <span class="math notranslate nohighlight">\(a \ne 1\)</span>.</p>
-<p>By defining a function like this, for each <span class="math notranslate nohighlight">\(x\)</span> we will have a number <span class="math notranslate nohighlight">\(y = f(x)\)</span> for which <span class="math notranslate nohighlight">\(a^y = x\)</span>. Therefore, the logarithmic function is only defined for <span class="math notranslate nohighlight">\(x&gt;0\)</span>. Like the exponential function, the behaviour of the logarithmic function for increasing <span class="math notranslate nohighlight">\(x\)</span> will depend on the value of the base <span class="math notranslate nohighlight">\(a\)</span>.</p>
-<div class="admonition tip">
-<p class="admonition-title">Tip</p>
-<p>The similarities between the graphs for exponential and logarithmic functions are not coincidental. In fact, the graph for a logarithmic function with base <span class="math notranslate nohighlight">\(a\)</span> can be obtained by drawing the graph for an exponential function with base <span class="math notranslate nohighlight">\(a\)</span> and reflecting it in relation to the line <span class="math notranslate nohighlight">\(y = x\)</span>.  [MAKE COMMENT ON INVERSE FUNCTIONS]</p>
-<img alt="../../_images/log_base_02.png" src="../../_images/log_base_02.png" />
-<p>(here we use the common nomenclature <span class="math notranslate nohighlight">\(\mbox{log}_e x = \mbox{ln}\ x\)</span>).</p>
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-    <h1>02   - Describing shapes and patterns</h1>
-    <!-- Table of contents -->
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-                <h2> Contents </h2>
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-            <nav aria-label="Page">
-                <ul class="visible nav section-nav flex-column">
-<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#measuring-angles">Measuring angles</a></li>
-<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#trigonometric-circle">Trigonometric circle</a><ul class="nav section-nav flex-column">
-<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#trigonometric-identities">Trigonometric identities</a></li>
-</ul>
-</li>
-<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#trigonometric-functions">Trigonometric functions</a><ul class="nav section-nav flex-column">
-<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#amplitude-and-period">Amplitude and Period</a></li>
-</ul>
-</li>
-<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#modelling-complex-periodic-phenomena">Modelling complex periodic phenomena</a></li>
-<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#transformation-on-graphs">Transformation on graphs</a><ul class="nav section-nav flex-column">
-<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#translations">Translations</a></li>
-<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#stretching-compression">Stretching/ Compression</a></li>
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-                  
-  <section class="tex2jax_ignore mathjax_ignore" id="describing-shapes-and-patterns">
-<h1>02   - Describing shapes and patterns<a class="headerlink" href="#describing-shapes-and-patterns" title="Link to this heading">#</a></h1>
-<section id="measuring-angles">
-<h2>Measuring angles<a class="headerlink" href="#measuring-angles" title="Link to this heading">#</a></h2>
-<p>An <strong>angle</strong> is a primitive geometrical element, such as the <em>point</em>, the <em>curve</em> or the <em>plane</em>. It is defined as the figure formed by two line segments that extend from a common point.</p>
-<br/>
-<img alt="../../_images/angle.png" src="../../_images/angle.png" />
-<br/>
-<br/>
-<p>The two most common units for measuring angles are the <em>degree</em> and the <em>radian</em>.</p>
-<p>For a given angle, transforming between units of degrees and radians can be simply done with:</p>
-<div class="math notranslate nohighlight">
-\[\frac{360^{\circ}}{\theta} = \frac{2\pi}{\phi},\]</div>
-<p>where <span class="math notranslate nohighlight">\(\theta\)</span> is the angle measured in degrees and <span class="math notranslate nohighlight">\(\phi\)</span>, measured in radians. In this lecture, we will consider the angles measured in radians (independent of the symbol used for the variable), unless stated otherwise.</p>
-<div class="admonition-example admonition">
-<p class="admonition-title">Example</p>
-<p>If <span class="math notranslate nohighlight">\(\theta = 90^{\circ}\)</span>, then using the previous expression we have <span class="math notranslate nohighlight">\(\phi = \displaystyle\frac{\pi}{2}\)</span>. Since <span class="math notranslate nohighlight">\(\theta\)</span> and $\phi are directly proportional, we also have:</p>
-<div class="math notranslate nohighlight">
-\[\theta = 180^{\circ} \longrightarrow \phi = 2 \cdot \frac{\pi}{2} = \pi \]</div>
-<div class="math notranslate nohighlight">
-\[\theta = 270^{\circ} \longrightarrow \phi = 3 \cdot \frac{\pi}{2} = \frac{3\pi}{2} \]</div>
-<div class="math notranslate nohighlight">
-\[\theta = 360^{\circ} \longrightarrow \phi = 4 \cdot \frac{\pi}{2} = 2\pi \]</div>
-</div>
-</section>
-<section id="trigonometric-circle">
-<h2>Trigonometric circle<a class="headerlink" href="#trigonometric-circle" title="Link to this heading">#</a></h2>
-<p>We obtain the trigonometric circle by superimposing a circle of radius <span class="math notranslate nohighlight">\(1\)</span> to the cartesian system of coordinates:</p>
-<img alt="../../_images/trig_circ.png" src="../../_images/trig_circ.png" />
-<p>The projections on the <span class="math notranslate nohighlight">\(x\)</span> and <span class="math notranslate nohighlight">\(y\)</span>-axes have special meanings, if we consider the angle <span class="math notranslate nohighlight">\(\theta\)</span> between the line segment in the radial direction and the <span class="math notranslate nohighlight">\(x\)</span>-axis.</p>
-<p>From the geometric relations on the right triangle (triangle in which one internal angle is a right angle):</p>
-<p>Back to the trigonometric circle, it is possible to construct a right triangle where the hypothenuse corresponds to the radius of the superimposed circle. In this case, since this radius equals to <span class="math notranslate nohighlight">\(1\)</span>, we have:</p>
-<div class="math notranslate nohighlight">
-\[\mbox{sin}(\theta) = y\]</div>
-<div class="math notranslate nohighlight">
-\[\mbox{cos}(\theta) = x.\]</div>
-<p>In other words, the projections of the end of the radial line segment on the <span class="math notranslate nohighlight">\(y\)</span> and <span class="math notranslate nohighlight">\(x\)</span>-axes correspond to the sine and cosine of the angle <span class="math notranslate nohighlight">\(\theta\)</span>. Sine and cosine for this angle can be obtained by the projections even if <span class="math notranslate nohighlight">\(\theta &gt; \displaystyle\frac{\pi}{2}\)</span>.</p>
-<section id="trigonometric-identities">
-<h3>Trigonometric identities<a class="headerlink" href="#trigonometric-identities" title="Link to this heading">#</a></h3>
-<p>With the aid of the projections on the trigonometric circle, some trigonometric identities become imediately clear. Consider <span class="math notranslate nohighlight">\(\theta &gt; 0\)</span> for counterclockwise angle measurements starting from the <span class="math notranslate nohighlight">\(x\)</span>-axis and <span class="math notranslate nohighlight">\(\theta &lt; 0\)</span> for clockwise measurements. We have:</p>
-<div class="math notranslate nohighlight">
-\[\mbox{sin}(0) = 0,\ \ \ \mbox{sin}\left(\frac{\pi}{2}\right) = 1,\ \ \ \mbox{sin}(\pi) = 0,\ \ \ \mbox{sin}\left(\frac{3\pi}{2}\right) = -1,\ \ \ \mbox{sin}(2\pi) = 0\]</div>
-<div class="math notranslate nohighlight">
-\[\mbox{cos}(0) = 1,\ \ \ \mbox{cos}\left(\frac{\pi}{2}\right) = 0,\ \ \ \mbox{cos}(\pi) = -1,\ \ \ \mbox{cos}\left(\frac{3\pi}{2}\right) = 0,\ \ \ \mbox{cos}(2\pi) = 1\]</div>
-<p>From the fact that a full rotation brings you back to the same point:</p>
-<div class="math notranslate nohighlight">
-\[\mbox{sin}(\theta) = \mbox{sin}(\theta + 2\pi)\]</div>
-<div class="math notranslate nohighlight">
-\[\mbox{cos}(\theta) = \mbox{cos}(\theta + 2\pi)\]</div>
-<p>or from Pythagoras’ theorem on the right triangle in the trigonometric circle:</p>
-<div class="math notranslate nohighlight">
-\[\begin{split}\begin{align}
-[\mbox{sin}(\theta)]^2 + [\mbox{cos}(\theta)]^2 =&amp; 1^2 \\
-\mbox{sin}^2(\theta) + \mbox{cos}^2(\theta) =&amp; 1
-\end{align}\end{split}\]</div>
-<div class="admonition tip">
-<p class="admonition-title">Tip</p>
-<p>The notation <span class="math notranslate nohighlight">\(\mbox{sin}^2(\theta)\)</span> is frequently used to express the square of <span class="math notranslate nohighlight">\(\mbox{sin}(\theta)\)</span>, or:</p>
-<div class="math notranslate nohighlight">
-\[ \mbox{sin}^2(\theta) = [\mbox{sin}(\theta)]^2 = [\mbox{sin}(\theta)]\cdot[\mbox{sin}(\theta)],\]</div>
-<p>which is different from <span class="math notranslate nohighlight">\(\mbox{sin}(\theta^2)\)</span>.</p>
-</div>
-<p>Now watch this <a class="reference external" href="https://web.microsoftstream.com/video/b3023026-7d45-4378-a8ee-c166a43eb2e7">video</a> for a different application of the same trigonometric construct.</p>
-</section>
-</section>
-<section id="trigonometric-functions">
-<h2>Trigonometric functions<a class="headerlink" href="#trigonometric-functions" title="Link to this heading">#</a></h2>
-<p>Since sine and cosine are defined for any real value of a given angle <span class="math notranslate nohighlight">\(x\)</span> (<span class="math notranslate nohighlight">\(x \in {\rm I\!R}\)</span>), we can define the functions:</p>
-<div class="math notranslate nohighlight">
-\[f(x) = \mbox{sin}(x)\]</div>
-<div class="math notranslate nohighlight">
-\[f(x) = \mbox{cos}(x).\]</div>
-<p>Thus, the association of sine and cosine with pure geometric ratios gives place for a more generic definition of functions of real variables. The graphs of these functions are given by:</p>
-<img alt="../../_images/sin_cos.png" src="../../_images/sin_cos.png" />
-<p>All the other known trigonometric relations can also be defined as functions in a similar way. Take for example the tangent of an angle <span class="math notranslate nohighlight">\(\theta\)</span>, which is:</p>
-<div class="math notranslate nohighlight">
-\[tan(\theta) = \displaystyle\frac{\mbox{sin}(\theta)}{\mbox{cos}(\theta)}.\]</div>
-<p>If <span class="math notranslate nohighlight">\(x \in {\rm I\!R}\)</span>, than we can define a function</p>
-<div class="math notranslate nohighlight">
-\[ f(x) = \mbox{tan}(x) = \displaystyle\frac{\mbox{sin}(x)}{cos(x)}, \ \ \ (\mbox{for }\mbox{cos}(x) \ne 0).\]</div>
-<p>The graph from the tangent function will be given by:</p>
-<img alt="../../_images/tan.png" src="../../_images/tan.png" />
-<p>Note by the above graphs that the functions sine and cosine have patterns that repeat themselves every interval of <span class="math notranslate nohighlight">\(2\pi\)</span>, while the tangent repeats itself every interval of <span class="math notranslate nohighlight">\(\pi\)</span>. We say that these three functions are <strong>periodic</strong>.</p>
-<p>A function <span class="math notranslate nohighlight">\(f(x)\)</span> is periodic if there is a positive constant <span class="math notranslate nohighlight">\(a\)</span>, so that:</p>
-<div class="math notranslate nohighlight">
-\[f(x + a) = f(x),\]</div>
-<p>for all values of <span class="math notranslate nohighlight">\(x\)</span> for which the function f(x) is defined.</p>
-<p>If <span class="math notranslate nohighlight">\(a\)</span> is the smallest number with this property, we call it the <em>period</em> of f(x).</p>
-<div class="admonition-example admonition">
-<p class="admonition-title">Example</p>
-<p>Since <span class="math notranslate nohighlight">\(\mbox{sin}(x + 2\pi) = \mbox{sin}(x)\)</span> for any value of <span class="math notranslate nohighlight">\(x\)</span> and since <span class="math notranslate nohighlight">\(2\pi\)</span> is the smallest number for which this property holds, then sine is a periodic function with period <span class="math notranslate nohighlight">\(2\pi\)</span>. With the same reasoning, it can be shown that the tangent is periodic with period <span class="math notranslate nohighlight">\(\pi\)</span>.</p>
-</div>
-<div class="admonition-example admonition">
-<p class="admonition-title">Example</p>
-<p>Many natural processes can be modelled as periodic. Think for example of a pendulum, the tides, the seasons. Often periodicity can be a good first order approximation, even if it does not strictly hold. For example when describimg the shape of the cloud cover in the picture below.</p>
-</div>
-<img alt="../../_images/cloud.png" src="../../_images/cloud.png" />
-<section id="amplitude-and-period">
-<h3>Amplitude and Period<a class="headerlink" href="#amplitude-and-period" title="Link to this heading">#</a></h3>
-<p>From the basic functions <span class="math notranslate nohighlight">\(\mbox{sin}(x)\)</span> or <span class="math notranslate nohighlight">\(\mbox{cos}(x)\)</span> we can write a more general function <span class="math notranslate nohighlight">\(f(x)\)</span> given by</p>
-<div class="math notranslate nohighlight">
-\[f(x) = A\mbox{sin}(\omega x),\ \ \ \ \ \ A, \omega \in {\rm I\!R}\ \ (A, \omega \ne 0)\]</div>
-<p>Since the sine ranges from <span class="math notranslate nohighlight">\(-1\)</span> to <span class="math notranslate nohighlight">\(1\)</span> (see the graph for <span class="math notranslate nohighlight">\(sin(x)\)</span> for reference), by its definition, <span class="math notranslate nohighlight">\(f(x)\)</span> will range from <span class="math notranslate nohighlight">\(-A\)</span> to <span class="math notranslate nohighlight">\(A\)</span>. We call <span class="math notranslate nohighlight">\(A\)</span> the <strong>amplitude</strong> of <span class="math notranslate nohighlight">\(f(x)\)</span> and it gives the extent of the variation of the oscillation of the function.</p>
-<p>Being sine a periodic function, <span class="math notranslate nohighlight">\(f(x)\)</span> will also be, possibly with a different period. The period <span class="math notranslate nohighlight">\(T\)</span> of <span class="math notranslate nohighlight">\(f(x)\)</span>, as discussed before, will be determined by the equation <span class="math notranslate nohighlight">\(f(x + T) = f(x)\)</span>, that should be valid for every <span class="math notranslate nohighlight">\(x\)</span>. Thus:</p>
-<div class="math notranslate nohighlight">
-\[\begin{split}\begin{align}
-    f(x + T) &amp;= f(x) \\
-    A\mbox{sin}(\omega(x+T)) &amp;= A\mbox{sin}(\omega x) \\
-    \mbox{sin}(\omega(x+T)) &amp;= \mbox{sin}(\omega x) \\
-    \mbox{sin}(\omega x+\omega T)) &amp;= \mbox{sin}(\omega x) \\
-    \mbox{sin}(z+\omega T)) &amp;= \mbox{sin}(z)\ \ \ \ \ \ (\mbox{where we change the variable to } z=\omega x)
-  \end{align}\end{split}\]</div>
-<p>But since we know that sine is periodic with period <span class="math notranslate nohighlight">\(2\pi\)</span>, we should have <span class="math notranslate nohighlight">\(|\omega|T = 2\pi\)</span>. Thus, the <strong>period</strong> of the function <span class="math notranslate nohighlight">\(f(x)\)</span> will be:</p>
-<div class="math notranslate nohighlight">
-\[T = \frac{2\pi}{\omega}.\]</div>
-<p>The constant <span class="math notranslate nohighlight">\(\omega = \displaystyle\frac{2\pi}{T}\)</span> is called <strong>angular frequency</strong> and measures the number of full rotations the function <span class="math notranslate nohighlight">\(f(x)\)</span> oscillates in one interval of <span class="math notranslate nohighlight">\(T\)</span>.</p>
-<div class="admonition-example admonition">
-<p class="admonition-title">Example</p>
-<p>The function <span class="math notranslate nohighlight">\(f(x) = 3.6\, \mbox{cos}\left(\displaystyle\frac{\pi}{3}x\right)\)</span> has an amplitude <span class="math notranslate nohighlight">\(A=3.6\)</span>, an angular frequency <span class="math notranslate nohighlight">\(\omega = \displaystyle\frac{\pi}{3}\)</span>, and a period <span class="math notranslate nohighlight">\(T = \displaystyle\frac{2\pi}{\omega} = \displaystyle\frac{2 \pi}{\frac{\pi}{3}} = 6\)</span>.</p>
-</div>
-</section>
-</section>
-<section id="modelling-complex-periodic-phenomena">
-<h2>Modelling complex periodic phenomena<a class="headerlink" href="#modelling-complex-periodic-phenomena" title="Link to this heading">#</a></h2>
-<p>Watch this <a class="reference external" href="https://web.microsoftstream.com/video/02f373a2-621b-4313-a18a-265f5fb65656">video</a> about periodic functions decomposition.</p>
-</section>
-<section id="transformation-on-graphs">
-<h2>Transformation on graphs<a class="headerlink" href="#transformation-on-graphs" title="Link to this heading">#</a></h2>
-<p>Starting from the elementary functions we have discussed</p>
-<div class="math notranslate nohighlight">
-\[ax+b,\ \ ax^2+bx+c,\ \ \sqrt{x},\ \ \mbox{e}^x,\ \ \mbox{log}(x),\ \ \mbox{sin}(x),\ \ \mbox{cos}(x)\]</div>
-<p>we can introduce several transformations to obtain new functions, by adding or multiplying constants to the argument (function variable) or to the whole function. These are the simplest cases of what is know as <em>function composition</em> with the resulting function called <em>composite function</em>.</p>
-<p>When working with mathematical models, this will help you to understand how the shape of a given function might change by changing one of the parameters, or how to construct a set of functions (that might be useful for your particular model) with small modifications of basic functions.</p>
-<section id="translations">
-<h3>Translations<a class="headerlink" href="#translations" title="Link to this heading">#</a></h3>
-<p>A given function can be dislocated, either vertically or horiontally, by <em>adding</em> a constant <span class="math notranslate nohighlight">\(c\)</span> to the whole function or to its argument. In general, starting from a function <span class="math notranslate nohighlight">\(f(x)\)</span>, translations are built following the rules:</p>
-<ul>
-<li><p><strong>Horizontal</strong> (Constant <span class="math notranslate nohighlight">\(c\)</span> summed to the argument of the function)</p>
-<p><span class="math notranslate nohighlight">\(y = f(x + c)\ \ \ \  -  \left\{
-\begin{array}{ll}
-    c &gt; 0 &amp; \rightarrow \ \  \mbox{function moves to the left} \\
-    c &lt; 0 &amp; \rightarrow \ \  \mbox{function moves to the right} \\
-\end{array} 
-\right. \)</span></p>
-</li>
-<li><p><strong>Vertical</strong> (Constant <span class="math notranslate nohighlight">\(c\)</span> summed to the whole function)</p>
-<p><span class="math notranslate nohighlight">\(y = f(x) + c\ \ \ \  -  \left\{
-\begin{array}{ll}
-    c &gt; 0 &amp; \rightarrow \ \  \mbox{function moves up} \\
-    c &lt; 0 &amp; \rightarrow \ \  \mbox{function moves down} \\
-\end{array} 
-\right. \)</span></p>
-</li>
-</ul>
-<div class="admonition-example admonition">
-<p class="admonition-title">Example</p>
-<p>If we start with the function <span class="math notranslate nohighlight">\(f(x) = x^3\)</span>, the function <span class="math notranslate nohighlight">\(y = f(x + 2) = (x+2)^3\)</span> will represent a horizontal translation, while the function <span class="math notranslate nohighlight">\(y = f(x) + 2 = x^3+2\)</span> will represent a vertical translation.</p>
-</div>
-<p>To see how this works, try to implement the following code in your own jupyter notebook. Try to use different functions <span class="math notranslate nohighlight">\(f(x)\)</span> (defined in ‘func(x)’) to see how different graphs behave to horizontal and vertical translations.</p>
-<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">ipywidgets</span> <span class="kn">import</span> <span class="n">interact</span>
-<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
-<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
-
-<span class="k">def</span> <span class="nf">func</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
-    <span class="k">return</span> <span class="n">x</span><span class="o">**</span><span class="mi">3</span>
-
-<span class="k">def</span> <span class="nf">htrans</span><span class="p">(</span><span class="n">c</span><span class="p">):</span>
-    <span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span><span class="mi">3</span><span class="p">)</span>
-    
-    <span class="n">y</span> <span class="o">=</span> <span class="n">func</span><span class="p">(</span><span class="n">x</span><span class="o">+</span><span class="n">c</span><span class="p">)</span> <span class="c1">#constant c in the argument of the function</span>
-    
-    <span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;$f(x+c)$&#39;</span><span class="p">)</span>
-    <span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s1">&#39;Horizontal translation&#39;</span><span class="p">)</span>
-    <span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">12</span><span class="p">)</span>
-    
-    <span class="n">plt</span><span class="o">.</span><span class="n">ylim</span><span class="p">(</span><span class="o">-</span><span class="mi">8</span><span class="p">,</span> <span class="mi">8</span><span class="p">)</span> <span class="c1">#remember to change the limits for better visualisation if necessary</span>
-    
-    <span class="n">plt</span><span class="o">.</span><span class="n">axhline</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;black&#39;</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
-    <span class="n">plt</span><span class="o">.</span><span class="n">axvline</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;black&#39;</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
-    
-    
-
-    <span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
-
-<span class="k">def</span> <span class="nf">vtrans</span><span class="p">(</span><span class="n">c</span><span class="p">):</span>
-    <span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span><span class="mi">3</span><span class="p">)</span>
-    
-    <span class="n">y</span> <span class="o">=</span> <span class="n">func</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">+</span><span class="n">c</span> <span class="c1">#constant c summing the whole function</span>
-    
-    <span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">,</span>  <span class="n">label</span><span class="o">=</span><span class="s1">&#39;$f(x)+c$&#39;</span><span class="p">)</span>
-    <span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s1">&#39;Vertical translation&#39;</span><span class="p">)</span>
-    
-    <span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s1">&#39;x&#39;</span><span class="p">)</span>
-    <span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s1">&#39;y&#39;</span><span class="p">)</span>
-    <span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">12</span><span class="p">)</span>
-    
-    <span class="n">plt</span><span class="o">.</span><span class="n">ylim</span><span class="p">(</span><span class="o">-</span><span class="mi">8</span><span class="p">,</span> <span class="mi">8</span><span class="p">)</span> <span class="c1">#remember to change the limits for better visualisation if necessary</span>
-        
-    <span class="n">plt</span><span class="o">.</span><span class="n">axhline</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;black&#39;</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
-    <span class="n">plt</span><span class="o">.</span><span class="n">axvline</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;black&#39;</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
-    
-    <span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
-    
-<span class="c1"># create a slider</span>
-
-<span class="n">interact</span><span class="p">(</span><span class="n">htrans</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="p">(</span><span class="o">-</span><span class="mf">2.0</span><span class="p">,</span><span class="mf">2.0</span><span class="p">))</span>
-<span class="n">interact</span><span class="p">(</span><span class="n">vtrans</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="p">(</span><span class="o">-</span><span class="mf">2.0</span><span class="p">,</span><span class="mf">2.0</span><span class="p">))</span>
-</pre></div>
-</div>
-</section>
-<section id="stretching-compression">
-<h3>Stretching/ Compression<a class="headerlink" href="#stretching-compression" title="Link to this heading">#</a></h3>
-<p>A given function can also be stretched or compressed, either vertically or horiontally, by <em>multiplying</em> a constant <span class="math notranslate nohighlight">\(c\)</span> to the whole function or to its argument. Starting from a function <span class="math notranslate nohighlight">\(f(x)\)</span>, stretchings/compressions are built following the rules:</p>
-<ul>
-<li><p><strong>Horizontal</strong> (Constant <span class="math notranslate nohighlight">\(c\)</span> multiplied to the argument of the function)</p>
-<p><span class="math notranslate nohighlight">\(y = f(cx)\ \ \ \  -  \left\{
-\begin{array}{ll}
-    c &gt; 1 &amp; \rightarrow \ \  \mbox{function is horizontally compressed} \\
-    0 &lt; c &lt; 1 &amp; \rightarrow \ \  \mbox{function is horizontally stretched} \\
-\end{array} 
-\right. \)</span></p>
-</li>
-<li><p><strong>Vertical</strong> (Constant <span class="math notranslate nohighlight">\(c\)</span> multiplied to the whole function)</p>
-<p><span class="math notranslate nohighlight">\(y = cf(x)\ \ \ \  -  \left\{
-\begin{array}{ll}
-    c &gt; 1 &amp; \rightarrow \ \  \mbox{function is vertically stretched} \\
-    0 &lt; c &lt; 1 &amp; \rightarrow \ \  \mbox{function is vertically compressed} \\
-\end{array} 
-\right. \)</span></p>
-</li>
-</ul>
-<div class="admonition-example admonition">
-<p class="admonition-title">Example</p>
-<p>For the function <span class="math notranslate nohighlight">\(f(x) = x^3-2x\)</span>, the function <span class="math notranslate nohighlight">\(y = f(2x) = (2x)^3-2(2x) = 8x^3-4x\)</span> will represent a horizontal compression, while the function <span class="math notranslate nohighlight">\(y = 2f(x) = 2x^3 - 4x\)</span> will represent a vertical stretching.</p>
-</div>
-<div class="admonition-example admonition">
-<p class="admonition-title">Example</p>
-<p>One example where we stretch or compress functions comes from chemical processes. Many chemical processes happen faster when the temperature is higher. If we describe the concentration of a certain chemical as a function of time <span class="math notranslate nohighlight">\(c(t)\)</span>, increasing the temperature will lead to a horizontal compression of the function.</p>
-</div>
-<p>Now play with the code below to see how different graphs behave to horizontal and vertical stretching/compression.</p>
-<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">ipywidgets</span> <span class="kn">import</span> <span class="n">interact</span>
-<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
-<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
-
-<span class="k">def</span> <span class="nf">func</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
-    <span class="k">return</span> <span class="n">x</span><span class="o">**</span><span class="mi">3</span><span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span>
-
-<span class="k">def</span> <span class="nf">hstetcom</span><span class="p">(</span><span class="n">c</span><span class="p">):</span>
-    <span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span><span class="mi">3</span><span class="p">)</span>
-    
-    <span class="n">y</span> <span class="o">=</span> <span class="n">func</span><span class="p">(</span><span class="n">c</span><span class="o">*</span><span class="n">x</span><span class="p">)</span> <span class="c1">#constant c multiplying the argument of the function</span>
-    
-    <span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;$f\, (cx)$&#39;</span><span class="p">)</span>
-    <span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s1">&#39;Horizontal stretching/compression&#39;</span><span class="p">)</span>
-        
-    <span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s1">&#39;x&#39;</span><span class="p">)</span>
-    <span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s1">&#39;y&#39;</span><span class="p">)</span>
-    <span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">12</span><span class="p">)</span>
-    
-    <span class="n">plt</span><span class="o">.</span><span class="n">ylim</span><span class="p">(</span><span class="o">-</span><span class="mi">4</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span> <span class="c1">#remember to change the limits for better visualisation if necessary</span>
-    
-    <span class="n">plt</span><span class="o">.</span><span class="n">axhline</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;black&#39;</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
-    <span class="n">plt</span><span class="o">.</span><span class="n">axvline</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;black&#39;</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
-    
-    
-
-    <span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
-
-<span class="k">def</span> <span class="nf">vstetcom</span><span class="p">(</span><span class="n">c</span><span class="p">):</span>
-    <span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span><span class="mi">3</span><span class="p">)</span>
-    
-    <span class="n">y</span> <span class="o">=</span> <span class="n">c</span><span class="o">*</span><span class="n">func</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="c1">#constant c multiplying the whole function</span>
-    
-    <span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">,</span>  <span class="n">label</span><span class="o">=</span><span class="s1">&#39;$c\, f\, (x)$&#39;</span><span class="p">)</span>
-    <span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s1">&#39;Vertical stretching/compression&#39;</span><span class="p">)</span>
-    
-    <span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s1">&#39;x&#39;</span><span class="p">)</span>
-    <span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s1">&#39;y&#39;</span><span class="p">)</span>
-    <span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">12</span><span class="p">)</span>
-    
-    <span class="n">plt</span><span class="o">.</span><span class="n">ylim</span><span class="p">(</span><span class="o">-</span><span class="mi">4</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span> <span class="c1">#remember to change the limits for better visualisation if necessary</span>
-        
-    <span class="n">plt</span><span class="o">.</span><span class="n">axhline</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;black&#39;</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
-    <span class="n">plt</span><span class="o">.</span><span class="n">axvline</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;black&#39;</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
-    
-    <span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
-    
-<span class="c1"># create a slider</span>
-
-<span class="n">interact</span><span class="p">(</span><span class="n">hstetcom</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="p">(</span><span class="mf">0.1</span><span class="p">,</span><span class="mf">3.0</span><span class="p">))</span>
-<span class="n">interact</span><span class="p">(</span><span class="n">vstetcom</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="p">(</span><span class="mf">0.1</span><span class="p">,</span><span class="mf">3.0</span><span class="p">))</span>
-</pre></div>
-</div>
-</section>
-<section id="reflection">
-<h3>Reflection<a class="headerlink" href="#reflection" title="Link to this heading">#</a></h3>
-<p>If the constant <span class="math notranslate nohighlight">\(c\)</span> multiplying the whole function or its argument has the particular value <span class="math notranslate nohighlight">\(c = -1\)</span>, we will have a <em>reflection</em>. Reflections can be done in relation to the <span class="math notranslate nohighlight">\(x\)</span> or <span class="math notranslate nohighlight">\(y\)</span>-axes and they are built, starting from a function <span class="math notranslate nohighlight">\(f(x)\)</span>, following the rules:</p>
-<ul class="simple">
-<li><p><strong>About the x-axis</strong> - <span class="math notranslate nohighlight">\(y = -f(x)\)</span></p></li>
-<li><p><strong>About the y-axis</strong> - <span class="math notranslate nohighlight">\(y = f(-x)\)</span></p></li>
-</ul>
-<p>The figure below shows these two reflections using the function <span class="math notranslate nohighlight">\(f(x) = (x-1)^2\)</span>.</p>
-<img alt="../../_images/reflection.png" src="../../_images/reflection.png" />
-</section>
-</section>
-</section>
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-<p>Imagine a colony of wasps is founded by a queen with the following pattern of eggs laid:</p>
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-</thead>
-<tbody>
-<tr class="row-even"><td><p><span class="math notranslate nohighlight">\(0\)</span></p></td>
-<td class="text-center"><p><span class="math notranslate nohighlight">\(100\)</span></p></td>
-</tr>
-<tr class="row-odd"><td><p><span class="math notranslate nohighlight">\(1\)</span></p></td>
-<td class="text-center"><p><span class="math notranslate nohighlight">\(135\)</span></p></td>
-</tr>
-<tr class="row-even"><td><p><span class="math notranslate nohighlight">\(2\)</span></p></td>
-<td class="text-center"><p><span class="math notranslate nohighlight">\(170\)</span></p></td>
-</tr>
-<tr class="row-odd"><td><p><span class="math notranslate nohighlight">\(3\)</span></p></td>
-<td class="text-center"><p><span class="math notranslate nohighlight">\(205\)</span></p></td>
-</tr>
-<tr class="row-even"><td><p><span class="math notranslate nohighlight">\(4\)</span></p></td>
-<td class="text-center"><p><span class="math notranslate nohighlight">\(240\)</span></p></td>
-</tr>
-</tbody>
-</table>
-</div>
-<p>Since there is an additive pattern for the increase in the total number of eggs laid, with the number of eggs in any given day, we can determine the number of eggs the colony will have in the following day. If we index the days by a sequence of natural numbers <em>n</em>, we have for the above colony:</p>
-<div class="math notranslate nohighlight">
-\[a_{n+1} = a_{n} + 35,\]</div>
-<p>where <span class="math notranslate nohighlight">\(a_n\)</span> represents the total amount of eggs in a given day <span class="math notranslate nohighlight">\(n\)</span> and <span class="math notranslate nohighlight">\(a_{n+1}\)</span> the same variable on the day following day <span class="math notranslate nohighlight">\(n\)</span>. The constant <span class="math notranslate nohighlight">\(35\)</span> represents the daily increment (or decrement) on this total number. A sequence of number built with constant additive increments receives the name of <em>arithmetic progression</em>.</p>
-<p>The above expression defines an iterative process (also known as a <em>recursion</em>) with which we can construct the whole sequence starting from a single initial value. If we want to know how many eggs will have been laid on a given day <span class="math notranslate nohighlight">\(n\)</span>, this shall be given by the expression:</p>
-<div class="math notranslate nohighlight">
-\[a_n = 100 + 35n.\]</div>
-<p>Thus, the total amount of eggs laid on day <span class="math notranslate nohighlight">\(n=100\)</span> is <span class="math notranslate nohighlight">\(a_{100} = 100 + 35 \cdot 100 = 3600\)</span>.</p>
-<br />
-<br />
-Now imagine a single *Escherichia coli* cell put on a Petri dish with a favorable growth medium. If given the right conditions, *E. coli* cells divide, generating two other cells, in a time of approximately $30$ minutes. For our experiment, we would have:  
-<br />  
-<div class="pst-scrollable-table-container"><table class="table">
-<thead>
-<tr class="row-odd"><th class="head"><p>Minutes</p></th>
-<th class="head text-center"><p># cells</p></th>
-</tr>
-</thead>
-<tbody>
-<tr class="row-even"><td><p><span class="math notranslate nohighlight">\(0\)</span></p></td>
-<td class="text-center"><p><span class="math notranslate nohighlight">\(1\)</span></p></td>
-</tr>
-<tr class="row-odd"><td><p><span class="math notranslate nohighlight">\(30\)</span></p></td>
-<td class="text-center"><p><span class="math notranslate nohighlight">\(2\)</span></p></td>
-</tr>
-<tr class="row-even"><td><p><span class="math notranslate nohighlight">\(60\)</span></p></td>
-<td class="text-center"><p><span class="math notranslate nohighlight">\(4\)</span></p></td>
-</tr>
-<tr class="row-odd"><td><p><span class="math notranslate nohighlight">\(90\)</span></p></td>
-<td class="text-center"><p><span class="math notranslate nohighlight">\(8\)</span></p></td>
-</tr>
-<tr class="row-even"><td><p><span class="math notranslate nohighlight">\(120\)</span></p></td>
-<td class="text-center"><p><span class="math notranslate nohighlight">\(16\)</span></p></td>
-</tr>
-</tbody>
-</table>
-</div>
-<p>If <span class="math notranslate nohighlight">\(t\)</span> represents the number of reproduction intervals since the start of the experiment (assuming values <span class="math notranslate nohighlight">\(t=0, 1, 2, \ldots\)</span>), then we can construct the recursion:</p>
-<div class="math notranslate nohighlight">
-\[N_{t+1} = 2N_t,\]</div>
-<p>where <span class="math notranslate nohighlight">\(N_t\)</span> represents the number of <em>E. coli</em> cells after <span class="math notranslate nohighlight">\(t\)</span> intervals of reproduction (each interval corresponding to <span class="math notranslate nohighlight">\(30\)</span> minutes) and <span class="math notranslate nohighlight">\(N_{t+1}\)</span> represents the number of cells on the interval following <span class="math notranslate nohighlight">\(t\)</span>. See that this time the constant <span class="math notranslate nohighlight">\(2\)</span> provides a multiplicative increment, so that the sequence formed by this multiplicative iteration is called <em>geometric progression</em>.</p>
-<p>If we want <span class="math notranslate nohighlight">\(N_t\)</span> for a given interval <span class="math notranslate nohighlight">\(t\)</span>, we then have:</p>
-<div class="math notranslate nohighlight">
-\[N_t = 2^t,\]</div>
-<p>leading to an exponential growth in discrete steps.</p>
-<p>In general, if we have a population growth described by a geometric progression, starting with a number of individuals <span class="math notranslate nohighlight">\(N_0\)</span>, the number of individuals on the subsequent generations can be described by the recursion:</p>
-<div class="math notranslate nohighlight">
-\[N_{t+1} = RN_t,\]</div>
-<p>where <span class="math notranslate nohighlight">\(t\)</span> indicates the number of generations and <span class="math notranslate nohighlight">\(R\)</span> is a positive constant (<span class="math notranslate nohighlight">\(R&gt;0\)</span>).</p>
-<p>With this recursion, starting from <span class="math notranslate nohighlight">\(N_0\)</span>, we can obtain the full sequence by construction:</p>
-<div class="math notranslate nohighlight">
-\[\begin{split}\begin{align}  
-N_1 &amp;= RN_0 \\
-N_2 &amp;= RN_1 = R(RN_0) = R^2N_0 \\
-N_3 &amp;= RN_2 = R(RN_1) = R[R(RN_0)] = R^3N_0 \\
-    &amp;\vdots \end{align}\end{split}\]</div>
-<p>From this process of construction, we can obtain what we call the <em>solution</em> of the above recursion, given by:</p>
-<div class="math notranslate nohighlight">
-\[N_t = N_0R^t.\]</div>
-<p>In problems of population dynamics, <span class="math notranslate nohighlight">\(R\)</span> is called <strong>growth constant</strong> and its value determines the long-term behaviour of the function:</p>
-<img alt="../../_images/expgrowth.png" src="../../_images/expgrowth.png" />
-<section id="sequences">
-<h2>Sequences<a class="headerlink" href="#sequences" title="Link to this heading">#</a></h2>
-<p>The two last problems are examples of <em>sequences</em> for which a formal definition is given as:</p>
-<div class="math notranslate nohighlight">
-\[\begin{split}\begin{align}
-    f\!:\ &amp;{\rm I\!N} \rightarrow {\rm I\!R} \\
-       &amp; n \rightarrow f(n),\end{align}\end{split}\]</div>
-<p>in other words, a sequence is a function <span class="math notranslate nohighlight">\(f\)</span> from the set of natural numbers to the set of real numbers. For each value of the independent variable <span class="math notranslate nohighlight">\(n\)</span> (a natural number), the sequence determines its image <span class="math notranslate nohighlight">\(f(n)\)</span>.</p>
-<div class="admonition-example admonition">
-<p class="admonition-title">Example</p>
-<p>The sequence defined by the rule <span class="math notranslate nohighlight">\(f(n) = \displaystyle\frac{1}{n+1}\)</span> defines the sequence of numbers (starting from <span class="math notranslate nohighlight">\(n=0\)</span>):</p>
-<div class="math notranslate nohighlight">
-\[1,\frac{1}{2},\frac{1}{3},\frac{1}{4},\frac{1}{5},\ldots,\]</div>
-<p>corresponding to the set of values <span class="math notranslate nohighlight">\(n=0,1,2,3,4,\ldots\)</span>.</p>
-</div>
-<p>Sequences are usually written as ordered lists of numbers <span class="math notranslate nohighlight">\(a_0, a_1, a_2, a_3, \ldots\)</span> where <span class="math notranslate nohighlight">\(a_n=f(n)\)</span> and referred to with the notation <span class="math notranslate nohighlight">\(\{a_n\}\)</span> (which is short for <span class="math notranslate nohighlight">\(\{a_n: n \in {\rm I\!N}\}\)</span>, making it explicit that <span class="math notranslate nohighlight">\(n\)</span> is a natural number).</p>
-<p>As we have seen before, two forms of representation are possible:</p>
-</section>
-<section id="limits">
-<h2>Limits<a class="headerlink" href="#limits" title="Link to this heading">#</a></h2>
-<p>In many applications, we are interested to know what is the <em>long-term behaviour</em> of a given sequence <span class="math notranslate nohighlight">\(\{a_n\}\)</span>. For example, is a given population tending towards a stable number of individuals as time passes? In other words, we want to know if there is a number <span class="math notranslate nohighlight">\(L\)</span> for which the sequence tends to as <span class="math notranslate nohighlight">\(n\)</span> increases. To represent this we use the following notation:</p>
-<div class="math notranslate nohighlight">
-\[\lim_{n\rightarrow \infty} a_n = L\ \ (\mbox{or simply}\ \lim a_n =L)\]</div>
-<p>If such a number exists, we say that the sequence <span class="math notranslate nohighlight">\(\{a_n\}\)</span> is <strong>convergent</strong>.</p>
-<div class="admonition-examples admonition">
-<p class="admonition-title">Examples</p>
-<ol class="arabic">
-<li><p><span class="math notranslate nohighlight">\(a_n = \displaystyle\frac{1}{n+1}: \ \ \ \left\{1,\displaystyle\frac{1}{2},\displaystyle\frac{1}{3},\displaystyle\frac{1}{4},\displaystyle\frac{1}{5},\ldots\right\}\)</span><br />
-<br />
-Note that each term is smaller than the previous and they are all positive. We can conclude that:</p>
-<div class="math notranslate nohighlight">
-\[\lim a_n = 0.\]</div>
-</li>
-<li><p><span class="math notranslate nohighlight">\(b_n = (-1)^n: \ \ \ \left\{1,-1,1,-1,1,-1,\ldots\right\}\)</span><br />
-<br />
-The terms in this sequence are alternating between <span class="math notranslate nohighlight">\(1\)</span> and <span class="math notranslate nohighlight">\(-1\)</span>, never convergeing for a specific value. Thus, the limit of sequence <span class="math notranslate nohighlight">\(\{b_n\}\)</span> does not exist.<br />
-<br /></p></li>
-<li><p><span class="math notranslate nohighlight">\(c_n = 2^n: \ \ \ \left\{1,2,4,8,16,32,\ldots\right\}\)</span><br />
-<br />
-For this sequence, the terms are ever increasing. We frequently say that the sequence tends to infinity (or tends to <span class="math notranslate nohighlight">\(\infty\)</span>). However, <span class="math notranslate nohighlight">\(\infty\)</span> represents an abstraction (“ever increasing”), not a specific number. Therefore, the limit for this sequence also does not exist.<br />
-<br /></p></li>
-<li><p><span class="math notranslate nohighlight">\(d_n = \displaystyle\frac{n+2}{n+1}: \ \ \ \left\{2,\displaystyle\frac{3}{2},\displaystyle\frac{4}{3},\displaystyle\frac{5}{4},\displaystyle\frac{6}{5},\ldots\right\}\)</span><br />
-<br />
-For this sequence, each term is smaller than the previous (look at the decimal forms of the fractions: <span class="math notranslate nohighlight">\(2, 1.5, 1.333\cdots, 1.25, 1.2 \ldots\)</span>) and they are greater than <span class="math notranslate nohighlight">\(1\)</span>. We can conclude that:</p>
-<div class="math notranslate nohighlight">
-\[\lim a_n = 1.\]</div>
-</li>
-</ol>
-</div>
-<p>Although there is a formal way to <em>prove</em> that a sequence <span class="math notranslate nohighlight">\(\{a_n\}\)</span> converges to a certain limit, in practice we use a set of rules to apply limits to sequences and composition of sequences.</p>
-<div class="admonition tip">
-<p class="admonition-title">Tip</p>
-<p>Consider two sequences <span class="math notranslate nohighlight">\(\{a_n\}\)</span> and <span class="math notranslate nohighlight">\(\{b_n\}\)</span>, and <span class="math notranslate nohighlight">\(C\)</span> as a real constant. If these two sequences are convergent with  <span class="math notranslate nohighlight">\(\lim a_n = L\)</span> and <span class="math notranslate nohighlight">\(\lim b_n = M\)</span>, we have:</p>
-<div class="math notranslate nohighlight">
-\[\lim (a_n + b_n) = \lim a_n + \lim b_n = L + M\]</div>
-<div class="math notranslate nohighlight">
-\[\lim (Ca_n) = C\lim a_n = CL\]</div>
-<div class="math notranslate nohighlight">
-\[\lim (a_nb_n) = (\lim a_n)(\lim b_n) = LM\]</div>
-<div class="math notranslate nohighlight">
-\[\lim \left(\frac{a_n}{b_n}\right) = \frac{\lim a_n}{\lim b_n} = \frac{L}{M},\ \ (\mbox{if}\ M \ne 0)\]</div>
-<p>In other other words, if you can decompose a given sequence into the sum, product or division of two other convergent sequences, the limit of this sequence will be the composition of the limits of the two others.</p>
-</div>
-<div class="admonition-examples admonition">
-<p class="admonition-title">Examples</p>
-<p>If <span class="math notranslate nohighlight">\(a_n = \displaystyle\frac{4n^2-1}{n^2}:\)</span></p>
-<div class="math notranslate nohighlight">
-\[\lim\left(\displaystyle\frac{4n^2-1}{n^2}\right) = \lim\left(4-\displaystyle\frac{1}{n^2}\right) = \lim 4 - \left(\lim\displaystyle\frac{1}{n}\right)\left(\lim\displaystyle\frac{1}{n}\right) = 4,\]</div>
-<p>since as we know: <span class="math notranslate nohighlight">\(\ \lim 4 = 4\ \)</span> and <span class="math notranslate nohighlight">\(\ \lim \displaystyle\frac{1}{n} = 0\)</span>.</p>
-</div>
-<p>For recursions, one way to find the limit is to find the solution (corresponding explicit description) and then calculate the limit of the sequence with the limit rules. However, finding the explicit form of a given sequence is not always possible.</p>
-<p>There is a procedure to find <em>candidates</em> for the long-term behaviour of a given sequence. We do this by calculating <strong>fixed points</strong>.</p>
-<p>Fixed points are specific values on a given sequence for which all subsequent values equal the first. That is, if <span class="math notranslate nohighlight">\(a\)</span> is a fixed point of the sequence <span class="math notranslate nohighlight">\(\{a_n\}\)</span>, and if <span class="math notranslate nohighlight">\(a_0 = a\)</span>, then <span class="math notranslate nohighlight">\(a_1 = a\)</span>, <span class="math notranslate nohighlight">\(a_2 = a\)</span>, and all the other values will be <span class="math notranslate nohighlight">\(a\)</span>. Thus, given the recursion <span class="math notranslate nohighlight">\(a_{n+1} = g(a_n)\)</span>, if <span class="math notranslate nohighlight">\(a\)</span> is a fixed point we will have:</p>
-<div class="math notranslate nohighlight">
-\[a = g(a)\]</div>
-<p>and the last equation provides a way of find possible values of <span class="math notranslate nohighlight">\(a\)</span>.</p>
-<div class="admonition-example admonition">
-<p class="admonition-title">Example</p>
-<p>Consider the following recursion</p>
-<div class="math notranslate nohighlight">
-\[a_{n+1} = \sqrt{3a_n},\ \ \ a_0 = 2.\]</div>
-<p>To obtain the fixed points we calculate:</p>
-<div class="math notranslate nohighlight">
-\[\begin{split}\begin{align}
-  a &amp;= \sqrt{3a} \\
-  a^2 &amp;= 3a \ \ \longrightarrow a = 0\ \mbox{ or }\ a = 3 \end{align}\end{split}\]</div>
-<p>Starting with <span class="math notranslate nohighlight">\(a_0 = 2\)</span>, we can use the recursion to construct the sequence as:</p>
-<div class="math notranslate nohighlight">
-\[\left\{\sqrt{6},\sqrt{3\sqrt{6}},\sqrt{3\sqrt{3\sqrt{6}}},\sqrt{3\sqrt{3\sqrt{3\sqrt{6}}}},\ldots \right\}\]</div>
-<p>or also: <span class="math notranslate nohighlight">\(\{2.449\cdots,2.711\cdots,2.852\cdots,2.925\cdots, \ldots\}\)</span>. We see that starting with <span class="math notranslate nohighlight">\(a_0=2\)</span>, we have <span class="math notranslate nohighlight">\(a_n&gt;2\)</span> for all <span class="math notranslate nohighlight">\(n\)</span>, with increasing values. Since <span class="math notranslate nohighlight">\(a=3\)</span> is a fixed point, we conclude that <span class="math notranslate nohighlight">\(\lim a_n = 3\)</span>.</p>
-</div>
-<div class="admonition-example admonition">
-<p class="admonition-title">Example</p>
-<p>Whenever we are asking questions like ‘where does this process stabilise’ or ‘how high/much will it be in the end,’ we are interested in the limit of some series. This could be a variety of things, like population size, ground water levels, global mean temperature, number of trees, etc.</p>
-</div>
-<div class="admonition warning">
-<p class="admonition-title">Warning</p>
-<p>Remember that fixed points are only <strong>candidates</strong> for long-term behaviour. See for example the recursion:</p>
-<div class="math notranslate nohighlight">
-\[a_{n+1} = \displaystyle\frac{3}{a_n}\]</div>
-<p>For the fixed points, we calculate:</p>
-<div class="math notranslate nohighlight">
-\[\begin{split}\begin{align}
-  a &amp;= \displaystyle\frac{3}{a} \\
-  a^2 &amp;= 3 \\
-  &amp;\longrightarrow a = -\sqrt{3}\ \mbox{ or }\ a = \sqrt{3}. \end{align}\end{split}\]</div>
-<p>Indeed, if we construct the sequence from the recursion starting with any of these two values, all the subsequent elements of the sequence will have the same value. However:</p>
-<div class="math notranslate nohighlight">
-\[\mbox{if } a_0 = 2 \ \mbox{the sequence will be given by: }\ \left\{2,\displaystyle\frac{3}{2},2,\displaystyle\frac{3}{2},\ldots \right\}\]</div>
-<p>or</p>
-<div class="math notranslate nohighlight">
-\[\mbox{If } a_0 = -3 \ \mbox{the sequence will be given by: }\ \left\{-3,-1,-3,-1,\ldots \right\}\]</div>
-<p>showing that for any initial value chosen (different from the fixed points), the sequence will oscillate between two values, without a clear tendency to any limit.</p>
-</div>
-</section>
-<section id="discrete-time-population-models">
-<h2>Discrete-time population models<a class="headerlink" href="#discrete-time-population-models" title="Link to this heading">#</a></h2>
-<p>Sequences are generally used in many applications in Biology. Important examples are models for <strong>seasonally breeding</strong> populations.</p>
-<section id="density-dependent-population-growth">
-<h3>Density-dependent population growth<a class="headerlink" href="#density-dependent-population-growth" title="Link to this heading">#</a></h3>
-<p>As we have seen before, the recursion relation for the exponential growth model is given by <span class="math notranslate nohighlight">\(N_{t+1} = RN_t\)</span>. Rearranging the terms of this equation we have:</p>
-<div class="math notranslate nohighlight">
-\[\displaystyle\frac{N_t}{N_{t+1}}=\frac{1}{R}.\]</div>
-<p>The left-hand term of the last equation is known as <em>parent-offspring ratio</em> as it denotes the ratio between the population size at time <span class="math notranslate nohighlight">\(t\)</span> (parents) and the population size at the following time <span class="math notranslate nohighlight">\(t+1\)</span> (offspring). Here we are considering a population that breeds without overlapping generations. Each generation breeds, dies, and is replaced by their offspring on the following generation.</p>
-<p>On the equation we also see that the parent-offspring ratio is equal to <span class="math notranslate nohighlight">\(\displaystyle\frac{1}{R}\)</span>, which is a constant. Therefore, we say that this growth is <em>density-independent</em> as the parent-offspring ratio does not depend on the density of individuals at any time step. Relating the parent-offspring ratio to the value of <span class="math notranslate nohighlight">\(R\)</span> allows us to predict the behaviour of this population in terms of growth or decline:</p>
-<div class="math notranslate nohighlight">
-\[\begin{split}\begin{align}
-R&gt;1 \ &amp;- \ \mbox{there will be more offspring than parents (population grows)} \\
-R&lt;1 \ &amp;- \ \mbox{there will be more parents than offspring (population declines)} \\
-R=1 \ &amp;- \ \mbox{there will be no change in population size} \end{align}\end{split}\]</div>
-<p>We know, however, than in real scenarios, that the growth factor of a given population should depend on the current size of that population, due to space or resources limitation. These are called <em>density-dependent</em> effects. One of the simplest methods to include density-dependent effects in this population model is to consider the parent-offspring ratio as a linear function of <span class="math notranslate nohighlight">\(N_t\)</span>, instead of a constant. Let us assume that the parent-offspring ratio show the following linear behaviour:</p>
-<img alt="../../_images/parent-offspring.png" src="../../_images/parent-offspring.png" />
-<p>This means that the parent-offspring ratio is smaller than <span class="math notranslate nohighlight">\(1\)</span> (population growth) for values of <span class="math notranslate nohighlight">\(N_t\)</span> smaller than a given population threshold <span class="math notranslate nohighlight">\(K\)</span>, and greater than <span class="math notranslate nohighlight">\(1\)</span> (population decline) if the population size is greater than this threshold. The graph above leads to the following expression:</p>
-<div class="math notranslate nohighlight">
-\[\begin{split}\begin{align}
-\displaystyle\frac{N_t}{N_{t+1}} = \frac{1}{R} + \frac{1-\frac{1}{R}}{K}N_t \\
-N_{t+1} = \frac{N_t}{\frac{1}{R} + \frac{1-\frac{1}{R}}{K}N_t}, \end{align}\end{split}\]</div>
-<p>which leads, after rearranging the terms, to:</p>
-<div class="math notranslate nohighlight">
-\[N_{t+1} = \frac{RN_t}{1 + \frac{(R-1)}{K}N_t}.\]</div>
-<p>The previous expression is known as <strong>Beverton-Holt recruitment curve</strong> (or Beverton-Holt model) and it describes a type of population growth called <em>logistic growth</em>.</p>
-<p>The fixed points <span class="math notranslate nohighlight">\(\bar{N}\)</span> for this recursion can be calculated as:</p>
-<div class="math notranslate nohighlight">
-\[\bar{N} = \frac{R\bar{N}}{1 + \frac{(R-1)}{K}\bar{N}}\ \longrightarrow \ \bar{N}\left(1 + \frac{(R-1)}{K}\bar{N}\right) = R\bar{N}\]</div>
-<p>From the last term we can show (try it as an exercise) that <span class="math notranslate nohighlight">\(\bar{N}=0\)</span> and <span class="math notranslate nohighlight">\(\bar{N}=K\)</span> are the two fixed points. Starting from different values for the initial population size <span class="math notranslate nohighlight">\(N_0\)</span> it can be shown that the population tends to <span class="math notranslate nohighlight">\(K\)</span> as time increases as:</p>
-<img alt="../../_images/logistic.png" src="../../_images/logistic.png" />
-<p>The constant <span class="math notranslate nohighlight">\(K\)</span> is know as <em>carrying capacity</em> and represents the maximum population size a given population can reach at a given place, due to intra-specific competition for resources or space.</p>
-<p>Watch this <a class="reference external" href="https://www.youtube.com/watch?v=Kas0tIxDvrg">video</a> about application of growth models to epidemics.</p>
-</section>
-<section id="annual-plants">
-<h3>Annual plants<a class="headerlink" href="#annual-plants" title="Link to this heading">#</a></h3>
-<p>Suppose a given population of plants produces, on average <span class="math notranslate nohighlight">\(f\)</span> seeds per individual in the reproductive period. Seeds survive winter with a probability <span class="math notranslate nohighlight">\(\sigma\)</span> and germinate on the next season with probability <span class="math notranslate nohighlight">\(\alpha\)</span> (these seeds reach reproductive maturity on the same season). If <span class="math notranslate nohighlight">\(p_n\)</span> is the plant population size on season <span class="math notranslate nohighlight">\(n\)</span> and <span class="math notranslate nohighlight">\(p_{n+1}\)</span> the population size on the season following <span class="math notranslate nohighlight">\(n\)</span>, we shall have:</p>
-<div class="math notranslate nohighlight">
-\[p_{n+1} = (\alpha\sigma f)p_n\]</div>
-<p>which correspond to the density-independent model since <span class="math notranslate nohighlight">\(R=\alpha\sigma f\)</span> is a constant.</p>
-<p>Now suppose that some of the seeds that were produced on the previous season and survived the previous winter, but did not germinate on the current season, might have a second chance of germinating. They might do so on the following season, provided they survive the winter and germinate on the following season. From the average number of seeds produced last season <span class="math notranslate nohighlight">\(fp_{n-1}\)</span>, a number <span class="math notranslate nohighlight">\(\sigma fp_{n-1}\)</span> of them survived last winter. From this number, a fraction <span class="math notranslate nohighlight">\((1-\alpha)\)</span> did not germinate this season (1 minus the probability of germinating). These seeds will have a second chance provided they survive next winter (with probability <span class="math notranslate nohighlight">\(\sigma\)</span>) and germinate on the next season (with probability <span class="math notranslate nohighlight">\(\alpha\)</span>). Thus, the number plant population size on the season following <span class="math notranslate nohighlight">\(n\)</span> will be:</p>
-<div class="math notranslate nohighlight">
-\[\begin{split}\begin{align}
-p_{n+1} = \alpha\sigma fp_n + \alpha\sigma(1-\alpha)\sigma fp_{n-1} \\  
-p_{n+1} = (\alpha\sigma f)p_n + [\alpha(1-\alpha)\sigma^2 f]p_{n-1} \\  
-p_{n+1} = \beta p_n + \gamma p_{n-1}\end{align}\end{split}\]</div>
-<p>where <span class="math notranslate nohighlight">\(\beta=\alpha\sigma f\)</span> and <span class="math notranslate nohighlight">\(\gamma = \alpha(1-\alpha)\sigma^2 f\)</span>.</p>
-<p>The previous model is an example of a discrete population model with overlapping populations. Since the term <span class="math notranslate nohighlight">\(n+1\)</span> depends not only on the term <span class="math notranslate nohighlight">\(n\)</span> but also on <span class="math notranslate nohighlight">\(n-1\)</span> this is called a <em>second-order difference equation</em> or <em>delay (or lagged) difference equation</em>.</p>
-</section>
-<section id="fibonacci-sequence">
-<h3>Fibonacci sequence<a class="headerlink" href="#fibonacci-sequence" title="Link to this heading">#</a></h3>
-<p>The Fibonacci sequence is a sequence defined by the following rule: each number is equal to the sum of the previous two numbers (starting with <span class="math notranslate nohighlight">\(0\)</span> and <span class="math notranslate nohighlight">\(1\)</span>). Thus the following sequence might be formed:</p>
-<div class="math notranslate nohighlight">
-\[\left\{ 0,1,1,2,3,5,8,13,21,\ldots \right\}. \]</div>
-<p>It can also be defined by the second-order difference equation:</p>
-<div class="math notranslate nohighlight">
-\[F_{n+1} = F_{n} + F_{n-1}.\]</div>
-<p>Although this is clearly not a convergent sequence (with sustained growth for each element), if we define a new sequence given by the ratio between successive elements of the Fibonacci sequence as <span class="math notranslate nohighlight">\(b_n = \displaystyle\frac{F_{n+1}}{F_n}\)</span>, it can be show that this sequence is convergent.</p>
-<p>Assuming that <span class="math notranslate nohighlight">\(\lim \displaystyle\frac{F_{n+1}}{F_n} = \lambda\)</span>, we have:</p>
-<div class="math notranslate nohighlight">
-\[\begin{split}\begin{align}
-F_{n+1} &amp;= F_{n} + F_{n-1}\quad \mbox{(Fibonacci seq.)} \\  
-\displaystyle\frac{F_{n+1}}{F_{n}} &amp;= 1 + \frac{F_{n-1}}{F_{n}}\quad \mbox{(divide both sides by}\ F_{n}\mbox{)} \\
-\lim \left(\displaystyle\frac{F_{n+1}}{F_{n}}\right) &amp;= \lim\left(1 + \frac{F_{n-1}}{F_{n}}\right) \\
-\lambda &amp;= 1 + \frac{1}{\lambda} \\
-\lambda^2 &amp;- \lambda -1 = 0 \end{align}\end{split}\]</div>
-<p>The previous equation is solved finding the roots of the second-degree polynomial, which gives <span class="math notranslate nohighlight">\(\lambda = \displaystyle\frac{1+\sqrt{5}}{2}\)</span> or <span class="math notranslate nohighlight">\(\lambda = \displaystyle\frac{1-\sqrt{5}}{2}\)</span>. Since the solution <span class="math notranslate nohighlight">\(\displaystyle\frac{1-\sqrt{5}}{2}\)</span> is negative, it can be the limit of the ratio of Fibonacci terms (all of them positive). Hence we conclude:</p>
-<div class="math notranslate nohighlight">
-\[\lim \left(\displaystyle\frac{F_{n+1}}{F_{n}}\right) = \displaystyle\frac{1+\sqrt{5}}{2} \approx 1.618.\]</div>
-<p>The previous number is known as the <em>Golden ratio</em> and it has fascinated mathematicians, phylosophers, and artists since ancient times. Fibonacci numbers and the Golden ratio can be found in several interesting natural patterns, as you can see in this <a class="reference external" href="https://www.youtube.com/watch?v=ahXIMUkSXX0">video</a>.</p>
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-  <section class="tex2jax_ignore mathjax_ignore" id="from-table-arrangements-to-powerful-tools">
-<h1>04 - From table arrangements to powerful tools<a class="headerlink" href="#from-table-arrangements-to-powerful-tools" title="Link to this heading">#</a></h1>
-<section id="systems-of-linear-equations">
-<h2>Systems of linear equations<a class="headerlink" href="#systems-of-linear-equations" title="Link to this heading">#</a></h2>
-<p>As we saw previously, the equation <span class="math notranslate nohighlight">\(ax + by = c\)</span>, with <span class="math notranslate nohighlight">\(a\)</span>, <span class="math notranslate nohighlight">\(b\)</span>, and <span class="math notranslate nohighlight">\(c\)</span> constants, defines a straight line on the <span class="math notranslate nohighlight">\(xy\)</span>-plane (note that it can also be written as <span class="math notranslate nohighlight">\(y = mx + k\)</span>, <span class="math notranslate nohighlight">\(m\)</span> and <span class="math notranslate nohighlight">\(k\)</span> constants, by a suitable change of the coeffients). Since the coefficients do not depend on the variables <span class="math notranslate nohighlight">\(x\)</span> and <span class="math notranslate nohighlight">\(y\)</span>, we call it a <em>linear equation</em>. A possible plot for this equation would be:</p>
-<img alt="../../_images/lineq.png" src="../../_images/lineq.png" />
-<p>Note that all the points <span class="math notranslate nohighlight">\((x,y)\)</span> on the blue line <em>satisfy</em> or <em>solve</em> the lienar equation <span class="math notranslate nohighlight">\(ax + by = c\)</span>.</p>
-<p>Now consider that we have the following:</p>
-<div class="math notranslate nohighlight">
-\[\begin{split}\begin{cases} ax + by = c \\ dx +ey = f \end{cases}\end{split}\]</div>
-<p>where <span class="math notranslate nohighlight">\(a, b, c, d, e,\ \mbox{and}\ f\)</span> are all <a class="reference external" href="http://constants.In">constants.In</a> this case we have a <em>system of linear equations</em>. Solving this system means to find all possible pairs <span class="math notranslate nohighlight">\((x,y)\)</span> that solve both equations <strong>simultaneously</strong>. For the previous system, we would have 3 options:</p>
-<img alt="../../_images/syslineq.png" src="../../_images/syslineq.png" />
-<p>On the left panel, the two lines intersect at a <em>single point</em>. This point is the only one that solves both linear equations at the same time, and, therefore, constitutes the <strong>only solution</strong> for the system of two equations. On the central panel, the two lines are <em>parallel</em>, so there is no point of intersection. In this case there is <strong>no solution</strong> and the system of equations is said to be <em>inconsistent</em>. On the right panel, the two lines coincide. Thus, every point in those lines correspond to a possible solution for the system, so that it has <strong>infinite solutions</strong>.</p>
-<div class="admonition-example admonition">
-<p class="admonition-title">Example</p>
-<p>Given the system of linear equations</p>
-<div class="math notranslate nohighlight">
-\[\begin{split}\begin{cases} 2x + 3y = 6 \\ 2x + y = 4 \end{cases}\end{split}\]</div>
-<p>we can obtain the solution by isolating one of the variables in one of the equations and substituting on the other equation. Isolating <span class="math notranslate nohighlight">\(x\)</span> on the first equation gives:</p>
-<div class="math notranslate nohighlight">
-\[x = 3 - \displaystyle\frac{3}{2}y\]</div>
-<p>Then by substituting on the second we have:</p>
-<div class="math notranslate nohighlight">
-\[\begin{split}\begin{align} 
-2\left(3 - \displaystyle\frac{3}{2}y \right) + y &amp;= 4 \\
-6 - 3y + y &amp;= 4 \\
--2y = -2 \\
-y = 1 \end{align}\end{split}\]</div>
-<p>The value of <span class="math notranslate nohighlight">\(x\)</span> shall then be given by:</p>
-<div class="math notranslate nohighlight">
-\[x = 3 - \displaystyle\frac{3}{2}\cdot 1 = 3 - \frac{3}{2} = \frac{3}{2}\]</div>
-<p>We see that the system has a single solution, given by <span class="math notranslate nohighlight">\(\left(\displaystyle\frac{3}{2},1\right)\)</span>.</p>
-</div>
-<div class="admonition-example admonition">
-<p class="admonition-title">Example</p>
-<p>Now consider the sytem</p>
-<div class="math notranslate nohighlight">
-\[\begin{split}\begin{cases} 2x + 3y &amp;= 6 \\ -4x + -6y &amp;= -12 \end{cases}\end{split}\]</div>
-<p>Isolating <span class="math notranslate nohighlight">\(x\)</span> on the first equation gives the same as before:</p>
-<div class="math notranslate nohighlight">
-\[x = 3 - \displaystyle\frac{3}{2}y\]</div>
-<p>Then by substituting on the second we have:</p>
-<div class="math notranslate nohighlight">
-\[\begin{split}\begin{align} 
--4\left(3 - \displaystyle\frac{3}{2}y \right) - 6y &amp;= -12 \\
--12 + 6y - 6y &amp;= -12 \\
-0 = 0 \end{align}\end{split}\]</div>
-<p>which is just a trivial equation.</p>
-<p>When we look at the equations in our system we see that the second equation is simply a constant multiplied by the first <span class="math notranslate nohighlight">\((\text{Eqn.} 2) = -2\cdot(\text{Eqn.} 1)\)</span>, and thus they should represent the same straight line.</p>
-<p>This system has then infinite solutions which are represented by the set of points <span class="math notranslate nohighlight">\(\left(3 - \displaystyle\frac{3}{2}y,y\right)\)</span></p>
-<div class="admonition tip">
-<p class="admonition-title">Tip</p>
-<p>Note that writing the solution as <span class="math notranslate nohighlight">\(\left(3 - \displaystyle\frac{3}{2}y,y\right)\)</span> represents an infinite set of points, since for every real value of <span class="math notranslate nohighlight">\(y\)</span> we would have a different ordered pair. This set of points forms the straight line corresponding to <span class="math notranslate nohighlight">\(2x+3y = 6\)</span>.</p>
-</div>
-</div>
-<p>Solving the system by this method of substitutions, however, can get much more complicated if we have more than two variables. For these cases, we need a simpler method.</p>
-<p>One strategy is to find equivalent systems, which have the same solution as the original one, but are much easier to solve. We can apply the following set of operations without changing the solutions of a given system:</p>
-<ul class="simple">
-<li><p>Multiplication of an entire row by a non-zero constant</p></li>
-<li><p>Adding/subtracting one row to another</p></li>
-<li><p>Rearranging the order of the rows</p></li>
-</ul>
-<p>The examples below show this procedure applied to two systems.</p>
-<div class="admonition-examples admonition">
-<p class="admonition-title">Examples</p>
-<p>(i) <span class="math notranslate nohighlight">\(\begin{cases} 3x + 5y - z = 10 \\ 2x –  y + 3z = 9 \\ 4x + 2y - 3z = -1 \end{cases}\)</span></p>
-<p>See the solution on this <a class="reference external" href="https://web.microsoftstream.com/video/194b0152-903a-4614-b608-306ce71d9a39">video</a></p>
-<p>(ii) <span class="math notranslate nohighlight">\(\begin{cases} x - 3y + z = 4 \\ x – 2y + 3z = 6 \\ 2x - 6y + 2z = 8 \end{cases}\)</span></p>
-<p>See the solution on this <a class="reference external" href="https://web.microsoftstream.com/video/067c8938-5e45-4f08-bf90-783cb125c6f5">video</a></p>
-</div>
-<p>Note that by applying the rules of matrix multiplication, the system <em>(ii)</em> can also be written in the form:</p>
-<div class="math notranslate nohighlight">
-\[\begin{split}\begin{pmatrix} 1 &amp; -3 &amp; 1\\ 1 &amp; -2 &amp; 3 \\ 2 &amp; -6 &amp; 2  \end{pmatrix} \begin{pmatrix} x \\ y \\ z   \end{pmatrix} = \begin{pmatrix} 4 \\ 6 \\ 8   \end{pmatrix} \quad \longrightarrow \quad A\mathbf{x}=\mathbf{b}\end{split}\]</div>
-<p>where <span class="math notranslate nohighlight">\(A\)</span> is the matrix of coefficients and <span class="math notranslate nohighlight">\(\mathbf{x}\)</span> is the vector of variables.</p>
-<div class="admonition tip">
-<p class="admonition-title">Tip</p>
-<p>We will be using the common notation of uppercase letters (<span class="math notranslate nohighlight">\(A\)</span>) to represent matrices and lowercase bold letters (<span class="math notranslate nohighlight">\(\mathbf{x}\)</span>) to represent vectors.</p>
-</div>
-<p>In general, the system with <span class="math notranslate nohighlight">\(n\)</span> variables <span class="math notranslate nohighlight">\(\{x_1, x_2, \ldots, x_n \}\)</span> and <span class="math notranslate nohighlight">\(m\)</span> equations can be written as</p>
-<div class="math notranslate nohighlight">
-\[\begin{split}\begin{cases} a_{11}x_1 + a_{12}x_2 + \ldots + a_{1n}x_n = b_1 \\ a_{21}x_1 + a_{22}x_2 + \ldots + a_{2n}x_n = b_2 \\ \dots \\ a_{m1}x_1 + a_{m2}x_2 + \ldots + a_{mn}x_n = b_m \end{cases}\end{split}\]</div>
-<p>can be written as <span class="math notranslate nohighlight">\(A\mathbf{x}=\mathbf{b}\)</span>, where</p>
-<div class="math notranslate nohighlight">
-\[\begin{split}A=\begin{pmatrix} a_{11} &amp; a_{12} &amp; \cdots &amp; a_{1n} \\ a_{21} &amp; a_{22} &amp; \cdots &amp; a_{2n} \\ \vdots &amp; &amp; \ddots &amp; \vdots \\ a_{m1} &amp; a_{m2} &amp; \cdots &amp; a_{mn} \end{pmatrix}\end{split}\]</div>
-<p>and</p>
-<div class="math notranslate nohighlight">
-\[\begin{split}\mathbf{x} = \begin{pmatrix} x_1 \\ x_2 \\ \vdots \\ x_n   \end{pmatrix}\quad \mbox{and} \quad \mathbf{b} = \begin{pmatrix} b_1 \\ b_2 \\ \vdots \\ b_m   \end{pmatrix}\end{split}\]</div>
-<p>The matrix <span class="math notranslate nohighlight">\(A\)</span> has size <span class="math notranslate nohighlight">\(m \times n\)</span> (since the number of equations can, in principle, be different from the number of variables), while the vectors <span class="math notranslate nohighlight">\(\mathbf{x}\)</span> and <span class="math notranslate nohighlight">\(\mathbf{b}\)</span> have sizes <span class="math notranslate nohighlight">\(n \times 1\)</span> and <span class="math notranslate nohighlight">\(m \times 1\)</span>, respectively.</p>
-</section>
-<section id="matrix-operations">
-<h2>Matrix operations<a class="headerlink" href="#matrix-operations" title="Link to this heading">#</a></h2>
-<p>Let us quickly remind the basic operations that can be performed with matrices. If <span class="math notranslate nohighlight">\(A\)</span> and <span class="math notranslate nohighlight">\(B\)</span> are both matrices of size <span class="math notranslate nohighlight">\(m \times n\)</span> (<span class="math notranslate nohighlight">\(m\)</span> rows and <span class="math notranslate nohighlight">\(n\)</span> columns), we have:</p>
-<ul>
-<li><p><span class="math notranslate nohighlight">\(A=B\ \)</span> if and only if <span class="math notranslate nohighlight">\(a_{ij} = b_{ij}\)</span> for every <span class="math notranslate nohighlight">\(1 \le i \le m,  1 \le j \le n\)</span> (elements at the same position have to be equal).</p></li>
-<li><p><span class="math notranslate nohighlight">\(C = A + B\)</span> is the matrix for which <span class="math notranslate nohighlight">\(c_{ij} = a_{ij} + b_{ij}\)</span> (sum is performed summing the elements at the same position)</p></li>
-<li><p>If <span class="math notranslate nohighlight">\(k \in \rm I\!R\)</span> than <span class="math notranslate nohighlight">\(kA\)</span> is the <span class="math notranslate nohighlight">\(m \times n\)</span> matrix with elements <span class="math notranslate nohighlight">\(ka_{ij}\)</span></p></li>
-<li><p>The following properties are valid:</p>
-<div class="math notranslate nohighlight">
-\[\begin{split}\begin{align}
-    A+B &amp;= B+A \\
-    (A+B)+C &amp;= A+(B+C) \\
-    A + 0 &amp;= A\ (0\ \mbox{here represents the matrix with null elements}) \end{align}\end{split}\]</div>
-<div class="admonition-example admonition">
-<p class="admonition-title">Example</p>
-<p>If <span class="math notranslate nohighlight">\(A=\begin{pmatrix} 1 &amp; 0  \\ -1 &amp; 2  \end{pmatrix}\)</span> and <span class="math notranslate nohighlight">\(B=\begin{pmatrix} 2 &amp; 3  \\ -1 &amp; 1  \end{pmatrix}\)</span>, then:</p>
-<div class="math notranslate nohighlight">
-\[\begin{split}C = 2A+B = \begin{pmatrix} 2 &amp; 0  \\ -2 &amp; 4  \end{pmatrix} + \begin{pmatrix} 2 &amp; 3  \\ -1 &amp; 1  \end{pmatrix} = \begin{pmatrix} 4 &amp; 3  \\ -3 &amp; 5  \end{pmatrix}\end{split}\]</div>
-</div>
-</li>
-<li><p>If <span class="math notranslate nohighlight">\(A\)</span> is the <span class="math notranslate nohighlight">\(m \times n\)</span> matrix with elements <span class="math notranslate nohighlight">\(a_{ij}\)</span>, the transpose of <span class="math notranslate nohighlight">\(A\)</span> (denoted by <span class="math notranslate nohighlight">\(A'\)</span>) is the <span class="math notranslate nohighlight">\(n \times m\)</span> matrix with elements <span class="math notranslate nohighlight">\(a_{ij}' = a_{ji}\)</span>.</p>
-<div class="admonition-example admonition">
-<p class="admonition-title">Example</p>
-<div class="math notranslate nohighlight">
-\[\begin{split}A=\begin{pmatrix} 1 &amp; 2 &amp; 3 \\ 4 &amp; 5 &amp; 6  \end{pmatrix} \longrightarrow A'=\begin{pmatrix} 1 &amp; 4 \\ 2 &amp; 5 \\ 3 &amp; 6   \end{pmatrix}\end{split}\]</div>
-<div class="math notranslate nohighlight">
-\[\begin{split}B=\begin{pmatrix} 1 &amp; 2 \end{pmatrix} \longrightarrow B' = \begin{pmatrix} 1 \\ 2 \end{pmatrix}\end{split}\]</div>
-</div>
-</li>
-<li><p><strong>(Matrix multiplication)</strong> Consider now that <span class="math notranslate nohighlight">\(A\)</span> and <span class="math notranslate nohighlight">\(B\)</span> are matrices with sizes <span class="math notranslate nohighlight">\(m \times l\)</span> and <span class="math notranslate nohighlight">\(l \times n\)</span> (number of columns of <span class="math notranslate nohighlight">\(A\)</span> is equal to the number of rows of <span class="math notranslate nohighlight">\(B\)</span>). Then we can define <span class="math notranslate nohighlight">\(C = AB\)</span>  as the <span class="math notranslate nohighlight">\(m \times n\)</span> matrix with elements <span class="math notranslate nohighlight">\(c_{ij}\)</span> formed by multiplying every element of the <span class="math notranslate nohighlight">\(i\)</span>-th row of <span class="math notranslate nohighlight">\(A\)</span> with the corresponding element of the <span class="math notranslate nohighlight">\(j\)</span>-th column of <span class="math notranslate nohighlight">\(B\)</span>.</p>
-<div class="admonition-example admonition">
-<p class="admonition-title">Example</p>
-<p>If <span class="math notranslate nohighlight">\(A=\begin{pmatrix} 1 &amp; 2  \\ -1 &amp; 0  \end{pmatrix}\)</span> and <span class="math notranslate nohighlight">\(B=\begin{pmatrix} 1 &amp; 2 &amp; 3  \\ 0 &amp; -1 &amp; 4  \end{pmatrix}\)</span>, then:</p>
-<div class="math notranslate nohighlight">
-\[\begin{split}C = AB = \begin{pmatrix} 1 &amp; 0 &amp; 11  \\ -1 &amp; -2 &amp; -3  \end{pmatrix}.\end{split}\]</div>
-<p>Note that the matrix <span class="math notranslate nohighlight">\(BA\)</span> cannot be defined here since the number of columns of <span class="math notranslate nohighlight">\(B\)</span> is different from the number of rows of <span class="math notranslate nohighlight">\(A\)</span>.</p>
-</div>
-<div class="admonition warning">
-<p class="admonition-title">Warning</p>
-<p>If matrices <span class="math notranslate nohighlight">\(A\)</span> and <span class="math notranslate nohighlight">\(B\)</span> are <em>square matrices</em> (same number of rows and columns) then we can define both <span class="math notranslate nohighlight">\(AB\)</span> and <span class="math notranslate nohighlight">\(BA\)</span>. But, in general, we <strong>cannot assume</strong> that <span class="math notranslate nohighlight">\(AB=BA\)</span>. For example, if  <span class="math notranslate nohighlight">\(A=\begin{pmatrix} 2 &amp; -1  \\ 4 &amp; -2  \end{pmatrix}\)</span> and <span class="math notranslate nohighlight">\(B=\begin{pmatrix} 1 &amp; -1  \\ 2 &amp; -2  \end{pmatrix}\)</span>, then:</p>
-<div class="math notranslate nohighlight">
-\[\begin{split}AB = \begin{pmatrix} 0 &amp; 0  \\ 0 &amp; 0  \end{pmatrix} \quad and \quad BA = \begin{pmatrix} -2 &amp; 1  \\ -4 &amp; 2  \end{pmatrix}\end{split}\]</div>
-</div>
-</li>
-<li><p>For matrix multiplication, the following properties are valid:</p>
-<div class="math notranslate nohighlight">
-\[\begin{split}\begin{align}
-    (A+B)C&amp;=AC+BC \\
-    A(B+C)&amp;=AB+AC \\
-    (AB)C &amp;= A(BC) \\
-    A0 &amp;= 0A = 0 \end{align}\end{split}\]</div>
-</li>
-<li><p>We can also define powers of <span class="math notranslate nohighlight">\(A\)</span>. If <span class="math notranslate nohighlight">\(k\)</span> is a positive integer:</p>
-<div class="math notranslate nohighlight">
-\[A^k = A^{k-1}A = AA^{k-1} = A\cdot A\cdot A \cdots A\ (\mbox{product of}\ k\ \mbox{matrices})\]</div>
-</li>
-</ul>
-</section>
-<section id="inverse-matrix">
-<h2>Inverse matrix<a class="headerlink" href="#inverse-matrix" title="Link to this heading">#</a></h2>
-<p>Suppose that <span class="math notranslate nohighlight">\(A\)</span> is a <span class="math notranslate nohighlight">\(n \times n\)</span> matrix. If there exists a <span class="math notranslate nohighlight">\(n \times n\)</span> matrix <span class="math notranslate nohighlight">\(B\)</span> so that</p>
-<div class="math notranslate nohighlight">
-\[ AB = BA = I_n,\]</div>
-<p>where <span class="math notranslate nohighlight">\(I=\begin{pmatrix} 1 &amp; 0 &amp; \cdots &amp; 0 \\ 0 &amp; 1 &amp; \cdots &amp; 0 \\ \vdots &amp; &amp; \ddots &amp; \vdots \\ 0 &amp; 0 &amp; \cdots &amp; 1 \end{pmatrix}\)</span> is the <span class="math notranslate nohighlight">\(n \times n\)</span> <em>identity matrix</em> (with elements <span class="math notranslate nohighlight">\(1\)</span> on the main diagonal and <span class="math notranslate nohighlight">\(0\)</span> otherwise).</p>
-<p>In this case we call the matrix <span class="math notranslate nohighlight">\(B\)</span> the <em>inverse</em> of <span class="math notranslate nohighlight">\(A\)</span> and denote <span class="math notranslate nohighlight">\(B=A^{-1}\)</span>. If <span class="math notranslate nohighlight">\(A\)</span> has an inverse than it is called <em>invertible</em> or <em>non-singular</em>. If it does not have an inverse, <span class="math notranslate nohighlight">\(A\)</span> is <em>singular</em>.</p>
-<p>Finding the inverse of <span class="math notranslate nohighlight">\(A\)</span> automatically gives us the solution to the system of linear equations <span class="math notranslate nohighlight">\(A\mathbf x = \mathbf x\)</span></p>
-<div class="math notranslate nohighlight">
-\[\begin{split}\begin{align}  
-A\mathbf{x}&amp;=\mathbf{b} \\
-A^{-1} A\mathbf{x} &amp;= A^{-1}\mathbf{b} \\
-\mathbf{x} &amp;= A^{-1}\mathbf{b} \end{align}\end{split}\]</div>
-<p>Finding ways of inverting matrices is therefore needed for a wide range of applications. A whole research field is dealing with optimising matrix inversions.</p>
-<div class="admonition-example admonition">
-<p class="admonition-title">Example</p>
-<p>To find the inverse of matrix <span class="math notranslate nohighlight">\(A= \begin{pmatrix} 2 &amp; 5  \\ 1 &amp; 3  \end{pmatrix}\)</span>, we have to find the matrix <span class="math notranslate nohighlight">\(B = \begin{pmatrix} b_{11} &amp; b_{12}  \\ b_{21} &amp; b_{22}  \end{pmatrix}\)</span> so that</p>
-<div class="math notranslate nohighlight">
-\[ AB = I \quad (\mbox{then check that }\ BA = I.\]</div>
-<p>Thus</p>
-<div class="math notranslate nohighlight">
-\[\begin{split}\begin{align}  
-  \begin{pmatrix} 2 &amp; 5  \\ 1 &amp; 3  \end{pmatrix}\begin{pmatrix} b_{11} &amp; b_{12}  \\ b_{21} &amp; b_{22}  \end{pmatrix} = \begin{pmatrix} 1 &amp; 0  \\ 0 &amp; 1  \end{pmatrix} \\  
-  \begin{pmatrix} 2b_{11} + 5b_{21}  &amp; 2b_{12} + 5b_{22} \\ b_{11} + 3b_{21} &amp; b_{12} + 3b_{22}  \end{pmatrix} = \begin{pmatrix} 1 &amp; 0  \\ 0 &amp; 1  \end{pmatrix}, \\
-\end{align}\end{split}\]</div>
-<p>which leads to two systems of equations:</p>
-<div class="math notranslate nohighlight">
-\[\begin{split}\begin{cases} 2b_{11} + 5b_{21} &amp;= 1 \\ b_{11} + 3b_{21} &amp;= 0 \end{cases}\quad \mbox{and}\quad \begin{cases} 2b_{12} + 5b_{22} &amp;= 0 \\ b_{12} + 3b_{22} &amp;= 1 \end{cases}\end{split}\]</div>
-<p>The solutions of these two systems, leads to <span class="math notranslate nohighlight">\(B= \begin{pmatrix} 3 &amp; -5  \\ -1 &amp; 2  \end{pmatrix}\)</span></p>
-</div>
-</section>
-<section id="determinant">
-<h2>Determinant<a class="headerlink" href="#determinant" title="Link to this heading">#</a></h2>
-<p>If <span class="math notranslate nohighlight">\(A = \begin{pmatrix} a_{11} &amp; a_{12}  \\ a_{21} &amp; a_{22}  \end{pmatrix}\)</span> then the expression:</p>
-<div class="math notranslate nohighlight">
-\[\mbox{det}A = a_{11}a_{22} - a_{21}a_{12}\]</div>
-<p>given by the product of the elements on the main diagonal minus the product of the elements on secondary diagonal. The term <span class="math notranslate nohighlight">\(\mbox{det}A\)</span> is called the <em>determinant</em> of <span class="math notranslate nohighlight">\(A\)</span>. The determinant can be calculated for matrices of any size, but the expression gets increasingly more complicated for larger matrices.</p>
-<p>The determinant establishes a clear criterium for the existence of <span class="math notranslate nohighlight">\(A^{-1}\)</span>. The matrix <span class="math notranslate nohighlight">\(A\)</span> will be invertible if and only if <span class="math notranslate nohighlight">\(\mbox{det}A \ne 0\)</span>.</p>
-<div class="admonition note">
-<p class="admonition-title">Note</p>
-<p>The previous criterium has an important implication. Consider the system of linear equations defined by <span class="math notranslate nohighlight">\(A\mathbf{x}=\mathbf{b}\)</span> with equal number of equations and variables. If <span class="math notranslate nohighlight">\(\mbox{det}A \ne 0\)</span> then the inverse of <span class="math notranslate nohighlight">\(A\)</span> exists. Thus, if we multiply <span class="math notranslate nohighlight">\(A^{-1}\)</span> to the left of each side of the equation defining the linear system, we have:</p>
-<div class="math notranslate nohighlight">
-\[\begin{split}\begin{align}  
-A\mathbf{x}&amp;=\mathbf{b} \\
-A^{-1}(A\mathbf{x}) &amp;= A^{-1}\mathbf{b} \\
-(A^{-1}A)\mathbf{x} &amp;= A^{-1}\mathbf{b} \\
-\mathbf{x} &amp;= A^{-1}\mathbf{b} \end{align}\end{split}\]</div>
-<p>which means that if <span class="math notranslate nohighlight">\(\mbox{det}A \ne 0\)</span> the system will have a <strong>single solution</strong> with the variables assuming the values <span class="math notranslate nohighlight">\(\mathbf{x} = A^{-1}\mathbf{b}\)</span>.</p>
-</div>
-</section>
-<section id="structured-models">
-<h2>Structured models<a class="headerlink" href="#structured-models" title="Link to this heading">#</a></h2>
-<p>In the models we have seen until now, all the individuals in a given population are identical. However, real populations have importance differences between individuals according to their life stages. For example, some insects have stages of eggs, larvae, pupae, and adults, while many plants pass through stages of seeds, seedlings, saplings, and mature individuals. Thus, understanding the differences in the dynamics of each life stage is important to add realism to our models (especially in applied models, such as pest control or harvesting management).</p>
-<p>Suppose our population is composed by two classes: <em>immature</em> and <em>mature</em> individuals. Let us assume non-overlapping breeding seasons and that the timescales for reproduction and maturation are similar. Thus, time can be counted in discrete steps corresponding to this reproduction/maturation interval. At time <span class="math notranslate nohighlight">\(t\)</span> we have <span class="math notranslate nohighlight">\(X_t\)</span> immature individuals and <span class="math notranslate nohighlight">\(Y_t\)</span> mature individuals. Immature individuals survive until maturity with a probability <span class="math notranslate nohighlight">\(p\)</span>, while mature individuals reproduce and generate, on average, <span class="math notranslate nohighlight">\(m\)</span> immature individuals for the next generation. Mature individuals die after reproduction. With these assumptions, we can calculate the sizes of each population class on the step <span class="math notranslate nohighlight">\(t+1\)</span> as:</p>
-<div class="math notranslate nohighlight">
-\[\begin{split}\begin{align}
-X_{t+1} = m Y_t \\
-Y_{t+1} = p X_t \end{align}\end{split}\]</div>
-<p>See that now we have a model with two equations describing the variable class sizes in our population. Moreover, the dynamics of each class depends on the other, so that we have a system of <em>coupled</em> difference equations.</p>
-<p>Interestingly, we can also write this system in matrix form, as:</p>
-<div class="math notranslate nohighlight">
-\[\begin{split}\begin{pmatrix} X_{t+1}  \\ Y_{t+1} \end{pmatrix} = \begin{pmatrix} 0 &amp; m  \\ p &amp; 0  \end{pmatrix} \begin{pmatrix} X_{t}  \\ Y_{t} \end{pmatrix}, \quad \mbox{or}\end{split}\]</div>
-<div class="math notranslate nohighlight">
-\[\mathbf{n_{t+1}} = P\mathbf{n_{t}}\]</div>
-<p>where the sum of the components of the population vector <span class="math notranslate nohighlight">\(\mathbf{n_{t}}\)</span>, gives the total population (immature plus mature individuals) at time <span class="math notranslate nohighlight">\(t\)</span>: <span class="math notranslate nohighlight">\(N_t = X_t + Y_t\)</span>. The matrix <span class="math notranslate nohighlight">\(P=\begin{pmatrix} 0 &amp; m  \\ p &amp; 0  \end{pmatrix}\)</span> is called <em>projection matrix</em> or <em>Leslie matrix</em>. Multiplying matrix <span class="math notranslate nohighlight">\(P\)</span> by the population vector at time <span class="math notranslate nohighlight">\(t\)</span>, <span class="math notranslate nohighlight">\(\mathbf{n_{t}}\)</span>, we can project the next population size at time <span class="math notranslate nohighlight">\(t+1\)</span>. Thus, from an initial population state, say <span class="math notranslate nohighlight">\(\mathbf{n_{0}}\)</span>, the projection matrix governs the dynamics of the model.</p>
-<br />
-<p>This model can be easily generalised for a larger number of age classes. Suppose we now have <span class="math notranslate nohighlight">\(q+1\)</span> classes, <span class="math notranslate nohighlight">\(X_0, X_1, X_2, \ldots, X_q\)</span>, from the youngest (meaningful) age <span class="math notranslate nohighlight">\(X_0\)</span> until the oldest age <span class="math notranslate nohighlight">\(X_q\)</span>. At each time step <span class="math notranslate nohighlight">\(t\)</span>, individuals from class <span class="math notranslate nohighlight">\(X_i\)</span> have a chance of surviving to the next age class <span class="math notranslate nohighlight">\(X_{i+1}\)</span>, with probability <span class="math notranslate nohighlight">\(p_{i+1\ i}\)</span> (imagine an arrow for direction of age class progession <span class="math notranslate nohighlight">\(p_{i+1\leftarrow i}\)</span>. We are going to omit the arrow for simplicity). Additionally, suppose that each age class <span class="math notranslate nohighlight">\(X_i\)</span> excluding the youngest, will, in principle, be able to reproduce, with average fecundity <span class="math notranslate nohighlight">\(m_i\)</span> generating new individuals for class <span class="math notranslate nohighlight">\(X_0\)</span>. Thus, with the age class sizes at time <span class="math notranslate nohighlight">\(t\)</span>, and the parameters for survival probabilities and average fecundities, we have:</p>
-<div class="math notranslate nohighlight">
-\[\begin{split}\begin{align}
-X_0(t+1) &amp;= m_1X_1(t) +  m_2X_2(t) + m_3X_3(t) + \ldots + m_qX_q(t) \\
-X_1(t+1) &amp;= p_{10}X_0(t) \\
-X_2(t+1) &amp;= p_{21}X_1(t) \\
-X_3(t+1) &amp;= p_{32}X_2(t) \\
-         &amp;\vdots \\
-X_q(t+1) &amp;= p_{q\ q-1}X_{q-1}(t)
-    \end{align}\end{split}\]</div>
-<p>This can also be written in matrix form as:</p>
-<div class="math notranslate nohighlight">
-\[\begin{split}\begin{pmatrix} X_0(t+1)  \\ X_1(t+1) \\X_2(t+1)  \\X_3(t+1) \\ \vdots \\X_q(t+1)  \\ \end{pmatrix} = \begin{pmatrix} 0 &amp; m_1 &amp; m_2 &amp; m_3 &amp; \cdots &amp; m_q \\ p_{10} &amp; 0 &amp; 0 &amp; 0 &amp; \cdots &amp;0 \\ 0 &amp; p_{21} &amp; 0 &amp; 0 &amp; \cdots &amp; 0 \\ 0 &amp; 0 &amp; p_{32} &amp; 0 &amp; \cdots &amp; 0 \\ \vdots &amp; \vdots &amp; \vdots &amp; \vdots &amp; \ddots &amp; \vdots \\ 0 &amp; 0 &amp; 0 &amp; \cdots &amp; p_{q\ q-1} &amp; 0\end{pmatrix} \begin{pmatrix} X_0(t)  \\ X_1(t) \\X_2(t) \\X_3(t) \\ \vdots \\X_q(t)  \\ \end{pmatrix}\end{split}\]</div>
-<div class="math notranslate nohighlight">
-\[\mathbf{n_{t+1}} = P\mathbf{n_{t}}\]</div>
-<p>The generalised projection matrix <span class="math notranslate nohighlight">\(P\)</span> has then all the surviving probabilities just below the main diagonal, and the fecundities for each class in its <span class="math notranslate nohighlight">\(0\)</span>-th row.</p>
-<div class="admonition note">
-<p class="admonition-title">Note</p>
-<p>Two important differences here. First, the rows and columns for the matrix are indexed starting from <span class="math notranslate nohighlight">\(0\)</span> until <span class="math notranslate nohighlight">\(q\)</span>, not starting from <span class="math notranslate nohighlight">\(1\)</span> as usual. Second, the time steps are written in parentheses for each age class, instead of a subscript as we have seen before for other population models, so that it does not conflict with the index for the age classes (going from <span class="math notranslate nohighlight">\(0\)</span> to <span class="math notranslate nohighlight">\(q\)</span>).</p>
-</div>
-<div class="admonition-example admonition">
-<p class="admonition-title">Example</p>
-<p>Consider a population with three age classes <span class="math notranslate nohighlight">\(X_0, X_1,\mbox{ and}\ X_2\)</span>, and the following projection matrix</p>
-<div class="math notranslate nohighlight">
-\[\begin{split}P=\begin{pmatrix} 0 &amp; 5 &amp; 10\\ 0.5 &amp; 0 &amp; 0  \\ 0 &amp; 0.2 &amp; 0 \end{pmatrix}\end{split}\]</div>
-<p>If the population starts with <span class="math notranslate nohighlight">\(10\)</span> individuals in the oldest category at time <span class="math notranslate nohighlight">\(t=0\)</span>, the population vector at time <span class="math notranslate nohighlight">\(t=1\)</span> will be:</p>
-<div class="math notranslate nohighlight">
-\[\begin{split}\mathbf{n_{1}} = P\mathbf{n_{0}} \longrightarrow \begin{pmatrix} X_0(1)  \\ X_1(1) \\X_2(1) \end{pmatrix}=\begin{pmatrix} 0 &amp; 5 &amp; 10\\ 0.5 &amp; 0 &amp; 0  \\ 0 &amp; 0.2 &amp; 0 \end{pmatrix}\begin{pmatrix} 0  \\ 0 \\ 10 \end{pmatrix} \longrightarrow \begin{pmatrix} 100  \\ 0 \\ 0 \end{pmatrix}\end{split}\]</div>
-<p>The population age distribution at each time step will then be given be the successive application of the matrix <span class="math notranslate nohighlight">\(P\)</span> to the previous population distribution. For the first 10 time steps we have:</p>
-<div class="math notranslate nohighlight">
-\[\begin{split}\begin{pmatrix}0 \\ 0 \\ 10 \end{pmatrix} \rightarrow \begin{pmatrix} 100  \\ 0 \\ 0 \end{pmatrix} \rightarrow \begin{pmatrix} 0  \\ 50 \\ 0 \end{pmatrix} \rightarrow \begin{pmatrix} 250  \\ 0 \\ 10 \end{pmatrix} \rightarrow \begin{pmatrix} 100  \\ 125 \\ 0 \end{pmatrix} \rightarrow \begin{pmatrix} 625  \\ 50 \\ 25 \end{pmatrix} \rightarrow \begin{pmatrix} 500  \\ 312 \\ 10 \end{pmatrix} \rightarrow \begin{pmatrix} 1662  \\ 250 \\ 62 \end{pmatrix} \rightarrow \begin{pmatrix} 1875  \\ 831 \\ 50 \end{pmatrix} \rightarrow \begin{pmatrix} 4656  \\ 937 \\ 166 \end{pmatrix}\end{split}\]</div>
-<p>Two important things are worth noting here. First, the total population size (sum accros all age classes), follows the sequence <span class="math notranslate nohighlight">\(N_t = \left\{ 10,100,50,260,225,700,822,1975,2756,5759,\ldots \right \}\)</span>. The reproduction process does not always lead to increasing population sizes in the beginning, but is subject to size fluctuations due to the demographic distribution of this population. After a few steps increases in population size become more consistent. Second, as the population size continues to increase, the population age distribution tends to a given proportion of the population classes given by: <span class="math notranslate nohighlight">\(X_0 \approx 76\%, X_1 \approx 22\%, \mbox{and}\ X_2 \approx 2\%\)</span>.</p>
-<p>Now consider that, instead of <span class="math notranslate nohighlight">\(10\)</span> individuals in the oldest class, we initiate the population with the following distribution <span class="math notranslate nohighlight">\(\mathbf{n_{0}} = \begin{pmatrix} 7.6  \\ 2.2 \\ 0.2 \end{pmatrix}\)</span>. The population progression by successive application of the projection matrix will be</p>
-<div class="math notranslate nohighlight">
-\[\begin{split}\begin{pmatrix} 7.6  \\ 2.2 \\ 0.2 \end{pmatrix} \longrightarrow \begin{pmatrix} 13  \\ 4 \\ 0.4 \end{pmatrix} \longrightarrow \begin{pmatrix} 23  \\ 6 \\ 0.76 \end{pmatrix} \longrightarrow \begin{pmatrix} 40  \\ 12 \\ 1 \end{pmatrix} \longrightarrow \begin{pmatrix} 72  \\ 20 \\ 2 \end{pmatrix} \longrightarrow \begin{pmatrix} 124  \\ 36 \\ 4 \end{pmatrix} \longrightarrow \begin{pmatrix} 219  \\ 62 \\ 7 \end{pmatrix} \longrightarrow \begin{pmatrix} 381  \\ 109 \\ 12 \end{pmatrix} \end{split}\]</div>
-<p>You can check that although each age cathegory is increasing, the percentage contribution of each class to the total population remains nearly constant. To this distribution of classes we give the name of <strong>stable age distribution</strong>.</p>
-<p>Moreover, at each time step, each age class increases with approximately the same factor in relation to the previous step, around <span class="math notranslate nohighlight">\(\lambda = 1.75\)</span>. If for each age class <span class="math notranslate nohighlight">\(X_i(t+1) = \lambda X_i(t)\)</span> is valid,then we have:</p>
-<div class="math notranslate nohighlight">
-\[\mathbf{n_{t+1}}=\lambda\mathbf{n_{t}},\]</div>
-<p>so that we recover the usual model for exponential growth for the total population size <span class="math notranslate nohighlight">\(N_{t+1} = \lambda N_t\)</span>.</p>
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-<h1>05 - From table arrangements to powerful tools - part 2<a class="headerlink" href="#from-table-arrangements-to-powerful-tools-part-2" title="Link to this heading">#</a></h1>
-<section id="linear-transformations">
-<h2>Linear transformations<a class="headerlink" href="#linear-transformations" title="Link to this heading">#</a></h2>
-<p>In general terms, a given matrix <span class="math notranslate nohighlight">\(A\)</span> is a representation of what we call a <em>linear transformation</em>, a function from a special kind of set called <em>vector space</em> into another vector space. Vector spaces are generalised sets with particular properties, but for the moment you can think of two-dimensional and three-dimensional cartesian spaces as usual vector spaces. Then vectors are gonna be represented as arrays of real numbers as before: <span class="math notranslate nohighlight">\(\mathbf{n_{0}} = \begin{pmatrix} 7.6  \\ 2.2  \end{pmatrix}\)</span> or <span class="math notranslate nohighlight">\(\mathbf{v} = \begin{pmatrix} x \\ y \\ z \end{pmatrix}\)</span>.</p>
-<p>Back to linear transformations, they are usually denoted as:</p>
-<div class="math notranslate nohighlight">
-\[\begin{split}\begin{align}
-A:V &amp;\longrightarrow W \\
-  \mathbf{v} &amp;\rightarrow A(\mathbf{v}) \end{align}\end{split}\]</div>
-<p>where <span class="math notranslate nohighlight">\(V\)</span> and <span class="math notranslate nohighlight">\(W\)</span> are the domain and codomain vector spaces, and the transformation <span class="math notranslate nohighlight">\(A\)</span> is a function that projects vectors <span class="math notranslate nohighlight">\(\mathbf{v}\)</span> into the images <span class="math notranslate nohighlight">\(A(\mathbf{v})\)</span>. The property that defines linear transformations is the following:</p>
-<div class="math notranslate nohighlight">
-\[A(a\mathbf{v}+b\mathbf{w})=aA(\mathbf{v})+bA(\mathbf{w}),\]</div>
-<p>where <span class="math notranslate nohighlight">\(a\)</span> and <span class="math notranslate nohighlight">\(b\)</span> are constants. In other words,for a linear transformation, the image of the transformation <span class="math notranslate nohighlight">\(A\)</span> of a linear combination of two vectors (<span class="math notranslate nohighlight">\(\mathbf{v}\)</span> and <span class="math notranslate nohighlight">\(\mathbf{w}\)</span>) is equal to the linear combination of the images of this vectors by <span class="math notranslate nohighlight">\(A\)</span>.</p>
-<p>If we represent vectors with their components in a cartesian system, the images of the transformation <span class="math notranslate nohighlight">\(A\)</span> will be given be the multiplication of the matrix representation of <span class="math notranslate nohighlight">\(A\)</span> on the vectors.</p>
-<div class="admonition-example admonition">
-<p class="admonition-title">Example</p>
-<p>Given the matrix <span class="math notranslate nohighlight">\(A = \begin{pmatrix} 1 &amp; 2 \\ 3 &amp; 2 \end{pmatrix}\)</span> and the vectors <span class="math notranslate nohighlight">\(\mathbf{v_1} = \begin{pmatrix} 3  \\ -1  \end{pmatrix}\)</span> and <span class="math notranslate nohighlight">\(\mathbf{v_2} = \begin{pmatrix} 2  \\ 3  \end{pmatrix}\)</span>, then we have:</p>
-<div class="math notranslate nohighlight">
-\[\begin{split}\begin{align} A\mathbf{v_1} &amp;= \begin{pmatrix} 1 &amp; 2 \\ 3 &amp; 2 \end{pmatrix}\begin{pmatrix} 3  \\ -1  \end{pmatrix} = \begin{pmatrix} 1  \\ 7  \end{pmatrix} \\  
-  A\mathbf{v_2} &amp;= \begin{pmatrix} 1 &amp; 2 \\ 3 &amp; 2 \end{pmatrix}\begin{pmatrix} 2  \\ 3  \end{pmatrix} = \begin{pmatrix} 8  \\ 12  \end{pmatrix} = 4\begin{pmatrix} 2  \\ 3  \end{pmatrix} \end{align}\end{split}\]</div>
-<p>The vectors and their respective transformations are given by:</p>
-<img alt="../../_images/lin_trans.png" src="../../_images/lin_trans.png" />
-<p>We see that under the transformation <span class="math notranslate nohighlight">\(A\)</span>, vector <span class="math notranslate nohighlight">\(\mathbf{v_1}\)</span> gets rotated (counterclockwise) and stretched, while vector <span class="math notranslate nohighlight">\(\mathbf{v_2}\)</span> only gets stretched without changing the direction.</p>
-</div>
-</section>
-<section id="eigenvalues-and-eigenvectors">
-<h2>Eigenvalues and eigenvectors<a class="headerlink" href="#eigenvalues-and-eigenvectors" title="Link to this heading">#</a></h2>
-<p>As seen on the previous example, there might be some vectors that upon application of <span class="math notranslate nohighlight">\(A\)</span> do not have their direction changed, only their magnitude (or <em>module</em> of the vector). For these vectors, we have, in general:</p>
-<div class="math notranslate nohighlight">
-\[A\mathbf{v} = \lambda\mathbf{v}, \quad \mbox{for some constant }\ \lambda\]</div>
-<p>But this is equivalent to:</p>
-<div class="math notranslate nohighlight">
-\[\begin{split}\begin{align}
-A\mathbf{v} - \lambda\mathbf{v} = 0 \\  
-A\mathbf{v} - \lambda I\mathbf{v} = 0 \\
-\left(A - \lambda I \right)\mathbf{v} = 0 \end{align},\end{split}\]</div>
-<p>where <span class="math notranslate nohighlight">\(I\)</span> is the identity matrix. The equation <span class="math notranslate nohighlight">\(\left(A - \lambda I \right)\mathbf{v} = 0\)</span> defines a system of linear equations where <span class="math notranslate nohighlight">\(A - \lambda I\)</span> is the coefficient matrix, <span class="math notranslate nohighlight">\(\mathbf{v}\)</span> is the vector of variables, and <span class="math notranslate nohighlight">\(\mathbf{b}=0\)</span>. See that since <span class="math notranslate nohighlight">\(\mathbf{b}=0\)</span> we already know one possible solution for this system, which is <span class="math notranslate nohighlight">\(\mathbf{v}=0\)</span> (all the variables equal to <span class="math notranslate nohighlight">\(0\)</span>). Therefore, either this system has an unique solution (the null solution) or it has infinite solutions, but definitely it is not inconsistent.  Thus, if we want to know all possible non-zero vectors <span class="math notranslate nohighlight">\(\mathbf{v}\)</span> that satisfy <span class="math notranslate nohighlight">\(A\mathbf{v} = \lambda\mathbf{v}\)</span>, we have to impose that the system assumes infinite solutions. This criterium should be defined by the determinant of the matrix of coefficients, as:</p>
-<div class="math notranslate nohighlight">
-\[\mbox{det}(A-\lambda I) = 0.\]</div>
-<p>The determinant then defines a polynomial in the variable <span class="math notranslate nohighlight">\(\lambda\)</span>, <span class="math notranslate nohighlight">\(\mbox{det}(A-\lambda I)=p(\lambda)\)</span>, called <em>characteristic polynomial</em>. Since we are interested in the condition <span class="math notranslate nohighlight">\(p(\lambda)=0\)</span>, what we want to know are the roots <span class="math notranslate nohighlight">\(\lambda\)</span> of this polynomial.</p>
-<p>The solutions <span class="math notranslate nohighlight">\(\lambda\)</span> for these equations are called <strong>eigenvalues</strong> and their corresponding vectors are <strong>eigenvectors</strong>. In summary, the eigenvectors are all the vectors that, upon application of <span class="math notranslate nohighlight">\(A\)</span>, remain on the same direction, only changing magnitude, with the eigenvalue being the stretching/compressing factor.</p>
-<div class="admonition-example admonition">
-<p class="admonition-title">Example</p>
-<p>Let us calculate the eigenvalues and eigenvectors of <span class="math notranslate nohighlight">\(A = \begin{pmatrix} 1 &amp; 2 \\ 3 &amp; 2 \end{pmatrix}\)</span>. First we need to obtain matrix <span class="math notranslate nohighlight">\((A-\lambda I)\)</span> and to calculate its determinant. We have:</p>
-<div class="math notranslate nohighlight">
-\[\begin{split}A-\lambda I = \begin{pmatrix} 1 &amp; 2 \\ 3 &amp; 2 \end{pmatrix} - \lambda \begin{pmatrix} 1 &amp; 0 \\ 0 &amp; 1 \end{pmatrix} = \begin{pmatrix} 1 -\lambda &amp; 2 \\ 3 &amp; 2 -\lambda  \end{pmatrix}\end{split}\]</div>
-<p>The characteristic polynomial will be given by <span class="math notranslate nohighlight">\(p(\lambda)=\mbox{det}(A-\lambda I)\)</span>, and applying the rule for the calculus of determinants of matrices size 2, we have:</p>
-<div class="math notranslate nohighlight">
-\[\begin{split}p(\lambda)=\mbox{det}(A-\lambda I) = \mbox{det}\begin{pmatrix} 1 -\lambda &amp; 2 \\ 3 &amp; 2 -\lambda  \end{pmatrix} = (1 -\lambda)(2 -\lambda) - 3\cdot 2 \end{split}\]</div>
-<div class="math notranslate nohighlight">
-\[p(\lambda) = 2 -\lambda -2\lambda +\lambda^2 - 6 = \lambda^2 - 3\lambda -4.\]</div>
-<p>Thus, calculating the roots of this 2nd-degree polynomial, we find <span class="math notranslate nohighlight">\(\lambda_1=-1\)</span> and <span class="math notranslate nohighlight">\(\lambda_2=4\)</span>. For each of the eigenvalues, to obtain the eigenvectors we have to solve the linear system  <span class="math notranslate nohighlight">\(A\mathbf{v} = \lambda\mathbf{v}\)</span> for some general vector <span class="math notranslate nohighlight">\(\mathbf{v} = \begin{pmatrix} x  \\ y  \end{pmatrix}\)</span> (with coordinates <span class="math notranslate nohighlight">\(x\)</span> and <span class="math notranslate nohighlight">\(y\)</span> to be specified). Thus:</p>
-<ul>
-<li><p><span class="math notranslate nohighlight">\(\mathbf{\lambda_1 = -1}\)</span>:</p>
-<p><span class="math notranslate nohighlight">\(\begin{align}
-A\mathbf{v} = \lambda\mathbf{v} \longrightarrow \begin{pmatrix} 1 &amp; 2 \\ 3 &amp; 2 \end{pmatrix}\begin{pmatrix} x  \\ y \end{pmatrix}= -1\begin{pmatrix} x  \\ y \end{pmatrix}\longrightarrow \cases{x + 2y = -x \\ 3x + 2y = -y} \longrightarrow \cases{2x + 2y = 0 \\ 3x + 3y = 0}
-\end{align}\)</span></p>
-<p>Note that both equations correspond to the same linear equation <span class="math notranslate nohighlight">\(x + y = 0\)</span>, which is exactly what we would expect since we built the system to be underdetermined when we did <span class="math notranslate nohighlight">\(\mbox{det}(A-\lambda I) = 0\)</span>. The eigenvector corresponding to <span class="math notranslate nohighlight">\(\lambda_1\)</span> will be such that <span class="math notranslate nohighlight">\(x + y = 0\)</span> or <span class="math notranslate nohighlight">\(x = -y\)</span>. Thus, we can choose <span class="math notranslate nohighlight">\(\mathbf{v_1} = \begin{pmatrix} 1  \\ -1  \end{pmatrix}\)</span>.</p>
-<div class="admonition note">
-<p class="admonition-title">Note</p>
-<p>Remember that infinite solutions are possible for the same system. The important is that the condition <span class="math notranslate nohighlight">\(x = -y\)</span> holds. In fact, multiplying a non-zero constant for the vector <span class="math notranslate nohighlight">\(\begin{pmatrix} 1  \\ -1  \end{pmatrix}\)</span> would still give a vector in the same direction. So if <span class="math notranslate nohighlight">\(\mathbf{v_1}\)</span> is an eigenvector, <span class="math notranslate nohighlight">\(k\mathbf{v_1}\ (k \in {\rm I\!R})\)</span> also is.</p>
-</div>
-  <br />  
-</li>
-<li><p><span class="math notranslate nohighlight">\(\mathbf{\lambda_1 = 4}\)</span>:</p>
-<p><span class="math notranslate nohighlight">\(\begin{align}
-A\mathbf{v} = \lambda\mathbf{v} \longrightarrow \begin{pmatrix} 1 &amp; 2 \\ 3 &amp; 2 \end{pmatrix}\begin{pmatrix} x  \\ y \end{pmatrix}= 4\begin{pmatrix} x  \\ y \end{pmatrix}\longrightarrow \cases{x + 2y = 4x \\ 3x + 2y = 4y} \longrightarrow \cases{-3x + 2y = 0 \\ 3x - 2y = 0}
-\end{align}\)</span></p>
-<p>Since the eigenvector needs to satisfy <span class="math notranslate nohighlight">\(3x=2y\)</span>, we choose <span class="math notranslate nohighlight">\(\mathbf{v_2} = \begin{pmatrix} 2/3  \\ 1  \end{pmatrix}\)</span>.</p>
-<p>The representation of these vectors on the cartesian plane will be:</p>
-</li>
-</ul>
-<img alt="../../_images/eigenvec_01.png" src="../../_images/eigenvec_01.png" />
-</div>
-<div class="admonition-example admonition">
-<p class="admonition-title">Example</p>
-<p>Let us calculate the eigenvalues and eigenvectors of matrix <span class="math notranslate nohighlight">\(A = \begin{pmatrix} -2 &amp; 1 \\ 0 &amp; -1 \end{pmatrix}\)</span>. We have:</p>
-<div class="math notranslate nohighlight">
-\[\begin{split}p(\lambda)=\mbox{det}(A-\lambda I) = \mbox{det}\begin{pmatrix} -2 -\lambda &amp; 1 \\ 0 &amp; -1 -\lambda  \end{pmatrix} = (-2 -\lambda)(-1 -\lambda) - 1\cdot 0 \end{split}\]</div>
-<div class="math notranslate nohighlight">
-\[p(\lambda)=(-2 -\lambda)(-1 -\lambda)\]</div>
-<p>Since we need <span class="math notranslate nohighlight">\(p(\lambda)=0\)</span> either one of the two factors has to be zero. So, <span class="math notranslate nohighlight">\(\lambda_1=-2\)</span> and <span class="math notranslate nohighlight">\(\lambda_2=-1\)</span>.<br />
-For the eigenvectors:</p>
-<ul>
-<li><p><span class="math notranslate nohighlight">\(\mathbf{\lambda_1 = -2}\)</span>:</p>
-<p><span class="math notranslate nohighlight">\(\begin{align}
-A\mathbf{v} = \lambda\mathbf{v} \longrightarrow \begin{pmatrix} -2 &amp; 1 \\ 0 &amp; -1 \end{pmatrix}\begin{pmatrix} x  \\ y \end{pmatrix}= -2\begin{pmatrix} x  \\ y \end{pmatrix}\longrightarrow \cases{-2x + y = -2x \\ -y = -2y} \longrightarrow \cases{y = 0 \\ y = 0}
-\end{align}\)</span></p>
-<p>Since <span class="math notranslate nohighlight">\(y=0\)</span>, we choose <span class="math notranslate nohighlight">\(\mathbf{v_1} = \begin{pmatrix} 1  \\ 0  \end{pmatrix}\)</span>.</p>
-</li>
-<li><p><span class="math notranslate nohighlight">\(\mathbf{\lambda_2 = -1}\)</span>:</p>
-<p><span class="math notranslate nohighlight">\(\begin{align}
-A\mathbf{v} = \lambda\mathbf{v} \longrightarrow \begin{pmatrix} -2 &amp; 1 \\ 0 &amp; -1 \end{pmatrix}\begin{pmatrix} x  \\ y \end{pmatrix}= -1\begin{pmatrix} x  \\ y \end{pmatrix}\longrightarrow \cases{-2x + y = -x \\ -y = -y} \longrightarrow \cases{-x + y = 0 \\ y = y}
-\end{align}\)</span></p>
-<p>The second equation is trivial and provides no information. From the first equation, since <span class="math notranslate nohighlight">\(x=y\)</span>, we choose <span class="math notranslate nohighlight">\(\mathbf{v_2} = \begin{pmatrix} 1  \\ 1  \end{pmatrix}\)</span>.</p>
-</li>
-</ul>
-<div class="admonition tip">
-<p class="admonition-title">Tip</p>
-<p>Note that the two eigenvalues <span class="math notranslate nohighlight">\(\lambda_1 = -2\)</span> and <span class="math notranslate nohighlight">\(\lambda_2 = -1\)</span> correspond to the elements of the main diagonal of the matrix <span class="math notranslate nohighlight">\(A\)</span>. Whenever the matrix <span class="math notranslate nohighlight">\(A\)</span> is <em>triangular</em>, meaning all the elements <em>below</em> the main diagonal are <span class="math notranslate nohighlight">\(0\)</span>, the eigenvalues will correspond to the elements of the main diagonal.</p>
-</div>
-</div>
-</section>
-<section id="vector-decomposition">
-<h2>Vector decomposition<a class="headerlink" href="#vector-decomposition" title="Link to this heading">#</a></h2>
-<p>Now suppose matrix <span class="math notranslate nohighlight">\(A\)</span> has eigenvalues <span class="math notranslate nohighlight">\(\lambda_1\)</span> and <span class="math notranslate nohighlight">\(\lambda_2\)</span>, with corresponding eigenvectors <span class="math notranslate nohighlight">\(\mathbf{u_1}\)</span> and <span class="math notranslate nohighlight">\(\mathbf{u_2}\)</span>. If <span class="math notranslate nohighlight">\(\lambda_1 \ne \lambda_2\)</span>, then the vectors <span class="math notranslate nohighlight">\(\mathbf{u_1}\)</span> and <span class="math notranslate nohighlight">\(\mathbf{u_2}\)</span> are <em>linearly independent</em> meaning one can not be written as a multiple (multiplied by a real constant) of the other. Geometrically, this means that the directions corresponding to this two eigenvectors are not on the same line.</p>
-<p>In two dimensions, this is enough to say that any given vector <span class="math notranslate nohighlight">\(\mathbf{v}\)</span> can be written as a linear combination of <span class="math notranslate nohighlight">\(\mathbf{u_1}\)</span> and <span class="math notranslate nohighlight">\(\mathbf{u_2}\)</span> (or that <span class="math notranslate nohighlight">\(\mathbf{u_1}\)</span> and <span class="math notranslate nohighlight">\(\mathbf{u_2}\)</span> consitute a <em>basis</em> of this space). We have:</p>
-<div class="math notranslate nohighlight">
-\[\mathbf{v}= a_1\mathbf{u_1} + a_2\mathbf{u_2},\]</div>
-<p>for constants <span class="math notranslate nohighlight">\(a_1\)</span> and <span class="math notranslate nohighlight">\(a_2\)</span> to be determined.</p>
-<p>But given that <span class="math notranslate nohighlight">\(A\)</span> is a linear transformation, it should hold that:</p>
-<div class="math notranslate nohighlight">
-\[A\mathbf{v} = A(a_1\mathbf{u_1} + a_2\mathbf{u_2}) = a_1A(\mathbf{u_1}) + a_2A(\mathbf{u_2}) = a_1\lambda_1\mathbf{u_1} + a_2\lambda_2\mathbf{u_2},\]</div>
-<p>with the last step coming from the fact that <span class="math notranslate nohighlight">\(\mathbf{u_1}\)</span> and <span class="math notranslate nohighlight">\(\mathbf{u_2}\)</span> are eigenvectors of <span class="math notranslate nohighlight">\(A\)</span>. Since <span class="math notranslate nohighlight">\(A\mathbf{v}\)</span> is also a vector in this space, we can go ahead and apply the transformation <span class="math notranslate nohighlight">\(A\)</span> again, and we will have:</p>
-<div class="math notranslate nohighlight">
-\[A(A\mathbf{v}) = A(a_1\lambda_1\mathbf{u_1} + a_2\lambda_2\mathbf{u_2})=a_1\lambda_1 A(\mathbf{u_1}) + a_2\lambda_2 A(\mathbf{u_2}) = a_1\lambda_1^2\mathbf{u_1} + a_2\lambda_2^2\mathbf{u_2}.\]</div>
-<p>So, in general:</p>
-<div class="math notranslate nohighlight">
-\[A(A(A \cdots A\mathbf{v}) \cdots )) = A^n\mathbf{v} = a_1\lambda_1^n\mathbf{u_1} + a_2\lambda_2^n\mathbf{u_2},\]</div>
-<p>or, in other words, if we know the eigenvectors and eigenvalues of <span class="math notranslate nohighlight">\(A\)</span> (with <span class="math notranslate nohighlight">\(\lambda_1 \ne \lambda_2\)</span>), and we know the coeffiencients <span class="math notranslate nohighlight">\(a_1\)</span> and <span class="math notranslate nohighlight">\(a_2\)</span> of the expansion of <span class="math notranslate nohighlight">\(\mathbf{v}\)</span> into the eigenvectors, then it is easy to calculate what is going to be the image ov vector <span class="math notranslate nohighlight">\(\mathbf{v}\)</span> by any number of successive applications of <span class="math notranslate nohighlight">\(A\)</span>. This is also equivalent of applying the <span class="math notranslate nohighlight">\(n\)</span>-th power of matrix <span class="math notranslate nohighlight">\(A\)</span> to <span class="math notranslate nohighlight">\(\mathbf{v}\)</span>.</p>
-<p>This fact will have many important applications, as we will se next.</p>
-</section>
-<section id="structured-models-revisited">
-<h2>Structured models revisited<a class="headerlink" href="#structured-models-revisited" title="Link to this heading">#</a></h2>
-<p>Let us considered a structured population model defined by the projection matrix <span class="math notranslate nohighlight">\(P=\begin{pmatrix} 1.5 &amp; 2  \\ 0.08 &amp; 0  \end{pmatrix}\)</span> and an initial population vector given by <span class="math notranslate nohighlight">\(\mathbf{n_0} = \begin{pmatrix} 105  \\ 1 \end{pmatrix}\)</span>. If we calculate the eigenvalues and eigenvectors, we will have:</p>
-<div class="math notranslate nohighlight">
-\[\begin{split}p(\lambda)=\mbox{det}(P-\lambda I) = \mbox{det}\begin{pmatrix} 1.5 -\lambda &amp; 2 \\ 0.08 &amp; -\lambda  \end{pmatrix} = (1.5 -\lambda)(-\lambda) - 0.08\cdot 2 \end{split}\]</div>
-<div class="math notranslate nohighlight">
-\[p(\lambda)=\lambda^2-1.5\lambda-0.16=(\lambda-1.6)(\lambda+0.1)\]</div>
-<p>Thus, we have <span class="math notranslate nohighlight">\(\lambda_1=1.6\)</span> and <span class="math notranslate nohighlight">\(\lambda_2=-0.1\)</span>.</p>
-<ul>
-<li><p><span class="math notranslate nohighlight">\(\mathbf{\lambda_1 = 1.6}\)</span>:</p>
-<p><span class="math notranslate nohighlight">\(\begin{align}
-A\mathbf{v} = \lambda\mathbf{v} \longrightarrow \begin{pmatrix} 1.5 &amp; 2  \\ 0.08 &amp; 0 \end{pmatrix}\begin{pmatrix} x  \\ y \end{pmatrix}= 1.6\begin{pmatrix} x  \\ y \end{pmatrix}\longrightarrow \cases{1.5x + 2y = 1.6x \\ 0.08x = 1.6y} \longrightarrow \cases{x - 20y = 0 \\ x - 20y = 0}
-\end{align}\)</span></p>
-<p>Since <span class="math notranslate nohighlight">\(x=20y\)</span>, we choose <span class="math notranslate nohighlight">\(\mathbf{u_1} = \begin{pmatrix} 20  \\ 1  \end{pmatrix}\)</span>.</p>
-<br />
-</li>
-<li><p><span class="math notranslate nohighlight">\(\mathbf{\lambda_2 = -0.1}\)</span>:</p>
-<p><span class="math notranslate nohighlight">\(\begin{align}
-A\mathbf{v} = \lambda\mathbf{v} \longrightarrow \begin{pmatrix} 1.5 &amp; 2  \\ 0.08 &amp; 0 \end{pmatrix}\begin{pmatrix} x  \\ y \end{pmatrix}= -0.1\begin{pmatrix} x  \\ y \end{pmatrix}\longrightarrow \cases{1.5x + 2y = -0.1x \\ 0.08x = -0.1y} \longrightarrow \cases{0.8x + y = 0 \\ 0.08x + 0.1y = 0}
-\end{align}\)</span></p>
-<p>Since <span class="math notranslate nohighlight">\(-\displaystyle\frac{4}{5}x=y\)</span>, we choose <span class="math notranslate nohighlight">\(\mathbf{u_2} = \begin{pmatrix} 5  \\ -4  \end{pmatrix}\)</span>.</p>
-</li>
-</ul>
-<br />
-<p>We can also note that the initial population vector can be decomposed into the eigenvectors <span class="math notranslate nohighlight">\(\mathbf{u_1}\)</span> and <span class="math notranslate nohighlight">\(\mathbf{u_2}\)</span> of the matrix <span class="math notranslate nohighlight">\(P\)</span> according to:</p>
-<div class="math notranslate nohighlight">
-\[\begin{split}\mathbf{n_0}=\begin{pmatrix} 105  \\ 1 \end{pmatrix} = 5\begin{pmatrix} 20  \\ 1 \end{pmatrix} + \begin{pmatrix} 5  \\ -4 \end{pmatrix} = 5\mathbf{u_1} + \mathbf{u_2}\end{split}\]</div>
-<p>We know that to find the population vector for the following step, we have to apply the projection matrix on the current population vector. Thus:</p>
-<div class="math notranslate nohighlight">
-\[\begin{split}\mathbf{n_1} = P\mathbf{n_0}= P(5\mathbf{u_1} + \mathbf{u_2})=5P(\mathbf{u_1}) + P(\mathbf{u_2})=5(1.6)\begin{pmatrix} 20  \\ 1 \end{pmatrix} + (-0.1)\begin{pmatrix} 5  \\ -4 \end{pmatrix} = \begin{pmatrix} 159.5  \\ 8.4 \end{pmatrix}\end{split}\]</div>
-<p>If we perform successive applications of <span class="math notranslate nohighlight">\(P\)</span>, we will finally have:</p>
-<div class="math notranslate nohighlight">
-\[\begin{split}\mathbf{n_t} = P^t\mathbf{n_0} = P^t(5\mathbf{u_1} + \mathbf{u_2})=5(1.6)^t\begin{pmatrix} 20  \\ 1 \end{pmatrix} + (-0.1)^t\begin{pmatrix} 5  \\ -4 \end{pmatrix}.\end{split}\]</div>
-<br />  
-<p>By decomposing into eigenvalues and eigenvectors, we know the behaviour of our population for any time <span class="math notranslate nohighlight">\(t\)</span>, which is much easier than applying <span class="math notranslate nohighlight">\(P\)</span> a number <span class="math notranslate nohighlight">\(t\)</span> of times, or than calculating <span class="math notranslate nohighlight">\(P^t\)</span>. Furthermore, as <span class="math notranslate nohighlight">\(t\rightarrow\infty\)</span> the first term will dominate (with the second term dying out) and the proportions for each age class in this population will be dictated by the vector <span class="math notranslate nohighlight">\(\begin{pmatrix} 20  \\ 1 \end{pmatrix}\)</span> (or, if we divide by the sum of the components: <span class="math notranslate nohighlight">\(\displaystyle\frac{1}{21}\begin{pmatrix} 20  \\ 1 \end{pmatrix} = \begin{pmatrix} 95\%  \\ 5\% \end{pmatrix}\)</span>.</p>
-<p>In general the eigenvector corresponding to the <strong>largest eigenvalue</strong> (also called <em>dominant eigenvalue</em>) of the projection matrix, which is always <em>real</em> and <em>positive</em>, will correspond to the stable age distribution, for which our population vector converges after we wait enough time.</p>
-<p>As we have seen, after enough time, the dominant eigenvalue will also correspond to the growth factor for each of the age classes. Thus, if <span class="math notranslate nohighlight">\(\lambda_1&gt;1\)</span> the population size will increase, whereas if <span class="math notranslate nohighlight">\(0&lt;\lambda_1&lt;1\)</span> the population size will decrease.</p>
-</section>
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-<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#instantaneous-rate-of-change">Instantaneous rate of change</a></li>
-<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#calculating-derivatives">Calculating derivatives</a></li>
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-  <section class="tex2jax_ignore mathjax_ignore" id="analysing-change-continuously">
-<h1>06 - Analysing change, continuously<a class="headerlink" href="#analysing-change-continuously" title="Link to this heading">#</a></h1>
-<section id="average-rate-of-change">
-<h2>Average rate of change<a class="headerlink" href="#average-rate-of-change" title="Link to this heading">#</a></h2>
-<p>Watch this <a class="reference external" href="https://web.microsoftstream.com/video/2953e661-e1c1-43ff-ba78-ca52099ed1ec">video</a>
-for a short introduction to this lecture.</p>
-</section>
-<section id="instantaneous-rate-of-change">
-<h2>Instantaneous rate of change<a class="headerlink" href="#instantaneous-rate-of-change" title="Link to this heading">#</a></h2>
-<p>If we have a continuous function <span class="math notranslate nohighlight">\(f(x)\)</span>, for each two points <span class="math notranslate nohighlight">\(x_{0}\)</span> and <span class="math notranslate nohighlight">\(x_{1}\)</span>, we can define the average rate of change between these points by the ratio between how much <span class="math notranslate nohighlight">\(f\)</span> has changed and how much <span class="math notranslate nohighlight">\(x\)</span> has changed:</p>
-<div class="math notranslate nohighlight">
-\[\frac{\Delta f(x)}{\Delta x} = \frac{f(x_{1}) - f(x_{0})}{x_{1} - x_{0}}\]</div>
-<p>For points <span class="math notranslate nohighlight">\(x_{0}\)</span> and <span class="math notranslate nohighlight">\(x_{1}\)</span> this can be graphically represented by:</p>
-<img alt="../../_images/sec_tan.png" src="../../_images/sec_tan.png" />
-<p>The line that passes through the points <span class="math notranslate nohighlight">\((x_0, f(x_0))\)</span> and <span class="math notranslate nohighlight">\((x_1, f(x_1))\)</span> (yellow line) is named <em>secant line</em>. As we have seen before in our treatment of linear functions, the slope of the secant line will be given by the average rate of change between this points (remember that the slope for linear functions is defined as <span class="math notranslate nohighlight">\(m=\displaystyle\frac{\Delta y}{\Delta x})\)</span>.</p>
-<p>Now consider that we change the value of <span class="math notranslate nohighlight">\(x_1\)</span> so that it slowly approaches <span class="math notranslate nohighlight">\(x_0\)</span> from the right. In this case, the secant lines also change (from yellow to red), as we change the image of <span class="math notranslate nohighlight">\(x_1\)</span> the function, <span class="math notranslate nohighlight">\(f(x_1)\)</span>. In the limit where <span class="math notranslate nohighlight">\(x_1\)</span> is very close to <span class="math notranslate nohighlight">\(x_0\)</span>, the secant line approaches the <em>tangent</em> (sark red line) of the function <span class="math notranslate nohighlight">\(f(x)\)</span> at the point <span class="math notranslate nohighlight">\(x_0\)</span> (the tangent is the straight line that “just touches” the curve at that point).</p>
-<p>If we represent the slope of the tangent line at the point <span class="math notranslate nohighlight">\(x_0\)</span> by the term <span class="math notranslate nohighlight">\(f'(x_0)\)</span>, we will have:</p>
-<div class="math notranslate nohighlight">
-\[\lim_{x_{1} \to x_{0}} \frac{\Delta f(x)}{\Delta x} = \frac{f(x_{1}) - f(x_{0})}{x_{1} - x_{0}} = f'(x_{0})\]</div>
-<p>Since we can do the same process as we change the value <span class="math notranslate nohighlight">\(x_0\)</span>, we define a new function <span class="math notranslate nohighlight">\(f'(x_{0})\)</span> which we call the <strong>derivative</strong> of the function <span class="math notranslate nohighlight">\(f\)</span> at the point <span class="math notranslate nohighlight">\(x_0\)</span>, and its value tells us about the <em>instantaneous</em> rate of change of <span class="math notranslate nohighlight">\(f\)</span> at the point <span class="math notranslate nohighlight">\(x_0\)</span>.</p>
-<p>If we look instead at the interval between <span class="math notranslate nohighlight">\(x_{0}\)</span> and <span class="math notranslate nohighlight">\(x_{1}\)</span> and define <span class="math notranslate nohighlight">\(h = x_1 - x_0\)</span>, if <span class="math notranslate nohighlight">\(x_1\)</span> approaches <span class="math notranslate nohighlight">\(x_0\)</span>, then <span class="math notranslate nohighlight">\(h\)</span> approaches <span class="math notranslate nohighlight">\(0\)</span>. and we would have for the derivative of <span class="math notranslate nohighlight">\(f(x)\)</span>:</p>
-<div class="math notranslate nohighlight">
-\[\lim_{h \to 0} \frac{f(x + h) - f(x)}{h} = f'(x)\]</div>
-<p>Visit this <a class="reference external" href="https://www.demonstrations.wolfram.com/SecantAndTangentLines/">link</a> for an interactive demonstration of the limit of the tangent curve.</p>
-<div class="admonition note">
-<p class="admonition-title">Note</p>
-<p>Note that in the previous expression the index <span class="math notranslate nohighlight">\(0\)</span> was omitted from <span class="math notranslate nohighlight">\(x_0\)</span>. We will assume that the derivative is defined for all the points in the domain of the function <span class="math notranslate nohighlight">\(f(x)\)</span> for which the limit in the expression exists.</p>
-<p>It is good to have a graphical intuition for when the derivative is not well defined. Some cases are shown below.</p>
-</div>
-<div class="admonition-example admonition">
-<p class="admonition-title">Example</p>
-<p>By the definition of the derivative of <span class="math notranslate nohighlight">\(f(x)\)</span>:</p>
-<div class="math notranslate nohighlight">
-\[f'(x)=\lim_{h \to 0} \frac{f(x + h) - f(x)}{h}\]</div>
-<p>Thus, if <span class="math notranslate nohighlight">\(f(x)=x^2\)</span>, we will have:</p>
-<div class="math notranslate nohighlight">
-\[f'(x)=\lim_{h \to 0} \frac{(x + h)^{2} - x^{2}}{h} = \lim_{h \to 0} \frac{x^{2} + 2xh + h^{2} - x^{2}}{h} = \lim_{h \to 0} (h + 2x) = 2x\]</div>
-<p>So that the derivative of <span class="math notranslate nohighlight">\(f(x)=x^2\)</span> is the function <span class="math notranslate nohighlight">\(f'(x)=2x\)</span>.</p>
-<div class="admonition note">
-<p class="admonition-title">Note</p>
-<p>Although limits of functions can be defined in a more rigorous way, in this course we will treat them more intuitively, and use, if required, the known rules for limits. Thus, if <span class="math notranslate nohighlight">\(C\)</span> is a constant, <span class="math notranslate nohighlight">\(\lim_{x \rightarrow a} f(x) = L\)</span>, and <span class="math notranslate nohighlight">\(\lim_{x \rightarrow a} g(x) = M\)</span>, it is generally valid that:</p>
-<div class="math notranslate nohighlight">
-\[\lim_{x \rightarrow a} [f(x) + g(x)] = \lim_{x \rightarrow a} f(x) + \lim_{x \rightarrow a} g(x) = L + M\]</div>
-<div class="math notranslate nohighlight">
-\[\lim_{x \rightarrow a} [Cf(x)] = C\lim_{x \rightarrow a} f(x) = CL\]</div>
-<div class="math notranslate nohighlight">
-\[\lim_{x \rightarrow a} [f(x)g(x)] = [\lim_{x \rightarrow a} f(x)][\lim_{x \rightarrow a} g(x)] = LM\]</div>
-<div class="math notranslate nohighlight">
-\[\lim_{x \rightarrow a} \left[\frac{f(x)}{g(x)}\right] = \frac{\lim_{x \rightarrow a} f(x)}{\lim_{x \rightarrow a} g(x)} = \frac{L}{M},\ \ (\mbox{if}\ M \ne 0)\]</div>
-<div class="math notranslate nohighlight">
-\[\lim_{x \rightarrow a} [C^{f(x)}] = C^{\lim_{x \rightarrow a} f(x)} = C^L\]</div>
-</div>
-</div>
-<p>Other common notations for the derivative are given by:</p>
-<div class="math notranslate nohighlight">
-\[f'(x) = \frac{df}{dx} = \frac{d}{dx}f(x) = \dot{f}(x)\]</div>
-<p>Visit this <a class="reference external" href="https://the-learning-machine.com/article/calculus/derivatives">link</a> for examples that provide demonstrations of the intuitive interpretation of the derivative.</p>
-</section>
-<section id="calculating-derivatives">
-<h2>Calculating derivatives<a class="headerlink" href="#calculating-derivatives" title="Link to this heading">#</a></h2>
-<p>It is possible to generalise the previous example for <span class="math notranslate nohighlight">\(f(x)=x^2\)</span> and obtain the derivative for any power function <span class="math notranslate nohighlight">\(f(x)=x^n\)</span> (<span class="math notranslate nohighlight">\(n \ne 0\)</span>):</p>
-<div class="math notranslate nohighlight">
-\[f(x) = x^{n} \implies f'(x) = nx^{n-1}\]</div>
-<div class="admonition-example admonition">
-<p class="admonition-title">Example</p>
-<div class="math notranslate nohighlight">
-\[x^{2} \implies 2x^{2-1} = 2x\]</div>
-<div class="math notranslate nohighlight">
-\[\ x^{3} \implies 3x^{3-1} = 3x^{2}\]</div>
-<div class="math notranslate nohighlight">
-\[\ x^{4.2} \implies 4.2x^{4.2-1} = 4.2x^{3.2}\]</div>
-</div>
-<p>It is possible to show, from the definition of derivative and the rules for limits, that there are simple rules for derivatives when calculating them for a sum of functions or a product by a constant. If <span class="math notranslate nohighlight">\(C\)</span> is a constant, we generally have:</p>
-<div class="math notranslate nohighlight">
-\[\begin{split}\begin{align}\frac{d}{dx}[f(x)+g(x)] &amp;= \frac{df}{dx} + \frac{dg}{dx} \\
-   \frac{d}{dx}[Cf(x)] &amp;= C\frac{df}{dx}\end{align}\end{split}\]</div>
-<p>Thus, differentiating a given polynomial p(x) stands for differentiating each term separately and summing the result.</p>
-<div class="admonition-example admonition">
-<p class="admonition-title">Example</p>
-<div class="math notranslate nohighlight">
-\[f(x) = 3x^{2} - 4x + 6 \implies f'(x) = 3(2x) - 4(x^{0}) + 0 = 6x - 4\]</div>
-<div class="math notranslate nohighlight">
-\[f(x) = x^{60} - 3x^{20} - 4x \implies f'(x) = 60x^{59} - 60x^{19} - 4\]</div>
-</div>
-<p>For products and quotients of functions, however, we have different rules:</p>
-<div class="math notranslate nohighlight">
-\[\frac{d}{dx}[f(x)g(x)] = \left(\frac{df}{dx}\right) g(x) + f(x)\left(\frac{dg}{dx}\right)\]</div>
-<div class="math notranslate nohighlight">
-\[\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{\left(\displaystyle\frac{df}{dx}\right)g(x) - f(x)\left(\displaystyle\frac{dg}{dx}\right)}{[g(x)]^{2}}\]</div>
-<div class="admonition tip">
-<p class="admonition-title">Tip</p>
-<p>If you do not want to remember the quotients rule, you can use the product rule on <span class="math notranslate nohighlight">\(f(x)j(x)\)</span>, with <span class="math notranslate nohighlight">\(j(x)=g(x)^{-1}\)</span> and use the chain rule to evaluate <span class="math notranslate nohighlight">\(\frac{d}{dx} j(x)\)</span>.</p>
-</div>
-<div class="admonition-examples admonition">
-<p class="admonition-title">Examples</p>
-<div class="math notranslate nohighlight">
-\[f(x) = (x + 2)(x - 4) \implies f'(x) = (1)(x-4) + (x+2)(1) = x - 4 + x + 2 = 2x - 2\]</div>
-<div class="math notranslate nohighlight">
-\[f(x) = \frac{\alpha x}{1 + \beta x} \implies f'(x) =  \frac{(\alpha)(1 + \beta x) - (\alpha x)(\beta)}{(1 + \beta x)^{2}} = \frac{\alpha + \alpha\beta x - \alpha \beta x}{(1 + \beta x)^{2}} =\frac{\alpha}{(1 + \beta x)^{2}}\]</div>
-</div>
-<p>Moreover, expressions for the derivatives of the other elementary functions can also be found:</p>
-<ul class="simple">
-<li><p><span class="math notranslate nohighlight">\(f(x) = a^{x} \implies f'(x) = a^{x} \cdot (\ln a) ,\ (a &gt; 0, a \neq 1)\)</span><br />
-<span class="math notranslate nohighlight">\(\left(\mbox{in particular: }\ (e^{x})' = e^{x}\right)\)</span></p></li>
-<li><p><span class="math notranslate nohighlight">\(f(x) = \log_{a}x \implies f'(x) = \displaystyle\frac{1}{(\ln a)x}\)</span><br />
-<span class="math notranslate nohighlight">\(\left(\mbox{in particular: }\ (\ln x)' = \displaystyle\frac{1}{x}\right)\)</span></p></li>
-<li><p><span class="math notranslate nohighlight">\(f(x) = \mbox{sin}(x) \implies f'(x) = \mbox{cos}(x)\)</span></p></li>
-<li><p><span class="math notranslate nohighlight">\(f(x) = \mbox{cos}(x) \implies f'(x) = -\mbox{sin}(x)\)</span></p></li>
-</ul>
-<div class="admonition-example admonition">
-<p class="admonition-title">Example</p>
-<p>If <span class="math notranslate nohighlight">\(f(x) = \mbox{tan}(x) = \displaystyle\frac{\mbox{sin}(x)}{\mbox{cos}(x)}\)</span>, then:</p>
-<div class="math notranslate nohighlight">
-\[f'(x) = \frac{(\mbox{cos}(x))\mbox{cos}(x) - \mbox{sin}(x)(-\mbox{sin}(x))}{(\mbox{cos}(x))^{2}} = \frac{\mbox{sin}^{2}(x) + \mbox{cos}^{2}(x)}{\mbox{cos}^{2}(x)} = \frac{1}{\mbox{cos}^{2}(x)} = \mbox{sec}^{2}(x)\]</div>
-</div>
-</section>
-<section id="chain-rule">
-<h2>Chain rule<a class="headerlink" href="#chain-rule" title="Link to this heading">#</a></h2>
-<p>The calculation of derivatives for more complicated functions can generally be made easier by breaking the function into components, for example:</p>
-<ul class="simple">
-<li><p><span class="math notranslate nohighlight">\(f(x) = (3x^{2} - 2x + 1)^{3} = (g(x))^{3}\)</span>, where <span class="math notranslate nohighlight">\(g(x) = 3x^{2} - 2x + 1\)</span></p></li>
-<li><p><span class="math notranslate nohighlight">\(f(x) = \displaystyle\frac{\sqrt{2x - 1}}{1 + \sqrt{2x - 1}} = \displaystyle\frac{g(x)}{1 + g(x)}\)</span>, where <span class="math notranslate nohighlight">\(g(x) = \sqrt{h(x)}\)</span>, and <span class="math notranslate nohighlight">\(h(x) = 2x - 1\)</span></p></li>
-</ul>
-<p>We have seen this before when discussing transformation of graphs.  When one function is nested within the other, we have a composite function, or as for the last example above:</p>
-<div class="math notranslate nohighlight">
-\[f(x) = f(g(h(x)))\]</div>
-<p>The derivative of composite functions in relation to the dependent variable is then given by the <em>chain rule</em>:</p>
-<ul class="simple">
-<li><p><span class="math notranslate nohighlight">\(\displaystyle\frac{d}{dx}f(x) = \displaystyle\frac{d}{dx}f(g(x)) = \left(\displaystyle\frac{df}{dg}\right)\left(\displaystyle\frac{dg}{dx}\right)\)</span></p></li>
-<li><p><span class="math notranslate nohighlight">\(\displaystyle\frac{d}{dx}f(x) = \displaystyle\frac{d}{dx}f(g(h(x))) = \left(\displaystyle\frac{df}{dg}\right) \left(\displaystyle\frac{dg}{dh}\right) \left(\displaystyle\frac{dh}{dx}\right)\)</span></p></li>
-</ul>
-<p>The strategy then consists in identifying the nested functions, calculating their derivatives and multiplying the results.</p>
-<div class="admonition-examples admonition">
-<p class="admonition-title">Examples</p>
-<ul>
-<li><p>If <span class="math notranslate nohighlight">\(f(x) = (3x^{2} - 2x + 1)^{3}\)</span>, we have:</p>
-<div class="math notranslate nohighlight">
-\[f(g) = g^{3} \implies \displaystyle\frac{df}{dg} = 3g^2\]</div>
-<div class="math notranslate nohighlight">
-\[g(x) = 3x^{2} - 2x + 1 \implies \displaystyle\frac{dg}{dx} = 6x - 2\]</div>
-<p>Thus:</p>
-<div class="math notranslate nohighlight">
-\[\displaystyle\frac{df}{dx} = \left(\displaystyle\frac{df}{dg}\right)\left(\displaystyle\frac{dg}{dx}\right) = 3g^{2} \cdot g' = 3(3x^{2} - 2x + 1)^2(6x - 2)\]</div>
-  <br />  
-</li>
-<li><p>If <span class="math notranslate nohighlight">\(f(x) = \displaystyle\frac{\sqrt{2x - 1}}{1 + \sqrt{2x - 1}}\)</span>, then:</p>
-<div class="math notranslate nohighlight">
-\[f(g) = \displaystyle\frac{g}{1+g} \implies \displaystyle\frac{df}{dg} = \frac{(1+g) - g}{(1+g)^2} = \frac{1}{(1+g)^2}\]</div>
-<div class="math notranslate nohighlight">
-\[g(h) = \sqrt{h} \implies \displaystyle\frac{dg}{dh} = \frac{1}{2}h^{-1/2}\]</div>
-<div class="math notranslate nohighlight">
-\[h(x) = 2x-1 \implies \displaystyle\frac{dh}{dx} = 2\]</div>
-<p>Thus:</p>
-<div class="math notranslate nohighlight">
-\[\displaystyle\frac{df}{dx} =  \left(\displaystyle\frac{df}{dg}\right) \left(\displaystyle\frac{dg}{dh}\right) \left(\displaystyle\frac{dh}{dx}\right) = \frac{1}{(1+\sqrt{2x - 1})^2}\cdot \frac{1}{2(\sqrt{2x - 1})}\cdot 2 = \frac{1}{(\sqrt{2x - 1})(1+\sqrt{2x - 1})^2}\]</div>
-  <br />
-</li>
-<li><p>If <span class="math notranslate nohighlight">\(f(t) = 5 + 5\mbox{cos}\left[\displaystyle\frac{\pi}{12}(t - 3)\right]\)</span>, we have:</p>
-<div class="math notranslate nohighlight">
-\[f(g) = 5 + 5g \implies \displaystyle\frac{df}{dg} = 5\]</div>
-<div class="math notranslate nohighlight">
-\[g(h) = \mbox{cos}(h) \implies \displaystyle\frac{dg}{dh} = -\mbox{sin}(h)\]</div>
-<div class="math notranslate nohighlight">
-\[h(t) = \frac{\pi}{12}(t - 3) \implies \displaystyle\frac{dh}{dt} = \frac{\pi}{12}\]</div>
-<p>Thus:</p>
-<div class="math notranslate nohighlight">
-\[\displaystyle\frac{df}{dx} =  \left(\displaystyle\frac{df}{dg}\right) \left(\displaystyle\frac{dg}{dh}\right) \left(\displaystyle\frac{dh}{dx}\right) = 5 \cdot \left(-\mbox{sin}(h)\right) \cdot \frac{\pi}{12} = -\frac{5\pi}{12}\mbox{sin}\left[\displaystyle\frac{\pi}{12}(t - 3)\right]\]</div>
-<p>For this last example if we plot the function <span class="math notranslate nohighlight">\(f(t)=5 + 5\mbox{cos}\left[\displaystyle\frac{\pi}{12}(t - 3)\right]\)</span> together with its derivative <span class="math notranslate nohighlight">\(f'(t)=-\displaystyle\frac{5\pi}{12}\mbox{sin}\left[\displaystyle\frac{\pi}{12}(t - 3)\right]\)</span> we will have:</p>
-<img alt="../../_images/func_diff.png" src="../../_images/func_diff.png" />
-<p>Try to compare the two graphs. What happens with the function <span class="math notranslate nohighlight">\(f(t)\)</span> on the intervals where the derivative <span class="math notranslate nohighlight">\(f'(t)\)</span> is positive or negative? What happens at the points where the derivative is zero?</p>
-</li>
-</ul>
-</div>
-</section>
-<section id="higher-order-derivatives">
-<h2>Higher order derivatives<a class="headerlink" href="#higher-order-derivatives" title="Link to this heading">#</a></h2>
-<p>If the derivative of a given function is also a “well-behaved” function (without kinks or discontinuities), then we can also calculate the derivative of the derivative. Thus:</p>
-<div class="math notranslate nohighlight">
-\[\frac{d}{dx}\left(\frac{df}{dx}\right) = \frac{d^{2}f}{dx^{2}} = f''(x)\]</div>
-<p>The function <span class="math notranslate nohighlight">\(f''(x)\)</span> is called the <em>second derivative</em> with respect to <span class="math notranslate nohighlight">\(x\)</span>.</p>
-<p>We can continue this process and get higher-order derivatives:</p>
-<div class="math notranslate nohighlight">
-\[\frac{d^{n}f}{dx^{n}} = f^{(n)}(x),\]</div>
-<p>assuming again that these functions are also well-behaved. Function <span class="math notranslate nohighlight">\(f^{(n)}(x)\)</span> is called the <span class="math notranslate nohighlight">\(n\)</span>-th derivative of <span class="math notranslate nohighlight">\(f\)</span> with respect to <span class="math notranslate nohighlight">\(x\)</span> (note the parenthesis on <span class="math notranslate nohighlight">\((n)\)</span> to distinguish from powers of <span class="math notranslate nohighlight">\(f\)</span>).</p>
-<div class="admonition-examples admonition">
-<p class="admonition-title">Examples</p>
-<p>The velocity is the derivative of the position, and the acceleration is the derivative of the velocity. Therefore, acceleration is the second-order derivative of the position</p>
-</div>
-<div class="admonition-examples admonition">
-<p class="admonition-title">Examples</p>
-<ul>
-<li><p><span class="math notranslate nohighlight">\(f(x) = 3x^{2} - 6x + 2\)</span>, then:</p>
-<div class="math notranslate nohighlight">
-\[f'(x) = 6x - 6\]</div>
-<div class="math notranslate nohighlight">
-\[f''(x) = 6\]</div>
-<p>Polynomials are always well-behaved and they are infinitely differentiable. However, if <span class="math notranslate nohighlight">\(n\)</span> is the degree of the polynomial (<em>i.e.</em> the largest exponent in the polynomial), we will have: <span class="math notranslate nohighlight">\(p^{(k)}(x) = 0,\ \mbox{for}\ k \ge n+1\)</span>.</p>
-  <br />
-</li>
-<li><p><span class="math notranslate nohighlight">\(f(x) = e^{-x^{2}}\)</span>, then:</p>
-<div class="math notranslate nohighlight">
-\[f'(x) = e^{-x^{2}} \cdot (-2x) = -2xe^{-x^{2}}\]</div>
-<div class="math notranslate nohighlight">
-\[\begin{split}\begin{align}f''(x)
-&amp;= -2[e^{-x^{2}} + x(-2x)(e^{-x^{2}})] \\
-&amp;= -2e^{-x^{2}} + 4x^{2}e^{-x^{2}} \\
-&amp;= -2e^{-x^{2}}(1 - 2x^{2})\end{align}\end{split}\]</div>
-</li>
-</ul>
-</div>
-</section>
-<section id="functions-of-several-variables">
-<h2>Functions of several variables<a class="headerlink" href="#functions-of-several-variables" title="Link to this heading">#</a></h2>
-<p>A function <span class="math notranslate nohighlight">\(z = f(x,y)\)</span> of two dependent variables <span class="math notranslate nohighlight">\(x\)</span> and <span class="math notranslate nohighlight">\(y\)</span> defines a surface on the three dimensional <span class="math notranslate nohighlight">\(xyz\)</span>-space. If this function is well-behave it will be smooth, like if we applied different deformations on a rubber sheet. For smooth functions, we can calculate derivatives with respect to each of the dependent variables. This is written as:</p>
-<div class="math notranslate nohighlight">
-\[ \frac{\partial f}{\partial x} \text{ or } \frac{\partial f}{\partial x}(x, y) \text{ or } \frac{\partial}{\partial x}f(x,y)\]</div>
-<div class="math notranslate nohighlight">
-\[\frac{\partial f}{\partial y}, \frac{\partial f}{\partial{y}}(x,y), \frac{\partial}{\partial y}f(x,y)\]</div>
-<p>The partial derivative informs us how the function <span class="math notranslate nohighlight">\(f\)</span> varies along the <span class="math notranslate nohighlight">\(x\)</span> (or the <span class="math notranslate nohighlight">\(y\)</span>) direction at a specific point.</p>
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-<p>Consider <span class="math notranslate nohighlight">\(f(x,y) = 4x^2y-2xy^3\)</span>. To compute the partial derivative with respect to <span class="math notranslate nohighlight">\(x\)</span> we consider all terms in <span class="math notranslate nohighlight">\(y\)</span> constants, and calculate the derivative as usual. For the partial derivative with respect to <span class="math notranslate nohighlight">\(y\)</span>, we consider the <span class="math notranslate nohighlight">\(x\)</span> terms constant. We have:</p>
-<div class="math notranslate nohighlight">
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-\[\frac{\partial f}{\partial y} = 4x^{2} - 2x(3y^{2}) = 4x^{2} - 6xy^{2}\]</div>
-</div>
-<p>Higher order partial derivatives can also be defined, and represented with the common notations:</p>
-<div class="math notranslate nohighlight">
-\[\frac{\partial}{\partial x}\left(\frac{\partial f}{\partial x}\right) = \frac{\partial^{2}f}{\partial x^{2}} = \partial_{xx} f = f_{xx}(x,y)\]</div>
-<div class="math notranslate nohighlight">
-\[\frac{\partial}{\partial y}\left(\frac{\partial f}{\partial y}\right) = \frac{\partial^{2}f}{\partial y^{2}} = 
-\partial_{yy} f = f_{yy}(x,y)\]</div>
-<p>with subscripts representing the variable and the order of differentiation.</p>
-<p>For well-behaved functions, the order of the variables chosen to differentiate does not matter (a special case of what is called Schwarz’s theorem):</p>
-<div class="math notranslate nohighlight">
-\[f_{yx}(x,y) = \frac{\partial}{\partial y}\left(\frac{\partial f}{\partial x}\right) = \frac{\partial^{2}f}{\partial y \partial x} = \frac{\partial^{2} f}{\partial x \partial y}= \frac{\partial}{\partial x}\left(\frac{\partial f}{\partial y}\right) = f_{xy}(x,y)\]</div>
-<div class="admonition-example admonition">
-<p class="admonition-title">Example</p>
-<p>Often, a quantity of interest does not depend only on one variable but on many. The population size of a species depends on various aspects, including the availability of food, the number of predators, and other environmental factors.</p>
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-    <h1>07 - Analysing change, continuously (Part 2)</h1>
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-<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#extreme-values-of-functions">Extreme values of functions</a></li>
-<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#monotonicity-and-concavity">Monotonicity and concavity</a></li>
-<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#finding-extrema">Finding extrema</a></li>
-<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#inflection-points">Inflection points</a></li>
-<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#series-expansions">Series expansions</a></li>
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-  <section class="tex2jax_ignore mathjax_ignore" id="analysing-change-continuously-part-2">
-<h1>07 - Analysing change, continuously (Part 2)<a class="headerlink" href="#analysing-change-continuously-part-2" title="Link to this heading">#</a></h1>
-<p>In many different contexts we may be interested in finding the maximum or minimum values a given function may assume. This process is referred to as <em>optimization</em>.</p>
-<p>In Ecology, some of the most common uses for optimization methods are:</p>
-<p>1 - <em>Statistical estimation and inference</em>: optimizing the likelihood of a model and its parameters being true, for a given set of observations.</p>
-<p>2 - <em>Resources management</em>: finding a balance between different wildlife and economic priorities (market, legal, natural constraints)</p>
-<p>3 - <em>Optimality theory</em>: how can the behavioural or life-history trade-offs result in optimal strategies for different organisms</p>
-<section id="extreme-values-of-functions">
-<h2>Extreme values of functions<a class="headerlink" href="#extreme-values-of-functions" title="Link to this heading">#</a></h2>
-<p>For a given function <span class="math notranslate nohighlight">\(f(x)\)</span> that is <strong>continuous</strong> in an interval <span class="math notranslate nohighlight">\(a \le x \le b\)</span>, there will always be a global maximum and a global minimum, points at which the function assumes its maximum and minimum values in the interval.</p>
-<img alt="../../_images/extrema.png" src="../../_images/extrema.png" />
-<p>A function <span class="math notranslate nohighlight">\(f(x)\)</span> presents a <em>local maximum</em> value if for a given vicinity of the point <span class="math notranslate nohighlight">\(x_0\)</span>, <span class="math notranslate nohighlight">\(f(x_0)\)</span> is greater than all the other <span class="math notranslate nohighlight">\(f(x)\)</span> for <span class="math notranslate nohighlight">\(x\)</span> in this vicinity. The analogous definition can be made for a <em>local minimum</em> value when <span class="math notranslate nohighlight">\(f(x_0)\)</span> is smaller than all the other <span class="math notranslate nohighlight">\(f(x)\)</span> in the vicinity. Local minima and local maxima constitue the <em>local extrema</em> for <span class="math notranslate nohighlight">\(f(x)\)</span>, and with the set of all extrema we can find the global maximum and the global minimum.</p>
-<p>An important result is that, if the function <span class="math notranslate nohighlight">\(f\)</span> has a local extremum at a given point <span class="math notranslate nohighlight">\(x=c\)</span> in that interval (<span class="math notranslate nohighlight">\(a&lt;c&lt;b\)</span>) and the derivative of the function exists at that point (in other words, if <span class="math notranslate nohighlight">\(f'(c)\)</span> is well defined), then we will have:</p>
-<p><span class="math notranslate nohighlight">\(f'(c)=0\)</span></p>
-<div class="admonition warning">
-<p class="admonition-title">Warning</p>
-</div>
-<p>Thus the points <span class="math notranslate nohighlight">\(x=c\)</span>, for which <span class="math notranslate nohighlight">\(f'(c)\)</span>, will constitute <strong>candidates</strong> for local extrema, and will have to be evaluated under other sets of conditions. Also important to have in mind that point for which the derivative is not definided (kinks or discontinuities) may also constitute local extrema (see on the first figure).</p>
-<p>In general, when searching for local extrema, we have to adopt the following rules:</p>
-<ul class="simple">
-<li><p>Do not assume points <span class="math notranslate nohighlight">\(x\)</span> for which <span class="math notranslate nohighlight">\(f'(x) = 0\)</span> are extrema. They are candidates.</p></li>
-<li><p>Check points where derivative is not defined.</p></li>
-<li><p>Check endpoints of domain.</p></li>
-</ul>
-<div class="admonition tip">
-<p class="admonition-title">Tip</p>
-<p>If possible, create a plot. Visuals are powerful tools and can help verify whether calculations have been done correctly.</p>
-</div>
-</section>
-<section id="monotonicity-and-concavity">
-<h2>Monotonicity and concavity<a class="headerlink" href="#monotonicity-and-concavity" title="Link to this heading">#</a></h2>
-<p>Imagine a function <span class="math notranslate nohighlight">\(f(x)\)</span> defined in a given interval <span class="math notranslate nohighlight">\(a \le x \le b\)</span>. For every <span class="math notranslate nohighlight">\(x_1\)</span> and <span class="math notranslate nohighlight">\(x_2\)</span> in this interval, if <span class="math notranslate nohighlight">\(x_{2} &gt; x_{1}\)</span> implies that <span class="math notranslate nohighlight">\(f(x_{2}) &gt; f(x_{1})\)</span> then the function is <strong>increasing</strong> in this interval. If on the other hand, <span class="math notranslate nohighlight">\(x_{2} &gt; x_{1}\)</span> implies that <span class="math notranslate nohighlight">\(f(x_{2}) &lt; f(x_{1})\)</span> then the function is <strong>decreasing</strong> in this interval.  This can be graphically shown as:</p>
-<img alt="../../_images/incr_decr.png" src="../../_images/incr_decr.png" />
-<div class="admonition note">
-<p class="admonition-title">Note</p>
-<p>Since we are using the stric inequalities (”<span class="math notranslate nohighlight">\(&gt;\)</span>” or “<span class="math notranslate nohighlight">\(&lt;\)</span>”) it can be added that the function is <strong>monotonic</strong>, or monotonically increasing or decreasing. If we had <span class="math notranslate nohighlight">\(x_{2} &gt; x_{1}\)</span> implying that <span class="math notranslate nohighlight">\(f(x_{2}) \ge f(x_{1})\)</span> (or <span class="math notranslate nohighlight">\(f(x_{2}) \le f(x_{1})\)</span>), the function would be <em>non-decreasing</em> (or <em>non-increasing</em>).</p>
-</div>
-<p>Now suppose that <span class="math notranslate nohighlight">\(f\)</span> is well-behaved in a given interval <span class="math notranslate nohighlight">\(a \le x \le b\)</span> (in mathematical terms, we say: <span class="math notranslate nohighlight">\(f\)</span> is continuous and differentiable in <span class="math notranslate nohighlight">\((a,b)\)</span>), then an important results is that</p>
-<div class="math notranslate nohighlight">
-\[\mbox{If }\ f'(x) &gt; 0\ \mbox{ for}\ a \le x \le b \implies f(x)\ \mbox{is increasing for}\ a \le x \le b\]</div>
-<div class="math notranslate nohighlight">
-\[\mbox{If }\ f'(x) &lt; 0\ \mbox{ for}\ a \le x \le b \implies f(x)\ \mbox{is decreasing for}\ a \le x \le b\]</div>
-<p>In other words, if the derivative of function <span class="math notranslate nohighlight">\(f\)</span> is positive (negative) for every point <span class="math notranslate nohighlight">\(x\)</span> in a given interval then the function is increasing (decreasing) in this interval.</p>
-<div class="admonition-example admonition">
-<p class="admonition-title">Example</p>
-<p>Let us calculate the intervals where the function <span class="math notranslate nohighlight">\(f(x) = x^{3} - \displaystyle\frac{3}{2}x^{2} - 6x + 3\)</span> is increasing and decreasing. Since <span class="math notranslate nohighlight">\(f(x)\)</span> is continuos and differentiable for all real values of <span class="math notranslate nohighlight">\(x\)</span> (remember that all polynomial functions are well-behved), we can use the first derivative test:</p>
-<div class="math notranslate nohighlight">
-\[f'(x) = 3x^{2}-3x-6 = 3(x^{2}-x-2) = 3(x+1)(x-2)\]</div>
-<p>and we see that <span class="math notranslate nohighlight">\(f'(x)\)</span> has roots in <span class="math notranslate nohighlight">\(x=-1\)</span> and <span class="math notranslate nohighlight">\(x=2\)</span>. Since <span class="math notranslate nohighlight">\(f'(x)\)</span> is a polynomial of degree 2 with the coefficient of the leading term positive, it is <em>concave up</em>. From this, we can conclude that:</p>
-<div class="math notranslate nohighlight">
-\[x&lt;-1\ \mbox{or}\ x&gt;2 \implies f'(x) &gt; 0 \implies f(x)\ \mbox{is}\ increasing\]</div>
-<div class="math notranslate nohighlight">
-\[-1&lt;x&lt;2 \implies f'(x) &lt; 0 \implies f(x)\ \mbox{is}\ decreasing\]</div>
-<p>The following plot shows the function <span class="math notranslate nohighlight">\(f(x)\)</span> and its derivative. Compare the intervals where the derivative is positive (negative) with the intervals where the function <span class="math notranslate nohighlight">\(f(x)\)</span> is increasing (decreasing).</p>
-<img alt="../../_images/func_diff1.png" src="../../_images/func_diff1.png" />
-</div>
-<p>When we discussed polynomials of degree 2, depending on the sign of the coefficient of the leading term, we had the two cases:</p>
-<img alt="../../_images/concav.png" src="../../_images/concav.png" />
-<p>Let us now generalise the notion of concavity for any arbitrary function. Suppose <span class="math notranslate nohighlight">\(f(x)\)</span> is differentiable in a given interval <span class="math notranslate nohighlight">\(a \le x \le b\)</span> (in other words, the derivative <span class="math notranslate nohighlight">\(f'(x)\)</span> is well-defined for every point in this interval). Then we have the following results:</p>
-<div class="math notranslate nohighlight">
-\[\mbox{If }\ f'(x)\ \mbox{is increasing for}\ a \le x \le b \implies f(x)\ \mbox{is}\ concave\ up\]</div>
-<div class="math notranslate nohighlight">
-\[\mbox{If }\ f'(x)\ \mbox{is decreasing for}\ a \le x \le b \implies f(x)\ \mbox{is}\ concave\ down\]</div>
-<p>This means that if for a given function <span class="math notranslate nohighlight">\(f(x)\)</span> the slopes of the tangent lines are increasing in a given interval of <span class="math notranslate nohighlight">\(x\)</span>, in this interval the function should resemble an “upwards U-shape”. If on the other hand, the slopes of the tangent lines are decreasing in this interval, the function should resemble a “downwards U-shape”. This can be visualised on the plots below.</p>
-<img alt="../../_images/slopes_inc_dec.png" src="../../_images/slopes_inc_dec.png" />
-<p>If the function <span class="math notranslate nohighlight">\(f\)</span> is twice differentiable in <span class="math notranslate nohighlight">\(a \le x \le b\)</span> (<span class="math notranslate nohighlight">\(f''(x)\)</span> is well-define for <span class="math notranslate nohighlight">\(a \le x \le b\)</span>), we can use the last two results to find a general criterium for concavity. This will be given by:</p>
-<div class="math notranslate nohighlight">
-\[\mbox{If }\ f''(x) &gt; 0\ \mbox{ for}\ a \le x \le b \implies f'(x)\ \mbox{is increasing for}\ a \le x \le b \implies f(x)\ \mbox{is}\ concave\ up\]</div>
-<div class="math notranslate nohighlight">
-\[\mbox{If }\ f''(x) &lt; 0\ \mbox{ for}\ a \le x \le b \implies f'(x)\ \mbox{is decreasing for}\ a \le x \le b \implies f(x)\ \mbox{is}\ concave\ down\]</div>
-<div class="admonition-example admonition">
-<p class="admonition-title">Example</p>
-<p>If you have a population curve, these tools will help you analyze questions such as ‘How fast is the population growing?’ ‘Is the growth accelerating?’ or ‘Are any of these factors changing over different periods?</p>
-</div>
-<div class="admonition-example admonition">
-<p class="admonition-title">Example</p>
-<p>Let us analyse the function <span class="math notranslate nohighlight">\(f(x) = x^{3}-\displaystyle\frac{3}{2}x^{2}-6x+3\)</span> again. We have:</p>
-<div class="math notranslate nohighlight">
-\[\begin{split}\begin{align} 
-f'(x) = 3x^{2}-3x-6 \\
-f''(x) = 6x -3 \end{align}\end{split}\]</div>
-<p>The function f’’(x) has a root in <span class="math notranslate nohighlight">\(x=\displaystyle\frac{1}{2}\)</span>. Since its slope is positive, we have:</p>
-<div class="math notranslate nohighlight">
-\[x&lt;\displaystyle\frac{1}{2} \implies f''(x) &lt; 0 \implies f(x)\ \mbox{is}\ concave\ down\]</div>
-<div class="math notranslate nohighlight">
-\[x&gt;\displaystyle\frac{1}{2} \implies f''(x) &gt; 0 \implies f(x)\ \mbox{is}\ concave\ up\]</div>
-<p>On the plots below, compare the intervals where the second derivative is positive (negative) with the intervals where the function <span class="math notranslate nohighlight">\(f(x)\)</span> is concave up (concave down).</p>
-<img alt="../../_images/plot_concav.png" src="../../_images/plot_concav.png" />
-</div>
-</section>
-<section id="finding-extrema">
-<h2>Finding extrema<a class="headerlink" href="#finding-extrema" title="Link to this heading">#</a></h2>
-<p>Back to our strategy for finding maxima or minima of functions, we first determine the set:</p>
-<ul class="simple">
-<li><p>find all points <span class="math notranslate nohighlight">\(c\)</span> for which <span class="math notranslate nohighlight">\(f'(c) = 0\)</span> or all point <span class="math notranslate nohighlight">\(c\)</span> where <span class="math notranslate nohighlight">\(f'(c)\)</span> does not exist. These are called <strong>critical points</strong>.</p></li>
-<li><p>find endpoints of domain of <span class="math notranslate nohighlight">\(f\)</span>.</p></li>
-</ul>
-<p>Additionally, if <span class="math notranslate nohighlight">\(f\)</span> is twice differentiable (in some interval containing <span class="math notranslate nohighlight">\(c\)</span>):</p>
-<ul class="simple">
-<li><p>If <span class="math notranslate nohighlight">\(f'(c) = 0\)</span> and <span class="math notranslate nohighlight">\(f''(c) &lt; 0 \implies c\)</span> is a local maximum (since the function is concave down)</p></li>
-<li><p>If <span class="math notranslate nohighlight">\(f'(c) = 0\)</span> and <span class="math notranslate nohighlight">\(f''(c) &gt; 0 \implies c\)</span> is a local minimum (since the function is concave up)</p></li>
-</ul>
-<div class="admonition-example admonition">
-<p class="admonition-title">Example</p>
-<p>Let us analyse the function <span class="math notranslate nohighlight">\(f(x) = \displaystyle\frac{3}{2}x^{4}-2x^{3}-6x^{2}+2\)</span>, and find its local extrema. We have:</p>
-<div class="math notranslate nohighlight">
-\[\begin{split}\begin{align}
-f'(x)&amp;= 6x^{3}-6x^{2}-12x=6x(x-2)(x+1) \\
-f''(x)&amp;=18x^{2}-12x-12=6(3x^{2}-2x-2)\end{align}\end{split}\]</div>
-<p>Since this is a polynomial function, the derivative is well-defined for all points. Also, for the critical points (roots of <span class="math notranslate nohighlight">\(f'(x)\)</span>) we have <span class="math notranslate nohighlight">\(x=0\)</span>, <span class="math notranslate nohighlight">\(x=-1\)</span>, and <span class="math notranslate nohighlight">\(x=2\)</span>. Calculating the value of the second derivative at the critical points:</p>
-<div class="math notranslate nohighlight">
-\[\begin{split}\begin{align}
-&amp;x=0 \implies f''(0)=-12 \text{ (local maximum)} \\
-&amp;x=-1 \implies f''(-1)=18 \text{ (local minimum)} \\
-&amp;x=2 \implies f''(2)=36 \text{ (local minimum)}\end{align}\end{split}\]</div>
-<p>Now we can check on the plot of <span class="math notranslate nohighlight">\(f(x)\)</span> that we have successfully determined the local extrema.</p>
-<img alt="../../_images/plot_extrema.png" src="../../_images/plot_extrema.png" />
-</div>
-</section>
-<section id="inflection-points">
-<h2>Inflection points<a class="headerlink" href="#inflection-points" title="Link to this heading">#</a></h2>
-<p>As we saw before, given a function <span class="math notranslate nohighlight">\(f(x)\)</span>, for some values of <span class="math notranslate nohighlight">\(x\)</span>, there might be a change in the concavity of the function:</p>
-<img alt="../../_images/inflection.png" src="../../_images/inflection.png" />
-<p>The points <span class="math notranslate nohighlight">\(x\)</span> for where the function changes its concavity are called <strong>inflection points</strong>. If <span class="math notranslate nohighlight">\(f\)</span> is twice differentiable and <span class="math notranslate nohighlight">\(c\)</span> is an inflection point, then <span class="math notranslate nohighlight">\(f''(c)=0\)</span>.</p>
-<p>Inflection points are the points where the function is (locally) steepest. It therefore helps us answer questions like when the growth/decline was strongest.</p>
-<div class="admonition note">
-<p class="admonition-title">Note</p>
-<p>Note again that the points <span class="math notranslate nohighlight">\(x=c\)</span> where <span class="math notranslate nohighlight">\(f''(c)=0\)</span> are only candidates for inflection points. See that if <span class="math notranslate nohighlight">\(f(x)=x^{4}\)</span>, <span class="math notranslate nohighlight">\(f''(x)=12x^{2}\)</span>. Then, <span class="math notranslate nohighlight">\(f''(0)=0\)</span>, but <span class="math notranslate nohighlight">\(x=0\)</span> is not an inflection point of <span class="math notranslate nohighlight">\(f\)</span>.</p>
-<p>After calculating the candidate points <span class="math notranslate nohighlight">\(x=c\)</span>, if the concavities of the function (given by the sign of the second derivative) to the left and to the right of <span class="math notranslate nohighlight">\(c\)</span> change, then <span class="math notranslate nohighlight">\(c\)</span> is an inflection point.</p>
-</div>
-<div class="admonition-example admonition">
-<p class="admonition-title">Example</p>
-<p>Consider the Holling Type III functional response</p>
-<div class="math notranslate nohighlight">
-\[f(N)=\frac{aN^{2}}{1+ahN^{2}},\]</div>
-<p>where <span class="math notranslate nohighlight">\(f(N)\)</span> is prey consumption rate per predator, <span class="math notranslate nohighlight">\(N\)</span> is the prey population density, <span class="math notranslate nohighlight">\(a\)</span> is the attack rate, and <span class="math notranslate nohighlight">\(h\)</span> is handling time. Let us check if this function presents any inflection point. We have:</p>
-<p><span class="math notranslate nohighlight">\(\begin{align}f'(N) 
-&amp;= \frac{2aN(1+ahN^{2})-aN^{2}(2ahN)}{(1+ahN^{2})^{2}} \\
-&amp;= \frac{2aN}{(1+ahN^{2})^{2}}\end{align}\)</span></p>
-<br />
-<p><span class="math notranslate nohighlight">\(\begin{align}f''(N)
-&amp;=\frac{2a(1+ahN^{2})^{2}-2aN[2(1+ahN^{2})\cdot2ahN]}{(1+ahN^{2})^{4}} \\
-&amp;=\frac{2a(1+2ahN^{2}+a^{2}h^{2}N^{4})-2a[4ahN^{2}+4a^{2}h^{2}N^{4}]}{(1+ahN^{2})^{4}} \\
-&amp;=\frac{2a(1-3ahN^{2})}{(1+ahN^{2})^{3}}\end{align}\)</span></p>
-<p>See that the function <span class="math notranslate nohighlight">\(f''(N)\)</span> has two roots: <span class="math notranslate nohighlight">\(N=-\sqrt{\displaystyle\frac{1}{3ah}}\)</span> and <span class="math notranslate nohighlight">\(N=\sqrt{\displaystyle\frac{1}{3ah}}\)</span>. Since <span class="math notranslate nohighlight">\(N\)</span> is a population density, it only makes sense to analyse <span class="math notranslate nohighlight">\(N \ge 0\)</span>. It is also possible to show that:</p>
-<div class="math notranslate nohighlight">
-\[N&lt;\sqrt{\displaystyle\frac{1}{3ah}} \implies f''(N) &gt; 0\]</div>
-<div class="math notranslate nohighlight">
-\[N&gt;\sqrt{\displaystyle\frac{1}{3ah}} \implies f''(N) &lt; 0\]</div>
-<p>This means that we have a change of concavity at the point <span class="math notranslate nohighlight">\(N=\sqrt{\displaystyle\frac{1}{3ah}}\)</span> and thus this is and inflection point of this curve.</p>
-<p>Analyse how the function <span class="math notranslate nohighlight">\(f(N)\)</span> changes with <span class="math notranslate nohighlight">\(N\)</span>.</p>
-<img alt="../../_images/hollingIII.png" src="../../_images/hollingIII.png" />
-<p>How does this function compares with the Holling type II functional response?</p>
-</div>
-<div class="admonition tip">
-<p class="admonition-title">Tip</p>
-<p>For the intervals where the function is concave up you can also think of it as representing “accelerated change” of that function (since the first derivative, the “velocity” is increasing), while the intervals where it is concave down represent “deccelerated change” of the function. Understanding this can be particularly important if you function has a specific biological meaning, as in the previous example.</p>
-</div>
-</section>
-<section id="series-expansions">
-<h2>Series expansions<a class="headerlink" href="#series-expansions" title="Link to this heading">#</a></h2>
-<p>From the formal definition of a derivative</p>
-<div class="math notranslate nohighlight">
-\[f'(a)= \lim_{x \to a} \frac{f(x)-f(a)}{(x-a)} \xrightarrow{\text{$x$ very close to $a$}} f'(a) \approx \frac{f(x)-f(a)}{(x-a)},\]</div>
-<p>which means that if <span class="math notranslate nohighlight">\(x\)</span> is sufficiently close to <span class="math notranslate nohighlight">\(a\)</span>, the average rate of change is sufficiently close to the value of the derivative at that point. From the previous approximation, we have:</p>
-<div class="math notranslate nohighlight">
-\[f(x) = f(a)+f'(a)(x-a),\]</div>
-<p>which provides a linear approximation of <span class="math notranslate nohighlight">\(f(x)\)</span> around <span class="math notranslate nohighlight">\(a\)</span> (see that if <span class="math notranslate nohighlight">\(\alpha= f(a)\)</span> and <span class="math notranslate nohighlight">\(\beta=f'(a)\)</span>, both constants, then <span class="math notranslate nohighlight">\(f(x) = \alpha+\beta(x-a)\)</span>, which can easily be arranged on the form <span class="math notranslate nohighlight">\(y=mx+b\)</span>).</p>
-<div class="admonition-example admonition">
-<p class="admonition-title">Example</p>
-</div>
-<p>One way of trying to make the approximation better is to add terms of higher order of <span class="math notranslate nohighlight">\(x\)</span>. Suppose we want to approximate the function <span class="math notranslate nohighlight">\(f(x)\)</span> at point <span class="math notranslate nohighlight">\(a\)</span> by a given polynomial of degree <span class="math notranslate nohighlight">\(n\)</span></p>
-<div class="math notranslate nohighlight">
-\[P(x)=a_{0}+a_{1}(x-a)+a_{2}(x-a)^{2}+\cdots+a_n(x-a)^{n}\]</div>
-<p>so that the derivatives of <span class="math notranslate nohighlight">\(f(x)\)</span> and <span class="math notranslate nohighlight">\(P(x)\)</span> are the same at <span class="math notranslate nohighlight">\(x=a\)</span>:</p>
-<div class="math notranslate nohighlight">
-\[\begin{split}\begin{align}f(a)&amp;=P(a) \\
-f'(a)&amp;=P'(a)\\
-&amp;\vdots \\
-f^{(n)}(a)&amp;=P^{(n)}(a)\end{align}.\end{split}\]</div>
-<p>But for the polynomial, we have:</p>
-<div class="math notranslate nohighlight">
-\[\begin{split}\begin{align}  
-P(a)&amp;=a_{0} \\
-P'(a)&amp;=1 \cdot a_{1} \\
-P''(a)&amp;=2 \cdot 1 \cdot a_{2} \\
-P'''(a)&amp;=3 \cdot 2 \cdot 1 \cdot a_{3} \\  
-P''''(a)&amp;=4 \cdot 3 \cdot 2 \cdot 1 \cdot a_{4} \\
-&amp;\vdots \\
-P^{(n)}(a)&amp;= \underbrace{n(n-1)(n-2) \cdots 3 \cdot 2 \cdot 1}_{\displaystyle\text{$$n!$$}}\cdot a_{n} \end{align}\end{split}\]</div>
-<p>where <span class="math notranslate nohighlight">\(n! = n\cdot(n-1)\cdot(n-2) \cdots 3 \cdot 2 \cdot 1\)</span> is the factorial of <span class="math notranslate nohighlight">\(n\)</span>.</p>
-<p>Thus, by making derivatives of <span class="math notranslate nohighlight">\(f(x)\)</span> and <span class="math notranslate nohighlight">\(P(x)\)</span> equal at <span class="math notranslate nohighlight">\(x=a\)</span>, we obtain:</p>
-<div class="math notranslate nohighlight">
-\[f^{(n)}(a) = P^{(n)}(a) \implies f^{(n)}(a) = n!\, a_n \implies a_{n} = \frac{f^{(n)}(a)}{n!}\]</div>
-<p>The polynomial with these coefficients is called <strong>Taylor polynomial</strong> of degree <span class="math notranslate nohighlight">\(n\)</span> at <span class="math notranslate nohighlight">\(x = a\)</span>:</p>
-<div class="math notranslate nohighlight">
-\[P(x)=f(a)+f'(a)(x-a)+\frac{f''(a)}{2!}(x-a)^{2}+\cdots+\frac{f^{(n)}(a)}{n!}(x-a)^{n}.\]</div>
-<div class="admonition-example admonition">
-<p class="admonition-title">Example</p>
-<p>To find approximation of degree 3 for <span class="math notranslate nohighlight">\(f(x)=e^{x}\)</span>, at <span class="math notranslate nohighlight">\(x=0\)</span>, we have:</p>
-<div class="math notranslate nohighlight">
-\[\begin{split}f(0)=1 \\
-f'(0)=1 \\
-f''(0)=1 \\
-f'''(0)=1\end{split}\]</div>
-<p>Thus,</p>
-<div class="math notranslate nohighlight">
-\[P_{3}(x)=1+(x-0)+\frac{1}{2!}(x-0)^2+\frac{1}{3!}(x-0)^3=1+x+\frac{x^2}{2}+\frac{x^3}{6}\]</div>
-<p>See how the approximation to the function <span class="math notranslate nohighlight">\(f(x)=e^{x}\)</span>  at <span class="math notranslate nohighlight">\(x=0\)</span> gets increasingly better when more terms are added to the polynomial.</p>
-<img alt="../../_images/exp_series.gif" src="../../_images/exp_series.gif" />
-</div>
-<div class="admonition-example admonition">
-<p class="admonition-title">Example</p>
-<p>For the function <span class="math notranslate nohighlight">\(f(x)=\displaystyle\frac{1}{1-x}\ (x\neq1)\)</span> at <span class="math notranslate nohighlight">\(x=0\)</span>, we have:</p>
-<div class="math notranslate nohighlight">
-\[\begin{split}f(x)=\frac{1}{1-x} \implies f(0)=1 \\
-f'(x)=\frac{1}{(1-x)^{2}} \implies f'(0)=1 \\
-f''(x)=\frac{2}{(1-x)^{3}} \implies f''(0)=2 \\
-f'''(x)=\frac{3\cdot2}{(1-x)^{4}} \implies f'''(0)=3! \\
-f^{(n)}(x)=\frac{n!}{(1-x)^{n+1}} \implies f^{(n)}(0)= n!\end{split}\]</div>
-<p>Thus the Taylor polynomial of degree <span class="math notranslate nohighlight">\(n\)</span> will be:</p>
-<div class="math notranslate nohighlight">
-\[\begin{split}\begin{align}P(x)
-&amp;=1+1(x-0)+\frac{2}{2}(x-0)^{2}+\frac{3!}{3!}(x-0)^{3}+\cdots+\frac{n!}{n!}(x-0)^{n} \\
-&amp;=1+x+x^{2}+x^{3}+\cdots+x^{n}\end{align}\end{split}\]</div>
-</div>
-<p>For some functions, the approximation <span class="math notranslate nohighlight">\(f(x)=P_n(x)\)</span> at a given reference point <span class="math notranslate nohighlight">\(x=a\)</span> can be made increasingly better by adding more terms to <span class="math notranslate nohighlight">\(P(x)\)</span>, so that in the limit <span class="math notranslate nohighlight">\(n\rightarrow \infty\)</span> the Taylor polynomials (called <strong>Taylor series</strong> in this limit) for the functions <span class="math notranslate nohighlight">\(e^{x}\)</span>, <span class="math notranslate nohighlight">\(\mbox{sin}(x)\)</span>, <span class="math notranslate nohighlight">\(\mbox{cos}(x)\)</span> converge exactly for all values of <span class="math notranslate nohighlight">\(x\)</span>. For other functions, such as <span class="math notranslate nohighlight">\(\mbox{ln}(1+x)\)</span> and <span class="math notranslate nohighlight">\(\displaystyle\frac{1}{1-x}\)</span>, there is a restricted range of values of <span class="math notranslate nohighlight">\(x\)</span> for which the approximation can be made better by adding terms.</p>
-<p>We have the following approximations by Taylor series, with ranges in <span class="math notranslate nohighlight">\(x\)</span>, at <span class="math notranslate nohighlight">\(x = 0\)</span>:</p>
-<div class="math notranslate nohighlight">
-\[e^x=\sum_{n=0}^\infty \frac{x^n}{n!} =1+x+\frac{x^2}{2}+\frac{x^3}{3!}+\cdots,\ -\infty&lt;x&lt;\infty\]</div>
-<div class="math notranslate nohighlight">
-\[\mbox{sin}(x)=\sum_{n=0}^\infty \frac{(-1)^nx^{1+2n}}{(1+2n)!} = x-\frac{x^3}{3}+\frac{x^5}{5!}-\frac{x^7}{7!}+\cdots,\ -\infty&lt;x&lt;\infty\]</div>
-<div class="math notranslate nohighlight">
-\[\mbox{cos}(x)=\sum_{n=0}^\infty \frac{(-1)^nx^{2n}}{(2n)!} = 1-\frac{x^2}{2}+\frac{x^4}{4!}-\frac{x^6}{6!}+\cdots,\ -\infty&lt;x&lt;\infty\]</div>
-<div class="math notranslate nohighlight">
-\[\mbox{ln}(1+x)=-\sum_{n=1}^\infty \frac{(-1)^n x^n}{n} = \frac{x^2}{2!}+\frac{x^3}{3!}-\frac{x^4}{4!}+\frac{x^5}{5!}+\cdots,\ -1 &lt; x \leq 1\]</div>
-<div class="math notranslate nohighlight">
-\[\frac{1}{1-x} = \sum_{n=0}^\infty x^n = 1+x+x^2+x^3+\cdots,\ -1 \leq x &lt; 1\]</div>
-<p>Knowing the Taylor series often gives insight into the properties of a function.</p>
-</section>
-</section>
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-  <section class="tex2jax_ignore mathjax_ignore" id="calculating-accumulated-change">
-<h1>08 - Calculating accumulated change<a class="headerlink" href="#calculating-accumulated-change" title="Link to this heading">#</a></h1>
-<section id="series">
-<h2>Series<a class="headerlink" href="#series" title="Link to this heading">#</a></h2>
-<p>For every sequence <span class="math notranslate nohighlight">\(\{a_n\}\)</span> of real numbers, it is possible to define another sequence <span class="math notranslate nohighlight">\(\{S_n\}\)</span> for which the elements are given by the sum of the first <span class="math notranslate nohighlight">\(n\)</span> terms of the sequence <span class="math notranslate nohighlight">\(\{a_n\}\)</span>:</p>
-<div class="math notranslate nohighlight">
-\[\begin{split}\begin{align}
-S_0 &amp;= a_0 \\
-S_1 &amp;= a_0 + a_1 \\
-S_2 &amp;= a_0 + a_1 + a_2 \\
-&amp;\vdots \\
-S_n &amp;= a_0 + a_1 + \cdots + a_{n-1} + a_n \\
-S_n &amp;= \sum_{i = 0}^n a_i \end{align}\end{split}\]</div>
-<p>The sequence <span class="math notranslate nohighlight">\(\{S_n\}\)</span> is called a <em>series</em> (or an infinite series, to emphasize the sum over increasing number of terms <span class="math notranslate nohighlight">\(\{a_n\}\)</span>).</p>
-<p>The same way as we asked about <span class="math notranslate nohighlight">\(\lim a_n\)</span>, we can ask if <span class="math notranslate nohighlight">\(\lim S_n\)</span> exists. If it exists, or, in other words, if the sum of infinite terms <span class="math notranslate nohighlight">\(\{a_n\}\)</span> is equal to a finite number, we say that the series <span class="math notranslate nohighlight">\(S_n\)</span> is convergent.</p>
-<div class="admonition-examples admonition">
-<p class="admonition-title">Examples</p>
-<p>The following expressions are example of convergent series:</p>
-<div class="math notranslate nohighlight">
-\[\begin{split}\begin{align}&amp;\sum_{n=0}^\infty \frac{x^n}{n!} \hspace{0.1cm}, \hspace{0.5cm} x \in \mathbb{R} \\
-&amp;\sum_{n=0}^\infty x^n \hspace{0.1cm}, \hspace{0.5cm} -1&lt;x&lt;1 \\
-&amp;\sum_{n=0}^\infty (-1)^n \frac{x^{2n+1}}{(2n+1)!} \hspace{0.1cm}, \hspace{0.5cm} x \in \mathbb{R}\end{align}\end{split}\]</div>
-</div>
-</section>
-<section id="calculating-areas">
-<h2>Calculating areas<a class="headerlink" href="#calculating-areas" title="Link to this heading">#</a></h2>
-<p>The idea of summing infinite quantities can be used to solve the problem of determining the area below a given curve. Let <span class="math notranslate nohighlight">\(f(x)\)</span> be a function defined on the interval <span class="math notranslate nohighlight">\(a \le x \le b\)</span> as:</p>
-<img alt="../../_images/sum_areas.png" src="../../_images/sum_areas.png" />
-<p>The area below the curve of <span class="math notranslate nohighlight">\(f(x)\)</span> can be found dividing the interval <span class="math notranslate nohighlight">\(a \le x \le b\)</span> into <span class="math notranslate nohighlight">\(n\)</span> subintervals:</p>
-<div class="math notranslate nohighlight">
-\[a=x_0&lt;x_1&lt;x_2&lt;\cdots&lt;x_{n-1}&lt;x_n=b\]</div>
-<p>where each interval defines a rectangle of height <span class="math notranslate nohighlight">\(f(x_i)\)</span> and width <span class="math notranslate nohighlight">\((x_{i+1}-x_i) = \Delta x_i = \displaystyle\frac{(b-a)}{n}\)</span>. The area below the curve will then be approximately the sum of the areas of the rectangles</p>
-<div class="math notranslate nohighlight">
-\[A \approx \sum_{i=0}^n f(x_i)\Delta x_i.\]</div>
-<p>As we increasethe number of intervals <span class="math notranslate nohighlight">\(n\)</span>, while decreasing the widths of each rectangle, the sum of the areas of the rectangles gives a better approximation of the area below the curve. The exact result is given by the limit <span class="math notranslate nohighlight">\(n \rightarrow \infty\)</span>. In this limit, the sum is represented by the following symbol:</p>
-<div class="math notranslate nohighlight">
-\[\lim_{n \to \infty} \sum_{i=0}^n f(x_i) \Delta x_i = \int_a^b f(x)dx\]</div>
-<p>If this limit exists, we say that the function <span class="math notranslate nohighlight">\(f\)</span> is <strong>integrable</strong> on the interval <span class="math notranslate nohighlight">\(a \lt x \lt b\)</span> and denote the limit by the <strong>definite integral</strong> <span class="math notranslate nohighlight">\(\int_a^b f(x)dx\)</span>.</p>
-<p>The interpretation of the integral as area can be generalized for when <span class="math notranslate nohighlight">\(f(x)\)</span> assumes negative values</p>
-<img alt="../../_images/defin_int.png" src="../../_images/defin_int.png" />
-<p>In the case of the previous figure, we will have:</p>
-<div class="math notranslate nohighlight">
-\[\begin{split}\begin{align} \int_a^b f(x)dx 
-&amp;= A_1 - A_2 + A_3 \hspace{0.1cm}, \hspace{0.5cm} (A_i \geq 0) \\
-&amp;= [\text{area above x-axis}] - [\text{area below x-axis}] \end{align}\end{split}\]</div>
-<p>Other intuitive results related to the calculation of the area are:</p>
-<div class="math notranslate nohighlight">
-\[\int_a^a f(x)dx = 0\quad \mbox{(size of the interval equal to zero)}\]</div>
-<div class="math notranslate nohighlight">
-\[\int_a^b f(x)dx = -\int_b^a f(x)dx\quad  \mbox{(orientation of the integral)}\]</div>
-<p>Furthermore, some of the properties of the integral follows directly from the properties of limits:</p>
-<div class="math notranslate nohighlight">
-\[\begin{split}\begin{align}
-\int_a^b kf(x)dx &amp;= k \int_a^b f(x)dx \\
-\int_a^b [f(x) + g(x)]dx &amp;= \int_a^b f(x)dx + \int_a^b g(x)dx \\
-\int_a^b f(x)dx &amp;= \int_a^c f(x)dx + \int_c^b f(x)dx ,\ \  (a \le c \le b) \end{align}\end{split}\]</div>
-</section>
-<section id="fundamental-theorem-of-calculus">
-<h2>Fundamental theorem of Calculus<a class="headerlink" href="#fundamental-theorem-of-calculus" title="Link to this heading">#</a></h2>
-<p>The problem of calculating integrals as limits of sums requires specific methods for each function and becomes quite impractical. Fortunately, this problem can be solved with the very important result that the area problem is intrinsically related to the tangent problem.</p>
-<p>Suppose we define a function <span class="math notranslate nohighlight">\(F(x)\)</span> that gives the area below the curve <span class="math notranslate nohighlight">\(f(u)\)</span> from <span class="math notranslate nohighlight">\(a\)</span> until a variable value <span class="math notranslate nohighlight">\(x\)</span>. By definition:</p>
-<div class="math notranslate nohighlight">
-\[F(x) = \int_a^x f(u)du\]</div>
-<img alt="../../_images/fund_calc.png" src="../../_images/fund_calc.png" />
-<p>If we consider a small increment <span class="math notranslate nohighlight">\(h\)</span> to <span class="math notranslate nohighlight">\(x\)</span>, the new area below the curve shall be given by:</p>
-<div class="math notranslate nohighlight">
-\[F(x+h) = \int_a^{x+h} f(u)du\]</div>
-<p>The difference between these two areas shall be approximately given by the area of the retangle with width <span class="math notranslate nohighlight">\(h\)</span> and height <span class="math notranslate nohighlight">\(f(x)\)</span>:</p>
-<div class="math notranslate nohighlight">
-\[F(x+h) - F(x) \approx hf(x) \implies \displaystyle\frac{F(x+h) - F(x)}{h} \approx f(x)\]</div>
-<p>But see that in the limit where the increment <span class="math notranslate nohighlight">\(h\)</span> gets very small, the approximation gets exactly equal to the increment in area, and we obtain the definition of the derivative of <span class="math notranslate nohighlight">\(F(x)\)</span>:</p>
-<div class="math notranslate nohighlight">
-\[\lim_{h\rightarrow 0} \displaystyle\frac{F(x+h) - F(x)}{h} = \displaystyle\frac{dF}{dx} = f(x).\]</div>
-<p>This results is known as the <em>Fundamental theorem of Calculus</em>, and it say that the instantaneous rate of change of the area below a given curve <span class="math notranslate nohighlight">\(f\)</span> at a point <span class="math notranslate nohighlight">\(x\)</span> is given by the value of the function at that point. In other words, the area <span class="math notranslate nohighlight">\(F(x)\)</span> is a function that, if differentiated, gives the function <span class="math notranslate nohighlight">\(f(x)\)</span>.</p>
-<p>The function <span class="math notranslate nohighlight">\(F(x)\)</span> is then called an <strong>antiderivative</strong> of <span class="math notranslate nohighlight">\(f(x)\)</span>. In general, there is a <em>family</em> of antiderivates of <span class="math notranslate nohighlight">\(f(x)\)</span>, since <span class="math notranslate nohighlight">\(\displaystyle\frac{d(F(x) + C)}{dx}=\displaystyle\frac{dF}{dx}=f(x)\)</span>, independent of the value of the constant <span class="math notranslate nohighlight">\(C\)</span>. We therefore call <strong>indefinite integral</strong>:</p>
-<div class="math notranslate nohighlight">
-\[\int f(x)dx = F(x) + C\]</div>
-<p>It also directly follows that</p>
-<div class="math notranslate nohighlight">
-\[\int_b^c f(x)dx = F(c) - F(b)\]</div>
-<p>Thus the problem of calculating the area below the curve <span class="math notranslate nohighlight">\(f(x)\)</span> from <span class="math notranslate nohighlight">\(b\)</span> to <span class="math notranslate nohighlight">\(c\)</span>, which, by definition, is given by the infinite sum of terms <span class="math notranslate nohighlight">\(f(x_i)\Delta x_i\)</span> (with <span class="math notranslate nohighlight">\(\Delta x_i \rightarrow 0\)</span>), can be reduced to obtaining the antiderivative of <span class="math notranslate nohighlight">\(f\)</span>, <span class="math notranslate nohighlight">\(F(x)\)</span>, and subtracting the values of this function at points <span class="math notranslate nohighlight">\(c\)</span> and <span class="math notranslate nohighlight">\(b\)</span>. The main problem then becomes how to calculate the antiderivates of a given function <span class="math notranslate nohighlight">\(f(x)\)</span>, for which different methods can be used.</p>
-</section>
-<section id="calculating-integrals">
-<h2>Calculating integrals<a class="headerlink" href="#calculating-integrals" title="Link to this heading">#</a></h2>
-<p>The simplest integrals can be calculated by inverting the differentiation of fundamental functions, as</p>
-<div class="math notranslate nohighlight">
-\[\int x^n \,dx = \frac{x^{n+1}}{n+1} + C\]</div>
-<div class="math notranslate nohighlight">
-\[\int e^x \,dx = e^x + C\]</div>
-<div class="math notranslate nohighlight">
-\[\int \frac{1}{x}\,dx = \ln |x| + C\]</div>
-<div class="math notranslate nohighlight">
-\[\int \mbox{cos}(x)\, dx = \mbox{sin}(x) + C\]</div>
-<div class="math notranslate nohighlight">
-\[\int \mbox{sin}(x)\, dx = -\mbox{cos}(x) + C\]</div>
-<p>This are obtained simply because we know that the derivative of the functions on the righ-hand sides of the equations give exactly the function that is being integrated. When the integrals are not so simple, we must apply other methods to turn them into integrals we know.</p>
-<section id="integration-by-substitution">
-<h3>Integration by substitution<a class="headerlink" href="#integration-by-substitution" title="Link to this heading">#</a></h3>
-<p>This method consists in finding an appropriate substitution of the variable of integration so that we end up with a simpler integral to calculate.</p>
-<div class="admonition-examples admonition">
-<p class="admonition-title">Examples</p>
-<ul>
-<li><p><span class="math notranslate nohighlight">\(\int\frac{1}{\sqrt{x+4}}dx\)</span></p>
-<p>Note that if <span class="math notranslate nohighlight">\(u=x+4 \implies \displaystyle\frac{du}{dx}=1 \implies du=dx\)</span>. Thus, the integral can be given by:</p>
-<div class="math notranslate nohighlight">
-\[\int u^{-\frac{1}{2}}du = \frac{1}{\left(-\frac{1}{2}+1\right)} \cdot u^{\frac{1}{2}} + C = 2\sqrt{x+4} + C.\]</div>
-<div class="admonition note">
-<p class="admonition-title">Note</p>
-<p>Note that three mathematicians die everytime we operate with <span class="math notranslate nohighlight">\(du\)</span> and <span class="math notranslate nohighlight">\(dx\)</span> as if they were numbers on a fraction. Formally, the correct substitution on the integral would be given by a function <span class="math notranslate nohighlight">\(u=u(x)\)</span>, so that <span class="math notranslate nohighlight">\( \int f(u(x)) u'(x)dx = \int f(u)du\)</span>. But we can procede with the heresy above as long as we have this in mind.</p>
-</div>
-  <br />
-</li>
-<li><p><span class="math notranslate nohighlight">\(\int xe^{(x^2-1)}dx\)</span></p>
-<p>See that <span class="math notranslate nohighlight">\(u=x^2-1 \implies \displaystyle\frac{du}{dx}=2x \implies  dx=\displaystyle\frac{du}{2x}\)</span>. Then:</p>
-<div class="math notranslate nohighlight">
-\[\int \frac{1}{2}e^{u}du = \frac{1}{2}e^u + C = \frac{e^{(x^2-1)}}{2} + C \]</div>
-</li>
-</ul>
-</div>
-</section>
-<section id="integration-by-parts">
-<h3>Integration by parts<a class="headerlink" href="#integration-by-parts" title="Link to this heading">#</a></h3>
-<p>When integrating by parts, we have to find an appropriate association <span class="math notranslate nohighlight">\((u=u(x), v=v(x))\)</span>, so that:</p>
-<div class="math notranslate nohighlight">
-\[\int uv'dx=uv-\int u'vdx\]</div>
-<div class="admonition-example admonition">
-<p class="admonition-title">Example</p>
-<ul>
-<li><p><span class="math notranslate nohighlight">\(\int x\ln x\, dx \)</span></p>
-<p>Note that</p>
-<div class="math notranslate nohighlight">
-\[\begin{split}\begin{align}
-    u=\ln x \implies u' = \displaystyle\frac{1}{x} \\
-    v'=x \implies v=\displaystyle\frac{x^2}{2} \end{align}\end{split}\]</div>
-<p>Thus:</p>
-<div class="math notranslate nohighlight">
-\[\begin{split}\begin{align}\int x\ln x\, dx 
-&amp;= \frac{x^2}{2}\ln x - \int \frac{x^2}{2} \cdot \frac{1}{x}dx \\
-&amp;= \frac{x^2 \ln x}{2} - \frac{1}{2}\int xdx \\
-&amp;= \frac{x^2 \ln x}{2} - \frac{1}{4}x^2 + C \\
-&amp;= \frac{1}{4} x^2 (2\ln x - 1) + C \end{align}\end{split}\]</div>
-</li>
-<li><p><span class="math notranslate nohighlight">\(\int xe^x \,dx\)</span></p>
-<p>Note that</p>
-<div class="math notranslate nohighlight">
-\[\begin{split}\begin{align}
-    u=x \implies u'=1 \\
-    v'=e^x \implies v=e^x \end{align}\end{split}\]</div>
-<p>Thus:</p>
-<div class="math notranslate nohighlight">
-\[\begin{split}\begin{align}\int xe^xdx  
-    &amp;= xe^x - \int e^x dx \hspace{0.1cm} \\
-    &amp;= xe^x-e^x + C \\
-    &amp;= e^x (x-1) + C \end{align}\end{split}\]</div>
-</li>
-</ul>
-</div>
-<div class="admonition tip">
-<p class="admonition-title">Tip</p>
-<p>For complicated integrals, computer algebra programs like Mathematica or Sympy can be helpful. An especially user-friendly option is the website WolframAlpha.</p>
-</div>
-</section>
-<section id="integration-by-partial-fractions">
-<h3>Integration by partial fractions<a class="headerlink" href="#integration-by-partial-fractions" title="Link to this heading">#</a></h3>
-<p>If the integral has the form</p>
-<div class="math notranslate nohighlight">
-\[\int \frac{P(x)}{Q(x)}dx\]</div>
-<p>where the degree of <span class="math notranslate nohighlight">\(P(x)\)</span> is smaller than the degree of <span class="math notranslate nohighlight">\(Q(x)\)</span>, and <span class="math notranslate nohighlight">\(Q(x)=a(x-x_1)(x-x_2)\cdots(x-x_n)\)</span>, we can write</p>
-<div class="math notranslate nohighlight">
-\[\frac{P(x)}{Q(x)} = \frac{1}{a}\left[\frac{A_1}{x-x_1}+\frac{A_2}{x-x_2}+\cdots+\frac{A_n}{x-x_n}\right]\]</div>
-<p>with <span class="math notranslate nohighlight">\(\{A_n\}\)</span> to be determined.</p>
-<div class="admonition-example admonition">
-<p class="admonition-title">Example</p>
-<ul>
-<li><p><span class="math notranslate nohighlight">\(\int \frac{1}{x(x-1)}\, dx\)</span></p>
-<p>First, note that <span class="math notranslate nohighlight">\(\displaystyle\frac{1}{x(x-1)} =  \displaystyle\frac{A}{x} + \displaystyle\frac{B}{(x-1)} \implies 1 = A(x-1) + Bx \implies 1 = x(A+B) - A\)</span>. By comparing the coeffients of the terms in <span class="math notranslate nohighlight">\(x\)</span> and the terms independent of <span class="math notranslate nohighlight">\(x\)</span>, we obtain:</p>
-<div class="math notranslate nohighlight">
-\[\begin{split}\begin{cases}
-    A + B &amp;= 0\\
-       -A &amp;= 1
-    \end{cases} \implies  
-    \begin{cases}
-    A &amp;= -1 \\
-    B &amp;= 1\end{cases}\end{split}\]</div>
-<p>Thus:</p>
-<div class="math notranslate nohighlight">
-\[\frac{1}{x(x-1)} = -\frac{1}{x} + \frac{1}{x-1}\]</div>
-<p>Then we will have for the integral:</p>
-<div class="math notranslate nohighlight">
-\[\begin{split}\begin{align} \int\frac{1}{x(x-1)}dx  
-    &amp;= \int\left(-\frac{1}{x}+\frac{1}{x-1}\right)dx \\
-    &amp;= -\int \frac{1}{x}dx + \int\frac{1}{x-1}dx \\
-    &amp;= - \ln|x| + \ln|x-1| + C \\
-    &amp;= \ln \left|\frac{x-1}{x}\right| + C\end{align}\end{split}\]</div>
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-  <section class="tex2jax_ignore mathjax_ignore" id="tutorial-01">
-<h1>Tutorial 01<a class="headerlink" href="#tutorial-01" title="Link to this heading">#</a></h1>
-<ol class="arabic">
-<li><p>An autocatalytic reaction uses its resulting product for the formation of a new product, as in the reaction</p>
-<div class="math notranslate nohighlight">
-\[A + X \rightarrow X\]</div>
-<p>If we assume that this reaction occurs in a closed vessel, then the reaction rate is given by</p>
-<div class="math notranslate nohighlight">
-\[
-       R(x) = kx(a - x)
-    \]</div>
-<p>for <span class="math notranslate nohighlight">\(0 \le x \le a\)</span>, where <span class="math notranslate nohighlight">\(a\)</span> is the initial concentration of <span class="math notranslate nohighlight">\(A\)</span> and <span class="math notranslate nohighlight">\(x\)</span> is the concentration of <span class="math notranslate nohighlight">\(X\)</span>.</p>
-<br/>
-<p>(a) Show that <span class="math notranslate nohighlight">\(R(x)\)</span> is a polynomial and determine its degree.
-<br/>
-(b) Without using any graphic visualiser, graph <span class="math notranslate nohighlight">\(R(x)\)</span> for <span class="math notranslate nohighlight">\(k = 2\)</span> and <span class="math notranslate nohighlight">\(a = 6\)</span>. Find the value of <span class="math notranslate nohighlight">\(x\)</span> at which the reaction rate is maximal.</p>
-<br/>
-</li>
-<li><p>Find the maximum/minimum value of the following polynomials</p>
- <br/>
-<p>(a) <span class="math notranslate nohighlight">\(p(x) = x^2 - 2x - 3\)</span></p>
-<p>(b) <span class="math notranslate nohighlight">\(p(x) = -x^2 - 6x + 8\)</span></p>
-<p>(<em>Hint:</em> Calculate the roots and consider the symmetry of the parabola)</p>
- <br/>
-</li>
-<li><p>There is a function that is frequently used to describe the per capita growth rate of organisms when the rate depends on the concentration of some nutrient and becomes saturated for large enough nutrient concentrations. If we denote the concentration of the nutrient by <span class="math notranslate nohighlight">\(N\)</span>, then the per capita growth rate <span class="math notranslate nohighlight">\(r(N)\)</span> is given by the <em>Monod growth function</em></p>
-<div class="math notranslate nohighlight">
-\[
-        r(N)=\frac{aN}{k+N},\ \ \ N \ge 0
-    \]</div>
-<p>where <span class="math notranslate nohighlight">\(a\)</span> and <span class="math notranslate nohighlight">\(k\)</span> are positive constants.</p>
- <br/>
-<p>(a) Use Python or R to investigate the Monod growth function. Graph <span class="math notranslate nohighlight">\(r(N)\)</span> for (i) <span class="math notranslate nohighlight">\(a = 5\)</span> and <span class="math notranslate nohighlight">\(k = 1\)</span>, (ii) <span class="math notranslate nohighlight">\(a = 5\)</span> and <span class="math notranslate nohighlight">\(k = 3\)</span>, and (iii) <span class="math notranslate nohighlight">\(a = 8\)</span> and k = 1. Place all three graphs in the same coordinate system.</p>
-<p>(b) On the basis of the graphs in (a), describe in words what happens when you change <span class="math notranslate nohighlight">\(a\)</span>.</p>
-<p>(c) On the basis of the graphs in (a), describe in words what happens when you change <span class="math notranslate nohighlight">\(k\)</span>.</p>
- <br/>
-</li>
-<li><p>The function</p>
-<div class="math notranslate nohighlight">
-\[
-        f(x) = \frac{x^n}{b^n + x^n},\ \ \ x \ge 0
-    \]</div>
-<p>where <span class="math notranslate nohighlight">\(n\)</span> is a positive integer and <span class="math notranslate nohighlight">\(b\)</span> is a positive real number, is used in biochemistry to model reaction rates as a function of the concentration of some reactants</p>
- <br/>
-<p>(a) Use Python or R to graph <span class="math notranslate nohighlight">\(f(x)\)</span> for <span class="math notranslate nohighlight">\(n = 1, 2,\ \mbox{and}\ 3\)</span> in the same coordinate system when <span class="math notranslate nohighlight">\(b = 2\)</span>.</p>
-<p>(b) Where do the three graphs in (a) intersect?</p>
-<p>(c) What happens to <span class="math notranslate nohighlight">\(f(x)\)</span> as <span class="math notranslate nohighlight">\(x\)</span> gets larger?</p>
-<p>(d) For an arbitrary positive value of <span class="math notranslate nohighlight">\(b\)</span>, show that <span class="math notranslate nohighlight">\(f (b) = 1/2\)</span>. On the basis of this demonstration and your answer in C. ,  explain why <span class="math notranslate nohighlight">\(b\)</span> is called the half-saturation constant.</p>
- <br/>
-</li>
-<li><p>The file <a class="reference external" href="https://imperialcollegelondon.box.com/s/940y521dq1owarx8pt4d4mu2d0f2cz85">words_moby_dick.csv</a> contains data on the cumulative distribution of the number of times specific words occur in the text of the novel <em>Moby Dick</em>, by Herman Melville. Assume that this distribution is a power law of the form</p>
-<div class="math notranslate nohighlight">
-\[
-        y = x^D
-    \]</div>
-<p>(a) Using the provided data, and R/Python, estimate the exponent of this power law.
-(<em>Hint:</em> If you plot this data in log-log scale, what should correspond to the exponent <span class="math notranslate nohighlight">\(D\)</span>?)</p>
-<p>(b) Can you think of a better way to estimate D?</p>
- <br/>
-</li>
-<li><p>Assume that a population size at time <span class="math notranslate nohighlight">\(t\)</span> is <span class="math notranslate nohighlight">\(N(t)\)</span> and that</p>
-<div class="math notranslate nohighlight">
-\[
-        N(t) = 40 \cdot 2^t,\ \ \ t \ge 0
-    \]</div>
-<p>(a) Find the population size at time t = 0.</p>
-<p>(b) Show that</p>
-<div class="math notranslate nohighlight">
-\[
-        N(t) = 40\, \mbox{e}^{t\, ln 2},\ \ \ t \ge 0
-    \]</div>
-<p>(c) How long will it take until the population size reaches 1000?</p>
-<p>(<em>Hint</em>: Find <span class="math notranslate nohighlight">\(t\)</span> so that <span class="math notranslate nohighlight">\(N(t) = 1000\)</span>.)</p>
-<br/>
-</li>
-<li><p>(<em>Adapted from Moss, 1980</em>) Hall (1964) investigated the change in population size of the zooplankton species <em>Daphnia galeata mendota</em> in Base Line Lake, Michigan. The population size <span class="math notranslate nohighlight">\(N(t)\)</span> at time <span class="math notranslate nohighlight">\(t\)</span> was modeled by the equation</p>
-<div class="math notranslate nohighlight">
-\[ 
-        N(t) = N_0\mbox{e}^{rt}
-    \]</div>
-<p>where <span class="math notranslate nohighlight">\(N_0\)</span> denotes the population size at time <span class="math notranslate nohighlight">\(0\)</span>. The constant <span class="math notranslate nohighlight">\(r\)</span> is called the <strong>intrinsic rate of growth</strong>.</p>
- <br/>
-<p>(a) Plot <span class="math notranslate nohighlight">\(N(t)\)</span> as a function of <span class="math notranslate nohighlight">\(t\)</span> if <span class="math notranslate nohighlight">\(N_0 = 100\)</span> and <span class="math notranslate nohighlight">\(r = 2\)</span>. Compare your graph against the graph of <span class="math notranslate nohighlight">\(N(t)\)</span> when <span class="math notranslate nohighlight">\(N_0 = 100\)</span> and <span class="math notranslate nohighlight">\(r = 3\)</span>. Which population grows faster?</p>
-<p>(b) The constant <span class="math notranslate nohighlight">\(r\)</span> is an important quantity because it describes how quickly the population changes. Suppose that you determine the size of the population at the beginning and at the end of a period of length <span class="math notranslate nohighlight">\(1\)</span>, and you find that at the beginning there were <span class="math notranslate nohighlight">\(200\)</span> individuals and after one unit of time there were <span class="math notranslate nohighlight">\(250\)</span> individuals. Determine <span class="math notranslate nohighlight">\(r\)</span>. (<em>Hint</em>: Consider the ratio <span class="math notranslate nohighlight">\(N(t+1)/N(t)\)</span>.)</p>
- <br/>
-</li>
-<li><p>Fish are indeterminate growers; that is, they grow throughout their lifetime. The growth of fish can be described by the von Bertalanffy function</p>
-<div class="math notranslate nohighlight">
-\[
-        L(x) = L_{\infty}(1-\textrm{e}^{-kx} )
-    \]</div>
-<p>for <span class="math notranslate nohighlight">\(x \ge 0\)</span>, where <span class="math notranslate nohighlight">\(L(x)\)</span> is the length of the fish at age <span class="math notranslate nohighlight">\(x\)</span> and <span class="math notranslate nohighlight">\(k\)</span> and <span class="math notranslate nohighlight">\(L_{\infty}\)</span> are positive</p>
- <br/>
-<p>(a) Using Python or R, graph <span class="math notranslate nohighlight">\(L(x)\)</span> for <span class="math notranslate nohighlight">\(L_{\infty} = 20\)</span>, for (i) <span class="math notranslate nohighlight">\(k = 1\)</span> and (ii) <span class="math notranslate nohighlight">\(k = 0.1\)</span>.</p>
-<p>(b) For <span class="math notranslate nohighlight">\(k = 1\)</span>, find <span class="math notranslate nohighlight">\(x\)</span> so that the length is 90% of <span class="math notranslate nohighlight">\(L_{\infty}\)</span>. Repeat for 99% of <span class="math notranslate nohighlight">\(L_{\infty}\)</span>. Can the fish ever attain length <span class="math notranslate nohighlight">\(L_{\infty}\)</span>? Interpret the meaning of <span class="math notranslate nohighlight">\(L_{\infty}\)</span>.</p>
-<p>(c) Compare the graphs obtained in A. Which growth curve reaches 90% of <span class="math notranslate nohighlight">\(L_{\infty}\)</span> faster? Can you explain what happens to the curve of <span class="math notranslate nohighlight">\(L(x)\)</span> when you vary <span class="math notranslate nohighlight">\(k\)</span> (for fixed <span class="math notranslate nohighlight">\(L_{\infty}\)</span>)?</p>
- <br/>
-</li>
-<li><p>For the following pairs of functions, plot the ratio (quotient) between the two using R or Python. Based on the behaviour of the ratio when <span class="math notranslate nohighlight">\(x \rightarrow \infty\)</span> (very large values of <span class="math notranslate nohighlight">\(x\)</span>), how does each of these functions compare to the other in the velocity that they grow?</p>
- <br/>  
-<p>(a) <span class="math notranslate nohighlight">\( f(x) = e^x \quad \text{ and } \quad g(x) = x^2 \)</span></p>
-<p>(b) <span class="math notranslate nohighlight">\( f(x) = e^x \quad \text{ and } \quad g(x) = x^{20} \)</span></p>
-<p>(c) <span class="math notranslate nohighlight">\( f(x) = x \quad \text{ and } \quad g(x) = \log(x) \)</span></p>
- <br/>
-</li>
-<li><p>A community measure that takes both species abundance and species richness into account is the Shannon diversity index <span class="math notranslate nohighlight">\(H\)</span>. To calculate <span class="math notranslate nohighlight">\(H\)</span>, the proportion <span class="math notranslate nohighlight">\(p_i\)</span> of species <span class="math notranslate nohighlight">\(i\)</span> in the community is used. Assume that the community consists of <span class="math notranslate nohighlight">\(S\)</span> species. Then</p>
-<div class="math notranslate nohighlight">
-\[
-        H = -\sum_{i=1}^S p_i\, \textrm{ln}\, p_i = -( p_1 \textrm{ln}\, p_1 + p_2 \textrm{ln}\, p_2 + \cdots + p_S \textrm{ln}\, p_S )
-    \]</div>
-<br/>
-<p>(a) Assume that <span class="math notranslate nohighlight">\(S = 5\)</span> and that all species are equally abundant; that is, <span class="math notranslate nohighlight">\(p_1 = p_2 = \cdots = p_5\)</span>. Compute <span class="math notranslate nohighlight">\(H\)</span>.</p>
-<p>(b) Assume that <span class="math notranslate nohighlight">\(S = 10\)</span> and that all species are equally abundant; that is, <span class="math notranslate nohighlight">\(p_1 = p_2 = \cdots = p_{10}\)</span>. Compute <span class="math notranslate nohighlight">\(H\)</span>.</p>
-<p>(c) A measure of equitability (or evenness) of the species distribution can be measured by dividing the diversity index <span class="math notranslate nohighlight">\(H\)</span> by <span class="math notranslate nohighlight">\(\textrm{ln}\, S\)</span>. Compute <span class="math notranslate nohighlight">\(H/\textrm{ln}\, S\)</span> for <span class="math notranslate nohighlight">\(S = 5\)</span> and <span class="math notranslate nohighlight">\(S = 10\)</span>.</p>
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- <br/>
-</li>
-<li><p>Three different species of insects are reared together in a laboratory cage. They are supplied with two different types of food each day. Each individual of species 1 consumes 3 units of food <span class="math notranslate nohighlight">\(A\)</span> and 5 units of food <span class="math notranslate nohighlight">\(B\)</span>, each individual of species 2 consumes 2 units of food <span class="math notranslate nohighlight">\(A\)</span> and 3 units of food <span class="math notranslate nohighlight">\(B\)</span>, and individual of species 3 consumes 1 unit of food <span class="math notranslate nohighlight">\(A\)</span> and 2 units of food <span class="math notranslate nohighlight">\(B\)</span>. Each day, 500 units of food <span class="math notranslate nohighlight">\(A\)</span> and 900 units of food <span class="math notranslate nohighlight">\(B\)</span> are supplied. How many individuals of each species can be reared together? Is there more than one solution? What happens if we add 550 units of a third type of food, called <span class="math notranslate nohighlight">\(C\)</span>, and each individual of species 1 consumes 2 units of food <span class="math notranslate nohighlight">\(C\)</span>, each individual of species 2 consumes 4 units of food <span class="math notranslate nohighlight">\(C\)</span>, and each individual of species 3 consumes 1 unit of food <span class="math notranslate nohighlight">\(C\)</span>?</p>
- <br/>
-</li>
-<li><p>Networks are abstractions of real systems used to identify and study patterns of connectivity between the individual units that compose those systems. Networks (often referred to as <em>graphs</em>) are mathematically defined as a set of <em>nodes</em>, or vertices, (representing the individual units) and a set of <em>links</em>, or connections, between these nodes, which account for the interaction patterns within the studied systems. In Ecology, graphs have been widely used in studies investigating food-webs, plant-pollinator interactions, and disease transmission in populations.</p>
- <br />
-<p>If the nodes of a network of size <span class="math notranslate nohighlight">\(N\)</span> are indexed with positive integers such as <span class="math notranslate nohighlight">\(i = 1, 2, \ldots, N\)</span>, an useful method for representing the structure of a network is to construct a matrix <span class="math notranslate nohighlight">\(A\)</span> for which the elements <span class="math notranslate nohighlight">\(a_{ij}\)</span> are given by:</p>
-<div class="math notranslate nohighlight">
-\[\begin{split}a_{ij} = \begin{cases}1,\ \mbox{if nodes}\ i \ \mbox{and}\ j \ \mbox{are linked} \\ 0,\ \mbox{otherwise}  \end{cases}\end{split}\]</div>
-<p>Matrix <span class="math notranslate nohighlight">\(A\)</span> is called the <em>adjacency matrix</em> of the network. One example of a network with <span class="math notranslate nohighlight">\(6\)</span> nodes is shown below.</p>
- <br/>  
-<img alt="../../../_images/graph_example.png" src="../../../_images/graph_example.png" />
-<p>(a) Based on the graphical representation of this network, write down its adjacency matrix.</p>
- <br/>
-<p>(b) By the definition of the adjacency matrix, the sum of the matrix elements on row <span class="math notranslate nohighlight">\(i\)</span>, <span class="math notranslate nohighlight">\(k_i = \sum_{j=1}^N a_{ij}\)</span>, corresponds to the number of links of node <span class="math notranslate nohighlight">\(i\)</span>. The quantity <span class="math notranslate nohighlight">\(k_i\)</span> is called the <em>degree</em> of node <span class="math notranslate nohighlight">\(i\)</span>. Calculate the corresponding degrees for each node on the previous network.</p>
- <br/>
-<p>(c) The Laplacian matrix is a mathematical object that can be used to find many useful properties of a network. By definition, <span class="math notranslate nohighlight">\(L\)</span> is constructed as</p>
- <br/>  
-<div class="math notranslate nohighlight">
-\[ L = D - A \]</div>
- <br/>  
-<p>Where <span class="math notranslate nohighlight">\(A\)</span> is the adjacency matrix, and <span class="math notranslate nohighlight">\(D\)</span> is the degree matrix. The degree matrix <span class="math notranslate nohighlight">\(D\)</span> is a diagonal matrix which contains information about the degree of each node. With your answer for itens (a) and (b), calculate <span class="math notranslate nohighlight">\(L\)</span>.</p>
- <br/>
-<p>(d) Use R/Python to numerically calculate the eigenvalues of the Laplacian matrix.</p>
- <br/>
-</li>
-<li><p>Write each system in matrix form
-<br/></p>
-<div class="math notranslate nohighlight">
-\[\begin{split}
-        \mbox{(a)}  \ 
-        \begin{cases}
-            2x_2-x_1 = x_3 \\
-            4x_1 + x_3 = 7x_2 \\
-            x_2 - x_1 = x_3\\
-        \end{cases}
-    \end{split}\]</div>
- <br/>
-<div class="math notranslate nohighlight">
-\[\begin{split}
-        \mbox{(b)}  \ 
-        \begin{cases}
-            2x_1 - 3x_2 = 4 \\
-            -x_1 + x_2 = 3 \\
-            3x_1 = 4\\
-        \end{cases}
-    \end{split}\]</div>
- <br/>
-</li>
-<li><p>Suppose that</p>
- <br/>
-<div class="math notranslate nohighlight">
-\[\begin{split}A = \begin{pmatrix}
-            2 &amp; 4 \\
-            3 &amp; 6 
-            \end{pmatrix}
-        \ \ \ \text{,}\ \ \
-        X = \begin{pmatrix}
-            x \\
-            y 
-            \end{pmatrix} \ \ \ \text{and}\ \ \
-            B = \begin{pmatrix}
-            b_1 \\
-            b_2 \\
-            \end{pmatrix}\end{split}\]</div>
-<p>(a) Write <span class="math notranslate nohighlight">\(AX = B\)</span> as a system of linear equation.</p>
- <br/>
-<p>(b) Show that if</p>
-<div class="math notranslate nohighlight">
-\[\begin{split}
-        B = \begin{pmatrix}
-            3 \\
-            \frac92 \\
-            \end{pmatrix}
-    \end{split}\]</div>
-<p>then <span class="math notranslate nohighlight">\(AX = B\)</span> has infinitely many solutions. Graph the two straight lines associated with the corresponding system of linear equations, and explain why the system has infinitely many solutions.</p>
- <br/>
-<p>(c) Find a column vector</p>
- <br/>
-<div class="math notranslate nohighlight">
-\[\begin{split}B = \begin{pmatrix}
-            b_1 \\
-            b_2 \\
-            \end{pmatrix}\end{split}\]</div>
-<p>so that <span class="math notranslate nohighlight">\(AX = B\)</span> has no solutions.</p>
- <br/>
-</li>
-<li><p>Assume that a population is divided into three age classes and that 80% of the individuals age 0 and 10% of the individuals age 1 survive until the end of the next breeding season. Assume further that individuals age 1 have an average of 1.6 offspring and individuals age 2 have an average of 3.9 offspring. If, at time 0, the population consists of 1000 individuals age 0, 100 individuals age 1, and 20 individuals age 2, find the projection matrix and the age distribution at time 3.</p>
- <br/>
-</li>
-<li><p>Consider the following projection matrix representing a population with five age classes:</p>
- <br/>  
-<div class="math notranslate nohighlight">
-\[\begin{split}
-        P = \begin{pmatrix}
-        0 &amp; 5 &amp; 3 &amp; 2 &amp; 1 \\
-        0.9 &amp; 0 &amp; 0 &amp; 0 &amp; 0 \\
-        0 &amp; 0.3 &amp; 0 &amp; 0 &amp; 0 \\
-        0 &amp; 0 &amp; 0.1 &amp; 0 &amp; 0 \\
-        0 &amp; 0 &amp; 0 &amp; 0.05 &amp; 0 \\
-    \end{pmatrix}
-    \end{split}\]</div>
- <br/>
-<p>(a) Making use of Python or R code, begin with a population distributed as <span class="math notranslate nohighlight">\((0,0,0,0,10)\)</span> individuals at time <span class="math notranslate nohighlight">\(t=0\)</span>, project the population at time <span class="math notranslate nohighlight">\(t=20\)</span>, and plot the total number over time. Plot the logarithm of the total population over time.</p>
- <br/>
-<p>(b) Repeat the above but begin with <span class="math notranslate nohighlight">\((80,16,5,1,1)\)</span> individuals. How do these results compare to the results in (a) ?</p>
- <br/>
-</li>
-<li><p>In the problems below, find the eigenvalues <span class="math notranslate nohighlight">\(\lambda_1\)</span> and <span class="math notranslate nohighlight">\(\lambda_2\)</span> and corresponding eigenvectors <span class="math notranslate nohighlight">\(\mathbf{v_1}\)</span> and <span class="math notranslate nohighlight">\(\mathbf{v_2}\)</span> for each matrix <span class="math notranslate nohighlight">\(B\)</span>. Determine the equations of the lines through the origin in the direction of the eigenvectors <span class="math notranslate nohighlight">\(\mathbf{v_1}\)</span> and <span class="math notranslate nohighlight">\(\mathbf{v_2}\)</span>, and graph the lines together with the eigenvectors <span class="math notranslate nohighlight">\(\mathbf{v_1}\)</span> and <span class="math notranslate nohighlight">\(\mathbf{v_2}\)</span> and the vectors <span class="math notranslate nohighlight">\(A\mathbf{v_1}\)</span> and <span class="math notranslate nohighlight">\(A\mathbf{v_2}\)</span>.</p>
- <br/>
-<div class="math notranslate nohighlight">
-\[\begin{split}
-    \text{(a)} \ \ B = \begin{pmatrix}
-    2 &amp; 3\\
-    0 &amp; -1
-    \end{pmatrix}
-    \end{split}\]</div>
- <br/>
-<div class="math notranslate nohighlight">
-\[\begin{split}
-    \text{(b)} \ \ B = \begin{pmatrix}
-    3 &amp; 6\\
-    -1 &amp; -4
-    \end{pmatrix}  
-    \end{split}\]</div>
- <br/>
-</li>
-<li><p>Suppse that</p>
- <br/>
-<div class="math notranslate nohighlight">
-\[\begin{split}
-    P = \begin{pmatrix}
-    0.5 &amp; 2.3\\
-    a &amp; 0
-    \end{pmatrix}
-    \end{split}\]</div>
- <br/>
-<p>is the projection matrix of a population with two age classes. For which values of <span class="math notranslate nohighlight">\(a\)</span> does this population grow?</p>
- <br/>
-</li>
-<li><p>Suppose that</p>
-<br/>
-<div class="math notranslate nohighlight">
-\[\begin{split}
-    P = \begin{pmatrix}
-    0.5 &amp; 2.0\\
-    0.1 &amp; 0
-    \end{pmatrix}
-    \end{split}\]</div>
-<br/>
-<p>is the projection matrix of a population with two age classes.</p>
-<br/>
-<p>(a) If you were to manage this population, would you need to be concerned about its long-term survival?</p>
-<br/>
-<p>(b) Suppose that you can improve either the fecundity or the survival of the zero-year-olds, but due to physiological and environmental constraints, the fecundity of zero-year-olds will not exceed 1.5 and the survival of zero-year-olds will not exceed 0.4. Investigate how the growth rate of the population is affected by changing either the survival or the fecundity of zero-year-olds, or both. What would be the maximum achievable growth rate?</p>
-<br/>
-<p>(c) In real situations, what other factors might you need to consider when you decide on management strategies?</p>
-</li>
-</ol>
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-<h1>Tutorial 06<a class="headerlink" href="#tutorial-06" title="Link to this heading">#</a></h1>
-<ol class="arabic">
-<li><p><strong>Island Model</strong> Assume that a species lives in a habitat that consists of many islands close to a mainland. The species occupies both the mainland and the islands, but, although it is present on the mainland at all times, it frequently goes extinct on the islands. Islands can be recolonized by migrants from the mainland. The following model keeps track of the fraction of islands occupied: Denote the fraction of islands occupied at time <span class="math notranslate nohighlight">\(t\)</span> by <span class="math notranslate nohighlight">\(p(t)\)</span>. Assume that each island experiences a constant risk of extinction and that vacant islands (the fraction <span class="math notranslate nohighlight">\(1-p\)</span>) are colonized from the mainland at a constant rate. Then</p>
-<div class="math notranslate nohighlight">
-\[
-        \frac{dp}{dt}=c(1-p)-ep
-    \]</div>
-<p>where <span class="math notranslate nohighlight">\(c\)</span> and <span class="math notranslate nohighlight">\(e\)</span> are positive constants.</p>
- <br/>
-<p>(a) The gain from colonization is <span class="math notranslate nohighlight">\(f (p) = c(1 - p)\)</span> and the loss from extinction is <span class="math notranslate nohighlight">\(g(p) = ep\)</span>. Graph (without graph visualisers) <span class="math notranslate nohighlight">\(f(p)\)</span> and <span class="math notranslate nohighlight">\(g(p)\)</span> for <span class="math notranslate nohighlight">\(0 \le p \le 1\)</span> in the same coordinate system. Explain why the two graphs intersect whenever <span class="math notranslate nohighlight">\(e\)</span> and <span class="math notranslate nohighlight">\(c\)</span> are both positive. Compute the point of intersection and interpret its biological meaning.</p>
-  <br/>
-<p>(b) The parameter <span class="math notranslate nohighlight">\(c\)</span> measures how quickly a vacant island becomes colonized from the mainland. The closer the islands, the larger is the value of <span class="math notranslate nohighlight">\(c\)</span>. Use your graph in (a) to explain what happens to the point of intersection of the two lines as c increases. Interpret your result in biological terms.</p>
- <br />  
-</li>
-<li><p><strong>Biotic Diversity</strong> (Adapted from Valentine, 1985.) Walker and Valentine (1984) suggested a model for species diversity which assumes that species extinction rates are independent of diversity but speciation rates are regulated by competition. Denoting the number of species at time <span class="math notranslate nohighlight">\(t\)</span> by <span class="math notranslate nohighlight">\(S(t)\)</span>, the speciation rate by <span class="math notranslate nohighlight">\(b\)</span>, and the extinction rate by <span class="math notranslate nohighlight">\(a\)</span>, they used the model</p>
-<div class="math notranslate nohighlight">
-\[
-        \frac{dS}{dt} = S\left[b\left(1-\frac{S}{K}\right)-a \right]
-    \]</div>
-<p>where <span class="math notranslate nohighlight">\(K\)</span> denotes the number of “niches”, or potential places for species in the ecosystem.</p>
- <br/>
-<p>(a) Find possible equilibria (<span class="math notranslate nohighlight">\(dS/dt = 0\)</span>) under the condition <span class="math notranslate nohighlight">\(a &lt; b\)</span>.</p>
- <br/>
-<p>(b) Use your result in (a) to explain the following statement by Valentine (1985):</p>
-<div class="admonition-quotation admonition">
-<p class="admonition-title">Quotation</p>
-<p>“In this situation, ecosystems are never ‘full’, with all potential niches occupied by species so long as the extinction rate is above zero.”</p>
-</div>
- <br/>
-<p>(c) What happens when <span class="math notranslate nohighlight">\(a \ge b\)</span>?</p>
- <br />  
-</li>
-<li><p>Find a point on the curve</p>
-<div class="math notranslate nohighlight">
-\[y = 1 - 3x^3\]</div>
-<p>whose tangent line is parallel to the line <span class="math notranslate nohighlight">\(y = -x\)</span>. Is there more than one such point? If so, find all other points with this property.</p>
- <br />  
-</li>
-<li><p>Find a point on the curve</p>
-<div class="math notranslate nohighlight">
-\[y = 2x^3 - 4x + 1\]</div>
-<p>whose tangent line is parallel to the line <span class="math notranslate nohighlight">\(y-2x = 1\)</span>. Is there more than one such point? If so, find all other points with this property.</p>
- <br />  
-</li>
-<li><p>In the following, use the product and quotient rules to find the derivative with respect to the independent variable.</p>
- <br/>
-<p>(a) <span class="math notranslate nohighlight">\(f(x) = (2x^3 - 1)(3 + 2x^2)\)</span></p>
- <br/>
-<p>(b) <span class="math notranslate nohighlight">\(h(s) = (4 - 3s^2 + 4s^3)^2\)</span></p>
- <br/>
-<p>(c) <span class="math notranslate nohighlight">\(f(x) =\sqrt{3x}(x^2 - 1)\)</span></p>
- <br/>
-<p>(d) <span class="math notranslate nohighlight">\(g(t) = 4(3t^2 - 1)(2t + 1)\)</span></p>
- <br/>
-<p>(e) <span class="math notranslate nohighlight">\(f(x) = 3(x^2 + 2)(4x^2 - 5x^4) - 3\)</span></p>
- <br/>
-<p>(f) <span class="math notranslate nohighlight">\(g(x) = \displaystyle\frac{1 - 4x^3}{1 - x}\)</span></p>
- <br/>
-<p>(g) <span class="math notranslate nohighlight">\(h(x) = \displaystyle\frac{1 + 2x^2 - 4x^4}{3x^3 - 5x^5}\)</span></p>
- <br/>
-<p>(h) <span class="math notranslate nohighlight">\(f(t) =\displaystyle\frac{3 - t^2}{(t + 1)^2}\)</span></p>
- <br/>
-<p>(i) <span class="math notranslate nohighlight">\(h(s) = \displaystyle\frac{2s^3 - 4s^2 + 5s - 7}{(s^2 - 3)^2}\)</span></p>
- <br/>
-<p>(j) <span class="math notranslate nohighlight">\(g(s) = \displaystyle\frac{s^{1/7} - s^{2/7}}{s^{3/7} + s^{4/7}}\)</span></p>
- <br />  
-</li>
-<li><p><strong>Logistic Growth</strong></p>
- <br/>
-<p>(a) Find the derivative of the logistic growth curve</p>
-<div class="math notranslate nohighlight">
-\[N(t)=\frac{K}{1+\left(\frac{K}{N(0)}-1\right)e^{-rt}}\]</div>
-<p>where <span class="math notranslate nohighlight">\(r\)</span> and <span class="math notranslate nohighlight">\(K\)</span> are positive constants and <span class="math notranslate nohighlight">\(N(0)\)</span> is the population size at time <span class="math notranslate nohighlight">\(0\)</span>.</p>
- <br/>
-<p>(b) Show that <span class="math notranslate nohighlight">\(N(t)\)</span> satisfies the equation</p>
-<div class="math notranslate nohighlight">
-\[\frac{dN}{dt}=rN\left(1-\frac{N}{K}\right)\]</div>
-<p>[<em>Hint:</em> Use the function <span class="math notranslate nohighlight">\(N(t)\)</span> given in (a) for the right-hand side, and simplify until you obtain the derivative of N(t) that you computed in A.]</p>
- <br/>
-<p>(c) What happens to the <em>per-capita</em> rate of growth <span class="math notranslate nohighlight">\(\frac{1}{N}\frac{dN}{dt}\)</span> as you increase N? What biological principle and what type of interaction is your model describing?</p>
- <br />  
-</li>
-<li><p>In the following, find the derivative with respect to the independent variable (<em>Hint</em>: use the chain rule).</p>
- <br/>
-<p>(a) <span class="math notranslate nohighlight">\(f(x) = \mbox{sin}(e^{2x} + x)\)</span></p>
- <br/>
-<p>(b) <span class="math notranslate nohighlight">\(h(x) = \mbox{exp}[x^2 - 2 \mbox{cos}\, x]\)</span></p>
- <br/>
-<p>(c) <span class="math notranslate nohighlight">\(f(x) =\displaystyle\frac{x-2^{−x}}
-{1+x2^{−x}}\)</span></p>
- <br/>
-<p>(d) <span class="math notranslate nohighlight">\(g(t) = \mbox{log}\left(\displaystyle\frac{1 - t}{1 + 2t}\right)\)</span></p>
- <br/>
-<p>(e) <span class="math notranslate nohighlight">\(f(t) = \mbox{sin}(\mbox{ln}(3t))\)</span></p>
- <br/>
-<p>(f) <span class="math notranslate nohighlight">\(g(x) = \mbox{ln}(\mbox{cos}(1 - x))\)</span></p>
- <br/>
-<p>(g) <span class="math notranslate nohighlight">\(h(x) = \displaystyle\frac{\sqrt{x^2 - 1}}{2+\sqrt{x^2+1}}\)</span></p>
- <br/>
-<p>(h) <span class="math notranslate nohighlight">\(f(t) =\sqrt[3]{t^2+\sqrt{1-t}}\)</span></p>
- <br/>
-</li>
-<li><p>The functional responses of some predators are sigmoidal; that is, the number of prey attacked per predator as a function of prey density has a sigmoidal shape. If we denote the prey density by <span class="math notranslate nohighlight">\(N\)</span>, the predator density by <span class="math notranslate nohighlight">\(P\)</span>, the time available for searching for prey by <span class="math notranslate nohighlight">\(T\)</span>, and the handling time of each prey item per predator by <span class="math notranslate nohighlight">\(T_h\)</span>, then the number of prey encounters per predator as a function of <span class="math notranslate nohighlight">\(N\)</span>, <span class="math notranslate nohighlight">\(T\)</span>, and <span class="math notranslate nohighlight">\(T_h\)</span> can be expressed as</p>
-<div class="math notranslate nohighlight">
-\[
-        f (N, T, T_h) = \frac{b^2N^2T}{1 + cN + bT_hN^2}
-    \]</div>
-<p>where <span class="math notranslate nohighlight">\(b\)</span> and <span class="math notranslate nohighlight">\(c\)</span> are positive constants.</p>
- <br/>
-<p>(a) Calculate the first-order partial derivative of <span class="math notranslate nohighlight">\(f(N, T, T_h)\)</span> with respect to the prey density <span class="math notranslate nohighlight">\(N\)</span>. If <span class="math notranslate nohighlight">\(N\)</span> increases, what happens to <span class="math notranslate nohighlight">\(f(N, T, T_h)\)</span>?</p>
- <br/>
-<p>(b) Calculate the first-order partial derivative of <span class="math notranslate nohighlight">\(f(N, T, T_h)\)</span> with respect to the the time <span class="math notranslate nohighlight">\(T\)</span> available for search. If <span class="math notranslate nohighlight">\(T\)</span> increases, what happens to <span class="math notranslate nohighlight">\(f(N, T, T_h)\)</span>?</p>
- <br/>
-<p>(c) Calculate the first-order partial derivative of <span class="math notranslate nohighlight">\(f(N, T, T_h)\)</span> with respect to the handling time <span class="math notranslate nohighlight">\(T_h\)</span>. If <span class="math notranslate nohighlight">\(T_h\)</span> increases, what happens to <span class="math notranslate nohighlight">\(f(N, T, T_h)\)</span>?</p>
-<p>[<em>Hint:</em> The function <span class="math notranslate nohighlight">\(f(N, T, T_h)\)</span> will be an increasing (decreasing) function of the independent variable if the first-order partial derivative of <span class="math notranslate nohighlight">\(f\)</span> with respect to this variable is positive (negative).]</p>
-</li>
-</ol>
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diff --git a/notebooks/Appendix-Assessment.html b/notebooks/Appendix-Assessment.html
index 0b3162d1..aa774504 100644
--- a/notebooks/Appendix-Assessment.html
+++ b/notebooks/Appendix-Assessment.html
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-    <title>TMQB Coursework Assessment &#8212; TheMulQuaBio</title>
+    <title>TMQB Coursework Assessment &#8212; MulQuaBio</title>
   
   
   
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-    <script>document.write(`<img src="../_static/logo.png" class="logo__image only-dark" alt="TheMulQuaBio - Home"/>`);</script>
+    <img src="../_static/logo.png" class="logo__image only-light" alt="MulQuaBio - Home"/>
+    <script>document.write(`<img src="../_static/logo.png" class="logo__image only-dark" alt="MulQuaBio - Home"/>`);</script>
   
   
 </a></div>
@@ -186,23 +186,9 @@
 <li class="toctree-l1"><a class="reference internal" href="Python.html">Biological Computing in Python</a></li>
 <li class="toctree-l1"><a class="reference internal" href="R.html">Biological Computing in R</a></li>
 </ul>
-<p aria-level="2" class="caption" role="heading"><span class="caption-text">Maths for Biologists</span></p>
-<ul class="nav bd-sidenav">
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture01/01_Making_mathematical_statements.html">01 - Making mathematical statements</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture02/02_Describing_shapes_and_patterns.html">02   - Describing shapes and patterns</a></li>
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-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture06/06_Analysing_change_continuously.html">06 - Analysing change, continuously</a></li>
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-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture08/08_Calculating_accumulated_change.html">08 - Calculating accumulated change</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/tutorials/lecture01/Tutorial_01.html">Tutorial 01</a></li>
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-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/tutorials/lecture06/Tutorial_06.html">Tutorial 06</a></li>
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 <p aria-level="2" class="caption" role="heading"><span class="caption-text">Basic Data Analyses and Statistics</span></p>
 <ul class="nav bd-sidenav">
-<li class="toctree-l1"><a class="reference internal" href="../Stats-Intro.html">Basic Data Science and Statistics: Introduction</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../intro-Stats.html">Introduction</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Data_R.html">Data Management and Visualization</a></li>
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@@ -222,6 +208,7 @@
 <li class="toctree-l1"><a class="reference internal" href="Appendix-JupyIntro.html">Introduction to Jupyter</a></li>
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 <li class="toctree-l1"><a class="reference internal" href="Appendix-Maths.html">Mathematical models in Jupyter</a></li>
+<li class="toctree-l1"><a class="reference internal" href="Appendix-MathsForBiologists/MathsForBiologists.html">Maths for Biologists</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-Databases.html">Databases <span class="tocSkip"></span></a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-NLLS-Python.html">Model fitting in Python</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-MiniProj.html">The Computing Miniproject</a></li>
@@ -281,7 +268,7 @@
       
       
       
-      <li><a href="https://github.com/mhasoba/TheMulQuaBio" target="_blank"
+      <li><a href="https://github.com/MulQuaBio/MQB" target="_blank"
    class="btn btn-sm btn-source-repository-button dropdown-item"
    title="Source repository"
    data-bs-placement="left" data-bs-toggle="tooltip"
@@ -298,7 +285,7 @@
       
       
       
-      <li><a href="https://github.com/mhasoba/TheMulQuaBio/issues/new?title=Issue%20on%20page%20%2Fnotebooks/Appendix-Assessment.html&body=Your%20issue%20content%20here." target="_blank"
+      <li><a href="https://github.com/MulQuaBio/MQB/issues/new?title=Issue%20on%20page%20%2Fnotebooks/Appendix-Assessment.html&body=Your%20issue%20content%20here." target="_blank"
    class="btn btn-sm btn-source-issues-button dropdown-item"
    title="Open an issue"
    data-bs-placement="left" data-bs-toggle="tooltip"
@@ -690,7 +677,7 @@ <h3>Appropriate usage of AI for coding<a class="headerlink" href="#appropriate-u
   <div class="footer-item">
     
 <p class="component-author">
-By TheMulQuaBio collective!
+By MulQuaBio collective!
 </p>
 
   </div>
diff --git a/notebooks/Appendix-Data-Python.html b/notebooks/Appendix-Data-Python.html
index 56388b8e..ccb2e829 100644
--- a/notebooks/Appendix-Data-Python.html
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-    <title>Data analyses with Python &amp; Jupyter &#8212; TheMulQuaBio</title>
+    <title>Data analyses with Python &amp; Jupyter &#8212; MulQuaBio</title>
   
   
   
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-    <script>document.write(`<img src="../_static/logo.png" class="logo__image only-dark" alt="TheMulQuaBio - Home"/>`);</script>
+    <img src="../_static/logo.png" class="logo__image only-light" alt="MulQuaBio - Home"/>
+    <script>document.write(`<img src="../_static/logo.png" class="logo__image only-dark" alt="MulQuaBio - Home"/>`);</script>
   
   
 </a></div>
@@ -186,23 +186,9 @@
 <li class="toctree-l1"><a class="reference internal" href="Python.html">Biological Computing in Python</a></li>
 <li class="toctree-l1"><a class="reference internal" href="R.html">Biological Computing in R</a></li>
 </ul>
-<p aria-level="2" class="caption" role="heading"><span class="caption-text">Maths for Biologists</span></p>
-<ul class="nav bd-sidenav">
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture01/01_Making_mathematical_statements.html">01 - Making mathematical statements</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture02/02_Describing_shapes_and_patterns.html">02   - Describing shapes and patterns</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture03/03_Analysing_change%2C_one_step_at_a_time.html">03   - Analysing change, one step at a time</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture04/04_From_table_arrangements_to_powerful_tools.html">04 - From table arrangements to powerful tools</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture05/05_From_table_arrangements_to_powerful_tools_part2.html">05 - From table arrangements to powerful tools - part 2</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture06/06_Analysing_change_continuously.html">06 - Analysing change, continuously</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture07/07_Analysing_change_continuously_part2.html">07 - Analysing change, continuously (Part 2)</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture08/08_Calculating_accumulated_change.html">08 - Calculating accumulated change</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/tutorials/lecture01/Tutorial_01.html">Tutorial 01</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/tutorials/lecture05/Tutorial_05.html">Tutorial 05</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/tutorials/lecture06/Tutorial_06.html">Tutorial 06</a></li>
-</ul>
 <p aria-level="2" class="caption" role="heading"><span class="caption-text">Basic Data Analyses and Statistics</span></p>
 <ul class="nav bd-sidenav">
-<li class="toctree-l1"><a class="reference internal" href="../Stats-Intro.html">Basic Data Science and Statistics: Introduction</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../intro-Stats.html">Introduction</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Data_R.html">Data Management and Visualization</a></li>
 <li class="toctree-l1"><a class="reference internal" href="ExpDesign.html">Experimental design and Data exploration</a></li>
 <li class="toctree-l1"><a class="reference internal" href="t_F_tests.html">Basic hypothesis testing: <span class="math notranslate nohighlight">\(t\)</span> and <span class="math notranslate nohighlight">\(F\)</span> tests</a></li>
@@ -215,13 +201,14 @@
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 <ul class="nav bd-sidenav">
 <li class="toctree-l1"><a class="reference internal" href="glm.html">Generalised Linear Models</a></li>
-<li class="toctree-l1"><a class="reference internal" href="ModelFitting-NLLS.html">Model Fitting using Non-linear Least-squares</a></li>
+<li class="toctree-l1"><a class="reference internal" href="NLLS.html">Model Fitting using Non-linear Least-squares</a></li>
 </ul>
 <p aria-level="2" class="caption" role="heading"><span class="caption-text">Appendices</span></p>
 <ul class="current nav bd-sidenav">
 <li class="toctree-l1"><a class="reference internal" href="Appendix-JupyIntro.html">Introduction to Jupyter</a></li>
 <li class="toctree-l1 current active"><a class="current reference internal" href="#">Data analyses  with Python &amp; Jupyter</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-Maths.html">Mathematical models in Jupyter</a></li>
+<li class="toctree-l1"><a class="reference internal" href="Appendix-MathsForBiologists/MathsForBiologists.html">Maths for Biologists</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-Databases.html">Databases <span class="tocSkip"></span></a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-NLLS-Python.html">Model fitting in Python</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-MiniProj.html">The Computing Miniproject</a></li>
@@ -281,7 +268,7 @@
       
       
       
-      <li><a href="https://github.com/mhasoba/TheMulQuaBio" target="_blank"
+      <li><a href="https://github.com/MulQuaBio/MQB" target="_blank"
    class="btn btn-sm btn-source-repository-button dropdown-item"
    title="Source repository"
    data-bs-placement="left" data-bs-toggle="tooltip"
@@ -298,7 +285,7 @@
       
       
       
-      <li><a href="https://github.com/mhasoba/TheMulQuaBio/issues/new?title=Issue%20on%20page%20%2Fnotebooks/Appendix-Data-Python.html&body=Your%20issue%20content%20here." target="_blank"
+      <li><a href="https://github.com/MulQuaBio/MQB/issues/new?title=Issue%20on%20page%20%2Fnotebooks/Appendix-Data-Python.html&body=Your%20issue%20content%20here." target="_blank"
    class="btn btn-sm btn-source-issues-button dropdown-item"
    title="Open an issue"
    data-bs-placement="left" data-bs-toggle="tooltip"
@@ -973,7 +960,7 @@ <h2>Readings and Resources<a class="headerlink" href="#readings-and-resources" t
   <div class="footer-item">
     
 <p class="component-author">
-By TheMulQuaBio collective!
+By MulQuaBio collective!
 </p>
 
   </div>
diff --git a/notebooks/Appendix-Databases.html b/notebooks/Appendix-Databases.html
index 390f5f3a..1e578f3f 100644
--- a/notebooks/Appendix-Databases.html
+++ b/notebooks/Appendix-Databases.html
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     <meta charset="utf-8" />
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-    <title>Databases  &#8212; TheMulQuaBio</title>
+    <title>Databases  &#8212; MulQuaBio</title>
   
   
   
@@ -32,7 +32,7 @@
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+    <link rel="stylesheet" type="text/css" href="../_static/mystnb.4510f1fc1dee50b3e5859aac5469c37c29e427902b24a333a5f9fcb2f0b3ac41.css?v=be8a1c11" />
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@@ -59,7 +59,7 @@
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     <link rel="search" title="Search" href="../search.html" />
     <link rel="next" title="Model fitting in Python" href="Appendix-NLLS-Python.html" />
-    <link rel="prev" title="Mathematical models in Jupyter" href="Appendix-Maths.html" />
+    <link rel="prev" title="Maths for Biologists" href="Appendix-MathsForBiologists/MathsForBiologists.html" />
   <meta name="viewport" content="width=device-width, initial-scale=1"/>
   <meta name="docsearch:language" content="en"/>
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-    <script>document.write(`<img src="../_static/logo.png" class="logo__image only-dark" alt="TheMulQuaBio - Home"/>`);</script>
+    <img src="../_static/logo.png" class="logo__image only-light" alt="MulQuaBio - Home"/>
+    <script>document.write(`<img src="../_static/logo.png" class="logo__image only-dark" alt="MulQuaBio - Home"/>`);</script>
   
   
 </a></div>
@@ -186,23 +186,9 @@
 <li class="toctree-l1"><a class="reference internal" href="Python.html">Biological Computing in Python</a></li>
 <li class="toctree-l1"><a class="reference internal" href="R.html">Biological Computing in R</a></li>
 </ul>
-<p aria-level="2" class="caption" role="heading"><span class="caption-text">Maths for Biologists</span></p>
-<ul class="nav bd-sidenav">
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture01/01_Making_mathematical_statements.html">01 - Making mathematical statements</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture02/02_Describing_shapes_and_patterns.html">02   - Describing shapes and patterns</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture03/03_Analysing_change%2C_one_step_at_a_time.html">03   - Analysing change, one step at a time</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture04/04_From_table_arrangements_to_powerful_tools.html">04 - From table arrangements to powerful tools</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture05/05_From_table_arrangements_to_powerful_tools_part2.html">05 - From table arrangements to powerful tools - part 2</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture06/06_Analysing_change_continuously.html">06 - Analysing change, continuously</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture07/07_Analysing_change_continuously_part2.html">07 - Analysing change, continuously (Part 2)</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture08/08_Calculating_accumulated_change.html">08 - Calculating accumulated change</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/tutorials/lecture01/Tutorial_01.html">Tutorial 01</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/tutorials/lecture05/Tutorial_05.html">Tutorial 05</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/tutorials/lecture06/Tutorial_06.html">Tutorial 06</a></li>
-</ul>
 <p aria-level="2" class="caption" role="heading"><span class="caption-text">Basic Data Analyses and Statistics</span></p>
 <ul class="nav bd-sidenav">
-<li class="toctree-l1"><a class="reference internal" href="../Stats-Intro.html">Basic Data Science and Statistics: Introduction</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../intro-Stats.html">Introduction</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Data_R.html">Data Management and Visualization</a></li>
 <li class="toctree-l1"><a class="reference internal" href="ExpDesign.html">Experimental design and Data exploration</a></li>
 <li class="toctree-l1"><a class="reference internal" href="t_F_tests.html">Basic hypothesis testing: <span class="math notranslate nohighlight">\(t\)</span> and <span class="math notranslate nohighlight">\(F\)</span> tests</a></li>
@@ -215,13 +201,14 @@
 <p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced Data Analyses and Statistics</span></p>
 <ul class="nav bd-sidenav">
 <li class="toctree-l1"><a class="reference internal" href="glm.html">Generalised Linear Models</a></li>
-<li class="toctree-l1"><a class="reference internal" href="ModelFitting-NLLS.html">Model Fitting using Non-linear Least-squares</a></li>
+<li class="toctree-l1"><a class="reference internal" href="NLLS.html">Model Fitting using Non-linear Least-squares</a></li>
 </ul>
 <p aria-level="2" class="caption" role="heading"><span class="caption-text">Appendices</span></p>
 <ul class="current nav bd-sidenav">
 <li class="toctree-l1"><a class="reference internal" href="Appendix-JupyIntro.html">Introduction to Jupyter</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-Data-Python.html">Data analyses  with Python &amp; Jupyter</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-Maths.html">Mathematical models in Jupyter</a></li>
+<li class="toctree-l1"><a class="reference internal" href="Appendix-MathsForBiologists/MathsForBiologists.html">Maths for Biologists</a></li>
 <li class="toctree-l1 current active"><a class="current reference internal" href="#">Databases <span class="tocSkip"></span></a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-NLLS-Python.html">Model fitting in Python</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-MiniProj.html">The Computing Miniproject</a></li>
@@ -281,7 +268,7 @@
       
       
       
-      <li><a href="https://github.com/mhasoba/TheMulQuaBio" target="_blank"
+      <li><a href="https://github.com/MulQuaBio/MQB" target="_blank"
    class="btn btn-sm btn-source-repository-button dropdown-item"
    title="Source repository"
    data-bs-placement="left" data-bs-toggle="tooltip"
@@ -298,7 +285,7 @@
       
       
       
-      <li><a href="https://github.com/mhasoba/TheMulQuaBio/issues/new?title=Issue%20on%20page%20%2Fnotebooks/Appendix-Databases.html&body=Your%20issue%20content%20here." target="_blank"
+      <li><a href="https://github.com/MulQuaBio/MQB/issues/new?title=Issue%20on%20page%20%2Fnotebooks/Appendix-Databases.html&body=Your%20issue%20content%20here." target="_blank"
    class="btn btn-sm btn-source-issues-button dropdown-item"
    title="Open an issue"
    data-bs-placement="left" data-bs-toggle="tooltip"
@@ -1128,12 +1115,12 @@ <h2>Readings and Resources<a class="headerlink" href="#readings-and-resources" t
                   
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-       href="Appendix-Maths.html"
+       href="Appendix-MathsForBiologists/MathsForBiologists.html"
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-        <p class="prev-next-title">Mathematical models in Jupyter</p>
+        <p class="prev-next-title">Maths for Biologists</p>
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@@ -1182,7 +1169,7 @@ <h2>Readings and Resources<a class="headerlink" href="#readings-and-resources" t
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+By MulQuaBio collective!
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diff --git a/notebooks/Appendix-HighPerformanceComputing.html b/notebooks/Appendix-HighPerformanceComputing.html
index 2d0f95f2..0c4eb55b 100644
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-    <title>Introduction to High-Performance Computing (HPC) &#8212; TheMulQuaBio</title>
+    <title>Introduction to High-Performance Computing (HPC) &#8212; MulQuaBio</title>
   
   
   
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-    <script>document.write(`<img src="../_static/logo.png" class="logo__image only-dark" alt="TheMulQuaBio - Home"/>`);</script>
+    <img src="../_static/logo.png" class="logo__image only-light" alt="MulQuaBio - Home"/>
+    <script>document.write(`<img src="../_static/logo.png" class="logo__image only-dark" alt="MulQuaBio - Home"/>`);</script>
   
   
 </a></div>
@@ -185,23 +185,9 @@
 <li class="toctree-l1"><a class="reference internal" href="Python.html">Biological Computing in Python</a></li>
 <li class="toctree-l1"><a class="reference internal" href="R.html">Biological Computing in R</a></li>
 </ul>
-<p aria-level="2" class="caption" role="heading"><span class="caption-text">Maths for Biologists</span></p>
-<ul class="nav bd-sidenav">
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture01/01_Making_mathematical_statements.html">01 - Making mathematical statements</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture02/02_Describing_shapes_and_patterns.html">02   - Describing shapes and patterns</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture03/03_Analysing_change%2C_one_step_at_a_time.html">03   - Analysing change, one step at a time</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture04/04_From_table_arrangements_to_powerful_tools.html">04 - From table arrangements to powerful tools</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture05/05_From_table_arrangements_to_powerful_tools_part2.html">05 - From table arrangements to powerful tools - part 2</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture06/06_Analysing_change_continuously.html">06 - Analysing change, continuously</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture07/07_Analysing_change_continuously_part2.html">07 - Analysing change, continuously (Part 2)</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture08/08_Calculating_accumulated_change.html">08 - Calculating accumulated change</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/tutorials/lecture01/Tutorial_01.html">Tutorial 01</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/tutorials/lecture05/Tutorial_05.html">Tutorial 05</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/tutorials/lecture06/Tutorial_06.html">Tutorial 06</a></li>
-</ul>
 <p aria-level="2" class="caption" role="heading"><span class="caption-text">Basic Data Analyses and Statistics</span></p>
 <ul class="nav bd-sidenav">
-<li class="toctree-l1"><a class="reference internal" href="../Stats-Intro.html">Basic Data Science and Statistics: Introduction</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../intro-Stats.html">Introduction</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Data_R.html">Data Management and Visualization</a></li>
 <li class="toctree-l1"><a class="reference internal" href="ExpDesign.html">Experimental design and Data exploration</a></li>
 <li class="toctree-l1"><a class="reference internal" href="t_F_tests.html">Basic hypothesis testing: <span class="math notranslate nohighlight">\(t\)</span> and <span class="math notranslate nohighlight">\(F\)</span> tests</a></li>
@@ -214,13 +200,14 @@
 <p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced Data Analyses and Statistics</span></p>
 <ul class="nav bd-sidenav">
 <li class="toctree-l1"><a class="reference internal" href="glm.html">Generalised Linear Models</a></li>
-<li class="toctree-l1"><a class="reference internal" href="ModelFitting-NLLS.html">Model Fitting using Non-linear Least-squares</a></li>
+<li class="toctree-l1"><a class="reference internal" href="NLLS.html">Model Fitting using Non-linear Least-squares</a></li>
 </ul>
 <p aria-level="2" class="caption" role="heading"><span class="caption-text">Appendices</span></p>
 <ul class="current nav bd-sidenav">
 <li class="toctree-l1"><a class="reference internal" href="Appendix-JupyIntro.html">Introduction to Jupyter</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-Data-Python.html">Data analyses  with Python &amp; Jupyter</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-Maths.html">Mathematical models in Jupyter</a></li>
+<li class="toctree-l1"><a class="reference internal" href="Appendix-MathsForBiologists/MathsForBiologists.html">Maths for Biologists</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-Databases.html">Databases <span class="tocSkip"></span></a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-NLLS-Python.html">Model fitting in Python</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-MiniProj.html">The Computing Miniproject</a></li>
@@ -280,7 +267,7 @@
       
       
       
-      <li><a href="https://github.com/mhasoba/TheMulQuaBio" target="_blank"
+      <li><a href="https://github.com/MulQuaBio/MQB" target="_blank"
    class="btn btn-sm btn-source-repository-button dropdown-item"
    title="Source repository"
    data-bs-placement="left" data-bs-toggle="tooltip"
@@ -297,7 +284,7 @@
       
       
       
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+      <li><a href="https://github.com/MulQuaBio/MQB/issues/new?title=Issue%20on%20page%20%2Fnotebooks/Appendix-HighPerformanceComputing.html&body=Your%20issue%20content%20here." target="_blank"
    class="btn btn-sm btn-source-issues-button dropdown-item"
    title="Open an issue"
    data-bs-placement="left" data-bs-toggle="tooltip"
@@ -1025,7 +1012,7 @@ <h2>Running a job on the Imperial HPC<a class="headerlink" href="#running-a-job-
   <div class="footer-item">
     
 <p class="component-author">
-By TheMulQuaBio collective!
+By MulQuaBio collective!
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diff --git a/notebooks/Appendix-JupyIntro.html b/notebooks/Appendix-JupyIntro.html
index f35c359c..3ad86392 100644
--- a/notebooks/Appendix-JupyIntro.html
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-    <title>Introduction to Jupyter &#8212; TheMulQuaBio</title>
+    <title>Introduction to Jupyter &#8212; MulQuaBio</title>
   
   
   
@@ -32,7 +32,7 @@
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@@ -61,7 +61,7 @@
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-    <link rel="prev" title="Model Fitting using Non-linear Least-squares" href="ModelFitting-NLLS.html" />
+    <link rel="prev" title="Model Fitting using Non-linear Least-squares" href="NLLS.html" />
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@@ -153,8 +153,8 @@
       
     
     
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-    <script>document.write(`<img src="../_static/logo.png" class="logo__image only-dark" alt="TheMulQuaBio - Home"/>`);</script>
+    <img src="../_static/logo.png" class="logo__image only-light" alt="MulQuaBio - Home"/>
+    <script>document.write(`<img src="../_static/logo.png" class="logo__image only-dark" alt="MulQuaBio - Home"/>`);</script>
   
   
 </a></div>
@@ -188,23 +188,9 @@
 <li class="toctree-l1"><a class="reference internal" href="Python.html">Biological Computing in Python</a></li>
 <li class="toctree-l1"><a class="reference internal" href="R.html">Biological Computing in R</a></li>
 </ul>
-<p aria-level="2" class="caption" role="heading"><span class="caption-text">Maths for Biologists</span></p>
-<ul class="nav bd-sidenav">
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture01/01_Making_mathematical_statements.html">01 - Making mathematical statements</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture02/02_Describing_shapes_and_patterns.html">02   - Describing shapes and patterns</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture03/03_Analysing_change%2C_one_step_at_a_time.html">03   - Analysing change, one step at a time</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture04/04_From_table_arrangements_to_powerful_tools.html">04 - From table arrangements to powerful tools</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture05/05_From_table_arrangements_to_powerful_tools_part2.html">05 - From table arrangements to powerful tools - part 2</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture06/06_Analysing_change_continuously.html">06 - Analysing change, continuously</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture07/07_Analysing_change_continuously_part2.html">07 - Analysing change, continuously (Part 2)</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture08/08_Calculating_accumulated_change.html">08 - Calculating accumulated change</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/tutorials/lecture01/Tutorial_01.html">Tutorial 01</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/tutorials/lecture05/Tutorial_05.html">Tutorial 05</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/tutorials/lecture06/Tutorial_06.html">Tutorial 06</a></li>
-</ul>
 <p aria-level="2" class="caption" role="heading"><span class="caption-text">Basic Data Analyses and Statistics</span></p>
 <ul class="nav bd-sidenav">
-<li class="toctree-l1"><a class="reference internal" href="../Stats-Intro.html">Basic Data Science and Statistics: Introduction</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../intro-Stats.html">Introduction</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Data_R.html">Data Management and Visualization</a></li>
 <li class="toctree-l1"><a class="reference internal" href="ExpDesign.html">Experimental design and Data exploration</a></li>
 <li class="toctree-l1"><a class="reference internal" href="t_F_tests.html">Basic hypothesis testing: <span class="math notranslate nohighlight">\(t\)</span> and <span class="math notranslate nohighlight">\(F\)</span> tests</a></li>
@@ -217,13 +203,14 @@
 <p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced Data Analyses and Statistics</span></p>
 <ul class="nav bd-sidenav">
 <li class="toctree-l1"><a class="reference internal" href="glm.html">Generalised Linear Models</a></li>
-<li class="toctree-l1"><a class="reference internal" href="ModelFitting-NLLS.html">Model Fitting using Non-linear Least-squares</a></li>
+<li class="toctree-l1"><a class="reference internal" href="NLLS.html">Model Fitting using Non-linear Least-squares</a></li>
 </ul>
 <p aria-level="2" class="caption" role="heading"><span class="caption-text">Appendices</span></p>
 <ul class="current nav bd-sidenav">
 <li class="toctree-l1 current active"><a class="current reference internal" href="#">Introduction to Jupyter</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-Data-Python.html">Data analyses  with Python &amp; Jupyter</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-Maths.html">Mathematical models in Jupyter</a></li>
+<li class="toctree-l1"><a class="reference internal" href="Appendix-MathsForBiologists/MathsForBiologists.html">Maths for Biologists</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-Databases.html">Databases <span class="tocSkip"></span></a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-NLLS-Python.html">Model fitting in Python</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-MiniProj.html">The Computing Miniproject</a></li>
@@ -283,7 +270,7 @@
       
       
       
-      <li><a href="https://github.com/mhasoba/TheMulQuaBio" target="_blank"
+      <li><a href="https://github.com/MulQuaBio/MQB" target="_blank"
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    title="Source repository"
    data-bs-placement="left" data-bs-toggle="tooltip"
@@ -300,7 +287,7 @@
       
       
       
-      <li><a href="https://github.com/mhasoba/TheMulQuaBio/issues/new?title=Issue%20on%20page%20%2Fnotebooks/Appendix-JupyIntro.html&body=Your%20issue%20content%20here." target="_blank"
+      <li><a href="https://github.com/MulQuaBio/MQB/issues/new?title=Issue%20on%20page%20%2Fnotebooks/Appendix-JupyIntro.html&body=Your%20issue%20content%20here." target="_blank"
    class="btn btn-sm btn-source-issues-button dropdown-item"
    title="Open an issue"
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@@ -816,7 +803,7 @@ <h2>Readings and Resources<a class="headerlink" href="#readings-and-resources" t
                   
 <div class="prev-next-area">
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-       href="ModelFitting-NLLS.html"
+       href="NLLS.html"
        title="previous page">
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@@ -900,7 +887,7 @@ <h2>Readings and Resources<a class="headerlink" href="#readings-and-resources" t
   <div class="footer-item">
     
 <p class="component-author">
-By TheMulQuaBio collective!
+By MulQuaBio collective!
 </p>
 
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diff --git a/notebooks/Appendix-Maths.html b/notebooks/Appendix-Maths.html
index 0bc0b1e8..40966726 100644
--- a/notebooks/Appendix-Maths.html
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-    <title>Mathematical models in Jupyter &#8212; TheMulQuaBio</title>
+    <title>Mathematical models in Jupyter &#8212; MulQuaBio</title>
   
   
   
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     <link rel="search" title="Search" href="../search.html" />
-    <link rel="next" title="Databases " href="Appendix-Databases.html" />
+    <link rel="next" title="Maths for Biologists" href="Appendix-MathsForBiologists/MathsForBiologists.html" />
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@@ -153,8 +153,8 @@
       
     
     
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-    <script>document.write(`<img src="../_static/logo.png" class="logo__image only-dark" alt="TheMulQuaBio - Home"/>`);</script>
+    <img src="../_static/logo.png" class="logo__image only-light" alt="MulQuaBio - Home"/>
+    <script>document.write(`<img src="../_static/logo.png" class="logo__image only-dark" alt="MulQuaBio - Home"/>`);</script>
   
   
 </a></div>
@@ -188,23 +188,9 @@
 <li class="toctree-l1"><a class="reference internal" href="Python.html">Biological Computing in Python</a></li>
 <li class="toctree-l1"><a class="reference internal" href="R.html">Biological Computing in R</a></li>
 </ul>
-<p aria-level="2" class="caption" role="heading"><span class="caption-text">Maths for Biologists</span></p>
-<ul class="nav bd-sidenav">
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture01/01_Making_mathematical_statements.html">01 - Making mathematical statements</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture02/02_Describing_shapes_and_patterns.html">02   - Describing shapes and patterns</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture03/03_Analysing_change%2C_one_step_at_a_time.html">03   - Analysing change, one step at a time</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture04/04_From_table_arrangements_to_powerful_tools.html">04 - From table arrangements to powerful tools</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture05/05_From_table_arrangements_to_powerful_tools_part2.html">05 - From table arrangements to powerful tools - part 2</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture06/06_Analysing_change_continuously.html">06 - Analysing change, continuously</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture07/07_Analysing_change_continuously_part2.html">07 - Analysing change, continuously (Part 2)</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture08/08_Calculating_accumulated_change.html">08 - Calculating accumulated change</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/tutorials/lecture01/Tutorial_01.html">Tutorial 01</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/tutorials/lecture05/Tutorial_05.html">Tutorial 05</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/tutorials/lecture06/Tutorial_06.html">Tutorial 06</a></li>
-</ul>
 <p aria-level="2" class="caption" role="heading"><span class="caption-text">Basic Data Analyses and Statistics</span></p>
 <ul class="nav bd-sidenav">
-<li class="toctree-l1"><a class="reference internal" href="../Stats-Intro.html">Basic Data Science and Statistics: Introduction</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../intro-Stats.html">Introduction</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Data_R.html">Data Management and Visualization</a></li>
 <li class="toctree-l1"><a class="reference internal" href="ExpDesign.html">Experimental design and Data exploration</a></li>
 <li class="toctree-l1"><a class="reference internal" href="t_F_tests.html">Basic hypothesis testing: <span class="math notranslate nohighlight">\(t\)</span> and <span class="math notranslate nohighlight">\(F\)</span> tests</a></li>
@@ -217,13 +203,14 @@
 <p aria-level="2" class="caption" role="heading"><span class="caption-text">Advanced Data Analyses and Statistics</span></p>
 <ul class="nav bd-sidenav">
 <li class="toctree-l1"><a class="reference internal" href="glm.html">Generalised Linear Models</a></li>
-<li class="toctree-l1"><a class="reference internal" href="ModelFitting-NLLS.html">Model Fitting using Non-linear Least-squares</a></li>
+<li class="toctree-l1"><a class="reference internal" href="NLLS.html">Model Fitting using Non-linear Least-squares</a></li>
 </ul>
 <p aria-level="2" class="caption" role="heading"><span class="caption-text">Appendices</span></p>
 <ul class="current nav bd-sidenav">
 <li class="toctree-l1"><a class="reference internal" href="Appendix-JupyIntro.html">Introduction to Jupyter</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-Data-Python.html">Data analyses  with Python &amp; Jupyter</a></li>
 <li class="toctree-l1 current active"><a class="current reference internal" href="#">Mathematical models in Jupyter</a></li>
+<li class="toctree-l1"><a class="reference internal" href="Appendix-MathsForBiologists/MathsForBiologists.html">Maths for Biologists</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-Databases.html">Databases <span class="tocSkip"></span></a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-NLLS-Python.html">Model fitting in Python</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-MiniProj.html">The Computing Miniproject</a></li>
@@ -283,7 +270,7 @@
       
       
       
-      <li><a href="https://github.com/mhasoba/TheMulQuaBio" target="_blank"
+      <li><a href="https://github.com/MulQuaBio/MQB" target="_blank"
    class="btn btn-sm btn-source-repository-button dropdown-item"
    title="Source repository"
    data-bs-placement="left" data-bs-toggle="tooltip"
@@ -300,7 +287,7 @@
       
       
       
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+      <li><a href="https://github.com/MulQuaBio/MQB/issues/new?title=Issue%20on%20page%20%2Fnotebooks/Appendix-Maths.html&body=Your%20issue%20content%20here." target="_blank"
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@@ -1376,8 +1363,8 @@ <h2>Some biological examples<a class="headerlink" href="#some-biological-example
 <section id="one-population-exponential-growth">
 <h3>One population: Exponential growth<a class="headerlink" href="#one-population-exponential-growth" title="Link to this heading">#</a></h3>
 <p>Let’s look at how populations grow when there are no environmental constraints. The differential equation model is:</p>
-<div class="amsmath math notranslate nohighlight" id="equation-dcf4db76-8a6b-4736-8117-f366c271f7df">
-<span class="eqno">(11)<a class="headerlink" href="#equation-dcf4db76-8a6b-4736-8117-f366c271f7df" title="Permalink to this equation">#</a></span>\[\begin{equation}\label{eq:exp_growth}
+<div class="amsmath math notranslate nohighlight" id="equation-16d40cd8-ff5f-489b-b820-354ee2fdb7b6">
+<span class="eqno">(11)<a class="headerlink" href="#equation-16d40cd8-ff5f-489b-b820-354ee2fdb7b6" title="Permalink to this equation">#</a></span>\[\begin{equation}\label{eq:exp_growth}
 	\frac{\text{d}N}{\text{d}t} = r_m N
 \end{equation}\]</div>
 <p>where <span class="math notranslate nohighlight">\(r_m\)</span> is the intrinsic, constant rate of population gowth (units of 1/time), and <span class="math notranslate nohighlight">\(N\)</span> is population size (or biomass abundance). I use the subscript <span class="math notranslate nohighlight">\(m\)</span> in <span class="math notranslate nohighlight">\(r_m\)</span> to denote both <a class="reference external" href="https://en.wikipedia.org/wiki/Malthusianism">Malthusian</a> and maximal population growth rate because, in theory, without any constraints, this growth rate is expected to at its theoretical/biological maximum.</p>
@@ -1438,8 +1425,8 @@ <h3>One population: Exponential growth<a class="headerlink" href="#one-populatio
 <div class="math notranslate nohighlight">
 \[C_1 = N_0\]</div>
 <p>That is,</p>
-<div class="amsmath math notranslate nohighlight" id="equation-3dc52961-d8be-4d0c-b6e5-12c1b28423ae">
-<span class="eqno">(12)<a class="headerlink" href="#equation-3dc52961-d8be-4d0c-b6e5-12c1b28423ae" title="Permalink to this equation">#</a></span>\[\begin{equation}\label{eq:exp_growth_sol}
+<div class="amsmath math notranslate nohighlight" id="equation-e76ca204-4288-4d22-91ea-87f22984c2f0">
+<span class="eqno">(12)<a class="headerlink" href="#equation-e76ca204-4288-4d22-91ea-87f22984c2f0" title="Permalink to this equation">#</a></span>\[\begin{equation}\label{eq:exp_growth_sol}
 N{\left (t \right )} = N_0 e^{r_m t}
 \end{equation}\]</div>
 <p>We could use Sympy for this last step as well (using <code class="docutils literal notranslate"><span class="pre">subs()</span></code> like you learned above), but that would be just plain silly – like using a sledge-hammer to drive in a nail!</p>
@@ -1547,8 +1534,8 @@ <h3>One population: Exponential growth<a class="headerlink" href="#one-populatio
 </div>
 </div>
 <p>This is a bit more complicated than the solution for exponential growth above. But we can solve it the same way. First substitute <span class="math notranslate nohighlight">\(t = 0\)</span>, and then solve the resulting equation for the initital condition of <span class="math notranslate nohighlight">\(N_0\)</span>, which then gives This the time-dependent solution:</p>
-<div class="amsmath math notranslate nohighlight" id="equation-21e34f5d-177b-478d-9a0a-58fe7c5b31f1">
-<span class="eqno">(14)<a class="headerlink" href="#equation-21e34f5d-177b-478d-9a0a-58fe7c5b31f1" title="Permalink to this equation">#</a></span>\[\begin{equation}
+<div class="amsmath math notranslate nohighlight" id="equation-37b95a46-71cd-452b-9f7c-6c64cd53511b">
+<span class="eqno">(14)<a class="headerlink" href="#equation-37b95a46-71cd-452b-9f7c-6c64cd53511b" title="Permalink to this equation">#</a></span>\[\begin{equation}
     N_t = \frac{N_0 K\mathrm{e}^{r_m t}}{K + N_0(\mathrm{e}^{r_m t}-1)}
 \end{equation}\]</div>
 <p>You can do the last steps to obtain this solution using Sympy as well (I leave it to you to try it).</p>
@@ -1592,8 +1579,8 @@ <h3>One population: Exponential growth<a class="headerlink" href="#one-populatio
 <section id="two-interacting-populations-the-lotka-volterra-predator-prey-model">
 <h3>Two interacting populations: The Lotka-Volterra predator-prey model<a class="headerlink" href="#two-interacting-populations-the-lotka-volterra-predator-prey-model" title="Link to this heading">#</a></h3>
 <p>Now for the classical Lotka-Volterra model that you encountered in the advanced Python week (without logistic growth for the consumer, <span class="math notranslate nohighlight">\(C\)</span>).</p>
-<div class="amsmath math notranslate nohighlight" id="equation-3a84a670-e30d-4a63-ba8d-50f490c63b45">
-<span class="eqno">(15)<a class="headerlink" href="#equation-3a84a670-e30d-4a63-ba8d-50f490c63b45" title="Permalink to this equation">#</a></span>\[\begin{align}
+<div class="amsmath math notranslate nohighlight" id="equation-8b249319-659b-42f6-9e94-69fe086b5422">
+<span class="eqno">(15)<a class="headerlink" href="#equation-8b249319-659b-42f6-9e94-69fe086b5422" title="Permalink to this equation">#</a></span>\[\begin{align}
 \frac{dN}{dt} &amp;= r_m N \left(1-\frac{N}{K}\right) - a N C\\
 \frac{dC}{dt} &amp;= e a N C - z C
 \end{align}\]</div>
@@ -1804,11 +1791,11 @@ <h2>Readings and Resources<a class="headerlink" href="#readings-and-resources" t
       </div>
     </a>
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-       href="Appendix-Databases.html"
+       href="Appendix-MathsForBiologists/MathsForBiologists.html"
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+        <p class="prev-next-title">Maths for Biologists</p>
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@@ -1869,7 +1856,7 @@ <h2>Readings and Resources<a class="headerlink" href="#readings-and-resources" t
   <div class="footer-item">
     
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-By TheMulQuaBio collective!
+By MulQuaBio collective!
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+<div id="jb-print-docs-body" class="onlyprint">
+    <h1>Maths for Biologists</h1>
+    <!-- Table of contents -->
+    <div id="print-main-content">
+        <div id="jb-print-toc">
+            
+            <div>
+                <h2> Contents </h2>
+            </div>
+            <nav aria-label="Page">
+                <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#making-mathematical-statements">Making mathematical statements</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#sets-and-relations">Sets and relations</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#polynomial-functions">Polynomial functions</a><ul class="nav section-nav flex-column">
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#degree-0">Degree 0</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#degree-1">Degree 1</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#degree-2">Degree 2</a><ul class="nav section-nav flex-column">
+<li class="toc-h5 nav-item toc-entry"><a class="reference internal nav-link" href="#concavity">Concavity</a></li>
+<li class="toc-h5 nav-item toc-entry"><a class="reference internal nav-link" href="#intercept-roots">Intercept &amp; Roots</a></li>
+<li class="toc-h5 nav-item toc-entry"><a class="reference internal nav-link" href="#minimum-maximum">Minimum / Maximum</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#rational-functions">Rational functions</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#power-functions">Power functions</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#exponential-functions">Exponential functions</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#logarithmic-functions">Logarithmic functions</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#describing-shapes-and-patterns">Describing shapes and patterns</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#measuring-angles">Measuring angles</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#trigonometric-circle">Trigonometric circle</a><ul class="nav section-nav flex-column">
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#trigonometric-identities">Trigonometric identities</a></li>
+</ul>
+</li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#trigonometric-functions">Trigonometric functions</a><ul class="nav section-nav flex-column">
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#amplitude-and-period">Amplitude and Period</a></li>
+</ul>
+</li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#modelling-complex-periodic-phenomena">Modelling complex periodic phenomena</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#transformation-on-graphs">Transformation on graphs</a><ul class="nav section-nav flex-column">
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#translations">Translations</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#stretching-compression">Stretching / Compression</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#reflection">Reflection</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#analysing-change-one-step-at-a-time">Analysing change, one step at a time</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#sequences">Sequences</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#limits">Limits</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#discrete-time-population-models">Discrete-time population models</a><ul class="nav section-nav flex-column">
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#density-dependent-population-growth">Density-dependent population growth</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#annual-plants">Annual plants</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#fibonacci-sequence">Fibonacci sequence</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#from-table-arrangements-to-powerful-tools">From table arrangements to powerful tools</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#systems-of-linear-equations">Systems of linear equations</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#matrix-operations">Matrix operations</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#inverse-matrix">Inverse matrix</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#determinant">Determinant</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#structured-models">Structured models</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#linear-transformations">Linear transformations</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#eigenvalues-and-eigenvectors">Eigenvalues and eigenvectors</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#vector-decomposition">Vector decomposition</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#structured-models-revisited">Structured models revisited</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#analysing-change-continuously">Analysing change, continuously</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#average-rate-of-change">Average rate of change</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#instantaneous-rate-of-change">Instantaneous rate of change</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#calculating-derivatives">Calculating derivatives</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#chain-rule">Chain rule</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#higher-order-derivatives">Higher order derivatives</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#functions-of-several-variables">Functions of several variables</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#analysing-change-continuously-part-2">Analysing change, continuously (Part 2)</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#extreme-values-of-functions">Extreme values of functions</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#monotonicity-and-concavity">Monotonicity and concavity</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#finding-extrema">Finding extrema</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#inflection-points">Inflection points</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#series-expansions">Series expansions</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#calculating-accumulated-change">Calculating accumulated change</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#series">Series</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#calculating-areas">Calculating areas</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#fundamental-theorem-of-calculus">Fundamental theorem of Calculus</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#calculating-integrals">Calculating integrals</a><ul class="nav section-nav flex-column">
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#integration-by-substitution">Integration by substitution</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#integration-by-parts">Integration by parts</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#integration-by-partial-fractions">Integration by partial fractions</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#exercise-set-1">Exercise Set 1</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#exercise-set-2">Exercise Set 2</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#exercise-set-3">Exercise Set 3</a></li>
+</ul>
+            </nav>
+        </div>
+    </div>
+</div>
+
+              
+                
+<div id="searchbox"></div>
+                <article class="bd-article">
+                  
+  <section class="tex2jax_ignore mathjax_ignore" id="maths-for-biologists">
+<h1>Maths for Biologists<a class="headerlink" href="#maths-for-biologists" title="Link to this heading">#</a></h1>
+<section id="making-mathematical-statements">
+<h2>Making mathematical statements<a class="headerlink" href="#making-mathematical-statements" title="Link to this heading">#</a></h2>
+<section id="sets-and-relations">
+<h3>Sets and relations<a class="headerlink" href="#sets-and-relations" title="Link to this heading">#</a></h3>
+<p>Start by watching this video.</p>
+<div style="padding:56.25% 0 0 0;position:relative;"><iframe src="https://player.vimeo.com/video/1046665765?title=0&amp;byline=0&amp;portrait=0&amp;badge=0&amp;autopause=0&amp;player_id=0&amp;app_id=58479" frameborder="0" allow="autoplay; fullscreen; picture-in-picture; clipboard-write; encrypted-media" style="position:absolute;top:0;left:0;width:100%;height:100%;" title="Making_Mathematical_Statements_01"></iframe></div><script src="https://player.vimeo.com/api/player.js"></script>
+</section>
+<section id="polynomial-functions">
+<h3>Polynomial functions<a class="headerlink" href="#polynomial-functions" title="Link to this heading">#</a></h3>
+<div style="padding:56.25% 0 0 0;position:relative;"><iframe src="https://player.vimeo.com/video/1046665782?title=0&amp;byline=0&amp;portrait=0&amp;badge=0&amp;autopause=0&amp;player_id=0&amp;app_id=58479" frameborder="0" allow="autoplay; fullscreen; picture-in-picture; clipboard-write; encrypted-media" style="position:absolute;top:0;left:0;width:100%;height:100%;" title="Making_Mathematical_Statements_02"></iframe></div><script src="https://player.vimeo.com/api/player.js"></script>
+<p>One of the simplest elemntary functions is given by:</p>
+<div class="math notranslate nohighlight">
+\[ f(x) = a_n x^n + a_{n-1} x^{n-1}+ \cdots + a_1 x^1 + a_0, \]</div>
+<p>where <span class="math notranslate nohighlight">\(n\)</span> is a <em>non-negative integer</em>, and the coefficients <span class="math notranslate nohighlight">\(a_n\)</span>, <span class="math notranslate nohighlight">\(a_{n-1}\)</span>, …, <span class="math notranslate nohighlight">\(a_0\)</span>. The highest value of <span class="math notranslate nohighlight">\(n\)</span> (for which the coefficient <span class="math notranslate nohighlight">\(a_n\)</span> is different from <span class="math notranslate nohighlight">\(0\)</span>) defines the <em>degree</em> of the polynomial.</p>
+<div class="admonition-examples admonition">
+<p class="admonition-title">Examples</p>
+<p><span class="math notranslate nohighlight">\(f(x) = 3x^5 + 12x^4 - 6x^3 - x^2 + x - 16\)</span> (Polynomial function of degree 5)<br />
+<span class="math notranslate nohighlight">\(g(x) = \frac{3}{4}x^2 - \frac{5}{2} \)</span> (Polynomial function of degree 2)<br />
+<span class="math notranslate nohighlight">\(h(x) = 1.6x^8 - 0.576x^4 + 1.89x^2 - 0.7\)</span> (Polynomial function of degree 8)</p>
+</div>
+<div class="admonition tip">
+<p class="admonition-title">Tip</p>
+<p>When the domain of the polynomial function is not explicitly given, it is frequently assumed to be all real numbers, <span class="math notranslate nohighlight">\({\rm I\!R}\)</span>.</p>
+</div>
+<div class="admonition tip">
+<p class="admonition-title">Tip</p>
+<p>When no application is specified, the function name ‘<span class="math notranslate nohighlight">\(f\)</span>’ and variable name ‘<span class="math notranslate nohighlight">\(x\)</span>’ are usually chosen. For application purposes, the function and variable are often named for what they represent; for example, <span class="math notranslate nohighlight">\(A(r)=\pi r^2\)</span> for the area of a circle dependent on its radius.</p>
+</div>
+<p>Let us discuss three of the most common forms of polynomial functions with degrees <span class="math notranslate nohighlight">\(0\)</span>, <span class="math notranslate nohighlight">\(1\)</span>, and <span class="math notranslate nohighlight">\(2\)</span>.</p>
+<section id="degree-0">
+<h4>Degree 0<a class="headerlink" href="#degree-0" title="Link to this heading">#</a></h4>
+<p>Polynomial functions of degree 0 assume the form:</p>
+<div class="math notranslate nohighlight">
+\[ f(x) = c, \]</div>
+<p>where <span class="math notranslate nohighlight">\(c\)</span> is a real number. This function is graphically represented by:</p>
+<img alt="../../_images/poly_deg0.png" src="../../_images/poly_deg0.png" />
+<div class="admonition-example admonition">
+<p class="admonition-title">Example</p>
+<p>For most mobile phone contracts nowadays a fixed fee is paid for the month with unlimited usage. So the cost as a function of used minutes would be a polynomial of degree zero, since the cost does not depend on the usage.</p>
+</div>
+</section>
+<section id="degree-1">
+<h4>Degree 1<a class="headerlink" href="#degree-1" title="Link to this heading">#</a></h4>
+<p>Polynomial functions of degree 1 are also called <em>linear functions</em> and are given by:</p>
+<div class="math notranslate nohighlight">
+\[ f(x) = mx + b,\]</div>
+<p>where again <span class="math notranslate nohighlight">\(m\)</span> and <span class="math notranslate nohighlight">\(b\)</span> are real numbers (with <span class="math notranslate nohighlight">\(m \ne 0\)</span>).</p>
+<p>The constants <span class="math notranslate nohighlight">\(m\)</span> and <span class="math notranslate nohighlight">\(b\)</span> are frequently referred to as <em>slope</em> and <em>intercept</em>. The intercept <span class="math notranslate nohighlight">\(b\)</span> shows where the function crosses the <span class="math notranslate nohighlight">\(y\)</span>-axis, while the slope <span class="math notranslate nohighlight">\(m\)</span> is related to the inclination of the function.</p>
+<p>The sign of the slope <span class="math notranslate nohighlight">\(m\)</span> also determines if the function increases or decreases as a function of <span class="math notranslate nohighlight">\(x\)</span>. We have:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split} \begin{align}
+      m&gt;0 &amp;\rightarrow \mbox{Increasing function of } x \\
+      m&lt;0 &amp;\rightarrow \mbox{Decreasing function of } x
+  \end{align} \end{split}\]</div>
+<p>One can also ask what is the value of <span class="math notranslate nohighlight">\(x\)</span> where the function <span class="math notranslate nohighlight">\(f(x)\)</span> crosses de <span class="math notranslate nohighlight">\(x\)</span>-axis, or, in mathematical notation, what is <span class="math notranslate nohighlight">\(x\ \mbox{so that}\ f(x) = 0\)</span>. This value of x is called a <strong>root of <span class="math notranslate nohighlight">\(f\)</span></strong> and it can be obtained by:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{align}
+    f(x) &amp;= 0 \\
+    mx + b &amp;= 0 \\
+    mx &amp;= -b \\
+    x &amp;= -\frac{b}{m}
+\end{align} \end{split}\]</div>
+<div class="admonition-example admonition">
+<p class="admonition-title">Example</p>
+</div>
+<div class="admonition-examples admonition">
+<p class="admonition-title">Examples</p>
+<p>The metabolic rate of some organisms depends (approximately) linearly on the body’s temperature, with higher rates at higher temperatures.</p>
+</div>
+</section>
+<section id="degree-2">
+<h4>Degree 2<a class="headerlink" href="#degree-2" title="Link to this heading">#</a></h4>
+<p>Now we look at polynomial functions of degree 2, given by the general form:</p>
+<div class="math notranslate nohighlight">
+\[f(x) = ax^2 + bx +c,\]</div>
+<p>the coefficients <span class="math notranslate nohighlight">\(a\)</span>, <span class="math notranslate nohighlight">\(b\)</span>, and <span class="math notranslate nohighlight">\(c\)</span> are real numbers (with <span class="math notranslate nohighlight">\(a \ne 0\)</span>). The graph of a polynomial function of degree 2 is a <em>parabola</em> and many of its properties can be readily known by analysing the coefficients.</p>
+<section id="concavity">
+<h5>Concavity<a class="headerlink" href="#concavity" title="Link to this heading">#</a></h5>
+</section>
+<section id="intercept-roots">
+<h5>Intercept &amp; Roots<a class="headerlink" href="#intercept-roots" title="Link to this heading">#</a></h5>
+<p>The expression inside the square root distinguish between different properties of this function and therefore receives the name of <em>discriminant</em>:</p>
+<div class="math notranslate nohighlight">
+\[ \Delta = b^2 -4ac.\]</div>
+<p>Depending on the value of the discriminant a quick conclusion can be reached regarding the roots of the function.</p>
+<div class="admonition-example admonition">
+<p class="admonition-title">Example</p>
+<p>If we throw a ball that is heavy enough so that air resistance is negligible (alternatively we can travel to the moon) the height of the ball as a function of time follows a quadratic equation. Knowing the roots we can calculate when and where the ball will land.</p>
+</div>
+</section>
+<section id="minimum-maximum">
+<h5>Minimum / Maximum<a class="headerlink" href="#minimum-maximum" title="Link to this heading">#</a></h5>
+<p>Depending on the concavity of the parabola (up or down), there will be a point where the function assumes a minimum or a maximum value. This point is called the <em>vertex</em> of the parabola. This extreme point can be easily determined by the symmetry properties of the function curve.</p>
+<div class="admonition-example admonition">
+<p class="admonition-title">Example</p>
+<p>Imagine the population of a specific region having a negative quadratic relationship - it undergoes initial growth, reaches its peak, and then declines. To determine both the maximum population size and the time when it peaked, we need to identify the extrema of this parabolic curve.</p>
+</div>
+<!-- Now watch this [video](https://web.microsoftstream.com/video/f64abd8a-9997-4b5e-b11f-0ca09b7603fe) for a short introduction to complex numbers.
+ -->
+</section>
+</section>
+</section>
+<section id="rational-functions">
+<h3>Rational functions<a class="headerlink" href="#rational-functions" title="Link to this heading">#</a></h3>
+<div style="padding:56.25% 0 0 0;position:relative;"><iframe src="https://player.vimeo.com/video/1046665803?title=0&amp;byline=0&amp;portrait=0&amp;badge=0&amp;autopause=0&amp;player_id=0&amp;app_id=58479" frameborder="0" allow="autoplay; fullscreen; picture-in-picture; clipboard-write; encrypted-media" style="position:absolute;top:0;left:0;width:100%;height:100%;" title="Making_Mathematical_Statements_03"></iframe></div><script src="https://player.vimeo.com/api/player.js"></script>
+<p>Rational functions are defined as quotients of polynomial functions:</p>
+<div class="math notranslate nohighlight">
+\[f(x) = \frac{p(x)}{q(x)},\ \ \ q(x) \ne 0\]</div>
+<div class="admonition-example admonition">
+<p class="admonition-title">Example</p>
+<p>Rational functions are often used to express ratios, for example of chemicals. Lets assume chemical A increases quadratically <span class="math notranslate nohighlight">\(A(t) = 2 t^2\)</span> and chemical B increases linearly <span class="math notranslate nohighlight">\(B(t)=t+5\)</span>, the ratio will be:</p>
+<div class="math notranslate nohighlight">
+\[r(t) = \frac{A(t)}{B(t)} = 2\frac{t^2}{t+5}\]</div>
+</div>
+</section>
+<section id="power-functions">
+<h3>Power functions<a class="headerlink" href="#power-functions" title="Link to this heading">#</a></h3>
+<p>Another type of function used in many applications are power functions (sometimes also called power laws). These functions have the functional form:</p>
+<div class="math notranslate nohighlight">
+\[f(x) = Cx^{\alpha}\]</div>
+<p>where <span class="math notranslate nohighlight">\(C\)</span> and <span class="math notranslate nohighlight">\(\alpha\)</span> are real constants (with <span class="math notranslate nohighlight">\(C \ne 0\)</span>).</p>
+<p>Many commonly used functions are examples of power functions, e.g. the square root <span class="math notranslate nohighlight">\(\sqrt{x}=x^{\frac 12}\)</span> or the inverse function <span class="math notranslate nohighlight">\(x^{-1}\)</span>.</p>
+<!-- Watch this [video](https://web.microsoftstream.com/video/bf7415e1-823f-449e-ace1-e379fac8fe0a) about additional properties of power law functions. --></section>
+<section id="exponential-functions">
+<h3>Exponential functions<a class="headerlink" href="#exponential-functions" title="Link to this heading">#</a></h3>
+<div style="padding:56.25% 0 0 0;position:relative;"><iframe src="https://player.vimeo.com/video/1046665831?title=0&amp;byline=0&amp;portrait=0&amp;badge=0&amp;autopause=0&amp;player_id=0&amp;app_id=58479" frameborder="0" allow="autoplay; fullscreen; picture-in-picture; clipboard-write; encrypted-media" style="position:absolute;top:0;left:0;width:100%;height:100%;" title="Making_Mathematical_Statements_04"></iframe></div><script src="https://player.vimeo.com/api/player.js"></script>
+<p>Similar to power functions, but with very different properties are the exponential functions:</p>
+<div class="math notranslate nohighlight">
+\[f(x) = a^x, \]</div>
+<p>where <span class="math notranslate nohighlight">\(a\)</span> is a positive constant. The parameter <span class="math notranslate nohighlight">\(a\)</span> is called <em>base</em> and the variable <span class="math notranslate nohighlight">\(x\)</span> is the exponent (remember that for power functions, the variable <span class="math notranslate nohighlight">\(x\)</span> is the base, raised to the power of the parameter <span class="math notranslate nohighlight">\(\alpha\)</span>).</p>
+<p>Exponential functions are used to model many different real-world applications, from bacterial growth to radioactive decay.</p>
+<div class="admonition-example admonition">
+<p class="admonition-title">Example</p>
+<p>Suppose we start a bacterial colony with a single individual on a Petri dish. At each time step, counted as an average reproduction interval, each individual bacterium duplicates, originating two other individuals. The number of bacteria in the colony as a function of time, <span class="math notranslate nohighlight">\(N(t)\)</span>, can be counted as:</p>
+<div class="pst-scrollable-table-container"><table class="table">
+<thead>
+<tr class="row-odd"><th class="head"><p><span class="math notranslate nohighlight">\(t\)</span></p></th>
+<th class="head text-center"><p><span class="math notranslate nohighlight">\(N(t)\)</span></p></th>
+</tr>
+</thead>
+<tbody>
+<tr class="row-even"><td><p><span class="math notranslate nohighlight">\(0\)</span></p></td>
+<td class="text-center"><p><span class="math notranslate nohighlight">\(1\ (= 2^0)\)</span></p></td>
+</tr>
+<tr class="row-odd"><td><p><span class="math notranslate nohighlight">\(1\)</span></p></td>
+<td class="text-center"><p><span class="math notranslate nohighlight">\(2\ (= 2^1)\)</span></p></td>
+</tr>
+<tr class="row-even"><td><p><span class="math notranslate nohighlight">\(2\)</span></p></td>
+<td class="text-center"><p><span class="math notranslate nohighlight">\(4\ (= 2^2)\)</span></p></td>
+</tr>
+<tr class="row-odd"><td><p><span class="math notranslate nohighlight">\(3\)</span></p></td>
+<td class="text-center"><p><span class="math notranslate nohighlight">\(8\ (= 2^3)\)</span></p></td>
+</tr>
+<tr class="row-even"><td><p><span class="math notranslate nohighlight">\(4\)</span></p></td>
+<td class="text-center"><p><span class="math notranslate nohighlight">\(16\ (= 2^4)\)</span></p></td>
+</tr>
+<tr class="row-odd"><td><p><span class="math notranslate nohighlight">\(5\)</span></p></td>
+<td class="text-center"><p><span class="math notranslate nohighlight">\(32\ (= 2^5)\)</span></p></td>
+</tr>
+</tbody>
+</table>
+</div>
+<p>If no limitation (on space or resources) is imposed to the colony, the number of bacteria at each time step <span class="math notranslate nohighlight">\(t\)</span> can be well modelled by an exponential function of the form:</p>
+<div class="math notranslate nohighlight">
+\[ N(t) = 2^t. \]</div>
+</div>
+<p>Exponential functions of the form <span class="math notranslate nohighlight">\(f(x) = a^x\)</span> (with <span class="math notranslate nohighlight">\(a&gt;0\)</span>) can never assume negative values and their behaviour depends on the value of <span class="math notranslate nohighlight">\(a\)</span>.</p>
+<img alt="../../_images/exponential_01.png" class="align-center" src="../../_images/exponential_01.png" />
+<div class="admonition tip">
+<p class="admonition-title">Tip</p>
+<p>Note that the function <em>increases</em> with <span class="math notranslate nohighlight">\(x\)</span> if <span class="math notranslate nohighlight">\(a&gt;1\)</span> (growth) and <em>decreases</em> with <span class="math notranslate nohighlight">\(x\)</span> if <span class="math notranslate nohighlight">\(0&lt;a&lt;1\)</span> (decay). To remember this, one can directly apply the properties of exponentiation. If <span class="math notranslate nohighlight">\(f(x) = (1/3)^x\)</span> (thus <span class="math notranslate nohighlight">\(a = (1/3) &lt; 1\)</span>), we have:</p>
+<div class="math notranslate nohighlight">
+\[f(x) = \left(\frac{1}{3}\right)^x = \frac{1}{3^x},\]</div>
+<p>and since <span class="math notranslate nohighlight">\(3^x\)</span> is an increasing function of <span class="math notranslate nohighlight">\(x\)</span>, then <span class="math notranslate nohighlight">\(1/3^x\)</span> should be a decreasing function of <span class="math notranslate nohighlight">\(x\)</span> (as the denominator increases, the fraction decreases).</p>
+</div>
+<div class="admonition-example admonition">
+<p class="admonition-title">Example</p>
+<p>In the early stages of a new desease, we can approximate the spread as exponential. In this case <span class="math notranslate nohighlight">\(a\)</span> is the number of people every infected person infects themselves. For <span class="math notranslate nohighlight">\(a&lt;1\)</span> every person infects less than one other person on average and the desease will slow down, for <span class="math notranslate nohighlight">\(a&gt;1\)</span> the desease will spread. At the beginning of the COVID-19 pandemic, an estimate for <span class="math notranslate nohighlight">\(a\)</span> (or in this case <span class="math notranslate nohighlight">\(R\)</span> for reproductive value) was <span class="math notranslate nohighlight">\(a=4\)</span>. If we start with 1 infected person, how many do we have after 10 infection cycles?</p>
+</div>
+<p>Two of the most used bases are <span class="math notranslate nohighlight">\(a=10\)</span> and <span class="math notranslate nohighlight">\(a=e\)</span>, where <span class="math notranslate nohighlight">\(e=2.71828\)</span> (called <em>Euler’s number</em> or <em>Napier’s constant</em>). Exponential functions with <span class="math notranslate nohighlight">\(a=e\)</span> are commonly written as:</p>
+<div class="math notranslate nohighlight">
+\[f(x) = e^x = \mbox{exp}(x) \]</div>
+<p>The unique charachteristic of the exponential of base <span class="math notranslate nohighlight">\(e\)</span> is, that it is its own derivative</p>
+<div class="math notranslate nohighlight">
+\[\frac{\text{d}}{\text{d}x} e^x = e^x \]</div>
+<p>The both formulations can be changed into one another via</p>
+<div class="math notranslate nohighlight">
+\[f(x) = e^{\lambda x} = ({\underbrace{e^\lambda}_a})^x.\]</div>
+<p>We have exponential growth when <span class="math notranslate nohighlight">\(a&gt;1\)</span> and therefore when <span class="math notranslate nohighlight">\(\lambda&gt;0\)</span> and decay when <span class="math notranslate nohighlight">\(0&lt;a&lt;1\)</span> and therefore <span class="math notranslate nohighlight">\(\lambda&lt;0\)</span>.</p>
+</section>
+<section id="logarithmic-functions">
+<h3>Logarithmic functions<a class="headerlink" href="#logarithmic-functions" title="Link to this heading">#</a></h3>
+<p>Let us start this section by understanding what the expression</p>
+<div class="math notranslate nohighlight">
+\[ \mbox{log}_a b = x \]</div>
+<p>actually means. One possible way to read it is: “<span class="math notranslate nohighlight">\(\mbox{log}_a b\)</span> is the number with which I have to exponentiate the base <span class="math notranslate nohighlight">\(a\)</span> so that I find the number <span class="math notranslate nohighlight">\(b\)</span>. Since “<span class="math notranslate nohighlight">\(\mbox{log}_a b = x\)</span>, the above expression is equivalent to:</p>
+<div class="math notranslate nohighlight">
+\[ a^x = b.\]</div>
+<p>The expression <span class="math notranslate nohighlight">\(\mbox{log}_a b\)</span> is called the <em>logarithm of b in the base a</em> and it is also possible to define a logarithmic function given by:</p>
+<div class="math notranslate nohighlight">
+\[f(x) = \mbox{log}_a x,\]</div>
+<p>where <span class="math notranslate nohighlight">\(a\)</span> is the base of the logarithm and it is assumed <span class="math notranslate nohighlight">\(a&gt;0\)</span> and <span class="math notranslate nohighlight">\(a \ne 1\)</span>.</p>
+<p>By defining a function like this, for each <span class="math notranslate nohighlight">\(x\)</span> we will have a number <span class="math notranslate nohighlight">\(y = f(x)\)</span> for which <span class="math notranslate nohighlight">\(a^y = x\)</span>. Therefore, the logarithmic function is only defined for <span class="math notranslate nohighlight">\(x&gt;0\)</span>. Like the exponential function, the behaviour of the logarithmic function for increasing <span class="math notranslate nohighlight">\(x\)</span> will depend on the value of the base <span class="math notranslate nohighlight">\(a\)</span>.</p>
+<div class="admonition tip">
+<p class="admonition-title">Tip</p>
+<p>The similarities between the graphs for exponential and logarithmic functions are not coincidental. In fact, the graph for a logarithmic function with base <span class="math notranslate nohighlight">\(a\)</span> can be obtained by drawing the graph for an exponential function with base <span class="math notranslate nohighlight">\(a\)</span> and reflecting it in relation to the line <span class="math notranslate nohighlight">\(y = x\)</span>.  [MAKE COMMENT ON INVERSE FUNCTIONS]</p>
+<img alt="../../_images/log_base_02.png" src="../../_images/log_base_02.png" />
+<p>(here we use the common nomenclature <span class="math notranslate nohighlight">\(\mbox{log}_e x = \mbox{ln}\ x\)</span>).</p>
+</div>
+<div style="padding:56.25% 0 0 0;position:relative;"><iframe src="https://player.vimeo.com/video/1046665847?badge=0&amp;autopause=0&amp;player_id=0&amp;app_id=58479" frameborder="0" allow="autoplay; fullscreen; picture-in-picture; clipboard-write; encrypted-media" style="position:absolute;top:0;left:0;width:100%;height:100%;" title="Making_Mathematical_Statements_05"></iframe></div><script src="https://player.vimeo.com/api/player.js"></script></section>
+</section>
+<section id="describing-shapes-and-patterns">
+<h2>Describing shapes and patterns<a class="headerlink" href="#describing-shapes-and-patterns" title="Link to this heading">#</a></h2>
+<section id="measuring-angles">
+<h3>Measuring angles<a class="headerlink" href="#measuring-angles" title="Link to this heading">#</a></h3>
+<p>An <strong>angle</strong> is a primitive geometrical element, such as the <em>point</em>, the <em>curve</em> or the <em>plane</em>. It is defined as the figure formed by two line segments that extend from a common point.</p>
+<br/>
+<img alt="../../_images/angle.png" src="../../_images/angle.png" />
+<br/>
+<br/>
+<p>The two most common units for measuring angles are the <em>degree</em> and the <em>radian</em>.</p>
+<p>For a given angle, transforming between units of degrees and radians can be simply done with:</p>
+<div class="math notranslate nohighlight">
+\[\frac{360^{\circ}}{\theta} = \frac{2\pi}{\phi},\]</div>
+<p>where <span class="math notranslate nohighlight">\(\theta\)</span> is the angle measured in degrees and <span class="math notranslate nohighlight">\(\phi\)</span>, measured in radians. In this lecture, we will consider the angles measured in radians (independent of the symbol used for the variable), unless stated otherwise.</p>
+<div class="admonition-example admonition">
+<p class="admonition-title">Example</p>
+<p>If <span class="math notranslate nohighlight">\(\theta = 90^{\circ}\)</span>, then using the previous expression we have <span class="math notranslate nohighlight">\(\phi = \displaystyle\frac{\pi}{2}\)</span>. Since <span class="math notranslate nohighlight">\(\theta\)</span> and $\phi are directly proportional, we also have:</p>
+<div class="math notranslate nohighlight">
+\[\theta = 180^{\circ} \longrightarrow \phi = 2 \cdot \frac{\pi}{2} = \pi \]</div>
+<div class="math notranslate nohighlight">
+\[\theta = 270^{\circ} \longrightarrow \phi = 3 \cdot \frac{\pi}{2} = \frac{3\pi}{2} \]</div>
+<div class="math notranslate nohighlight">
+\[\theta = 360^{\circ} \longrightarrow \phi = 4 \cdot \frac{\pi}{2} = 2\pi \]</div>
+</div>
+</section>
+<section id="trigonometric-circle">
+<h3>Trigonometric circle<a class="headerlink" href="#trigonometric-circle" title="Link to this heading">#</a></h3>
+<p>We obtain the trigonometric circle by superimposing a circle of radius <span class="math notranslate nohighlight">\(1\)</span> to the cartesian system of coordinates:</p>
+<img alt="../../_images/trig_circ.png" src="../../_images/trig_circ.png" />
+<p>The projections on the <span class="math notranslate nohighlight">\(x\)</span> and <span class="math notranslate nohighlight">\(y\)</span>-axes have special meanings, if we consider the angle <span class="math notranslate nohighlight">\(\theta\)</span> between the line segment in the radial direction and the <span class="math notranslate nohighlight">\(x\)</span>-axis.</p>
+<p>From the geometric relations on the right triangle (triangle in which one internal angle is a right angle):</p>
+<p>Back to the trigonometric circle, it is possible to construct a right triangle where the hypothenuse corresponds to the radius of the superimposed circle. In this case, since this radius equals to <span class="math notranslate nohighlight">\(1\)</span>, we have:</p>
+<div class="math notranslate nohighlight">
+\[\mbox{sin}(\theta) = y\]</div>
+<div class="math notranslate nohighlight">
+\[\mbox{cos}(\theta) = x.\]</div>
+<p>In other words, the projections of the end of the radial line segment on the <span class="math notranslate nohighlight">\(y\)</span> and <span class="math notranslate nohighlight">\(x\)</span>-axes correspond to the sine and cosine of the angle <span class="math notranslate nohighlight">\(\theta\)</span>. Sine and cosine for this angle can be obtained by the projections even if <span class="math notranslate nohighlight">\(\theta &gt; \displaystyle\frac{\pi}{2}\)</span>.</p>
+<section id="trigonometric-identities">
+<h4>Trigonometric identities<a class="headerlink" href="#trigonometric-identities" title="Link to this heading">#</a></h4>
+<p>With the aid of the projections on the trigonometric circle, some trigonometric identities become imediately clear. Consider <span class="math notranslate nohighlight">\(\theta &gt; 0\)</span> for counterclockwise angle measurements starting from the <span class="math notranslate nohighlight">\(x\)</span>-axis and <span class="math notranslate nohighlight">\(\theta &lt; 0\)</span> for clockwise measurements. We have:</p>
+<div class="math notranslate nohighlight">
+\[\mbox{sin}(0) = 0,\ \ \ \mbox{sin}\left(\frac{\pi}{2}\right) = 1,\ \ \ \mbox{sin}(\pi) = 0,\ \ \ \mbox{sin}\left(\frac{3\pi}{2}\right) = -1,\ \ \ \mbox{sin}(2\pi) = 0\]</div>
+<div class="math notranslate nohighlight">
+\[\mbox{cos}(0) = 1,\ \ \ \mbox{cos}\left(\frac{\pi}{2}\right) = 0,\ \ \ \mbox{cos}(\pi) = -1,\ \ \ \mbox{cos}\left(\frac{3\pi}{2}\right) = 0,\ \ \ \mbox{cos}(2\pi) = 1\]</div>
+<p>From the fact that a full rotation brings you back to the same point:</p>
+<div class="math notranslate nohighlight">
+\[\mbox{sin}(\theta) = \mbox{sin}(\theta + 2\pi)\]</div>
+<div class="math notranslate nohighlight">
+\[\mbox{cos}(\theta) = \mbox{cos}(\theta + 2\pi)\]</div>
+<p>or from Pythagoras’ theorem on the right triangle in the trigonometric circle:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{align}
+[\mbox{sin}(\theta)]^2 + [\mbox{cos}(\theta)]^2 =&amp; 1^2 \\
+\mbox{sin}^2(\theta) + \mbox{cos}^2(\theta) =&amp; 1
+\end{align}\end{split}\]</div>
+<div class="admonition tip">
+<p class="admonition-title">Tip</p>
+<p>The notation <span class="math notranslate nohighlight">\(\mbox{sin}^2(\theta)\)</span> is frequently used to express the square of <span class="math notranslate nohighlight">\(\mbox{sin}(\theta)\)</span>, or:</p>
+<div class="math notranslate nohighlight">
+\[ \mbox{sin}^2(\theta) = [\mbox{sin}(\theta)]^2 = [\mbox{sin}(\theta)]\cdot[\mbox{sin}(\theta)],\]</div>
+<p>which is different from <span class="math notranslate nohighlight">\(\mbox{sin}(\theta^2)\)</span>.</p>
+</div>
+<!-- Now watch this [video](https://web.microsoftstream.com/video/b3023026-7d45-4378-a8ee-c166a43eb2e7) for a different application of the same trigonometric construct. -->
+</section>
+</section>
+<section id="trigonometric-functions">
+<h3>Trigonometric functions<a class="headerlink" href="#trigonometric-functions" title="Link to this heading">#</a></h3>
+<p>Since sine and cosine are defined for any real value of a given angle <span class="math notranslate nohighlight">\(x\)</span> (<span class="math notranslate nohighlight">\(x \in {\rm I\!R}\)</span>), we can define the functions:</p>
+<div class="math notranslate nohighlight">
+\[f(x) = \mbox{sin}(x)\]</div>
+<div class="math notranslate nohighlight">
+\[f(x) = \mbox{cos}(x).\]</div>
+<p>Thus, the association of sine and cosine with pure geometric ratios gives place for a more generic definition of functions of real variables. The graphs of these functions are given by:</p>
+<img alt="../../_images/sin_cos.png" src="../../_images/sin_cos.png" />
+<p>All the other known trigonometric relations can also be defined as functions in a similar way. Take for example the tangent of an angle <span class="math notranslate nohighlight">\(\theta\)</span>, which is:</p>
+<div class="math notranslate nohighlight">
+\[tan(\theta) = \displaystyle\frac{\mbox{sin}(\theta)}{\mbox{cos}(\theta)}.\]</div>
+<p>If <span class="math notranslate nohighlight">\(x \in {\rm I\!R}\)</span>, than we can define a function</p>
+<div class="math notranslate nohighlight">
+\[ f(x) = \mbox{tan}(x) = \displaystyle\frac{\mbox{sin}(x)}{cos(x)}, \ \ \ (\mbox{for }\mbox{cos}(x) \ne 0).\]</div>
+<p>The graph from the tangent function will be given by:</p>
+<img alt="../../_images/tan.png" src="../../_images/tan.png" />
+<p>Note by the above graphs that the functions sine and cosine have patterns that repeat themselves every interval of <span class="math notranslate nohighlight">\(2\pi\)</span>, while the tangent repeats itself every interval of <span class="math notranslate nohighlight">\(\pi\)</span>. We say that these three functions are <strong>periodic</strong>.</p>
+<p>A function <span class="math notranslate nohighlight">\(f(x)\)</span> is periodic if there is a positive constant <span class="math notranslate nohighlight">\(a\)</span>, so that:</p>
+<div class="math notranslate nohighlight">
+\[f(x + a) = f(x),\]</div>
+<p>for all values of <span class="math notranslate nohighlight">\(x\)</span> for which the function f(x) is defined.</p>
+<p>If <span class="math notranslate nohighlight">\(a\)</span> is the smallest number with this property, we call it the <em>period</em> of f(x).</p>
+<div class="admonition-example admonition">
+<p class="admonition-title">Example</p>
+<p>Since <span class="math notranslate nohighlight">\(\mbox{sin}(x + 2\pi) = \mbox{sin}(x)\)</span> for any value of <span class="math notranslate nohighlight">\(x\)</span> and since <span class="math notranslate nohighlight">\(2\pi\)</span> is the smallest number for which this property holds, then sine is a periodic function with period <span class="math notranslate nohighlight">\(2\pi\)</span>. With the same reasoning, it can be shown that the tangent is periodic with period <span class="math notranslate nohighlight">\(\pi\)</span>.</p>
+</div>
+<div class="admonition-example admonition">
+<p class="admonition-title">Example</p>
+<p>Many natural processes can be modelled as periodic. Think for example of a pendulum, the tides, the seasons. Often periodicity can be a good first order approximation, even if it does not strictly hold. For example when describimg the shape of the cloud cover in the picture below.</p>
+</div>
+<img alt="../../_images/cloud.png" src="../../_images/cloud.png" />
+<section id="amplitude-and-period">
+<h4>Amplitude and Period<a class="headerlink" href="#amplitude-and-period" title="Link to this heading">#</a></h4>
+<p>From the basic functions <span class="math notranslate nohighlight">\(\mbox{sin}(x)\)</span> or <span class="math notranslate nohighlight">\(\mbox{cos}(x)\)</span> we can write a more general function <span class="math notranslate nohighlight">\(f(x)\)</span> given by</p>
+<div class="math notranslate nohighlight">
+\[f(x) = A\mbox{sin}(\omega x),\ \ \ \ \ \ A, \omega \in {\rm I\!R}\ \ (A, \omega \ne 0)\]</div>
+<p>Since the sine ranges from <span class="math notranslate nohighlight">\(-1\)</span> to <span class="math notranslate nohighlight">\(1\)</span> (see the graph for <span class="math notranslate nohighlight">\(sin(x)\)</span> for reference), by its definition, <span class="math notranslate nohighlight">\(f(x)\)</span> will range from <span class="math notranslate nohighlight">\(-A\)</span> to <span class="math notranslate nohighlight">\(A\)</span>. We call <span class="math notranslate nohighlight">\(A\)</span> the <strong>amplitude</strong> of <span class="math notranslate nohighlight">\(f(x)\)</span> and it gives the extent of the variation of the oscillation of the function.</p>
+<p>Being sine a periodic function, <span class="math notranslate nohighlight">\(f(x)\)</span> will also be, possibly with a different period. The period <span class="math notranslate nohighlight">\(T\)</span> of <span class="math notranslate nohighlight">\(f(x)\)</span>, as discussed before, will be determined by the equation <span class="math notranslate nohighlight">\(f(x + T) = f(x)\)</span>, that should be valid for every <span class="math notranslate nohighlight">\(x\)</span>. Thus:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{align}
+    f(x + T) &amp;= f(x) \\
+    A\mbox{sin}(\omega(x+T)) &amp;= A\mbox{sin}(\omega x) \\
+    \mbox{sin}(\omega(x+T)) &amp;= \mbox{sin}(\omega x) \\
+    \mbox{sin}(\omega x+\omega T)) &amp;= \mbox{sin}(\omega x) \\
+    \mbox{sin}(z+\omega T)) &amp;= \mbox{sin}(z)\ \ \ \ \ \ (\mbox{where we change the variable to } z=\omega x)
+  \end{align}\end{split}\]</div>
+<p>But since we know that sine is periodic with period <span class="math notranslate nohighlight">\(2\pi\)</span>, we should have <span class="math notranslate nohighlight">\(|\omega|T = 2\pi\)</span>. Thus, the <strong>period</strong> of the function <span class="math notranslate nohighlight">\(f(x)\)</span> will be:</p>
+<div class="math notranslate nohighlight">
+\[T = \frac{2\pi}{\omega}.\]</div>
+<p>The constant <span class="math notranslate nohighlight">\(\omega = \displaystyle\frac{2\pi}{T}\)</span> is called <strong>angular frequency</strong> and measures the number of full rotations the function <span class="math notranslate nohighlight">\(f(x)\)</span> oscillates in one interval of <span class="math notranslate nohighlight">\(T\)</span>.</p>
+<div class="admonition-example admonition">
+<p class="admonition-title">Example</p>
+<p>The function <span class="math notranslate nohighlight">\(f(x) = 3.6\, \mbox{cos}\left(\displaystyle\frac{\pi}{3}x\right)\)</span> has an amplitude <span class="math notranslate nohighlight">\(A=3.6\)</span>, an angular frequency <span class="math notranslate nohighlight">\(\omega = \displaystyle\frac{\pi}{3}\)</span>, and a period <span class="math notranslate nohighlight">\(T = \displaystyle\frac{2\pi}{\omega} = \displaystyle\frac{2 \pi}{\frac{\pi}{3}} = 6\)</span>.</p>
+</div>
+</section>
+</section>
+<section id="modelling-complex-periodic-phenomena">
+<h3>Modelling complex periodic phenomena<a class="headerlink" href="#modelling-complex-periodic-phenomena" title="Link to this heading">#</a></h3>
+<!-- Watch this [video](https://web.microsoftstream.com/video/02f373a2-621b-4313-a18a-265f5fb65656) about periodic functions decomposition. --></section>
+<section id="transformation-on-graphs">
+<h3>Transformation on graphs<a class="headerlink" href="#transformation-on-graphs" title="Link to this heading">#</a></h3>
+<p>Starting from the elementary functions we have discussed</p>
+<div class="math notranslate nohighlight">
+\[ax+b,\ \ ax^2+bx+c,\ \ \sqrt{x},\ \ \mbox{e}^x,\ \ \mbox{log}(x),\ \ \mbox{sin}(x),\ \ \mbox{cos}(x)\]</div>
+<p>we can introduce several transformations to obtain new functions, by adding or multiplying constants to the argument (function variable) or to the whole function. These are the simplest cases of what is know as <em>function composition</em> with the resulting function called <em>composite function</em>.</p>
+<p>When working with mathematical models, this will help you to understand how the shape of a given function might change by changing one of the parameters, or how to construct a set of functions (that might be useful for your particular model) with small modifications of basic functions.</p>
+<section id="translations">
+<h4>Translations<a class="headerlink" href="#translations" title="Link to this heading">#</a></h4>
+<p>A given function can be dislocated, either vertically or horiontally, by <em>adding</em> a constant <span class="math notranslate nohighlight">\(c\)</span> to the whole function or to its argument. In general, starting from a function <span class="math notranslate nohighlight">\(f(x)\)</span>, translations are built following the rules:</p>
+<ul>
+<li><p><strong>Horizontal</strong> (Constant <span class="math notranslate nohighlight">\(c\)</span> summed to the argument of the function)</p>
+<p><span class="math notranslate nohighlight">\(y = f(x + c)\ \ \ \  -  \left\{
+\begin{array}{ll}
+    c &gt; 0 &amp; \rightarrow \ \  \mbox{function moves to the left} \\
+    c &lt; 0 &amp; \rightarrow \ \  \mbox{function moves to the right} \\
+\end{array} 
+\right. \)</span></p>
+</li>
+<li><p><strong>Vertical</strong> (Constant <span class="math notranslate nohighlight">\(c\)</span> summed to the whole function)</p>
+<p><span class="math notranslate nohighlight">\(y = f(x) + c\ \ \ \  -  \left\{
+\begin{array}{ll}
+    c &gt; 0 &amp; \rightarrow \ \  \mbox{function moves up} \\
+    c &lt; 0 &amp; \rightarrow \ \  \mbox{function moves down} \\
+\end{array} 
+\right. \)</span></p>
+</li>
+</ul>
+<div class="admonition-example admonition">
+<p class="admonition-title">Example</p>
+<p>If we start with the function <span class="math notranslate nohighlight">\(f(x) = x^3\)</span>, the function <span class="math notranslate nohighlight">\(y = f(x + 2) = (x+2)^3\)</span> will represent a horizontal translation, while the function <span class="math notranslate nohighlight">\(y = f(x) + 2 = x^3+2\)</span> will represent a vertical translation.</p>
+</div>
+<p>To see how this works, try to implement the following code in your own jupyter notebook. Try to use different functions <span class="math notranslate nohighlight">\(f(x)\)</span> (defined in ‘func(x)’) to see how different graphs behave to horizontal and vertical translations.</p>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">ipywidgets</span> <span class="kn">import</span> <span class="n">interact</span>
+<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
+<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+
+<span class="k">def</span> <span class="nf">func</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
+    <span class="k">return</span> <span class="n">x</span><span class="o">**</span><span class="mi">3</span>
+
+<span class="k">def</span> <span class="nf">htrans</span><span class="p">(</span><span class="n">c</span><span class="p">):</span>
+    <span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span><span class="mi">3</span><span class="p">)</span>
+    
+    <span class="n">y</span> <span class="o">=</span> <span class="n">func</span><span class="p">(</span><span class="n">x</span><span class="o">+</span><span class="n">c</span><span class="p">)</span> <span class="c1">#constant c in the argument of the function</span>
+    
+    <span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;$f(x+c)$&#39;</span><span class="p">)</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s1">&#39;Horizontal translation&#39;</span><span class="p">)</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">12</span><span class="p">)</span>
+    
+    <span class="n">plt</span><span class="o">.</span><span class="n">ylim</span><span class="p">(</span><span class="o">-</span><span class="mi">8</span><span class="p">,</span> <span class="mi">8</span><span class="p">)</span> <span class="c1">#remember to change the limits for better visualisation if necessary</span>
+    
+    <span class="n">plt</span><span class="o">.</span><span class="n">axhline</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;black&#39;</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">axvline</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;black&#39;</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
+    
+    
+
+    <span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
+
+<span class="k">def</span> <span class="nf">vtrans</span><span class="p">(</span><span class="n">c</span><span class="p">):</span>
+    <span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span><span class="mi">3</span><span class="p">)</span>
+    
+    <span class="n">y</span> <span class="o">=</span> <span class="n">func</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">+</span><span class="n">c</span> <span class="c1">#constant c summing the whole function</span>
+    
+    <span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">,</span>  <span class="n">label</span><span class="o">=</span><span class="s1">&#39;$f(x)+c$&#39;</span><span class="p">)</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s1">&#39;Vertical translation&#39;</span><span class="p">)</span>
+    
+    <span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s1">&#39;x&#39;</span><span class="p">)</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s1">&#39;y&#39;</span><span class="p">)</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">12</span><span class="p">)</span>
+    
+    <span class="n">plt</span><span class="o">.</span><span class="n">ylim</span><span class="p">(</span><span class="o">-</span><span class="mi">8</span><span class="p">,</span> <span class="mi">8</span><span class="p">)</span> <span class="c1">#remember to change the limits for better visualisation if necessary</span>
+        
+    <span class="n">plt</span><span class="o">.</span><span class="n">axhline</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;black&#39;</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">axvline</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;black&#39;</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
+    
+    <span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
+    
+<span class="c1"># create a slider</span>
+
+<span class="n">interact</span><span class="p">(</span><span class="n">htrans</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="p">(</span><span class="o">-</span><span class="mf">2.0</span><span class="p">,</span><span class="mf">2.0</span><span class="p">))</span>
+<span class="n">interact</span><span class="p">(</span><span class="n">vtrans</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="p">(</span><span class="o">-</span><span class="mf">2.0</span><span class="p">,</span><span class="mf">2.0</span><span class="p">))</span>
+</pre></div>
+</div>
+</section>
+<section id="stretching-compression">
+<h4>Stretching / Compression<a class="headerlink" href="#stretching-compression" title="Link to this heading">#</a></h4>
+<p>A given function can also be stretched or compressed, either vertically or horiontally, by <em>multiplying</em> a constant <span class="math notranslate nohighlight">\(c\)</span> to the whole function or to its argument. Starting from a function <span class="math notranslate nohighlight">\(f(x)\)</span>, stretchings/compressions are built following the rules:</p>
+<ul>
+<li><p><strong>Horizontal</strong> (Constant <span class="math notranslate nohighlight">\(c\)</span> multiplied to the argument of the function)</p>
+<p><span class="math notranslate nohighlight">\(y = f(cx)\ \ \ \  -  \left\{
+\begin{array}{ll}
+    c &gt; 1 &amp; \rightarrow \ \  \mbox{function is horizontally compressed} \\
+    0 &lt; c &lt; 1 &amp; \rightarrow \ \  \mbox{function is horizontally stretched} \\
+\end{array} 
+\right. \)</span></p>
+</li>
+<li><p><strong>Vertical</strong> (Constant <span class="math notranslate nohighlight">\(c\)</span> multiplied to the whole function)</p>
+<p><span class="math notranslate nohighlight">\(y = cf(x)\ \ \ \  -  \left\{
+\begin{array}{ll}
+    c &gt; 1 &amp; \rightarrow \ \  \mbox{function is vertically stretched} \\
+    0 &lt; c &lt; 1 &amp; \rightarrow \ \  \mbox{function is vertically compressed} \\
+\end{array} 
+\right. \)</span></p>
+</li>
+</ul>
+<div class="admonition-example admonition">
+<p class="admonition-title">Example</p>
+<p>For the function <span class="math notranslate nohighlight">\(f(x) = x^3-2x\)</span>, the function <span class="math notranslate nohighlight">\(y = f(2x) = (2x)^3-2(2x) = 8x^3-4x\)</span> will represent a horizontal compression, while the function <span class="math notranslate nohighlight">\(y = 2f(x) = 2x^3 - 4x\)</span> will represent a vertical stretching.</p>
+</div>
+<div class="admonition-example admonition">
+<p class="admonition-title">Example</p>
+<p>One example where we stretch or compress functions comes from chemical processes. Many chemical processes happen faster when the temperature is higher. If we describe the concentration of a certain chemical as a function of time <span class="math notranslate nohighlight">\(c(t)\)</span>, increasing the temperature will lead to a horizontal compression of the function.</p>
+</div>
+<p>Now play with the code below to see how different graphs behave to horizontal and vertical stretching/compression.</p>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">ipywidgets</span> <span class="kn">import</span> <span class="n">interact</span>
+<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
+<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+
+<span class="k">def</span> <span class="nf">func</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
+    <span class="k">return</span> <span class="n">x</span><span class="o">**</span><span class="mi">3</span><span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span>
+
+<span class="k">def</span> <span class="nf">hstetcom</span><span class="p">(</span><span class="n">c</span><span class="p">):</span>
+    <span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span><span class="mi">3</span><span class="p">)</span>
+    
+    <span class="n">y</span> <span class="o">=</span> <span class="n">func</span><span class="p">(</span><span class="n">c</span><span class="o">*</span><span class="n">x</span><span class="p">)</span> <span class="c1">#constant c multiplying the argument of the function</span>
+    
+    <span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;$f\, (cx)$&#39;</span><span class="p">)</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s1">&#39;Horizontal stretching/compression&#39;</span><span class="p">)</span>
+        
+    <span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s1">&#39;x&#39;</span><span class="p">)</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s1">&#39;y&#39;</span><span class="p">)</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">12</span><span class="p">)</span>
+    
+    <span class="n">plt</span><span class="o">.</span><span class="n">ylim</span><span class="p">(</span><span class="o">-</span><span class="mi">4</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span> <span class="c1">#remember to change the limits for better visualisation if necessary</span>
+    
+    <span class="n">plt</span><span class="o">.</span><span class="n">axhline</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;black&#39;</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">axvline</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;black&#39;</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
+    
+    
+
+    <span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
+
+<span class="k">def</span> <span class="nf">vstetcom</span><span class="p">(</span><span class="n">c</span><span class="p">):</span>
+    <span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span><span class="mi">3</span><span class="p">)</span>
+    
+    <span class="n">y</span> <span class="o">=</span> <span class="n">c</span><span class="o">*</span><span class="n">func</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="c1">#constant c multiplying the whole function</span>
+    
+    <span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">,</span>  <span class="n">label</span><span class="o">=</span><span class="s1">&#39;$c\, f\, (x)$&#39;</span><span class="p">)</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s1">&#39;Vertical stretching/compression&#39;</span><span class="p">)</span>
+    
+    <span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s1">&#39;x&#39;</span><span class="p">)</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s1">&#39;y&#39;</span><span class="p">)</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">12</span><span class="p">)</span>
+    
+    <span class="n">plt</span><span class="o">.</span><span class="n">ylim</span><span class="p">(</span><span class="o">-</span><span class="mi">4</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span> <span class="c1">#remember to change the limits for better visualisation if necessary</span>
+        
+    <span class="n">plt</span><span class="o">.</span><span class="n">axhline</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;black&#39;</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">axvline</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;black&#39;</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
+    
+    <span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
+    
+<span class="c1"># create a slider</span>
+
+<span class="n">interact</span><span class="p">(</span><span class="n">hstetcom</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="p">(</span><span class="mf">0.1</span><span class="p">,</span><span class="mf">3.0</span><span class="p">))</span>
+<span class="n">interact</span><span class="p">(</span><span class="n">vstetcom</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="p">(</span><span class="mf">0.1</span><span class="p">,</span><span class="mf">3.0</span><span class="p">))</span>
+</pre></div>
+</div>
+</section>
+<section id="reflection">
+<h4>Reflection<a class="headerlink" href="#reflection" title="Link to this heading">#</a></h4>
+<p>If the constant <span class="math notranslate nohighlight">\(c\)</span> multiplying the whole function or its argument has the particular value <span class="math notranslate nohighlight">\(c = -1\)</span>, we will have a <em>reflection</em>. Reflections can be done in relation to the <span class="math notranslate nohighlight">\(x\)</span> or <span class="math notranslate nohighlight">\(y\)</span>-axes and they are built, starting from a function <span class="math notranslate nohighlight">\(f(x)\)</span>, following the rules:</p>
+<ul class="simple">
+<li><p><strong>About the x-axis</strong> - <span class="math notranslate nohighlight">\(y = -f(x)\)</span></p></li>
+<li><p><strong>About the y-axis</strong> - <span class="math notranslate nohighlight">\(y = f(-x)\)</span></p></li>
+</ul>
+<p>The figure below shows these two reflections using the function <span class="math notranslate nohighlight">\(f(x) = (x-1)^2\)</span>.</p>
+<img alt="../../_images/reflection.png" src="../../_images/reflection.png" />
+</section>
+</section>
+</section>
+<section id="analysing-change-one-step-at-a-time">
+<h2>Analysing change, one step at a time<a class="headerlink" href="#analysing-change-one-step-at-a-time" title="Link to this heading">#</a></h2>
+<p>Imagine a colony of wasps is founded by a queen with the following pattern of eggs laid:</p>
+<br />  
+<div class="pst-scrollable-table-container"><table class="table">
+<thead>
+<tr class="row-odd"><th class="head"><p>Days</p></th>
+<th class="head text-center"><p>Eggs</p></th>
+</tr>
+</thead>
+<tbody>
+<tr class="row-even"><td><p><span class="math notranslate nohighlight">\(0\)</span></p></td>
+<td class="text-center"><p><span class="math notranslate nohighlight">\(100\)</span></p></td>
+</tr>
+<tr class="row-odd"><td><p><span class="math notranslate nohighlight">\(1\)</span></p></td>
+<td class="text-center"><p><span class="math notranslate nohighlight">\(135\)</span></p></td>
+</tr>
+<tr class="row-even"><td><p><span class="math notranslate nohighlight">\(2\)</span></p></td>
+<td class="text-center"><p><span class="math notranslate nohighlight">\(170\)</span></p></td>
+</tr>
+<tr class="row-odd"><td><p><span class="math notranslate nohighlight">\(3\)</span></p></td>
+<td class="text-center"><p><span class="math notranslate nohighlight">\(205\)</span></p></td>
+</tr>
+<tr class="row-even"><td><p><span class="math notranslate nohighlight">\(4\)</span></p></td>
+<td class="text-center"><p><span class="math notranslate nohighlight">\(240\)</span></p></td>
+</tr>
+</tbody>
+</table>
+</div>
+<p>Since there is an additive pattern for the increase in the total number of eggs laid, with the number of eggs in any given day, we can determine the number of eggs the colony will have in the following day. If we index the days by a sequence of natural numbers <em>n</em>, we have for the above colony:</p>
+<div class="math notranslate nohighlight">
+\[a_{n+1} = a_{n} + 35,\]</div>
+<p>where <span class="math notranslate nohighlight">\(a_n\)</span> represents the total amount of eggs in a given day <span class="math notranslate nohighlight">\(n\)</span> and <span class="math notranslate nohighlight">\(a_{n+1}\)</span> the same variable on the day following day <span class="math notranslate nohighlight">\(n\)</span>. The constant <span class="math notranslate nohighlight">\(35\)</span> represents the daily increment (or decrement) on this total number. A sequence of number built with constant additive increments receives the name of <em>arithmetic progression</em>.</p>
+<p>The above expression defines an iterative process (also known as a <em>recursion</em>) with which we can construct the whole sequence starting from a single initial value. If we want to know how many eggs will have been laid on a given day <span class="math notranslate nohighlight">\(n\)</span>, this shall be given by the expression:</p>
+<div class="math notranslate nohighlight">
+\[a_n = 100 + 35n.\]</div>
+<p>Thus, the total amount of eggs laid on day <span class="math notranslate nohighlight">\(n=100\)</span> is <span class="math notranslate nohighlight">\(a_{100} = 100 + 35 \cdot 100 = 3600\)</span>.</p>
+<br />
+<br />
+Now imagine a single *Escherichia coli* cell put on a Petri dish with a favorable growth medium. If given the right conditions, *E. coli* cells divide, generating two other cells, in a time of approximately $30$ minutes. For our experiment, we would have:  
+<br />  
+<div class="pst-scrollable-table-container"><table class="table">
+<thead>
+<tr class="row-odd"><th class="head"><p>Minutes</p></th>
+<th class="head text-center"><p># cells</p></th>
+</tr>
+</thead>
+<tbody>
+<tr class="row-even"><td><p><span class="math notranslate nohighlight">\(0\)</span></p></td>
+<td class="text-center"><p><span class="math notranslate nohighlight">\(1\)</span></p></td>
+</tr>
+<tr class="row-odd"><td><p><span class="math notranslate nohighlight">\(30\)</span></p></td>
+<td class="text-center"><p><span class="math notranslate nohighlight">\(2\)</span></p></td>
+</tr>
+<tr class="row-even"><td><p><span class="math notranslate nohighlight">\(60\)</span></p></td>
+<td class="text-center"><p><span class="math notranslate nohighlight">\(4\)</span></p></td>
+</tr>
+<tr class="row-odd"><td><p><span class="math notranslate nohighlight">\(90\)</span></p></td>
+<td class="text-center"><p><span class="math notranslate nohighlight">\(8\)</span></p></td>
+</tr>
+<tr class="row-even"><td><p><span class="math notranslate nohighlight">\(120\)</span></p></td>
+<td class="text-center"><p><span class="math notranslate nohighlight">\(16\)</span></p></td>
+</tr>
+</tbody>
+</table>
+</div>
+<p>If <span class="math notranslate nohighlight">\(t\)</span> represents the number of reproduction intervals since the start of the experiment (assuming values <span class="math notranslate nohighlight">\(t=0, 1, 2, \ldots\)</span>), then we can construct the recursion:</p>
+<div class="math notranslate nohighlight">
+\[N_{t+1} = 2N_t,\]</div>
+<p>where <span class="math notranslate nohighlight">\(N_t\)</span> represents the number of <em>E. coli</em> cells after <span class="math notranslate nohighlight">\(t\)</span> intervals of reproduction (each interval corresponding to <span class="math notranslate nohighlight">\(30\)</span> minutes) and <span class="math notranslate nohighlight">\(N_{t+1}\)</span> represents the number of cells on the interval following <span class="math notranslate nohighlight">\(t\)</span>. See that this time the constant <span class="math notranslate nohighlight">\(2\)</span> provides a multiplicative increment, so that the sequence formed by this multiplicative iteration is called <em>geometric progression</em>.</p>
+<p>If we want <span class="math notranslate nohighlight">\(N_t\)</span> for a given interval <span class="math notranslate nohighlight">\(t\)</span>, we then have:</p>
+<div class="math notranslate nohighlight">
+\[N_t = 2^t,\]</div>
+<p>leading to an exponential growth in discrete steps.</p>
+<p>In general, if we have a population growth described by a geometric progression, starting with a number of individuals <span class="math notranslate nohighlight">\(N_0\)</span>, the number of individuals on the subsequent generations can be described by the recursion:</p>
+<div class="math notranslate nohighlight">
+\[N_{t+1} = RN_t,\]</div>
+<p>where <span class="math notranslate nohighlight">\(t\)</span> indicates the number of generations and <span class="math notranslate nohighlight">\(R\)</span> is a positive constant (<span class="math notranslate nohighlight">\(R&gt;0\)</span>).</p>
+<p>With this recursion, starting from <span class="math notranslate nohighlight">\(N_0\)</span>, we can obtain the full sequence by construction:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{align}  
+N_1 &amp;= RN_0 \\
+N_2 &amp;= RN_1 = R(RN_0) = R^2N_0 \\
+N_3 &amp;= RN_2 = R(RN_1) = R[R(RN_0)] = R^3N_0 \\
+    &amp;\vdots \end{align}\end{split}\]</div>
+<p>From this process of construction, we can obtain what we call the <em>solution</em> of the above recursion, given by:</p>
+<div class="math notranslate nohighlight">
+\[N_t = N_0R^t.\]</div>
+<p>In problems of population dynamics, <span class="math notranslate nohighlight">\(R\)</span> is called <strong>growth constant</strong> and its value determines the long-term behaviour of the function:</p>
+<img alt="../../_images/expgrowth.png" src="../../_images/expgrowth.png" />
+<section id="sequences">
+<h3>Sequences<a class="headerlink" href="#sequences" title="Link to this heading">#</a></h3>
+<p>The two last problems are examples of <em>sequences</em> for which a formal definition is given as:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{align}
+    f\!:\ &amp;{\rm I\!N} \rightarrow {\rm I\!R} \\
+       &amp; n \rightarrow f(n),\end{align}\end{split}\]</div>
+<p>in other words, a sequence is a function <span class="math notranslate nohighlight">\(f\)</span> from the set of natural numbers to the set of real numbers. For each value of the independent variable <span class="math notranslate nohighlight">\(n\)</span> (a natural number), the sequence determines its image <span class="math notranslate nohighlight">\(f(n)\)</span>.</p>
+<div class="admonition-example admonition">
+<p class="admonition-title">Example</p>
+<p>The sequence defined by the rule <span class="math notranslate nohighlight">\(f(n) = \displaystyle\frac{1}{n+1}\)</span> defines the sequence of numbers (starting from <span class="math notranslate nohighlight">\(n=0\)</span>):</p>
+<div class="math notranslate nohighlight">
+\[1,\frac{1}{2},\frac{1}{3},\frac{1}{4},\frac{1}{5},\ldots,\]</div>
+<p>corresponding to the set of values <span class="math notranslate nohighlight">\(n=0,1,2,3,4,\ldots\)</span>.</p>
+</div>
+<p>Sequences are usually written as ordered lists of numbers <span class="math notranslate nohighlight">\(a_0, a_1, a_2, a_3, \ldots\)</span> where <span class="math notranslate nohighlight">\(a_n=f(n)\)</span> and referred to with the notation <span class="math notranslate nohighlight">\(\{a_n\}\)</span> (which is short for <span class="math notranslate nohighlight">\(\{a_n: n \in {\rm I\!N}\}\)</span>, making it explicit that <span class="math notranslate nohighlight">\(n\)</span> is a natural number).</p>
+<p>As we have seen before, two forms of representation are possible:</p>
+</section>
+<section id="limits">
+<h3>Limits<a class="headerlink" href="#limits" title="Link to this heading">#</a></h3>
+<p>In many applications, we are interested to know what is the <em>long-term behaviour</em> of a given sequence <span class="math notranslate nohighlight">\(\{a_n\}\)</span>. For example, is a given population tending towards a stable number of individuals as time passes? In other words, we want to know if there is a number <span class="math notranslate nohighlight">\(L\)</span> for which the sequence tends to as <span class="math notranslate nohighlight">\(n\)</span> increases. To represent this we use the following notation:</p>
+<div class="math notranslate nohighlight">
+\[\lim_{n\rightarrow \infty} a_n = L\ \ (\mbox{or simply}\ \lim a_n =L)\]</div>
+<p>If such a number exists, we say that the sequence <span class="math notranslate nohighlight">\(\{a_n\}\)</span> is <strong>convergent</strong>.</p>
+<div class="admonition-examples admonition">
+<p class="admonition-title">Examples</p>
+<ol class="arabic">
+<li><p><span class="math notranslate nohighlight">\(a_n = \displaystyle\frac{1}{n+1}: \ \ \ \left\{1,\displaystyle\frac{1}{2},\displaystyle\frac{1}{3},\displaystyle\frac{1}{4},\displaystyle\frac{1}{5},\ldots\right\}\)</span><br />
+<br />
+Note that each term is smaller than the previous and they are all positive. We can conclude that:</p>
+<div class="math notranslate nohighlight">
+\[\lim a_n = 0.\]</div>
+</li>
+<li><p><span class="math notranslate nohighlight">\(b_n = (-1)^n: \ \ \ \left\{1,-1,1,-1,1,-1,\ldots\right\}\)</span><br />
+<br />
+The terms in this sequence are alternating between <span class="math notranslate nohighlight">\(1\)</span> and <span class="math notranslate nohighlight">\(-1\)</span>, never convergeing for a specific value. Thus, the limit of sequence <span class="math notranslate nohighlight">\(\{b_n\}\)</span> does not exist.<br />
+<br /></p></li>
+<li><p><span class="math notranslate nohighlight">\(c_n = 2^n: \ \ \ \left\{1,2,4,8,16,32,\ldots\right\}\)</span><br />
+<br />
+For this sequence, the terms are ever increasing. We frequently say that the sequence tends to infinity (or tends to <span class="math notranslate nohighlight">\(\infty\)</span>). However, <span class="math notranslate nohighlight">\(\infty\)</span> represents an abstraction (“ever increasing”), not a specific number. Therefore, the limit for this sequence also does not exist.<br />
+<br /></p></li>
+<li><p><span class="math notranslate nohighlight">\(d_n = \displaystyle\frac{n+2}{n+1}: \ \ \ \left\{2,\displaystyle\frac{3}{2},\displaystyle\frac{4}{3},\displaystyle\frac{5}{4},\displaystyle\frac{6}{5},\ldots\right\}\)</span><br />
+<br />
+For this sequence, each term is smaller than the previous (look at the decimal forms of the fractions: <span class="math notranslate nohighlight">\(2, 1.5, 1.333\cdots, 1.25, 1.2 \ldots\)</span>) and they are greater than <span class="math notranslate nohighlight">\(1\)</span>. We can conclude that:</p>
+<div class="math notranslate nohighlight">
+\[\lim a_n = 1.\]</div>
+</li>
+</ol>
+</div>
+<p>Although there is a formal way to <em>prove</em> that a sequence <span class="math notranslate nohighlight">\(\{a_n\}\)</span> converges to a certain limit, in practice we use a set of rules to apply limits to sequences and composition of sequences.</p>
+<div class="admonition tip">
+<p class="admonition-title">Tip</p>
+<p>Consider two sequences <span class="math notranslate nohighlight">\(\{a_n\}\)</span> and <span class="math notranslate nohighlight">\(\{b_n\}\)</span>, and <span class="math notranslate nohighlight">\(C\)</span> as a real constant. If these two sequences are convergent with  <span class="math notranslate nohighlight">\(\lim a_n = L\)</span> and <span class="math notranslate nohighlight">\(\lim b_n = M\)</span>, we have:</p>
+<div class="math notranslate nohighlight">
+\[\lim (a_n + b_n) = \lim a_n + \lim b_n = L + M\]</div>
+<div class="math notranslate nohighlight">
+\[\lim (Ca_n) = C\lim a_n = CL\]</div>
+<div class="math notranslate nohighlight">
+\[\lim (a_nb_n) = (\lim a_n)(\lim b_n) = LM\]</div>
+<div class="math notranslate nohighlight">
+\[\lim \left(\frac{a_n}{b_n}\right) = \frac{\lim a_n}{\lim b_n} = \frac{L}{M},\ \ (\mbox{if}\ M \ne 0)\]</div>
+<p>In other other words, if you can decompose a given sequence into the sum, product or division of two other convergent sequences, the limit of this sequence will be the composition of the limits of the two others.</p>
+</div>
+<div class="admonition-examples admonition">
+<p class="admonition-title">Examples</p>
+<p>If <span class="math notranslate nohighlight">\(a_n = \displaystyle\frac{4n^2-1}{n^2}:\)</span></p>
+<div class="math notranslate nohighlight">
+\[\lim\left(\displaystyle\frac{4n^2-1}{n^2}\right) = \lim\left(4-\displaystyle\frac{1}{n^2}\right) = \lim 4 - \left(\lim\displaystyle\frac{1}{n}\right)\left(\lim\displaystyle\frac{1}{n}\right) = 4,\]</div>
+<p>since as we know: <span class="math notranslate nohighlight">\(\ \lim 4 = 4\ \)</span> and <span class="math notranslate nohighlight">\(\ \lim \displaystyle\frac{1}{n} = 0\)</span>.</p>
+</div>
+<p>For recursions, one way to find the limit is to find the solution (corresponding explicit description) and then calculate the limit of the sequence with the limit rules. However, finding the explicit form of a given sequence is not always possible.</p>
+<p>There is a procedure to find <em>candidates</em> for the long-term behaviour of a given sequence. We do this by calculating <strong>fixed points</strong>.</p>
+<p>Fixed points are specific values on a given sequence for which all subsequent values equal the first. That is, if <span class="math notranslate nohighlight">\(a\)</span> is a fixed point of the sequence <span class="math notranslate nohighlight">\(\{a_n\}\)</span>, and if <span class="math notranslate nohighlight">\(a_0 = a\)</span>, then <span class="math notranslate nohighlight">\(a_1 = a\)</span>, <span class="math notranslate nohighlight">\(a_2 = a\)</span>, and all the other values will be <span class="math notranslate nohighlight">\(a\)</span>. Thus, given the recursion <span class="math notranslate nohighlight">\(a_{n+1} = g(a_n)\)</span>, if <span class="math notranslate nohighlight">\(a\)</span> is a fixed point we will have:</p>
+<div class="math notranslate nohighlight">
+\[a = g(a)\]</div>
+<p>and the last equation provides a way of find possible values of <span class="math notranslate nohighlight">\(a\)</span>.</p>
+<div class="admonition-example admonition">
+<p class="admonition-title">Example</p>
+<p>Consider the following recursion</p>
+<div class="math notranslate nohighlight">
+\[a_{n+1} = \sqrt{3a_n},\ \ \ a_0 = 2.\]</div>
+<p>To obtain the fixed points we calculate:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{align}
+  a &amp;= \sqrt{3a} \\
+  a^2 &amp;= 3a \ \ \longrightarrow a = 0\ \mbox{ or }\ a = 3 \end{align}\end{split}\]</div>
+<p>Starting with <span class="math notranslate nohighlight">\(a_0 = 2\)</span>, we can use the recursion to construct the sequence as:</p>
+<div class="math notranslate nohighlight">
+\[\left\{\sqrt{6},\sqrt{3\sqrt{6}},\sqrt{3\sqrt{3\sqrt{6}}},\sqrt{3\sqrt{3\sqrt{3\sqrt{6}}}},\ldots \right\}\]</div>
+<p>or also: <span class="math notranslate nohighlight">\(\{2.449\cdots,2.711\cdots,2.852\cdots,2.925\cdots, \ldots\}\)</span>. We see that starting with <span class="math notranslate nohighlight">\(a_0=2\)</span>, we have <span class="math notranslate nohighlight">\(a_n&gt;2\)</span> for all <span class="math notranslate nohighlight">\(n\)</span>, with increasing values. Since <span class="math notranslate nohighlight">\(a=3\)</span> is a fixed point, we conclude that <span class="math notranslate nohighlight">\(\lim a_n = 3\)</span>.</p>
+</div>
+<div class="admonition-example admonition">
+<p class="admonition-title">Example</p>
+<p>Whenever we are asking questions like ‘where does this process stabilise’ or ‘how high/much will it be in the end,’ we are interested in the limit of some series. This could be a variety of things, like population size, ground water levels, global mean temperature, number of trees, etc.</p>
+</div>
+<div class="admonition warning">
+<p class="admonition-title">Warning</p>
+<p>Remember that fixed points are only <strong>candidates</strong> for long-term behaviour. See for example the recursion:</p>
+<div class="math notranslate nohighlight">
+\[a_{n+1} = \displaystyle\frac{3}{a_n}\]</div>
+<p>For the fixed points, we calculate:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{align}
+  a &amp;= \displaystyle\frac{3}{a} \\
+  a^2 &amp;= 3 \\
+  &amp;\longrightarrow a = -\sqrt{3}\ \mbox{ or }\ a = \sqrt{3}. \end{align}\end{split}\]</div>
+<p>Indeed, if we construct the sequence from the recursion starting with any of these two values, all the subsequent elements of the sequence will have the same value. However:</p>
+<div class="math notranslate nohighlight">
+\[\mbox{if } a_0 = 2 \ \mbox{the sequence will be given by: }\ \left\{2,\displaystyle\frac{3}{2},2,\displaystyle\frac{3}{2},\ldots \right\}\]</div>
+<p>or</p>
+<div class="math notranslate nohighlight">
+\[\mbox{If } a_0 = -3 \ \mbox{the sequence will be given by: }\ \left\{-3,-1,-3,-1,\ldots \right\}\]</div>
+<p>showing that for any initial value chosen (different from the fixed points), the sequence will oscillate between two values, without a clear tendency to any limit.</p>
+</div>
+</section>
+<section id="discrete-time-population-models">
+<h3>Discrete-time population models<a class="headerlink" href="#discrete-time-population-models" title="Link to this heading">#</a></h3>
+<p>Sequences are generally used in many applications in Biology. Important examples are models for <strong>seasonally breeding</strong> populations.</p>
+<section id="density-dependent-population-growth">
+<h4>Density-dependent population growth<a class="headerlink" href="#density-dependent-population-growth" title="Link to this heading">#</a></h4>
+<p>As we have seen before, the recursion relation for the exponential growth model is given by <span class="math notranslate nohighlight">\(N_{t+1} = RN_t\)</span>. Rearranging the terms of this equation we have:</p>
+<div class="math notranslate nohighlight">
+\[\displaystyle\frac{N_t}{N_{t+1}}=\frac{1}{R}.\]</div>
+<p>The left-hand term of the last equation is known as <em>parent-offspring ratio</em> as it denotes the ratio between the population size at time <span class="math notranslate nohighlight">\(t\)</span> (parents) and the population size at the following time <span class="math notranslate nohighlight">\(t+1\)</span> (offspring). Here we are considering a population that breeds without overlapping generations. Each generation breeds, dies, and is replaced by their offspring on the following generation.</p>
+<p>On the equation we also see that the parent-offspring ratio is equal to <span class="math notranslate nohighlight">\(\displaystyle\frac{1}{R}\)</span>, which is a constant. Therefore, we say that this growth is <em>density-independent</em> as the parent-offspring ratio does not depend on the density of individuals at any time step. Relating the parent-offspring ratio to the value of <span class="math notranslate nohighlight">\(R\)</span> allows us to predict the behaviour of this population in terms of growth or decline:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{align}
+R&gt;1 \ &amp;- \ \mbox{there will be more offspring than parents (population grows)} \\
+R&lt;1 \ &amp;- \ \mbox{there will be more parents than offspring (population declines)} \\
+R=1 \ &amp;- \ \mbox{there will be no change in population size} \end{align}\end{split}\]</div>
+<p>We know, however, than in real scenarios, that the growth factor of a given population should depend on the current size of that population, due to space or resources limitation. These are called <em>density-dependent</em> effects. One of the simplest methods to include density-dependent effects in this population model is to consider the parent-offspring ratio as a linear function of <span class="math notranslate nohighlight">\(N_t\)</span>, instead of a constant. Let us assume that the parent-offspring ratio show the following linear behaviour:</p>
+<img alt="../../_images/parent-offspring.png" src="../../_images/parent-offspring.png" />
+<p>This means that the parent-offspring ratio is smaller than <span class="math notranslate nohighlight">\(1\)</span> (population growth) for values of <span class="math notranslate nohighlight">\(N_t\)</span> smaller than a given population threshold <span class="math notranslate nohighlight">\(K\)</span>, and greater than <span class="math notranslate nohighlight">\(1\)</span> (population decline) if the population size is greater than this threshold. The graph above leads to the following expression:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{align}
+\displaystyle\frac{N_t}{N_{t+1}} = \frac{1}{R} + \frac{1-\frac{1}{R}}{K}N_t \\
+N_{t+1} = \frac{N_t}{\frac{1}{R} + \frac{1-\frac{1}{R}}{K}N_t}, \end{align}\end{split}\]</div>
+<p>which leads, after rearranging the terms, to:</p>
+<div class="math notranslate nohighlight">
+\[N_{t+1} = \frac{RN_t}{1 + \frac{(R-1)}{K}N_t}.\]</div>
+<p>The previous expression is known as <strong>Beverton-Holt recruitment curve</strong> (or Beverton-Holt model) and it describes a type of population growth called <em>logistic growth</em>.</p>
+<p>The fixed points <span class="math notranslate nohighlight">\(\bar{N}\)</span> for this recursion can be calculated as:</p>
+<div class="math notranslate nohighlight">
+\[\bar{N} = \frac{R\bar{N}}{1 + \frac{(R-1)}{K}\bar{N}}\ \longrightarrow \ \bar{N}\left(1 + \frac{(R-1)}{K}\bar{N}\right) = R\bar{N}\]</div>
+<p>From the last term we can show (try it as an exercise) that <span class="math notranslate nohighlight">\(\bar{N}=0\)</span> and <span class="math notranslate nohighlight">\(\bar{N}=K\)</span> are the two fixed points. Starting from different values for the initial population size <span class="math notranslate nohighlight">\(N_0\)</span> it can be shown that the population tends to <span class="math notranslate nohighlight">\(K\)</span> as time increases as:</p>
+<img alt="../../_images/logistic.png" src="../../_images/logistic.png" />
+<p>The constant <span class="math notranslate nohighlight">\(K\)</span> is know as <em>carrying capacity</em> and represents the maximum population size a given population can reach at a given place, due to intra-specific competition for resources or space.</p>
+<!-- Watch this [video](https://www.youtube.com/watch?v=Kas0tIxDvrg) about application of growth models to epidemics.   -->
+</section>
+<section id="annual-plants">
+<h4>Annual plants<a class="headerlink" href="#annual-plants" title="Link to this heading">#</a></h4>
+<p>Suppose a given population of plants produces, on average <span class="math notranslate nohighlight">\(f\)</span> seeds per individual in the reproductive period. Seeds survive winter with a probability <span class="math notranslate nohighlight">\(\sigma\)</span> and germinate on the next season with probability <span class="math notranslate nohighlight">\(\alpha\)</span> (these seeds reach reproductive maturity on the same season). If <span class="math notranslate nohighlight">\(p_n\)</span> is the plant population size on season <span class="math notranslate nohighlight">\(n\)</span> and <span class="math notranslate nohighlight">\(p_{n+1}\)</span> the population size on the season following <span class="math notranslate nohighlight">\(n\)</span>, we shall have:</p>
+<div class="math notranslate nohighlight">
+\[p_{n+1} = (\alpha\sigma f)p_n\]</div>
+<p>which correspond to the density-independent model since <span class="math notranslate nohighlight">\(R=\alpha\sigma f\)</span> is a constant.</p>
+<p>Now suppose that some of the seeds that were produced on the previous season and survived the previous winter, but did not germinate on the current season, might have a second chance of germinating. They might do so on the following season, provided they survive the winter and germinate on the following season. From the average number of seeds produced last season <span class="math notranslate nohighlight">\(fp_{n-1}\)</span>, a number <span class="math notranslate nohighlight">\(\sigma fp_{n-1}\)</span> of them survived last winter. From this number, a fraction <span class="math notranslate nohighlight">\((1-\alpha)\)</span> did not germinate this season (1 minus the probability of germinating). These seeds will have a second chance provided they survive next winter (with probability <span class="math notranslate nohighlight">\(\sigma\)</span>) and germinate on the next season (with probability <span class="math notranslate nohighlight">\(\alpha\)</span>). Thus, the number plant population size on the season following <span class="math notranslate nohighlight">\(n\)</span> will be:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{align}
+p_{n+1} = \alpha\sigma fp_n + \alpha\sigma(1-\alpha)\sigma fp_{n-1} \\  
+p_{n+1} = (\alpha\sigma f)p_n + [\alpha(1-\alpha)\sigma^2 f]p_{n-1} \\  
+p_{n+1} = \beta p_n + \gamma p_{n-1}\end{align}\end{split}\]</div>
+<p>where <span class="math notranslate nohighlight">\(\beta=\alpha\sigma f\)</span> and <span class="math notranslate nohighlight">\(\gamma = \alpha(1-\alpha)\sigma^2 f\)</span>.</p>
+<p>The previous model is an example of a discrete population model with overlapping populations. Since the term <span class="math notranslate nohighlight">\(n+1\)</span> depends not only on the term <span class="math notranslate nohighlight">\(n\)</span> but also on <span class="math notranslate nohighlight">\(n-1\)</span> this is called a <em>second-order difference equation</em> or <em>delay (or lagged) difference equation</em>.</p>
+</section>
+<section id="fibonacci-sequence">
+<h4>Fibonacci sequence<a class="headerlink" href="#fibonacci-sequence" title="Link to this heading">#</a></h4>
+<p>The Fibonacci sequence is a sequence defined by the following rule: each number is equal to the sum of the previous two numbers (starting with <span class="math notranslate nohighlight">\(0\)</span> and <span class="math notranslate nohighlight">\(1\)</span>). Thus the following sequence might be formed:</p>
+<div class="math notranslate nohighlight">
+\[\left\{ 0,1,1,2,3,5,8,13,21,\ldots \right\}. \]</div>
+<p>It can also be defined by the second-order difference equation:</p>
+<div class="math notranslate nohighlight">
+\[F_{n+1} = F_{n} + F_{n-1}.\]</div>
+<p>Although this is clearly not a convergent sequence (with sustained growth for each element), if we define a new sequence given by the ratio between successive elements of the Fibonacci sequence as <span class="math notranslate nohighlight">\(b_n = \displaystyle\frac{F_{n+1}}{F_n}\)</span>, it can be show that this sequence is convergent.</p>
+<p>Assuming that <span class="math notranslate nohighlight">\(\lim \displaystyle\frac{F_{n+1}}{F_n} = \lambda\)</span>, we have:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{align}
+F_{n+1} &amp;= F_{n} + F_{n-1}\quad \mbox{(Fibonacci seq.)} \\  
+\displaystyle\frac{F_{n+1}}{F_{n}} &amp;= 1 + \frac{F_{n-1}}{F_{n}}\quad \mbox{(divide both sides by}\ F_{n}\mbox{)} \\
+\lim \left(\displaystyle\frac{F_{n+1}}{F_{n}}\right) &amp;= \lim\left(1 + \frac{F_{n-1}}{F_{n}}\right) \\
+\lambda &amp;= 1 + \frac{1}{\lambda} \\
+\lambda^2 &amp;- \lambda -1 = 0 \end{align}\end{split}\]</div>
+<p>The previous equation is solved finding the roots of the second-degree polynomial, which gives <span class="math notranslate nohighlight">\(\lambda = \displaystyle\frac{1+\sqrt{5}}{2}\)</span> or <span class="math notranslate nohighlight">\(\lambda = \displaystyle\frac{1-\sqrt{5}}{2}\)</span>. Since the solution <span class="math notranslate nohighlight">\(\displaystyle\frac{1-\sqrt{5}}{2}\)</span> is negative, it can be the limit of the ratio of Fibonacci terms (all of them positive). Hence we conclude:</p>
+<div class="math notranslate nohighlight">
+\[\lim \left(\displaystyle\frac{F_{n+1}}{F_{n}}\right) = \displaystyle\frac{1+\sqrt{5}}{2} \approx 1.618.\]</div>
+<p>The previous number is known as the <em>Golden ratio</em> and it has fascinated mathematicians, phylosophers, and artists since ancient times. Fibonacci numbers and the Golden ratio can be found in several interesting natural patterns, as you can see in this <a class="reference external" href="https://www.youtube.com/watch?v=ahXIMUkSXX0">video</a>.</p>
+</section>
+</section>
+</section>
+<section id="from-table-arrangements-to-powerful-tools">
+<h2>From table arrangements to powerful tools<a class="headerlink" href="#from-table-arrangements-to-powerful-tools" title="Link to this heading">#</a></h2>
+<section id="systems-of-linear-equations">
+<h3>Systems of linear equations<a class="headerlink" href="#systems-of-linear-equations" title="Link to this heading">#</a></h3>
+<p>As we saw previously, the equation <span class="math notranslate nohighlight">\(ax + by = c\)</span>, with <span class="math notranslate nohighlight">\(a\)</span>, <span class="math notranslate nohighlight">\(b\)</span>, and <span class="math notranslate nohighlight">\(c\)</span> constants, defines a straight line on the <span class="math notranslate nohighlight">\(xy\)</span>-plane (note that it can also be written as <span class="math notranslate nohighlight">\(y = mx + k\)</span>, <span class="math notranslate nohighlight">\(m\)</span> and <span class="math notranslate nohighlight">\(k\)</span> constants, by a suitable change of the coeffients). Since the coefficients do not depend on the variables <span class="math notranslate nohighlight">\(x\)</span> and <span class="math notranslate nohighlight">\(y\)</span>, we call it a <em>linear equation</em>. A possible plot for this equation would be:</p>
+<img alt="../../_images/lineq.png" src="../../_images/lineq.png" />
+<p>Note that all the points <span class="math notranslate nohighlight">\((x,y)\)</span> on the blue line <em>satisfy</em> or <em>solve</em> the lienar equation <span class="math notranslate nohighlight">\(ax + by = c\)</span>.</p>
+<p>Now consider that we have the following:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{cases} ax + by = c \\ dx +ey = f \end{cases}\end{split}\]</div>
+<p>where <span class="math notranslate nohighlight">\(a, b, c, d, e,\ \mbox{and}\ f\)</span> are all <a class="reference external" href="http://constants.In">constants.In</a> this case we have a <em>system of linear equations</em>. Solving this system means to find all possible pairs <span class="math notranslate nohighlight">\((x,y)\)</span> that solve both equations <strong>simultaneously</strong>. For the previous system, we would have 3 options:</p>
+<img alt="../../_images/syslineq.png" src="../../_images/syslineq.png" />
+<p>On the left panel, the two lines intersect at a <em>single point</em>. This point is the only one that solves both linear equations at the same time, and, therefore, constitutes the <strong>only solution</strong> for the system of two equations. On the central panel, the two lines are <em>parallel</em>, so there is no point of intersection. In this case there is <strong>no solution</strong> and the system of equations is said to be <em>inconsistent</em>. On the right panel, the two lines coincide. Thus, every point in those lines correspond to a possible solution for the system, so that it has <strong>infinite solutions</strong>.</p>
+<div class="admonition-example admonition">
+<p class="admonition-title">Example</p>
+<p>Given the system of linear equations</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{cases} 2x + 3y = 6 \\ 2x + y = 4 \end{cases}\end{split}\]</div>
+<p>we can obtain the solution by isolating one of the variables in one of the equations and substituting on the other equation. Isolating <span class="math notranslate nohighlight">\(x\)</span> on the first equation gives:</p>
+<div class="math notranslate nohighlight">
+\[x = 3 - \displaystyle\frac{3}{2}y\]</div>
+<p>Then by substituting on the second we have:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{align} 
+2\left(3 - \displaystyle\frac{3}{2}y \right) + y &amp;= 4 \\
+6 - 3y + y &amp;= 4 \\
+-2y = -2 \\
+y = 1 \end{align}\end{split}\]</div>
+<p>The value of <span class="math notranslate nohighlight">\(x\)</span> shall then be given by:</p>
+<div class="math notranslate nohighlight">
+\[x = 3 - \displaystyle\frac{3}{2}\cdot 1 = 3 - \frac{3}{2} = \frac{3}{2}\]</div>
+<p>We see that the system has a single solution, given by <span class="math notranslate nohighlight">\(\left(\displaystyle\frac{3}{2},1\right)\)</span>.</p>
+</div>
+<div class="admonition-example admonition">
+<p class="admonition-title">Example</p>
+<p>Now consider the sytem</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{cases} 2x + 3y &amp;= 6 \\ -4x + -6y &amp;= -12 \end{cases}\end{split}\]</div>
+<p>Isolating <span class="math notranslate nohighlight">\(x\)</span> on the first equation gives the same as before:</p>
+<div class="math notranslate nohighlight">
+\[x = 3 - \displaystyle\frac{3}{2}y\]</div>
+<p>Then by substituting on the second we have:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{align} 
+-4\left(3 - \displaystyle\frac{3}{2}y \right) - 6y &amp;= -12 \\
+-12 + 6y - 6y &amp;= -12 \\
+0 = 0 \end{align}\end{split}\]</div>
+<p>which is just a trivial equation.</p>
+<p>When we look at the equations in our system we see that the second equation is simply a constant multiplied by the first <span class="math notranslate nohighlight">\((\text{Eqn.} 2) = -2\cdot(\text{Eqn.} 1)\)</span>, and thus they should represent the same straight line.</p>
+<p>This system has then infinite solutions which are represented by the set of points <span class="math notranslate nohighlight">\(\left(3 - \displaystyle\frac{3}{2}y,y\right)\)</span></p>
+<div class="admonition tip">
+<p class="admonition-title">Tip</p>
+<p>Note that writing the solution as <span class="math notranslate nohighlight">\(\left(3 - \displaystyle\frac{3}{2}y,y\right)\)</span> represents an infinite set of points, since for every real value of <span class="math notranslate nohighlight">\(y\)</span> we would have a different ordered pair. This set of points forms the straight line corresponding to <span class="math notranslate nohighlight">\(2x+3y = 6\)</span>.</p>
+</div>
+</div>
+<p>Solving the system by this method of substitutions, however, can get much more complicated if we have more than two variables. For these cases, we need a simpler method.</p>
+<p>One strategy is to find equivalent systems, which have the same solution as the original one, but are much easier to solve. We can apply the following set of operations without changing the solutions of a given system:</p>
+<ul class="simple">
+<li><p>Multiplication of an entire row by a non-zero constant</p></li>
+<li><p>Adding/subtracting one row to another</p></li>
+<li><p>Rearranging the order of the rows</p></li>
+</ul>
+<p>The examples below show this procedure applied to two systems.</p>
+<div class="admonition-examples admonition">
+<p class="admonition-title">Examples</p>
+<p>(i) <span class="math notranslate nohighlight">\(\begin{cases} 3x + 5y - z = 10 \\ 2x –  y + 3z = 9 \\ 4x + 2y - 3z = -1 \end{cases}\)</span></p>
+<p>(ii) <span class="math notranslate nohighlight">\(\begin{cases} x - 3y + z = 4 \\ x – 2y + 3z = 6 \\ 2x - 6y + 2z = 8 \end{cases}\)</span></p>
+</div>
+<!-- See the solution on this [video](https://web.microsoftstream.com/video/194b0152-903a-4614-b608-306ce71d9a39) -->
+<!-- See the solution on this [video](https://web.microsoftstream.com/video/067c8938-5e45-4f08-bf90-783cb125c6f5) -->
+<p>Note that by applying the rules of matrix multiplication, the system <em>(ii)</em> can also be written in the form:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{pmatrix} 1 &amp; -3 &amp; 1\\ 1 &amp; -2 &amp; 3 \\ 2 &amp; -6 &amp; 2  \end{pmatrix} \begin{pmatrix} x \\ y \\ z   \end{pmatrix} = \begin{pmatrix} 4 \\ 6 \\ 8   \end{pmatrix} \quad \longrightarrow \quad A\mathbf{x}=\mathbf{b}\end{split}\]</div>
+<p>where <span class="math notranslate nohighlight">\(A\)</span> is the matrix of coefficients and <span class="math notranslate nohighlight">\(\mathbf{x}\)</span> is the vector of variables.</p>
+<div class="admonition tip">
+<p class="admonition-title">Tip</p>
+<p>We will be using the common notation of uppercase letters (<span class="math notranslate nohighlight">\(A\)</span>) to represent matrices and lowercase bold letters (<span class="math notranslate nohighlight">\(\mathbf{x}\)</span>) to represent vectors.</p>
+</div>
+<p>In general, the system with <span class="math notranslate nohighlight">\(n\)</span> variables <span class="math notranslate nohighlight">\(\{x_1, x_2, \ldots, x_n \}\)</span> and <span class="math notranslate nohighlight">\(m\)</span> equations can be written as</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{cases} a_{11}x_1 + a_{12}x_2 + \ldots + a_{1n}x_n = b_1 \\ a_{21}x_1 + a_{22}x_2 + \ldots + a_{2n}x_n = b_2 \\ \dots \\ a_{m1}x_1 + a_{m2}x_2 + \ldots + a_{mn}x_n = b_m \end{cases}\end{split}\]</div>
+<p>can be written as <span class="math notranslate nohighlight">\(A\mathbf{x}=\mathbf{b}\)</span>, where</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}A=\begin{pmatrix} a_{11} &amp; a_{12} &amp; \cdots &amp; a_{1n} \\ a_{21} &amp; a_{22} &amp; \cdots &amp; a_{2n} \\ \vdots &amp; &amp; \ddots &amp; \vdots \\ a_{m1} &amp; a_{m2} &amp; \cdots &amp; a_{mn} \end{pmatrix}\end{split}\]</div>
+<p>and</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\mathbf{x} = \begin{pmatrix} x_1 \\ x_2 \\ \vdots \\ x_n   \end{pmatrix}\quad \mbox{and} \quad \mathbf{b} = \begin{pmatrix} b_1 \\ b_2 \\ \vdots \\ b_m   \end{pmatrix}\end{split}\]</div>
+<p>The matrix <span class="math notranslate nohighlight">\(A\)</span> has size <span class="math notranslate nohighlight">\(m \times n\)</span> (since the number of equations can, in principle, be different from the number of variables), while the vectors <span class="math notranslate nohighlight">\(\mathbf{x}\)</span> and <span class="math notranslate nohighlight">\(\mathbf{b}\)</span> have sizes <span class="math notranslate nohighlight">\(n \times 1\)</span> and <span class="math notranslate nohighlight">\(m \times 1\)</span>, respectively.</p>
+</section>
+<section id="matrix-operations">
+<h3>Matrix operations<a class="headerlink" href="#matrix-operations" title="Link to this heading">#</a></h3>
+<p>Let us quickly remind the basic operations that can be performed with matrices. If <span class="math notranslate nohighlight">\(A\)</span> and <span class="math notranslate nohighlight">\(B\)</span> are both matrices of size <span class="math notranslate nohighlight">\(m \times n\)</span> (<span class="math notranslate nohighlight">\(m\)</span> rows and <span class="math notranslate nohighlight">\(n\)</span> columns), we have:</p>
+<ul>
+<li><p><span class="math notranslate nohighlight">\(A=B\ \)</span> if and only if <span class="math notranslate nohighlight">\(a_{ij} = b_{ij}\)</span> for every <span class="math notranslate nohighlight">\(1 \le i \le m,  1 \le j \le n\)</span> (elements at the same position have to be equal).</p></li>
+<li><p><span class="math notranslate nohighlight">\(C = A + B\)</span> is the matrix for which <span class="math notranslate nohighlight">\(c_{ij} = a_{ij} + b_{ij}\)</span> (sum is performed summing the elements at the same position)</p></li>
+<li><p>If <span class="math notranslate nohighlight">\(k \in \rm I\!R\)</span> than <span class="math notranslate nohighlight">\(kA\)</span> is the <span class="math notranslate nohighlight">\(m \times n\)</span> matrix with elements <span class="math notranslate nohighlight">\(ka_{ij}\)</span></p></li>
+<li><p>The following properties are valid:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{align}
+    A+B &amp;= B+A \\
+    (A+B)+C &amp;= A+(B+C) \\
+    A + 0 &amp;= A\ (0\ \mbox{here represents the matrix with null elements}) \end{align}\end{split}\]</div>
+<div class="admonition-example admonition">
+<p class="admonition-title">Example</p>
+<p>If <span class="math notranslate nohighlight">\(A=\begin{pmatrix} 1 &amp; 0  \\ -1 &amp; 2  \end{pmatrix}\)</span> and <span class="math notranslate nohighlight">\(B=\begin{pmatrix} 2 &amp; 3  \\ -1 &amp; 1  \end{pmatrix}\)</span>, then:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}C = 2A+B = \begin{pmatrix} 2 &amp; 0  \\ -2 &amp; 4  \end{pmatrix} + \begin{pmatrix} 2 &amp; 3  \\ -1 &amp; 1  \end{pmatrix} = \begin{pmatrix} 4 &amp; 3  \\ -3 &amp; 5  \end{pmatrix}\end{split}\]</div>
+</div>
+</li>
+<li><p>If <span class="math notranslate nohighlight">\(A\)</span> is the <span class="math notranslate nohighlight">\(m \times n\)</span> matrix with elements <span class="math notranslate nohighlight">\(a_{ij}\)</span>, the transpose of <span class="math notranslate nohighlight">\(A\)</span> (denoted by <span class="math notranslate nohighlight">\(A'\)</span>) is the <span class="math notranslate nohighlight">\(n \times m\)</span> matrix with elements <span class="math notranslate nohighlight">\(a_{ij}' = a_{ji}\)</span>.</p>
+<div class="admonition-example admonition">
+<p class="admonition-title">Example</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}A=\begin{pmatrix} 1 &amp; 2 &amp; 3 \\ 4 &amp; 5 &amp; 6  \end{pmatrix} \longrightarrow A'=\begin{pmatrix} 1 &amp; 4 \\ 2 &amp; 5 \\ 3 &amp; 6   \end{pmatrix}\end{split}\]</div>
+<div class="math notranslate nohighlight">
+\[\begin{split}B=\begin{pmatrix} 1 &amp; 2 \end{pmatrix} \longrightarrow B' = \begin{pmatrix} 1 \\ 2 \end{pmatrix}\end{split}\]</div>
+</div>
+</li>
+<li><p><strong>(Matrix multiplication)</strong> Consider now that <span class="math notranslate nohighlight">\(A\)</span> and <span class="math notranslate nohighlight">\(B\)</span> are matrices with sizes <span class="math notranslate nohighlight">\(m \times l\)</span> and <span class="math notranslate nohighlight">\(l \times n\)</span> (number of columns of <span class="math notranslate nohighlight">\(A\)</span> is equal to the number of rows of <span class="math notranslate nohighlight">\(B\)</span>). Then we can define <span class="math notranslate nohighlight">\(C = AB\)</span>  as the <span class="math notranslate nohighlight">\(m \times n\)</span> matrix with elements <span class="math notranslate nohighlight">\(c_{ij}\)</span> formed by multiplying every element of the <span class="math notranslate nohighlight">\(i\)</span>-th row of <span class="math notranslate nohighlight">\(A\)</span> with the corresponding element of the <span class="math notranslate nohighlight">\(j\)</span>-th column of <span class="math notranslate nohighlight">\(B\)</span>.</p>
+<div class="admonition-example admonition">
+<p class="admonition-title">Example</p>
+<p>If <span class="math notranslate nohighlight">\(A=\begin{pmatrix} 1 &amp; 2  \\ -1 &amp; 0  \end{pmatrix}\)</span> and <span class="math notranslate nohighlight">\(B=\begin{pmatrix} 1 &amp; 2 &amp; 3  \\ 0 &amp; -1 &amp; 4  \end{pmatrix}\)</span>, then:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}C = AB = \begin{pmatrix} 1 &amp; 0 &amp; 11  \\ -1 &amp; -2 &amp; -3  \end{pmatrix}.\end{split}\]</div>
+<p>Note that the matrix <span class="math notranslate nohighlight">\(BA\)</span> cannot be defined here since the number of columns of <span class="math notranslate nohighlight">\(B\)</span> is different from the number of rows of <span class="math notranslate nohighlight">\(A\)</span>.</p>
+</div>
+<div class="admonition warning">
+<p class="admonition-title">Warning</p>
+<p>If matrices <span class="math notranslate nohighlight">\(A\)</span> and <span class="math notranslate nohighlight">\(B\)</span> are <em>square matrices</em> (same number of rows and columns) then we can define both <span class="math notranslate nohighlight">\(AB\)</span> and <span class="math notranslate nohighlight">\(BA\)</span>. But, in general, we <strong>cannot assume</strong> that <span class="math notranslate nohighlight">\(AB=BA\)</span>. For example, if  <span class="math notranslate nohighlight">\(A=\begin{pmatrix} 2 &amp; -1  \\ 4 &amp; -2  \end{pmatrix}\)</span> and <span class="math notranslate nohighlight">\(B=\begin{pmatrix} 1 &amp; -1  \\ 2 &amp; -2  \end{pmatrix}\)</span>, then:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}AB = \begin{pmatrix} 0 &amp; 0  \\ 0 &amp; 0  \end{pmatrix} \quad and \quad BA = \begin{pmatrix} -2 &amp; 1  \\ -4 &amp; 2  \end{pmatrix}\end{split}\]</div>
+</div>
+</li>
+<li><p>For matrix multiplication, the following properties are valid:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{align}
+    (A+B)C&amp;=AC+BC \\
+    A(B+C)&amp;=AB+AC \\
+    (AB)C &amp;= A(BC) \\
+    A0 &amp;= 0A = 0 \end{align}\end{split}\]</div>
+</li>
+<li><p>We can also define powers of <span class="math notranslate nohighlight">\(A\)</span>. If <span class="math notranslate nohighlight">\(k\)</span> is a positive integer:</p>
+<div class="math notranslate nohighlight">
+\[A^k = A^{k-1}A = AA^{k-1} = A\cdot A\cdot A \cdots A\ (\mbox{product of}\ k\ \mbox{matrices})\]</div>
+</li>
+</ul>
+</section>
+<section id="inverse-matrix">
+<h3>Inverse matrix<a class="headerlink" href="#inverse-matrix" title="Link to this heading">#</a></h3>
+<p>Suppose that <span class="math notranslate nohighlight">\(A\)</span> is a <span class="math notranslate nohighlight">\(n \times n\)</span> matrix. If there exists a <span class="math notranslate nohighlight">\(n \times n\)</span> matrix <span class="math notranslate nohighlight">\(B\)</span> so that</p>
+<div class="math notranslate nohighlight">
+\[ AB = BA = I_n,\]</div>
+<p>where <span class="math notranslate nohighlight">\(I=\begin{pmatrix} 1 &amp; 0 &amp; \cdots &amp; 0 \\ 0 &amp; 1 &amp; \cdots &amp; 0 \\ \vdots &amp; &amp; \ddots &amp; \vdots \\ 0 &amp; 0 &amp; \cdots &amp; 1 \end{pmatrix}\)</span> is the <span class="math notranslate nohighlight">\(n \times n\)</span> <em>identity matrix</em> (with elements <span class="math notranslate nohighlight">\(1\)</span> on the main diagonal and <span class="math notranslate nohighlight">\(0\)</span> otherwise).</p>
+<p>In this case we call the matrix <span class="math notranslate nohighlight">\(B\)</span> the <em>inverse</em> of <span class="math notranslate nohighlight">\(A\)</span> and denote <span class="math notranslate nohighlight">\(B=A^{-1}\)</span>. If <span class="math notranslate nohighlight">\(A\)</span> has an inverse than it is called <em>invertible</em> or <em>non-singular</em>. If it does not have an inverse, <span class="math notranslate nohighlight">\(A\)</span> is <em>singular</em>.</p>
+<p>Finding the inverse of <span class="math notranslate nohighlight">\(A\)</span> automatically gives us the solution to the system of linear equations <span class="math notranslate nohighlight">\(A\mathbf x = \mathbf x\)</span></p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{align}  
+A\mathbf{x}&amp;=\mathbf{b} \\
+A^{-1} A\mathbf{x} &amp;= A^{-1}\mathbf{b} \\
+\mathbf{x} &amp;= A^{-1}\mathbf{b} \end{align}\end{split}\]</div>
+<p>Finding ways of inverting matrices is therefore needed for a wide range of applications. A whole research field is dealing with optimising matrix inversions.</p>
+<div class="admonition-example admonition">
+<p class="admonition-title">Example</p>
+<p>To find the inverse of matrix <span class="math notranslate nohighlight">\(A= \begin{pmatrix} 2 &amp; 5  \\ 1 &amp; 3  \end{pmatrix}\)</span>, we have to find the matrix <span class="math notranslate nohighlight">\(B = \begin{pmatrix} b_{11} &amp; b_{12}  \\ b_{21} &amp; b_{22}  \end{pmatrix}\)</span> so that</p>
+<div class="math notranslate nohighlight">
+\[ AB = I \quad (\mbox{then check that }\ BA = I.\]</div>
+<p>Thus</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{align}  
+  \begin{pmatrix} 2 &amp; 5  \\ 1 &amp; 3  \end{pmatrix}\begin{pmatrix} b_{11} &amp; b_{12}  \\ b_{21} &amp; b_{22}  \end{pmatrix} = \begin{pmatrix} 1 &amp; 0  \\ 0 &amp; 1  \end{pmatrix} \\  
+  \begin{pmatrix} 2b_{11} + 5b_{21}  &amp; 2b_{12} + 5b_{22} \\ b_{11} + 3b_{21} &amp; b_{12} + 3b_{22}  \end{pmatrix} = \begin{pmatrix} 1 &amp; 0  \\ 0 &amp; 1  \end{pmatrix}, \\
+\end{align}\end{split}\]</div>
+<p>which leads to two systems of equations:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{cases} 2b_{11} + 5b_{21} &amp;= 1 \\ b_{11} + 3b_{21} &amp;= 0 \end{cases}\quad \mbox{and}\quad \begin{cases} 2b_{12} + 5b_{22} &amp;= 0 \\ b_{12} + 3b_{22} &amp;= 1 \end{cases}\end{split}\]</div>
+<p>The solutions of these two systems, leads to <span class="math notranslate nohighlight">\(B= \begin{pmatrix} 3 &amp; -5  \\ -1 &amp; 2  \end{pmatrix}\)</span></p>
+</div>
+</section>
+<section id="determinant">
+<h3>Determinant<a class="headerlink" href="#determinant" title="Link to this heading">#</a></h3>
+<p>If <span class="math notranslate nohighlight">\(A = \begin{pmatrix} a_{11} &amp; a_{12}  \\ a_{21} &amp; a_{22}  \end{pmatrix}\)</span> then the expression:</p>
+<div class="math notranslate nohighlight">
+\[\mbox{det}A = a_{11}a_{22} - a_{21}a_{12}\]</div>
+<p>given by the product of the elements on the main diagonal minus the product of the elements on secondary diagonal. The term <span class="math notranslate nohighlight">\(\mbox{det}A\)</span> is called the <em>determinant</em> of <span class="math notranslate nohighlight">\(A\)</span>. The determinant can be calculated for matrices of any size, but the expression gets increasingly more complicated for larger matrices.</p>
+<p>The determinant establishes a clear criterium for the existence of <span class="math notranslate nohighlight">\(A^{-1}\)</span>. The matrix <span class="math notranslate nohighlight">\(A\)</span> will be invertible if and only if <span class="math notranslate nohighlight">\(\mbox{det}A \ne 0\)</span>.</p>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>The previous criterium has an important implication. Consider the system of linear equations defined by <span class="math notranslate nohighlight">\(A\mathbf{x}=\mathbf{b}\)</span> with equal number of equations and variables. If <span class="math notranslate nohighlight">\(\mbox{det}A \ne 0\)</span> then the inverse of <span class="math notranslate nohighlight">\(A\)</span> exists. Thus, if we multiply <span class="math notranslate nohighlight">\(A^{-1}\)</span> to the left of each side of the equation defining the linear system, we have:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{align}  
+A\mathbf{x}&amp;=\mathbf{b} \\
+A^{-1}(A\mathbf{x}) &amp;= A^{-1}\mathbf{b} \\
+(A^{-1}A)\mathbf{x} &amp;= A^{-1}\mathbf{b} \\
+\mathbf{x} &amp;= A^{-1}\mathbf{b} \end{align}\end{split}\]</div>
+<p>which means that if <span class="math notranslate nohighlight">\(\mbox{det}A \ne 0\)</span> the system will have a <strong>single solution</strong> with the variables assuming the values <span class="math notranslate nohighlight">\(\mathbf{x} = A^{-1}\mathbf{b}\)</span>.</p>
+</div>
+</section>
+<section id="structured-models">
+<h3>Structured models<a class="headerlink" href="#structured-models" title="Link to this heading">#</a></h3>
+<p>In the models we have seen until now, all the individuals in a given population are identical. However, real populations have importance differences between individuals according to their life stages. For example, some insects have stages of eggs, larvae, pupae, and adults, while many plants pass through stages of seeds, seedlings, saplings, and mature individuals. Thus, understanding the differences in the dynamics of each life stage is important to add realism to our models (especially in applied models, such as pest control or harvesting management).</p>
+<p>Suppose our population is composed by two classes: <em>immature</em> and <em>mature</em> individuals. Let us assume non-overlapping breeding seasons and that the timescales for reproduction and maturation are similar. Thus, time can be counted in discrete steps corresponding to this reproduction/maturation interval. At time <span class="math notranslate nohighlight">\(t\)</span> we have <span class="math notranslate nohighlight">\(X_t\)</span> immature individuals and <span class="math notranslate nohighlight">\(Y_t\)</span> mature individuals. Immature individuals survive until maturity with a probability <span class="math notranslate nohighlight">\(p\)</span>, while mature individuals reproduce and generate, on average, <span class="math notranslate nohighlight">\(m\)</span> immature individuals for the next generation. Mature individuals die after reproduction. With these assumptions, we can calculate the sizes of each population class on the step <span class="math notranslate nohighlight">\(t+1\)</span> as:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{align}
+X_{t+1} = m Y_t \\
+Y_{t+1} = p X_t \end{align}\end{split}\]</div>
+<p>See that now we have a model with two equations describing the variable class sizes in our population. Moreover, the dynamics of each class depends on the other, so that we have a system of <em>coupled</em> difference equations.</p>
+<p>Interestingly, we can also write this system in matrix form, as:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{pmatrix} X_{t+1}  \\ Y_{t+1} \end{pmatrix} = \begin{pmatrix} 0 &amp; m  \\ p &amp; 0  \end{pmatrix} \begin{pmatrix} X_{t}  \\ Y_{t} \end{pmatrix}, \quad \mbox{or}\end{split}\]</div>
+<div class="math notranslate nohighlight">
+\[\mathbf{n_{t+1}} = P\mathbf{n_{t}}\]</div>
+<p>where the sum of the components of the population vector <span class="math notranslate nohighlight">\(\mathbf{n_{t}}\)</span>, gives the total population (immature plus mature individuals) at time <span class="math notranslate nohighlight">\(t\)</span>: <span class="math notranslate nohighlight">\(N_t = X_t + Y_t\)</span>. The matrix <span class="math notranslate nohighlight">\(P=\begin{pmatrix} 0 &amp; m  \\ p &amp; 0  \end{pmatrix}\)</span> is called <em>projection matrix</em> or <em>Leslie matrix</em>. Multiplying matrix <span class="math notranslate nohighlight">\(P\)</span> by the population vector at time <span class="math notranslate nohighlight">\(t\)</span>, <span class="math notranslate nohighlight">\(\mathbf{n_{t}}\)</span>, we can project the next population size at time <span class="math notranslate nohighlight">\(t+1\)</span>. Thus, from an initial population state, say <span class="math notranslate nohighlight">\(\mathbf{n_{0}}\)</span>, the projection matrix governs the dynamics of the model.</p>
+<br />
+<p>This model can be easily generalised for a larger number of age classes. Suppose we now have <span class="math notranslate nohighlight">\(q+1\)</span> classes, <span class="math notranslate nohighlight">\(X_0, X_1, X_2, \ldots, X_q\)</span>, from the youngest (meaningful) age <span class="math notranslate nohighlight">\(X_0\)</span> until the oldest age <span class="math notranslate nohighlight">\(X_q\)</span>. At each time step <span class="math notranslate nohighlight">\(t\)</span>, individuals from class <span class="math notranslate nohighlight">\(X_i\)</span> have a chance of surviving to the next age class <span class="math notranslate nohighlight">\(X_{i+1}\)</span>, with probability <span class="math notranslate nohighlight">\(p_{i+1\ i}\)</span> (imagine an arrow for direction of age class progession <span class="math notranslate nohighlight">\(p_{i+1\leftarrow i}\)</span>. We are going to omit the arrow for simplicity). Additionally, suppose that each age class <span class="math notranslate nohighlight">\(X_i\)</span> excluding the youngest, will, in principle, be able to reproduce, with average fecundity <span class="math notranslate nohighlight">\(m_i\)</span> generating new individuals for class <span class="math notranslate nohighlight">\(X_0\)</span>. Thus, with the age class sizes at time <span class="math notranslate nohighlight">\(t\)</span>, and the parameters for survival probabilities and average fecundities, we have:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{align}
+X_0(t+1) &amp;= m_1X_1(t) +  m_2X_2(t) + m_3X_3(t) + \ldots + m_qX_q(t) \\
+X_1(t+1) &amp;= p_{10}X_0(t) \\
+X_2(t+1) &amp;= p_{21}X_1(t) \\
+X_3(t+1) &amp;= p_{32}X_2(t) \\
+         &amp;\vdots \\
+X_q(t+1) &amp;= p_{q\ q-1}X_{q-1}(t)
+    \end{align}\end{split}\]</div>
+<p>This can also be written in matrix form as:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{pmatrix} X_0(t+1)  \\ X_1(t+1) \\X_2(t+1)  \\X_3(t+1) \\ \vdots \\X_q(t+1)  \\ \end{pmatrix} = \begin{pmatrix} 0 &amp; m_1 &amp; m_2 &amp; m_3 &amp; \cdots &amp; m_q \\ p_{10} &amp; 0 &amp; 0 &amp; 0 &amp; \cdots &amp;0 \\ 0 &amp; p_{21} &amp; 0 &amp; 0 &amp; \cdots &amp; 0 \\ 0 &amp; 0 &amp; p_{32} &amp; 0 &amp; \cdots &amp; 0 \\ \vdots &amp; \vdots &amp; \vdots &amp; \vdots &amp; \ddots &amp; \vdots \\ 0 &amp; 0 &amp; 0 &amp; \cdots &amp; p_{q\ q-1} &amp; 0\end{pmatrix} \begin{pmatrix} X_0(t)  \\ X_1(t) \\X_2(t) \\X_3(t) \\ \vdots \\X_q(t)  \\ \end{pmatrix}\end{split}\]</div>
+<div class="math notranslate nohighlight">
+\[\mathbf{n_{t+1}} = P\mathbf{n_{t}}\]</div>
+<p>The generalised projection matrix <span class="math notranslate nohighlight">\(P\)</span> has then all the surviving probabilities just below the main diagonal, and the fecundities for each class in its <span class="math notranslate nohighlight">\(0\)</span>-th row.</p>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>Two important differences here. First, the rows and columns for the matrix are indexed starting from <span class="math notranslate nohighlight">\(0\)</span> until <span class="math notranslate nohighlight">\(q\)</span>, not starting from <span class="math notranslate nohighlight">\(1\)</span> as usual. Second, the time steps are written in parentheses for each age class, instead of a subscript as we have seen before for other population models, so that it does not conflict with the index for the age classes (going from <span class="math notranslate nohighlight">\(0\)</span> to <span class="math notranslate nohighlight">\(q\)</span>).</p>
+</div>
+<div class="admonition-example admonition">
+<p class="admonition-title">Example</p>
+<p>Consider a population with three age classes <span class="math notranslate nohighlight">\(X_0, X_1,\mbox{ and}\ X_2\)</span>, and the following projection matrix</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}P=\begin{pmatrix} 0 &amp; 5 &amp; 10\\ 0.5 &amp; 0 &amp; 0  \\ 0 &amp; 0.2 &amp; 0 \end{pmatrix}\end{split}\]</div>
+<p>If the population starts with <span class="math notranslate nohighlight">\(10\)</span> individuals in the oldest category at time <span class="math notranslate nohighlight">\(t=0\)</span>, the population vector at time <span class="math notranslate nohighlight">\(t=1\)</span> will be:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\mathbf{n_{1}} = P\mathbf{n_{0}} \longrightarrow \begin{pmatrix} X_0(1)  \\ X_1(1) \\X_2(1) \end{pmatrix}=\begin{pmatrix} 0 &amp; 5 &amp; 10\\ 0.5 &amp; 0 &amp; 0  \\ 0 &amp; 0.2 &amp; 0 \end{pmatrix}\begin{pmatrix} 0  \\ 0 \\ 10 \end{pmatrix} \longrightarrow \begin{pmatrix} 100  \\ 0 \\ 0 \end{pmatrix}\end{split}\]</div>
+<p>The population age distribution at each time step will then be given be the successive application of the matrix <span class="math notranslate nohighlight">\(P\)</span> to the previous population distribution. For the first 10 time steps we have:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{pmatrix}0 \\ 0 \\ 10 \end{pmatrix} \rightarrow \begin{pmatrix} 100  \\ 0 \\ 0 \end{pmatrix} \rightarrow \begin{pmatrix} 0  \\ 50 \\ 0 \end{pmatrix} \rightarrow \begin{pmatrix} 250  \\ 0 \\ 10 \end{pmatrix} \rightarrow \begin{pmatrix} 100  \\ 125 \\ 0 \end{pmatrix} \rightarrow \begin{pmatrix} 625  \\ 50 \\ 25 \end{pmatrix} \rightarrow \begin{pmatrix} 500  \\ 312 \\ 10 \end{pmatrix} \rightarrow \begin{pmatrix} 1662  \\ 250 \\ 62 \end{pmatrix} \rightarrow \begin{pmatrix} 1875  \\ 831 \\ 50 \end{pmatrix} \rightarrow \begin{pmatrix} 4656  \\ 937 \\ 166 \end{pmatrix}\end{split}\]</div>
+<p>Two important things are worth noting here. First, the total population size (sum accros all age classes), follows the sequence <span class="math notranslate nohighlight">\(N_t = \left\{ 10,100,50,260,225,700,822,1975,2756,5759,\ldots \right \}\)</span>. The reproduction process does not always lead to increasing population sizes in the beginning, but is subject to size fluctuations due to the demographic distribution of this population. After a few steps increases in population size become more consistent. Second, as the population size continues to increase, the population age distribution tends to a given proportion of the population classes given by: <span class="math notranslate nohighlight">\(X_0 \approx 76\%, X_1 \approx 22\%, \mbox{and}\ X_2 \approx 2\%\)</span>.</p>
+<p>Now consider that, instead of <span class="math notranslate nohighlight">\(10\)</span> individuals in the oldest class, we initiate the population with the following distribution <span class="math notranslate nohighlight">\(\mathbf{n_{0}} = \begin{pmatrix} 7.6  \\ 2.2 \\ 0.2 \end{pmatrix}\)</span>. The population progression by successive application of the projection matrix will be</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{pmatrix} 7.6  \\ 2.2 \\ 0.2 \end{pmatrix} \longrightarrow \begin{pmatrix} 13  \\ 4 \\ 0.4 \end{pmatrix} \longrightarrow \begin{pmatrix} 23  \\ 6 \\ 0.76 \end{pmatrix} \longrightarrow \begin{pmatrix} 40  \\ 12 \\ 1 \end{pmatrix} \longrightarrow \begin{pmatrix} 72  \\ 20 \\ 2 \end{pmatrix} \longrightarrow \begin{pmatrix} 124  \\ 36 \\ 4 \end{pmatrix} \longrightarrow \begin{pmatrix} 219  \\ 62 \\ 7 \end{pmatrix} \longrightarrow \begin{pmatrix} 381  \\ 109 \\ 12 \end{pmatrix} \end{split}\]</div>
+<p>You can check that although each age cathegory is increasing, the percentage contribution of each class to the total population remains nearly constant. To this distribution of classes we give the name of <strong>stable age distribution</strong>.</p>
+<p>Moreover, at each time step, each age class increases with approximately the same factor in relation to the previous step, around <span class="math notranslate nohighlight">\(\lambda = 1.75\)</span>. If for each age class <span class="math notranslate nohighlight">\(X_i(t+1) = \lambda X_i(t)\)</span> is valid,then we have:</p>
+<div class="math notranslate nohighlight">
+\[\mathbf{n_{t+1}}=\lambda\mathbf{n_{t}},\]</div>
+<p>so that we recover the usual model for exponential growth for the total population size <span class="math notranslate nohighlight">\(N_{t+1} = \lambda N_t\)</span>.</p>
+</div>
+<!-- ## From table arrangements to powerful tools - part 2 --></section>
+<section id="linear-transformations">
+<h3>Linear transformations<a class="headerlink" href="#linear-transformations" title="Link to this heading">#</a></h3>
+<p>In general terms, a given matrix <span class="math notranslate nohighlight">\(A\)</span> is a representation of what we call a <em>linear transformation</em>, a function from a special kind of set called <em>vector space</em> into another vector space. Vector spaces are generalised sets with particular properties, but for the moment you can think of two-dimensional and three-dimensional cartesian spaces as usual vector spaces. Then vectors are gonna be represented as arrays of real numbers as before: <span class="math notranslate nohighlight">\(\mathbf{n_{0}} = \begin{pmatrix} 7.6  \\ 2.2  \end{pmatrix}\)</span> or <span class="math notranslate nohighlight">\(\mathbf{v} = \begin{pmatrix} x \\ y \\ z \end{pmatrix}\)</span>.</p>
+<p>Back to linear transformations, they are usually denoted as:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{align}
+A:V &amp;\longrightarrow W \\
+  \mathbf{v} &amp;\rightarrow A(\mathbf{v}) \end{align}\end{split}\]</div>
+<p>where <span class="math notranslate nohighlight">\(V\)</span> and <span class="math notranslate nohighlight">\(W\)</span> are the domain and codomain vector spaces, and the transformation <span class="math notranslate nohighlight">\(A\)</span> is a function that projects vectors <span class="math notranslate nohighlight">\(\mathbf{v}\)</span> into the images <span class="math notranslate nohighlight">\(A(\mathbf{v})\)</span>. The property that defines linear transformations is the following:</p>
+<div class="math notranslate nohighlight">
+\[A(a\mathbf{v}+b\mathbf{w})=aA(\mathbf{v})+bA(\mathbf{w}),\]</div>
+<p>where <span class="math notranslate nohighlight">\(a\)</span> and <span class="math notranslate nohighlight">\(b\)</span> are constants. In other words,for a linear transformation, the image of the transformation <span class="math notranslate nohighlight">\(A\)</span> of a linear combination of two vectors (<span class="math notranslate nohighlight">\(\mathbf{v}\)</span> and <span class="math notranslate nohighlight">\(\mathbf{w}\)</span>) is equal to the linear combination of the images of this vectors by <span class="math notranslate nohighlight">\(A\)</span>.</p>
+<p>If we represent vectors with their components in a cartesian system, the images of the transformation <span class="math notranslate nohighlight">\(A\)</span> will be given be the multiplication of the matrix representation of <span class="math notranslate nohighlight">\(A\)</span> on the vectors.</p>
+<div class="admonition-example admonition">
+<p class="admonition-title">Example</p>
+<p>Given the matrix <span class="math notranslate nohighlight">\(A = \begin{pmatrix} 1 &amp; 2 \\ 3 &amp; 2 \end{pmatrix}\)</span> and the vectors <span class="math notranslate nohighlight">\(\mathbf{v_1} = \begin{pmatrix} 3  \\ -1  \end{pmatrix}\)</span> and <span class="math notranslate nohighlight">\(\mathbf{v_2} = \begin{pmatrix} 2  \\ 3  \end{pmatrix}\)</span>, then we have:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{align} A\mathbf{v_1} &amp;= \begin{pmatrix} 1 &amp; 2 \\ 3 &amp; 2 \end{pmatrix}\begin{pmatrix} 3  \\ -1  \end{pmatrix} = \begin{pmatrix} 1  \\ 7  \end{pmatrix} \\  
+  A\mathbf{v_2} &amp;= \begin{pmatrix} 1 &amp; 2 \\ 3 &amp; 2 \end{pmatrix}\begin{pmatrix} 2  \\ 3  \end{pmatrix} = \begin{pmatrix} 8  \\ 12  \end{pmatrix} = 4\begin{pmatrix} 2  \\ 3  \end{pmatrix} \end{align}\end{split}\]</div>
+<p>The vectors and their respective transformations are given by:</p>
+<img alt="../../_images/lin_trans.png" src="../../_images/lin_trans.png" />
+<p>We see that under the transformation <span class="math notranslate nohighlight">\(A\)</span>, vector <span class="math notranslate nohighlight">\(\mathbf{v_1}\)</span> gets rotated (counterclockwise) and stretched, while vector <span class="math notranslate nohighlight">\(\mathbf{v_2}\)</span> only gets stretched without changing the direction.</p>
+</div>
+</section>
+<section id="eigenvalues-and-eigenvectors">
+<h3>Eigenvalues and eigenvectors<a class="headerlink" href="#eigenvalues-and-eigenvectors" title="Link to this heading">#</a></h3>
+<p>As seen on the previous example, there might be some vectors that upon application of <span class="math notranslate nohighlight">\(A\)</span> do not have their direction changed, only their magnitude (or <em>module</em> of the vector). For these vectors, we have, in general:</p>
+<div class="math notranslate nohighlight">
+\[A\mathbf{v} = \lambda\mathbf{v}, \quad \mbox{for some constant }\ \lambda\]</div>
+<p>But this is equivalent to:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{align}
+A\mathbf{v} - \lambda\mathbf{v} = 0 \\  
+A\mathbf{v} - \lambda I\mathbf{v} = 0 \\
+\left(A - \lambda I \right)\mathbf{v} = 0 \end{align},\end{split}\]</div>
+<p>where <span class="math notranslate nohighlight">\(I\)</span> is the identity matrix. The equation <span class="math notranslate nohighlight">\(\left(A - \lambda I \right)\mathbf{v} = 0\)</span> defines a system of linear equations where <span class="math notranslate nohighlight">\(A - \lambda I\)</span> is the coefficient matrix, <span class="math notranslate nohighlight">\(\mathbf{v}\)</span> is the vector of variables, and <span class="math notranslate nohighlight">\(\mathbf{b}=0\)</span>. See that since <span class="math notranslate nohighlight">\(\mathbf{b}=0\)</span> we already know one possible solution for this system, which is <span class="math notranslate nohighlight">\(\mathbf{v}=0\)</span> (all the variables equal to <span class="math notranslate nohighlight">\(0\)</span>). Therefore, either this system has an unique solution (the null solution) or it has infinite solutions, but definitely it is not inconsistent.  Thus, if we want to know all possible non-zero vectors <span class="math notranslate nohighlight">\(\mathbf{v}\)</span> that satisfy <span class="math notranslate nohighlight">\(A\mathbf{v} = \lambda\mathbf{v}\)</span>, we have to impose that the system assumes infinite solutions. This criterium should be defined by the determinant of the matrix of coefficients, as:</p>
+<div class="math notranslate nohighlight">
+\[\mbox{det}(A-\lambda I) = 0.\]</div>
+<p>The determinant then defines a polynomial in the variable <span class="math notranslate nohighlight">\(\lambda\)</span>, <span class="math notranslate nohighlight">\(\mbox{det}(A-\lambda I)=p(\lambda)\)</span>, called <em>characteristic polynomial</em>. Since we are interested in the condition <span class="math notranslate nohighlight">\(p(\lambda)=0\)</span>, what we want to know are the roots <span class="math notranslate nohighlight">\(\lambda\)</span> of this polynomial.</p>
+<p>The solutions <span class="math notranslate nohighlight">\(\lambda\)</span> for these equations are called <strong>eigenvalues</strong> and their corresponding vectors are <strong>eigenvectors</strong>. In summary, the eigenvectors are all the vectors that, upon application of <span class="math notranslate nohighlight">\(A\)</span>, remain on the same direction, only changing magnitude, with the eigenvalue being the stretching/compressing factor.</p>
+<div class="admonition-example admonition">
+<p class="admonition-title">Example</p>
+<p>Let us calculate the eigenvalues and eigenvectors of <span class="math notranslate nohighlight">\(A = \begin{pmatrix} 1 &amp; 2 \\ 3 &amp; 2 \end{pmatrix}\)</span>. First we need to obtain matrix <span class="math notranslate nohighlight">\((A-\lambda I)\)</span> and to calculate its determinant. We have:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}A-\lambda I = \begin{pmatrix} 1 &amp; 2 \\ 3 &amp; 2 \end{pmatrix} - \lambda \begin{pmatrix} 1 &amp; 0 \\ 0 &amp; 1 \end{pmatrix} = \begin{pmatrix} 1 -\lambda &amp; 2 \\ 3 &amp; 2 -\lambda  \end{pmatrix}\end{split}\]</div>
+<p>The characteristic polynomial will be given by <span class="math notranslate nohighlight">\(p(\lambda)=\mbox{det}(A-\lambda I)\)</span>, and applying the rule for the calculus of determinants of matrices size 2, we have:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}p(\lambda)=\mbox{det}(A-\lambda I) = \mbox{det}\begin{pmatrix} 1 -\lambda &amp; 2 \\ 3 &amp; 2 -\lambda  \end{pmatrix} = (1 -\lambda)(2 -\lambda) - 3\cdot 2 \end{split}\]</div>
+<div class="math notranslate nohighlight">
+\[p(\lambda) = 2 -\lambda -2\lambda +\lambda^2 - 6 = \lambda^2 - 3\lambda -4.\]</div>
+<p>Thus, calculating the roots of this 2nd-degree polynomial, we find <span class="math notranslate nohighlight">\(\lambda_1=-1\)</span> and <span class="math notranslate nohighlight">\(\lambda_2=4\)</span>. For each of the eigenvalues, to obtain the eigenvectors we have to solve the linear system  <span class="math notranslate nohighlight">\(A\mathbf{v} = \lambda\mathbf{v}\)</span> for some general vector <span class="math notranslate nohighlight">\(\mathbf{v} = \begin{pmatrix} x  \\ y  \end{pmatrix}\)</span> (with coordinates <span class="math notranslate nohighlight">\(x\)</span> and <span class="math notranslate nohighlight">\(y\)</span> to be specified). Thus:</p>
+<ul>
+<li><p><span class="math notranslate nohighlight">\(\mathbf{\lambda_1 = -1}\)</span>:</p>
+<p><span class="math notranslate nohighlight">\(\begin{align}
+A\mathbf{v} = \lambda\mathbf{v} \longrightarrow \begin{pmatrix} 1 &amp; 2 \\ 3 &amp; 2 \end{pmatrix}\begin{pmatrix} x  \\ y \end{pmatrix}= -1\begin{pmatrix} x  \\ y \end{pmatrix}\longrightarrow \cases{x + 2y = -x \\ 3x + 2y = -y} \longrightarrow \cases{2x + 2y = 0 \\ 3x + 3y = 0}
+\end{align}\)</span></p>
+<p>Note that both equations correspond to the same linear equation <span class="math notranslate nohighlight">\(x + y = 0\)</span>, which is exactly what we would expect since we built the system to be underdetermined when we did <span class="math notranslate nohighlight">\(\mbox{det}(A-\lambda I) = 0\)</span>. The eigenvector corresponding to <span class="math notranslate nohighlight">\(\lambda_1\)</span> will be such that <span class="math notranslate nohighlight">\(x + y = 0\)</span> or <span class="math notranslate nohighlight">\(x = -y\)</span>. Thus, we can choose <span class="math notranslate nohighlight">\(\mathbf{v_1} = \begin{pmatrix} 1  \\ -1  \end{pmatrix}\)</span>.</p>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>Remember that infinite solutions are possible for the same system. The important is that the condition <span class="math notranslate nohighlight">\(x = -y\)</span> holds. In fact, multiplying a non-zero constant for the vector <span class="math notranslate nohighlight">\(\begin{pmatrix} 1  \\ -1  \end{pmatrix}\)</span> would still give a vector in the same direction. So if <span class="math notranslate nohighlight">\(\mathbf{v_1}\)</span> is an eigenvector, <span class="math notranslate nohighlight">\(k\mathbf{v_1}\ (k \in {\rm I\!R})\)</span> also is.</p>
+</div>
+  <br />  
+</li>
+<li><p><span class="math notranslate nohighlight">\(\mathbf{\lambda_1 = 4}\)</span>:</p>
+<p><span class="math notranslate nohighlight">\(\begin{align}
+A\mathbf{v} = \lambda\mathbf{v} \longrightarrow \begin{pmatrix} 1 &amp; 2 \\ 3 &amp; 2 \end{pmatrix}\begin{pmatrix} x  \\ y \end{pmatrix}= 4\begin{pmatrix} x  \\ y \end{pmatrix}\longrightarrow \cases{x + 2y = 4x \\ 3x + 2y = 4y} \longrightarrow \cases{-3x + 2y = 0 \\ 3x - 2y = 0}
+\end{align}\)</span></p>
+<p>Since the eigenvector needs to satisfy <span class="math notranslate nohighlight">\(3x=2y\)</span>, we choose <span class="math notranslate nohighlight">\(\mathbf{v_2} = \begin{pmatrix} 2/3  \\ 1  \end{pmatrix}\)</span>.</p>
+<p>The representation of these vectors on the cartesian plane will be:</p>
+</li>
+</ul>
+<img alt="../../_images/eigenvec_01.png" src="../../_images/eigenvec_01.png" />
+</div>
+<div class="admonition-example admonition">
+<p class="admonition-title">Example</p>
+<p>Let us calculate the eigenvalues and eigenvectors of matrix <span class="math notranslate nohighlight">\(A = \begin{pmatrix} -2 &amp; 1 \\ 0 &amp; -1 \end{pmatrix}\)</span>. We have:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}p(\lambda)=\mbox{det}(A-\lambda I) = \mbox{det}\begin{pmatrix} -2 -\lambda &amp; 1 \\ 0 &amp; -1 -\lambda  \end{pmatrix} = (-2 -\lambda)(-1 -\lambda) - 1\cdot 0 \end{split}\]</div>
+<div class="math notranslate nohighlight">
+\[p(\lambda)=(-2 -\lambda)(-1 -\lambda)\]</div>
+<p>Since we need <span class="math notranslate nohighlight">\(p(\lambda)=0\)</span> either one of the two factors has to be zero. So, <span class="math notranslate nohighlight">\(\lambda_1=-2\)</span> and <span class="math notranslate nohighlight">\(\lambda_2=-1\)</span>.<br />
+For the eigenvectors:</p>
+<ul>
+<li><p><span class="math notranslate nohighlight">\(\mathbf{\lambda_1 = -2}\)</span>:</p>
+<p><span class="math notranslate nohighlight">\(\begin{align}
+A\mathbf{v} = \lambda\mathbf{v} \longrightarrow \begin{pmatrix} -2 &amp; 1 \\ 0 &amp; -1 \end{pmatrix}\begin{pmatrix} x  \\ y \end{pmatrix}= -2\begin{pmatrix} x  \\ y \end{pmatrix}\longrightarrow \cases{-2x + y = -2x \\ -y = -2y} \longrightarrow \cases{y = 0 \\ y = 0}
+\end{align}\)</span></p>
+<p>Since <span class="math notranslate nohighlight">\(y=0\)</span>, we choose <span class="math notranslate nohighlight">\(\mathbf{v_1} = \begin{pmatrix} 1  \\ 0  \end{pmatrix}\)</span>.</p>
+</li>
+<li><p><span class="math notranslate nohighlight">\(\mathbf{\lambda_2 = -1}\)</span>:</p>
+<p><span class="math notranslate nohighlight">\(\begin{align}
+A\mathbf{v} = \lambda\mathbf{v} \longrightarrow \begin{pmatrix} -2 &amp; 1 \\ 0 &amp; -1 \end{pmatrix}\begin{pmatrix} x  \\ y \end{pmatrix}= -1\begin{pmatrix} x  \\ y \end{pmatrix}\longrightarrow \cases{-2x + y = -x \\ -y = -y} \longrightarrow \cases{-x + y = 0 \\ y = y}
+\end{align}\)</span></p>
+<p>The second equation is trivial and provides no information. From the first equation, since <span class="math notranslate nohighlight">\(x=y\)</span>, we choose <span class="math notranslate nohighlight">\(\mathbf{v_2} = \begin{pmatrix} 1  \\ 1  \end{pmatrix}\)</span>.</p>
+</li>
+</ul>
+<div class="admonition tip">
+<p class="admonition-title">Tip</p>
+<p>Note that the two eigenvalues <span class="math notranslate nohighlight">\(\lambda_1 = -2\)</span> and <span class="math notranslate nohighlight">\(\lambda_2 = -1\)</span> correspond to the elements of the main diagonal of the matrix <span class="math notranslate nohighlight">\(A\)</span>. Whenever the matrix <span class="math notranslate nohighlight">\(A\)</span> is <em>triangular</em>, meaning all the elements <em>below</em> the main diagonal are <span class="math notranslate nohighlight">\(0\)</span>, the eigenvalues will correspond to the elements of the main diagonal.</p>
+</div>
+</div>
+</section>
+<section id="vector-decomposition">
+<h3>Vector decomposition<a class="headerlink" href="#vector-decomposition" title="Link to this heading">#</a></h3>
+<p>Now suppose matrix <span class="math notranslate nohighlight">\(A\)</span> has eigenvalues <span class="math notranslate nohighlight">\(\lambda_1\)</span> and <span class="math notranslate nohighlight">\(\lambda_2\)</span>, with corresponding eigenvectors <span class="math notranslate nohighlight">\(\mathbf{u_1}\)</span> and <span class="math notranslate nohighlight">\(\mathbf{u_2}\)</span>. If <span class="math notranslate nohighlight">\(\lambda_1 \ne \lambda_2\)</span>, then the vectors <span class="math notranslate nohighlight">\(\mathbf{u_1}\)</span> and <span class="math notranslate nohighlight">\(\mathbf{u_2}\)</span> are <em>linearly independent</em> meaning one can not be written as a multiple (multiplied by a real constant) of the other. Geometrically, this means that the directions corresponding to this two eigenvectors are not on the same line.</p>
+<p>In two dimensions, this is enough to say that any given vector <span class="math notranslate nohighlight">\(\mathbf{v}\)</span> can be written as a linear combination of <span class="math notranslate nohighlight">\(\mathbf{u_1}\)</span> and <span class="math notranslate nohighlight">\(\mathbf{u_2}\)</span> (or that <span class="math notranslate nohighlight">\(\mathbf{u_1}\)</span> and <span class="math notranslate nohighlight">\(\mathbf{u_2}\)</span> consitute a <em>basis</em> of this space). We have:</p>
+<div class="math notranslate nohighlight">
+\[\mathbf{v}= a_1\mathbf{u_1} + a_2\mathbf{u_2},\]</div>
+<p>for constants <span class="math notranslate nohighlight">\(a_1\)</span> and <span class="math notranslate nohighlight">\(a_2\)</span> to be determined.</p>
+<p>But given that <span class="math notranslate nohighlight">\(A\)</span> is a linear transformation, it should hold that:</p>
+<div class="math notranslate nohighlight">
+\[A\mathbf{v} = A(a_1\mathbf{u_1} + a_2\mathbf{u_2}) = a_1A(\mathbf{u_1}) + a_2A(\mathbf{u_2}) = a_1\lambda_1\mathbf{u_1} + a_2\lambda_2\mathbf{u_2},\]</div>
+<p>with the last step coming from the fact that <span class="math notranslate nohighlight">\(\mathbf{u_1}\)</span> and <span class="math notranslate nohighlight">\(\mathbf{u_2}\)</span> are eigenvectors of <span class="math notranslate nohighlight">\(A\)</span>. Since <span class="math notranslate nohighlight">\(A\mathbf{v}\)</span> is also a vector in this space, we can go ahead and apply the transformation <span class="math notranslate nohighlight">\(A\)</span> again, and we will have:</p>
+<div class="math notranslate nohighlight">
+\[A(A\mathbf{v}) = A(a_1\lambda_1\mathbf{u_1} + a_2\lambda_2\mathbf{u_2})=a_1\lambda_1 A(\mathbf{u_1}) + a_2\lambda_2 A(\mathbf{u_2}) = a_1\lambda_1^2\mathbf{u_1} + a_2\lambda_2^2\mathbf{u_2}.\]</div>
+<p>So, in general:</p>
+<div class="math notranslate nohighlight">
+\[A(A(A \cdots A\mathbf{v}) \cdots )) = A^n\mathbf{v} = a_1\lambda_1^n\mathbf{u_1} + a_2\lambda_2^n\mathbf{u_2},\]</div>
+<p>or, in other words, if we know the eigenvectors and eigenvalues of <span class="math notranslate nohighlight">\(A\)</span> (with <span class="math notranslate nohighlight">\(\lambda_1 \ne \lambda_2\)</span>), and we know the coeffiencients <span class="math notranslate nohighlight">\(a_1\)</span> and <span class="math notranslate nohighlight">\(a_2\)</span> of the expansion of <span class="math notranslate nohighlight">\(\mathbf{v}\)</span> into the eigenvectors, then it is easy to calculate what is going to be the image ov vector <span class="math notranslate nohighlight">\(\mathbf{v}\)</span> by any number of successive applications of <span class="math notranslate nohighlight">\(A\)</span>. This is also equivalent of applying the <span class="math notranslate nohighlight">\(n\)</span>-th power of matrix <span class="math notranslate nohighlight">\(A\)</span> to <span class="math notranslate nohighlight">\(\mathbf{v}\)</span>.</p>
+<p>This fact will have many important applications, as we will se next.</p>
+</section>
+<section id="structured-models-revisited">
+<h3>Structured models revisited<a class="headerlink" href="#structured-models-revisited" title="Link to this heading">#</a></h3>
+<p>Let us considered a structured population model defined by the projection matrix <span class="math notranslate nohighlight">\(P=\begin{pmatrix} 1.5 &amp; 2  \\ 0.08 &amp; 0  \end{pmatrix}\)</span> and an initial population vector given by <span class="math notranslate nohighlight">\(\mathbf{n_0} = \begin{pmatrix} 105  \\ 1 \end{pmatrix}\)</span>. If we calculate the eigenvalues and eigenvectors, we will have:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}p(\lambda)=\mbox{det}(P-\lambda I) = \mbox{det}\begin{pmatrix} 1.5 -\lambda &amp; 2 \\ 0.08 &amp; -\lambda  \end{pmatrix} = (1.5 -\lambda)(-\lambda) - 0.08\cdot 2 \end{split}\]</div>
+<div class="math notranslate nohighlight">
+\[p(\lambda)=\lambda^2-1.5\lambda-0.16=(\lambda-1.6)(\lambda+0.1)\]</div>
+<p>Thus, we have <span class="math notranslate nohighlight">\(\lambda_1=1.6\)</span> and <span class="math notranslate nohighlight">\(\lambda_2=-0.1\)</span>.</p>
+<ul>
+<li><p><span class="math notranslate nohighlight">\(\mathbf{\lambda_1 = 1.6}\)</span>:</p>
+<p><span class="math notranslate nohighlight">\(\begin{align}
+A\mathbf{v} = \lambda\mathbf{v} \longrightarrow \begin{pmatrix} 1.5 &amp; 2  \\ 0.08 &amp; 0 \end{pmatrix}\begin{pmatrix} x  \\ y \end{pmatrix}= 1.6\begin{pmatrix} x  \\ y \end{pmatrix}\longrightarrow \cases{1.5x + 2y = 1.6x \\ 0.08x = 1.6y} \longrightarrow \cases{x - 20y = 0 \\ x - 20y = 0}
+\end{align}\)</span></p>
+<p>Since <span class="math notranslate nohighlight">\(x=20y\)</span>, we choose <span class="math notranslate nohighlight">\(\mathbf{u_1} = \begin{pmatrix} 20  \\ 1  \end{pmatrix}\)</span>.</p>
+<br />
+</li>
+<li><p><span class="math notranslate nohighlight">\(\mathbf{\lambda_2 = -0.1}\)</span>:</p>
+<p><span class="math notranslate nohighlight">\(\begin{align}
+A\mathbf{v} = \lambda\mathbf{v} \longrightarrow \begin{pmatrix} 1.5 &amp; 2  \\ 0.08 &amp; 0 \end{pmatrix}\begin{pmatrix} x  \\ y \end{pmatrix}= -0.1\begin{pmatrix} x  \\ y \end{pmatrix}\longrightarrow \cases{1.5x + 2y = -0.1x \\ 0.08x = -0.1y} \longrightarrow \cases{0.8x + y = 0 \\ 0.08x + 0.1y = 0}
+\end{align}\)</span></p>
+<p>Since <span class="math notranslate nohighlight">\(-\displaystyle\frac{4}{5}x=y\)</span>, we choose <span class="math notranslate nohighlight">\(\mathbf{u_2} = \begin{pmatrix} 5  \\ -4  \end{pmatrix}\)</span>.</p>
+</li>
+</ul>
+<br />
+<p>We can also note that the initial population vector can be decomposed into the eigenvectors <span class="math notranslate nohighlight">\(\mathbf{u_1}\)</span> and <span class="math notranslate nohighlight">\(\mathbf{u_2}\)</span> of the matrix <span class="math notranslate nohighlight">\(P\)</span> according to:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\mathbf{n_0}=\begin{pmatrix} 105  \\ 1 \end{pmatrix} = 5\begin{pmatrix} 20  \\ 1 \end{pmatrix} + \begin{pmatrix} 5  \\ -4 \end{pmatrix} = 5\mathbf{u_1} + \mathbf{u_2}\end{split}\]</div>
+<p>We know that to find the population vector for the following step, we have to apply the projection matrix on the current population vector. Thus:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\mathbf{n_1} = P\mathbf{n_0}= P(5\mathbf{u_1} + \mathbf{u_2})=5P(\mathbf{u_1}) + P(\mathbf{u_2})=5(1.6)\begin{pmatrix} 20  \\ 1 \end{pmatrix} + (-0.1)\begin{pmatrix} 5  \\ -4 \end{pmatrix} = \begin{pmatrix} 159.5  \\ 8.4 \end{pmatrix}\end{split}\]</div>
+<p>If we perform successive applications of <span class="math notranslate nohighlight">\(P\)</span>, we will finally have:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\mathbf{n_t} = P^t\mathbf{n_0} = P^t(5\mathbf{u_1} + \mathbf{u_2})=5(1.6)^t\begin{pmatrix} 20  \\ 1 \end{pmatrix} + (-0.1)^t\begin{pmatrix} 5  \\ -4 \end{pmatrix}.\end{split}\]</div>
+<br />  
+<p>By decomposing into eigenvalues and eigenvectors, we know the behaviour of our population for any time <span class="math notranslate nohighlight">\(t\)</span>, which is much easier than applying <span class="math notranslate nohighlight">\(P\)</span> a number <span class="math notranslate nohighlight">\(t\)</span> of times, or than calculating <span class="math notranslate nohighlight">\(P^t\)</span>. Furthermore, as <span class="math notranslate nohighlight">\(t\rightarrow\infty\)</span> the first term will dominate (with the second term dying out) and the proportions for each age class in this population will be dictated by the vector <span class="math notranslate nohighlight">\(\begin{pmatrix} 20  \\ 1 \end{pmatrix}\)</span> (or, if we divide by the sum of the components: <span class="math notranslate nohighlight">\(\displaystyle\frac{1}{21}\begin{pmatrix} 20  \\ 1 \end{pmatrix} = \begin{pmatrix} 95\%  \\ 5\% \end{pmatrix}\)</span>.</p>
+<p>In general the eigenvector corresponding to the <strong>largest eigenvalue</strong> (also called <em>dominant eigenvalue</em>) of the projection matrix, which is always <em>real</em> and <em>positive</em>, will correspond to the stable age distribution, for which our population vector converges after we wait enough time.</p>
+<p>As we have seen, after enough time, the dominant eigenvalue will also correspond to the growth factor for each of the age classes. Thus, if <span class="math notranslate nohighlight">\(\lambda_1&gt;1\)</span> the population size will increase, whereas if <span class="math notranslate nohighlight">\(0&lt;\lambda_1&lt;1\)</span> the population size will decrease.</p>
+</section>
+</section>
+<section id="analysing-change-continuously">
+<h2>Analysing change, continuously<a class="headerlink" href="#analysing-change-continuously" title="Link to this heading">#</a></h2>
+<section id="average-rate-of-change">
+<h3>Average rate of change<a class="headerlink" href="#average-rate-of-change" title="Link to this heading">#</a></h3>
+<p>Watch this <a class="reference external" href="https://web.microsoftstream.com/video/2953e661-e1c1-43ff-ba78-ca52099ed1ec">video</a>
+for a short introduction to this lecture.</p>
+</section>
+<section id="instantaneous-rate-of-change">
+<h3>Instantaneous rate of change<a class="headerlink" href="#instantaneous-rate-of-change" title="Link to this heading">#</a></h3>
+<p>If we have a continuous function <span class="math notranslate nohighlight">\(f(x)\)</span>, for each two points <span class="math notranslate nohighlight">\(x_{0}\)</span> and <span class="math notranslate nohighlight">\(x_{1}\)</span>, we can define the average rate of change between these points by the ratio between how much <span class="math notranslate nohighlight">\(f\)</span> has changed and how much <span class="math notranslate nohighlight">\(x\)</span> has changed:</p>
+<div class="math notranslate nohighlight">
+\[\frac{\Delta f(x)}{\Delta x} = \frac{f(x_{1}) - f(x_{0})}{x_{1} - x_{0}}\]</div>
+<p>For points <span class="math notranslate nohighlight">\(x_{0}\)</span> and <span class="math notranslate nohighlight">\(x_{1}\)</span> this can be graphically represented by:</p>
+<img alt="../../_images/sec_tan.png" src="../../_images/sec_tan.png" />
+<p>The line that passes through the points <span class="math notranslate nohighlight">\((x_0, f(x_0))\)</span> and <span class="math notranslate nohighlight">\((x_1, f(x_1))\)</span> (yellow line) is named <em>secant line</em>. As we have seen before in our treatment of linear functions, the slope of the secant line will be given by the average rate of change between this points (remember that the slope for linear functions is defined as <span class="math notranslate nohighlight">\(m=\displaystyle\frac{\Delta y}{\Delta x})\)</span>.</p>
+<p>Now consider that we change the value of <span class="math notranslate nohighlight">\(x_1\)</span> so that it slowly approaches <span class="math notranslate nohighlight">\(x_0\)</span> from the right. In this case, the secant lines also change (from yellow to red), as we change the image of <span class="math notranslate nohighlight">\(x_1\)</span> the function, <span class="math notranslate nohighlight">\(f(x_1)\)</span>. In the limit where <span class="math notranslate nohighlight">\(x_1\)</span> is very close to <span class="math notranslate nohighlight">\(x_0\)</span>, the secant line approaches the <em>tangent</em> (sark red line) of the function <span class="math notranslate nohighlight">\(f(x)\)</span> at the point <span class="math notranslate nohighlight">\(x_0\)</span> (the tangent is the straight line that “just touches” the curve at that point).</p>
+<p>If we represent the slope of the tangent line at the point <span class="math notranslate nohighlight">\(x_0\)</span> by the term <span class="math notranslate nohighlight">\(f'(x_0)\)</span>, we will have:</p>
+<div class="math notranslate nohighlight">
+\[\lim_{x_{1} \to x_{0}} \frac{\Delta f(x)}{\Delta x} = \frac{f(x_{1}) - f(x_{0})}{x_{1} - x_{0}} = f'(x_{0})\]</div>
+<p>Since we can do the same process as we change the value <span class="math notranslate nohighlight">\(x_0\)</span>, we define a new function <span class="math notranslate nohighlight">\(f'(x_{0})\)</span> which we call the <strong>derivative</strong> of the function <span class="math notranslate nohighlight">\(f\)</span> at the point <span class="math notranslate nohighlight">\(x_0\)</span>, and its value tells us about the <em>instantaneous</em> rate of change of <span class="math notranslate nohighlight">\(f\)</span> at the point <span class="math notranslate nohighlight">\(x_0\)</span>.</p>
+<p>If we look instead at the interval between <span class="math notranslate nohighlight">\(x_{0}\)</span> and <span class="math notranslate nohighlight">\(x_{1}\)</span> and define <span class="math notranslate nohighlight">\(h = x_1 - x_0\)</span>, if <span class="math notranslate nohighlight">\(x_1\)</span> approaches <span class="math notranslate nohighlight">\(x_0\)</span>, then <span class="math notranslate nohighlight">\(h\)</span> approaches <span class="math notranslate nohighlight">\(0\)</span>. and we would have for the derivative of <span class="math notranslate nohighlight">\(f(x)\)</span>:</p>
+<div class="math notranslate nohighlight">
+\[\lim_{h \to 0} \frac{f(x + h) - f(x)}{h} = f'(x)\]</div>
+<p>Visit this <a class="reference external" href="https://www.demonstrations.wolfram.com/SecantAndTangentLines/">link</a> for an interactive demonstration of the limit of the tangent curve.</p>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>Note that in the previous expression the index <span class="math notranslate nohighlight">\(0\)</span> was omitted from <span class="math notranslate nohighlight">\(x_0\)</span>. We will assume that the derivative is defined for all the points in the domain of the function <span class="math notranslate nohighlight">\(f(x)\)</span> for which the limit in the expression exists.</p>
+<p>It is good to have a graphical intuition for when the derivative is not well defined. Some cases are shown below.</p>
+</div>
+<div class="admonition-example admonition">
+<p class="admonition-title">Example</p>
+<p>By the definition of the derivative of <span class="math notranslate nohighlight">\(f(x)\)</span>:</p>
+<div class="math notranslate nohighlight">
+\[f'(x)=\lim_{h \to 0} \frac{f(x + h) - f(x)}{h}\]</div>
+<p>Thus, if <span class="math notranslate nohighlight">\(f(x)=x^2\)</span>, we will have:</p>
+<div class="math notranslate nohighlight">
+\[f'(x)=\lim_{h \to 0} \frac{(x + h)^{2} - x^{2}}{h} = \lim_{h \to 0} \frac{x^{2} + 2xh + h^{2} - x^{2}}{h} = \lim_{h \to 0} (h + 2x) = 2x\]</div>
+<p>So that the derivative of <span class="math notranslate nohighlight">\(f(x)=x^2\)</span> is the function <span class="math notranslate nohighlight">\(f'(x)=2x\)</span>.</p>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>Although limits of functions can be defined in a more rigorous way, in this course we will treat them more intuitively, and use, if required, the known rules for limits. Thus, if <span class="math notranslate nohighlight">\(C\)</span> is a constant, <span class="math notranslate nohighlight">\(\lim_{x \rightarrow a} f(x) = L\)</span>, and <span class="math notranslate nohighlight">\(\lim_{x \rightarrow a} g(x) = M\)</span>, it is generally valid that:</p>
+<div class="math notranslate nohighlight">
+\[\lim_{x \rightarrow a} [f(x) + g(x)] = \lim_{x \rightarrow a} f(x) + \lim_{x \rightarrow a} g(x) = L + M\]</div>
+<div class="math notranslate nohighlight">
+\[\lim_{x \rightarrow a} [Cf(x)] = C\lim_{x \rightarrow a} f(x) = CL\]</div>
+<div class="math notranslate nohighlight">
+\[\lim_{x \rightarrow a} [f(x)g(x)] = [\lim_{x \rightarrow a} f(x)][\lim_{x \rightarrow a} g(x)] = LM\]</div>
+<div class="math notranslate nohighlight">
+\[\lim_{x \rightarrow a} \left[\frac{f(x)}{g(x)}\right] = \frac{\lim_{x \rightarrow a} f(x)}{\lim_{x \rightarrow a} g(x)} = \frac{L}{M},\ \ (\mbox{if}\ M \ne 0)\]</div>
+<div class="math notranslate nohighlight">
+\[\lim_{x \rightarrow a} [C^{f(x)}] = C^{\lim_{x \rightarrow a} f(x)} = C^L\]</div>
+</div>
+</div>
+<p>Other common notations for the derivative are given by:</p>
+<div class="math notranslate nohighlight">
+\[f'(x) = \frac{df}{dx} = \frac{d}{dx}f(x) = \dot{f}(x)\]</div>
+<p>Visit this <a class="reference external" href="https://the-learning-machine.com/article/calculus/derivatives">link</a> for examples that provide demonstrations of the intuitive interpretation of the derivative.</p>
+</section>
+<section id="calculating-derivatives">
+<h3>Calculating derivatives<a class="headerlink" href="#calculating-derivatives" title="Link to this heading">#</a></h3>
+<p>It is possible to generalise the previous example for <span class="math notranslate nohighlight">\(f(x)=x^2\)</span> and obtain the derivative for any power function <span class="math notranslate nohighlight">\(f(x)=x^n\)</span> (<span class="math notranslate nohighlight">\(n \ne 0\)</span>):</p>
+<div class="math notranslate nohighlight">
+\[f(x) = x^{n} \implies f'(x) = nx^{n-1}\]</div>
+<div class="admonition-example admonition">
+<p class="admonition-title">Example</p>
+<div class="math notranslate nohighlight">
+\[x^{2} \implies 2x^{2-1} = 2x\]</div>
+<div class="math notranslate nohighlight">
+\[\ x^{3} \implies 3x^{3-1} = 3x^{2}\]</div>
+<div class="math notranslate nohighlight">
+\[\ x^{4.2} \implies 4.2x^{4.2-1} = 4.2x^{3.2}\]</div>
+</div>
+<p>It is possible to show, from the definition of derivative and the rules for limits, that there are simple rules for derivatives when calculating them for a sum of functions or a product by a constant. If <span class="math notranslate nohighlight">\(C\)</span> is a constant, we generally have:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{align}\frac{d}{dx}[f(x)+g(x)] &amp;= \frac{df}{dx} + \frac{dg}{dx} \\
+   \frac{d}{dx}[Cf(x)] &amp;= C\frac{df}{dx}\end{align}\end{split}\]</div>
+<p>Thus, differentiating a given polynomial p(x) stands for differentiating each term separately and summing the result.</p>
+<div class="admonition-example admonition">
+<p class="admonition-title">Example</p>
+<div class="math notranslate nohighlight">
+\[f(x) = 3x^{2} - 4x + 6 \implies f'(x) = 3(2x) - 4(x^{0}) + 0 = 6x - 4\]</div>
+<div class="math notranslate nohighlight">
+\[f(x) = x^{60} - 3x^{20} - 4x \implies f'(x) = 60x^{59} - 60x^{19} - 4\]</div>
+</div>
+<p>For products and quotients of functions, however, we have different rules:</p>
+<div class="math notranslate nohighlight">
+\[\frac{d}{dx}[f(x)g(x)] = \left(\frac{df}{dx}\right) g(x) + f(x)\left(\frac{dg}{dx}\right)\]</div>
+<div class="math notranslate nohighlight">
+\[\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{\left(\displaystyle\frac{df}{dx}\right)g(x) - f(x)\left(\displaystyle\frac{dg}{dx}\right)}{[g(x)]^{2}}\]</div>
+<div class="admonition tip">
+<p class="admonition-title">Tip</p>
+<p>If you do not want to remember the quotients rule, you can use the product rule on <span class="math notranslate nohighlight">\(f(x)j(x)\)</span>, with <span class="math notranslate nohighlight">\(j(x)=g(x)^{-1}\)</span> and use the chain rule to evaluate <span class="math notranslate nohighlight">\(\frac{d}{dx} j(x)\)</span>.</p>
+</div>
+<div class="admonition-examples admonition">
+<p class="admonition-title">Examples</p>
+<div class="math notranslate nohighlight">
+\[f(x) = (x + 2)(x - 4) \implies f'(x) = (1)(x-4) + (x+2)(1) = x - 4 + x + 2 = 2x - 2\]</div>
+<div class="math notranslate nohighlight">
+\[f(x) = \frac{\alpha x}{1 + \beta x} \implies f'(x) =  \frac{(\alpha)(1 + \beta x) - (\alpha x)(\beta)}{(1 + \beta x)^{2}} = \frac{\alpha + \alpha\beta x - \alpha \beta x}{(1 + \beta x)^{2}} =\frac{\alpha}{(1 + \beta x)^{2}}\]</div>
+</div>
+<p>Moreover, expressions for the derivatives of the other elementary functions can also be found:</p>
+<ul class="simple">
+<li><p><span class="math notranslate nohighlight">\(f(x) = a^{x} \implies f'(x) = a^{x} \cdot (\ln a) ,\ (a &gt; 0, a \neq 1)\)</span><br />
+<span class="math notranslate nohighlight">\(\left(\mbox{in particular: }\ (e^{x})' = e^{x}\right)\)</span></p></li>
+<li><p><span class="math notranslate nohighlight">\(f(x) = \log_{a}x \implies f'(x) = \displaystyle\frac{1}{(\ln a)x}\)</span><br />
+<span class="math notranslate nohighlight">\(\left(\mbox{in particular: }\ (\ln x)' = \displaystyle\frac{1}{x}\right)\)</span></p></li>
+<li><p><span class="math notranslate nohighlight">\(f(x) = \mbox{sin}(x) \implies f'(x) = \mbox{cos}(x)\)</span></p></li>
+<li><p><span class="math notranslate nohighlight">\(f(x) = \mbox{cos}(x) \implies f'(x) = -\mbox{sin}(x)\)</span></p></li>
+</ul>
+<div class="admonition-example admonition">
+<p class="admonition-title">Example</p>
+<p>If <span class="math notranslate nohighlight">\(f(x) = \mbox{tan}(x) = \displaystyle\frac{\mbox{sin}(x)}{\mbox{cos}(x)}\)</span>, then:</p>
+<div class="math notranslate nohighlight">
+\[f'(x) = \frac{(\mbox{cos}(x))\mbox{cos}(x) - \mbox{sin}(x)(-\mbox{sin}(x))}{(\mbox{cos}(x))^{2}} = \frac{\mbox{sin}^{2}(x) + \mbox{cos}^{2}(x)}{\mbox{cos}^{2}(x)} = \frac{1}{\mbox{cos}^{2}(x)} = \mbox{sec}^{2}(x)\]</div>
+</div>
+</section>
+<section id="chain-rule">
+<h3>Chain rule<a class="headerlink" href="#chain-rule" title="Link to this heading">#</a></h3>
+<p>The calculation of derivatives for more complicated functions can generally be made easier by breaking the function into components, for example:</p>
+<ul class="simple">
+<li><p><span class="math notranslate nohighlight">\(f(x) = (3x^{2} - 2x + 1)^{3} = (g(x))^{3}\)</span>, where <span class="math notranslate nohighlight">\(g(x) = 3x^{2} - 2x + 1\)</span></p></li>
+<li><p><span class="math notranslate nohighlight">\(f(x) = \displaystyle\frac{\sqrt{2x - 1}}{1 + \sqrt{2x - 1}} = \displaystyle\frac{g(x)}{1 + g(x)}\)</span>, where <span class="math notranslate nohighlight">\(g(x) = \sqrt{h(x)}\)</span>, and <span class="math notranslate nohighlight">\(h(x) = 2x - 1\)</span></p></li>
+</ul>
+<p>We have seen this before when discussing transformation of graphs.  When one function is nested within the other, we have a composite function, or as for the last example above:</p>
+<div class="math notranslate nohighlight">
+\[f(x) = f(g(h(x)))\]</div>
+<p>The derivative of composite functions in relation to the dependent variable is then given by the <em>chain rule</em>:</p>
+<ul class="simple">
+<li><p><span class="math notranslate nohighlight">\(\displaystyle\frac{d}{dx}f(x) = \displaystyle\frac{d}{dx}f(g(x)) = \left(\displaystyle\frac{df}{dg}\right)\left(\displaystyle\frac{dg}{dx}\right)\)</span></p></li>
+<li><p><span class="math notranslate nohighlight">\(\displaystyle\frac{d}{dx}f(x) = \displaystyle\frac{d}{dx}f(g(h(x))) = \left(\displaystyle\frac{df}{dg}\right) \left(\displaystyle\frac{dg}{dh}\right) \left(\displaystyle\frac{dh}{dx}\right)\)</span></p></li>
+</ul>
+<p>The strategy then consists in identifying the nested functions, calculating their derivatives and multiplying the results.</p>
+<div class="admonition-examples admonition">
+<p class="admonition-title">Examples</p>
+<ul>
+<li><p>If <span class="math notranslate nohighlight">\(f(x) = (3x^{2} - 2x + 1)^{3}\)</span>, we have:</p>
+<div class="math notranslate nohighlight">
+\[f(g) = g^{3} \implies \displaystyle\frac{df}{dg} = 3g^2\]</div>
+<div class="math notranslate nohighlight">
+\[g(x) = 3x^{2} - 2x + 1 \implies \displaystyle\frac{dg}{dx} = 6x - 2\]</div>
+<p>Thus:</p>
+<div class="math notranslate nohighlight">
+\[\displaystyle\frac{df}{dx} = \left(\displaystyle\frac{df}{dg}\right)\left(\displaystyle\frac{dg}{dx}\right) = 3g^{2} \cdot g' = 3(3x^{2} - 2x + 1)^2(6x - 2)\]</div>
+  <br />  
+</li>
+<li><p>If <span class="math notranslate nohighlight">\(f(x) = \displaystyle\frac{\sqrt{2x - 1}}{1 + \sqrt{2x - 1}}\)</span>, then:</p>
+<div class="math notranslate nohighlight">
+\[f(g) = \displaystyle\frac{g}{1+g} \implies \displaystyle\frac{df}{dg} = \frac{(1+g) - g}{(1+g)^2} = \frac{1}{(1+g)^2}\]</div>
+<div class="math notranslate nohighlight">
+\[g(h) = \sqrt{h} \implies \displaystyle\frac{dg}{dh} = \frac{1}{2}h^{-1/2}\]</div>
+<div class="math notranslate nohighlight">
+\[h(x) = 2x-1 \implies \displaystyle\frac{dh}{dx} = 2\]</div>
+<p>Thus:</p>
+<div class="math notranslate nohighlight">
+\[\displaystyle\frac{df}{dx} =  \left(\displaystyle\frac{df}{dg}\right) \left(\displaystyle\frac{dg}{dh}\right) \left(\displaystyle\frac{dh}{dx}\right) = \frac{1}{(1+\sqrt{2x - 1})^2}\cdot \frac{1}{2(\sqrt{2x - 1})}\cdot 2 = \frac{1}{(\sqrt{2x - 1})(1+\sqrt{2x - 1})^2}\]</div>
+  <br />
+</li>
+<li><p>If <span class="math notranslate nohighlight">\(f(t) = 5 + 5\mbox{cos}\left[\displaystyle\frac{\pi}{12}(t - 3)\right]\)</span>, we have:</p>
+<div class="math notranslate nohighlight">
+\[f(g) = 5 + 5g \implies \displaystyle\frac{df}{dg} = 5\]</div>
+<div class="math notranslate nohighlight">
+\[g(h) = \mbox{cos}(h) \implies \displaystyle\frac{dg}{dh} = -\mbox{sin}(h)\]</div>
+<div class="math notranslate nohighlight">
+\[h(t) = \frac{\pi}{12}(t - 3) \implies \displaystyle\frac{dh}{dt} = \frac{\pi}{12}\]</div>
+<p>Thus:</p>
+<div class="math notranslate nohighlight">
+\[\displaystyle\frac{df}{dx} =  \left(\displaystyle\frac{df}{dg}\right) \left(\displaystyle\frac{dg}{dh}\right) \left(\displaystyle\frac{dh}{dx}\right) = 5 \cdot \left(-\mbox{sin}(h)\right) \cdot \frac{\pi}{12} = -\frac{5\pi}{12}\mbox{sin}\left[\displaystyle\frac{\pi}{12}(t - 3)\right]\]</div>
+<p>For this last example if we plot the function <span class="math notranslate nohighlight">\(f(t)=5 + 5\mbox{cos}\left[\displaystyle\frac{\pi}{12}(t - 3)\right]\)</span> together with its derivative <span class="math notranslate nohighlight">\(f'(t)=-\displaystyle\frac{5\pi}{12}\mbox{sin}\left[\displaystyle\frac{\pi}{12}(t - 3)\right]\)</span> we will have:</p>
+<img alt="../../_images/func_diff.png" src="../../_images/func_diff.png" />
+<p>Try to compare the two graphs. What happens with the function <span class="math notranslate nohighlight">\(f(t)\)</span> on the intervals where the derivative <span class="math notranslate nohighlight">\(f'(t)\)</span> is positive or negative? What happens at the points where the derivative is zero?</p>
+</li>
+</ul>
+</div>
+</section>
+<section id="higher-order-derivatives">
+<h3>Higher order derivatives<a class="headerlink" href="#higher-order-derivatives" title="Link to this heading">#</a></h3>
+<p>If the derivative of a given function is also a “well-behaved” function (without kinks or discontinuities), then we can also calculate the derivative of the derivative. Thus:</p>
+<div class="math notranslate nohighlight">
+\[\frac{d}{dx}\left(\frac{df}{dx}\right) = \frac{d^{2}f}{dx^{2}} = f''(x)\]</div>
+<p>The function <span class="math notranslate nohighlight">\(f''(x)\)</span> is called the <em>second derivative</em> with respect to <span class="math notranslate nohighlight">\(x\)</span>.</p>
+<p>We can continue this process and get higher-order derivatives:</p>
+<div class="math notranslate nohighlight">
+\[\frac{d^{n}f}{dx^{n}} = f^{(n)}(x),\]</div>
+<p>assuming again that these functions are also well-behaved. Function <span class="math notranslate nohighlight">\(f^{(n)}(x)\)</span> is called the <span class="math notranslate nohighlight">\(n\)</span>-th derivative of <span class="math notranslate nohighlight">\(f\)</span> with respect to <span class="math notranslate nohighlight">\(x\)</span> (note the parenthesis on <span class="math notranslate nohighlight">\((n)\)</span> to distinguish from powers of <span class="math notranslate nohighlight">\(f\)</span>).</p>
+<div class="admonition-examples admonition">
+<p class="admonition-title">Examples</p>
+<p>The velocity is the derivative of the position, and the acceleration is the derivative of the velocity. Therefore, acceleration is the second-order derivative of the position</p>
+</div>
+<div class="admonition-examples admonition">
+<p class="admonition-title">Examples</p>
+<ul>
+<li><p><span class="math notranslate nohighlight">\(f(x) = 3x^{2} - 6x + 2\)</span>, then:</p>
+<div class="math notranslate nohighlight">
+\[f'(x) = 6x - 6\]</div>
+<div class="math notranslate nohighlight">
+\[f''(x) = 6\]</div>
+<p>Polynomials are always well-behaved and they are infinitely differentiable. However, if <span class="math notranslate nohighlight">\(n\)</span> is the degree of the polynomial (<em>i.e.</em> the largest exponent in the polynomial), we will have: <span class="math notranslate nohighlight">\(p^{(k)}(x) = 0,\ \mbox{for}\ k \ge n+1\)</span>.</p>
+  <br />
+</li>
+<li><p><span class="math notranslate nohighlight">\(f(x) = e^{-x^{2}}\)</span>, then:</p>
+<div class="math notranslate nohighlight">
+\[f'(x) = e^{-x^{2}} \cdot (-2x) = -2xe^{-x^{2}}\]</div>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{align}f''(x)
+&amp;= -2[e^{-x^{2}} + x(-2x)(e^{-x^{2}})] \\
+&amp;= -2e^{-x^{2}} + 4x^{2}e^{-x^{2}} \\
+&amp;= -2e^{-x^{2}}(1 - 2x^{2})\end{align}\end{split}\]</div>
+</li>
+</ul>
+</div>
+</section>
+<section id="functions-of-several-variables">
+<h3>Functions of several variables<a class="headerlink" href="#functions-of-several-variables" title="Link to this heading">#</a></h3>
+<p>A function <span class="math notranslate nohighlight">\(z = f(x,y)\)</span> of two dependent variables <span class="math notranslate nohighlight">\(x\)</span> and <span class="math notranslate nohighlight">\(y\)</span> defines a surface on the three dimensional <span class="math notranslate nohighlight">\(xyz\)</span>-space. If this function is well-behave it will be smooth, like if we applied different deformations on a rubber sheet. For smooth functions, we can calculate derivatives with respect to each of the dependent variables. This is written as:</p>
+<div class="math notranslate nohighlight">
+\[ \frac{\partial f}{\partial x} \text{ or } \frac{\partial f}{\partial x}(x, y) \text{ or } \frac{\partial}{\partial x}f(x,y)\]</div>
+<div class="math notranslate nohighlight">
+\[\frac{\partial f}{\partial y}, \frac{\partial f}{\partial{y}}(x,y), \frac{\partial}{\partial y}f(x,y)\]</div>
+<p>The partial derivative informs us how the function <span class="math notranslate nohighlight">\(f\)</span> varies along the <span class="math notranslate nohighlight">\(x\)</span> (or the <span class="math notranslate nohighlight">\(y\)</span>) direction at a specific point.</p>
+<div class="admonition-example admonition">
+<p class="admonition-title">Example</p>
+<p>Consider <span class="math notranslate nohighlight">\(f(x,y) = 4x^2y-2xy^3\)</span>. To compute the partial derivative with respect to <span class="math notranslate nohighlight">\(x\)</span> we consider all terms in <span class="math notranslate nohighlight">\(y\)</span> constants, and calculate the derivative as usual. For the partial derivative with respect to <span class="math notranslate nohighlight">\(y\)</span>, we consider the <span class="math notranslate nohighlight">\(x\)</span> terms constant. We have:</p>
+<div class="math notranslate nohighlight">
+\[\frac{\partial f}{\partial x} = (4y) \cdot (2x) - 2y^{3} \cdot (1) = 8xy - 2y^{3}\]</div>
+<div class="math notranslate nohighlight">
+\[\frac{\partial f}{\partial y} = 4x^{2} - 2x(3y^{2}) = 4x^{2} - 6xy^{2}\]</div>
+</div>
+<p>Higher order partial derivatives can also be defined, and represented with the common notations:</p>
+<div class="math notranslate nohighlight">
+\[\frac{\partial}{\partial x}\left(\frac{\partial f}{\partial x}\right) = \frac{\partial^{2}f}{\partial x^{2}} = \partial_{xx} f = f_{xx}(x,y)\]</div>
+<div class="math notranslate nohighlight">
+\[\frac{\partial}{\partial y}\left(\frac{\partial f}{\partial y}\right) = \frac{\partial^{2}f}{\partial y^{2}} = 
+\partial_{yy} f = f_{yy}(x,y)\]</div>
+<p>with subscripts representing the variable and the order of differentiation.</p>
+<p>For well-behaved functions, the order of the variables chosen to differentiate does not matter (a special case of what is called Schwarz’s theorem):</p>
+<div class="math notranslate nohighlight">
+\[f_{yx}(x,y) = \frac{\partial}{\partial y}\left(\frac{\partial f}{\partial x}\right) = \frac{\partial^{2}f}{\partial y \partial x} = \frac{\partial^{2} f}{\partial x \partial y}= \frac{\partial}{\partial x}\left(\frac{\partial f}{\partial y}\right) = f_{xy}(x,y)\]</div>
+<div class="admonition-example admonition">
+<p class="admonition-title">Example</p>
+<p>Often, a quantity of interest does not depend only on one variable but on many. The population size of a species depends on various aspects, including the availability of food, the number of predators, and other environmental factors.</p>
+</div>
+</section>
+</section>
+<section id="analysing-change-continuously-part-2">
+<h2>Analysing change, continuously (Part 2)<a class="headerlink" href="#analysing-change-continuously-part-2" title="Link to this heading">#</a></h2>
+<p>In many different contexts we may be interested in finding the maximum or minimum values a given function may assume. This process is referred to as <em>optimization</em>.</p>
+<p>In Ecology, some of the most common uses for optimization methods are:</p>
+<p>1 - <em>Statistical estimation and inference</em>: optimizing the likelihood of a model and its parameters being true, for a given set of observations.</p>
+<p>2 - <em>Resources management</em>: finding a balance between different wildlife and economic priorities (market, legal, natural constraints)</p>
+<p>3 - <em>Optimality theory</em>: how can the behavioural or life-history trade-offs result in optimal strategies for different organisms</p>
+<section id="extreme-values-of-functions">
+<h3>Extreme values of functions<a class="headerlink" href="#extreme-values-of-functions" title="Link to this heading">#</a></h3>
+<p>For a given function <span class="math notranslate nohighlight">\(f(x)\)</span> that is <strong>continuous</strong> in an interval <span class="math notranslate nohighlight">\(a \le x \le b\)</span>, there will always be a global maximum and a global minimum, points at which the function assumes its maximum and minimum values in the interval.</p>
+<img alt="../../_images/extrema.png" src="../../_images/extrema.png" />
+<p>A function <span class="math notranslate nohighlight">\(f(x)\)</span> presents a <em>local maximum</em> value if for a given vicinity of the point <span class="math notranslate nohighlight">\(x_0\)</span>, <span class="math notranslate nohighlight">\(f(x_0)\)</span> is greater than all the other <span class="math notranslate nohighlight">\(f(x)\)</span> for <span class="math notranslate nohighlight">\(x\)</span> in this vicinity. The analogous definition can be made for a <em>local minimum</em> value when <span class="math notranslate nohighlight">\(f(x_0)\)</span> is smaller than all the other <span class="math notranslate nohighlight">\(f(x)\)</span> in the vicinity. Local minima and local maxima constitue the <em>local extrema</em> for <span class="math notranslate nohighlight">\(f(x)\)</span>, and with the set of all extrema we can find the global maximum and the global minimum.</p>
+<p>An important result is that, if the function <span class="math notranslate nohighlight">\(f\)</span> has a local extremum at a given point <span class="math notranslate nohighlight">\(x=c\)</span> in that interval (<span class="math notranslate nohighlight">\(a&lt;c&lt;b\)</span>) and the derivative of the function exists at that point (in other words, if <span class="math notranslate nohighlight">\(f'(c)\)</span> is well defined), then we will have:</p>
+<p><span class="math notranslate nohighlight">\(f'(c)=0\)</span></p>
+<div class="admonition warning">
+<p class="admonition-title">Warning</p>
+</div>
+<p>Thus the points <span class="math notranslate nohighlight">\(x=c\)</span>, for which <span class="math notranslate nohighlight">\(f'(c)\)</span>, will constitute <strong>candidates</strong> for local extrema, and will have to be evaluated under other sets of conditions. Also important to have in mind that point for which the derivative is not definided (kinks or discontinuities) may also constitute local extrema (see on the first figure).</p>
+<p>In general, when searching for local extrema, we have to adopt the following rules:</p>
+<ul class="simple">
+<li><p>Do not assume points <span class="math notranslate nohighlight">\(x\)</span> for which <span class="math notranslate nohighlight">\(f'(x) = 0\)</span> are extrema. They are candidates.</p></li>
+<li><p>Check points where derivative is not defined.</p></li>
+<li><p>Check endpoints of domain.</p></li>
+</ul>
+<div class="admonition tip">
+<p class="admonition-title">Tip</p>
+<p>If possible, create a plot. Visuals are powerful tools and can help verify whether calculations have been done correctly.</p>
+</div>
+</section>
+<section id="monotonicity-and-concavity">
+<h3>Monotonicity and concavity<a class="headerlink" href="#monotonicity-and-concavity" title="Link to this heading">#</a></h3>
+<p>Imagine a function <span class="math notranslate nohighlight">\(f(x)\)</span> defined in a given interval <span class="math notranslate nohighlight">\(a \le x \le b\)</span>. For every <span class="math notranslate nohighlight">\(x_1\)</span> and <span class="math notranslate nohighlight">\(x_2\)</span> in this interval, if <span class="math notranslate nohighlight">\(x_{2} &gt; x_{1}\)</span> implies that <span class="math notranslate nohighlight">\(f(x_{2}) &gt; f(x_{1})\)</span> then the function is <strong>increasing</strong> in this interval. If on the other hand, <span class="math notranslate nohighlight">\(x_{2} &gt; x_{1}\)</span> implies that <span class="math notranslate nohighlight">\(f(x_{2}) &lt; f(x_{1})\)</span> then the function is <strong>decreasing</strong> in this interval.  This can be graphically shown as:</p>
+<img alt="../../_images/incr_decr.png" src="../../_images/incr_decr.png" />
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>Since we are using the stric inequalities (”<span class="math notranslate nohighlight">\(&gt;\)</span>” or “<span class="math notranslate nohighlight">\(&lt;\)</span>”) it can be added that the function is <strong>monotonic</strong>, or monotonically increasing or decreasing. If we had <span class="math notranslate nohighlight">\(x_{2} &gt; x_{1}\)</span> implying that <span class="math notranslate nohighlight">\(f(x_{2}) \ge f(x_{1})\)</span> (or <span class="math notranslate nohighlight">\(f(x_{2}) \le f(x_{1})\)</span>), the function would be <em>non-decreasing</em> (or <em>non-increasing</em>).</p>
+</div>
+<p>Now suppose that <span class="math notranslate nohighlight">\(f\)</span> is well-behaved in a given interval <span class="math notranslate nohighlight">\(a \le x \le b\)</span> (in mathematical terms, we say: <span class="math notranslate nohighlight">\(f\)</span> is continuous and differentiable in <span class="math notranslate nohighlight">\((a,b)\)</span>), then an important results is that</p>
+<div class="math notranslate nohighlight">
+\[\mbox{If }\ f'(x) &gt; 0\ \mbox{ for}\ a \le x \le b \implies f(x)\ \mbox{is increasing for}\ a \le x \le b\]</div>
+<div class="math notranslate nohighlight">
+\[\mbox{If }\ f'(x) &lt; 0\ \mbox{ for}\ a \le x \le b \implies f(x)\ \mbox{is decreasing for}\ a \le x \le b\]</div>
+<p>In other words, if the derivative of function <span class="math notranslate nohighlight">\(f\)</span> is positive (negative) for every point <span class="math notranslate nohighlight">\(x\)</span> in a given interval then the function is increasing (decreasing) in this interval.</p>
+<div class="admonition-example admonition">
+<p class="admonition-title">Example</p>
+<p>Let us calculate the intervals where the function <span class="math notranslate nohighlight">\(f(x) = x^{3} - \displaystyle\frac{3}{2}x^{2} - 6x + 3\)</span> is increasing and decreasing. Since <span class="math notranslate nohighlight">\(f(x)\)</span> is continuos and differentiable for all real values of <span class="math notranslate nohighlight">\(x\)</span> (remember that all polynomial functions are well-behved), we can use the first derivative test:</p>
+<div class="math notranslate nohighlight">
+\[f'(x) = 3x^{2}-3x-6 = 3(x^{2}-x-2) = 3(x+1)(x-2)\]</div>
+<p>and we see that <span class="math notranslate nohighlight">\(f'(x)\)</span> has roots in <span class="math notranslate nohighlight">\(x=-1\)</span> and <span class="math notranslate nohighlight">\(x=2\)</span>. Since <span class="math notranslate nohighlight">\(f'(x)\)</span> is a polynomial of degree 2 with the coefficient of the leading term positive, it is <em>concave up</em>. From this, we can conclude that:</p>
+<div class="math notranslate nohighlight">
+\[x&lt;-1\ \mbox{or}\ x&gt;2 \implies f'(x) &gt; 0 \implies f(x)\ \mbox{is}\ increasing\]</div>
+<div class="math notranslate nohighlight">
+\[-1&lt;x&lt;2 \implies f'(x) &lt; 0 \implies f(x)\ \mbox{is}\ decreasing\]</div>
+<p>The following plot shows the function <span class="math notranslate nohighlight">\(f(x)\)</span> and its derivative. Compare the intervals where the derivative is positive (negative) with the intervals where the function <span class="math notranslate nohighlight">\(f(x)\)</span> is increasing (decreasing).</p>
+<img alt="notebooks/Appendix-MathsForBiologists/func_diff2.png" src="notebooks/Appendix-MathsForBiologists/func_diff2.png" />
+</div>
+<p>When we discussed polynomials of degree 2, depending on the sign of the coefficient of the leading term, we had the two cases:</p>
+<img alt="../../_images/concav.png" src="../../_images/concav.png" />
+<p>Let us now generalise the notion of concavity for any arbitrary function. Suppose <span class="math notranslate nohighlight">\(f(x)\)</span> is differentiable in a given interval <span class="math notranslate nohighlight">\(a \le x \le b\)</span> (in other words, the derivative <span class="math notranslate nohighlight">\(f'(x)\)</span> is well-defined for every point in this interval). Then we have the following results:</p>
+<div class="math notranslate nohighlight">
+\[\mbox{If }\ f'(x)\ \mbox{is increasing for}\ a \le x \le b \implies f(x)\ \mbox{is}\ concave\ up\]</div>
+<div class="math notranslate nohighlight">
+\[\mbox{If }\ f'(x)\ \mbox{is decreasing for}\ a \le x \le b \implies f(x)\ \mbox{is}\ concave\ down\]</div>
+<p>This means that if for a given function <span class="math notranslate nohighlight">\(f(x)\)</span> the slopes of the tangent lines are increasing in a given interval of <span class="math notranslate nohighlight">\(x\)</span>, in this interval the function should resemble an “upwards U-shape”. If on the other hand, the slopes of the tangent lines are decreasing in this interval, the function should resemble a “downwards U-shape”. This can be visualised on the plots below.</p>
+<img alt="../../_images/slopes_inc_dec.png" src="../../_images/slopes_inc_dec.png" />
+<p>If the function <span class="math notranslate nohighlight">\(f\)</span> is twice differentiable in <span class="math notranslate nohighlight">\(a \le x \le b\)</span> (<span class="math notranslate nohighlight">\(f''(x)\)</span> is well-define for <span class="math notranslate nohighlight">\(a \le x \le b\)</span>), we can use the last two results to find a general criterium for concavity. This will be given by:</p>
+<div class="math notranslate nohighlight">
+\[\mbox{If }\ f''(x) &gt; 0\ \mbox{ for}\ a \le x \le b \implies f'(x)\ \mbox{is increasing for}\ a \le x \le b \implies f(x)\ \mbox{is}\ concave\ up\]</div>
+<div class="math notranslate nohighlight">
+\[\mbox{If }\ f''(x) &lt; 0\ \mbox{ for}\ a \le x \le b \implies f'(x)\ \mbox{is decreasing for}\ a \le x \le b \implies f(x)\ \mbox{is}\ concave\ down\]</div>
+<div class="admonition-example admonition">
+<p class="admonition-title">Example</p>
+<p>If you have a population curve, these tools will help you analyze questions such as ‘How fast is the population growing?’ ‘Is the growth accelerating?’ or ‘Are any of these factors changing over different periods?</p>
+</div>
+<div class="admonition-example admonition">
+<p class="admonition-title">Example</p>
+<p>Let us analyse the function <span class="math notranslate nohighlight">\(f(x) = x^{3}-\displaystyle\frac{3}{2}x^{2}-6x+3\)</span> again. We have:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{align} 
+f'(x) = 3x^{2}-3x-6 \\
+f''(x) = 6x -3 \end{align}\end{split}\]</div>
+<p>The function f’’(x) has a root in <span class="math notranslate nohighlight">\(x=\displaystyle\frac{1}{2}\)</span>. Since its slope is positive, we have:</p>
+<div class="math notranslate nohighlight">
+\[x&lt;\displaystyle\frac{1}{2} \implies f''(x) &lt; 0 \implies f(x)\ \mbox{is}\ concave\ down\]</div>
+<div class="math notranslate nohighlight">
+\[x&gt;\displaystyle\frac{1}{2} \implies f''(x) &gt; 0 \implies f(x)\ \mbox{is}\ concave\ up\]</div>
+<p>On the plots below, compare the intervals where the second derivative is positive (negative) with the intervals where the function <span class="math notranslate nohighlight">\(f(x)\)</span> is concave up (concave down).</p>
+<img alt="../../_images/plot_concav.png" src="../../_images/plot_concav.png" />
+</div>
+</section>
+<section id="finding-extrema">
+<h3>Finding extrema<a class="headerlink" href="#finding-extrema" title="Link to this heading">#</a></h3>
+<p>Back to our strategy for finding maxima or minima of functions, we first determine the set:</p>
+<ul class="simple">
+<li><p>find all points <span class="math notranslate nohighlight">\(c\)</span> for which <span class="math notranslate nohighlight">\(f'(c) = 0\)</span> or all point <span class="math notranslate nohighlight">\(c\)</span> where <span class="math notranslate nohighlight">\(f'(c)\)</span> does not exist. These are called <strong>critical points</strong>.</p></li>
+<li><p>find endpoints of domain of <span class="math notranslate nohighlight">\(f\)</span>.</p></li>
+</ul>
+<p>Additionally, if <span class="math notranslate nohighlight">\(f\)</span> is twice differentiable (in some interval containing <span class="math notranslate nohighlight">\(c\)</span>):</p>
+<ul class="simple">
+<li><p>If <span class="math notranslate nohighlight">\(f'(c) = 0\)</span> and <span class="math notranslate nohighlight">\(f''(c) &lt; 0 \implies c\)</span> is a local maximum (since the function is concave down)</p></li>
+<li><p>If <span class="math notranslate nohighlight">\(f'(c) = 0\)</span> and <span class="math notranslate nohighlight">\(f''(c) &gt; 0 \implies c\)</span> is a local minimum (since the function is concave up)</p></li>
+</ul>
+<div class="admonition-example admonition">
+<p class="admonition-title">Example</p>
+<p>Let us analyse the function <span class="math notranslate nohighlight">\(f(x) = \displaystyle\frac{3}{2}x^{4}-2x^{3}-6x^{2}+2\)</span>, and find its local extrema. We have:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{align}
+f'(x)&amp;= 6x^{3}-6x^{2}-12x=6x(x-2)(x+1) \\
+f''(x)&amp;=18x^{2}-12x-12=6(3x^{2}-2x-2)\end{align}\end{split}\]</div>
+<p>Since this is a polynomial function, the derivative is well-defined for all points. Also, for the critical points (roots of <span class="math notranslate nohighlight">\(f'(x)\)</span>) we have <span class="math notranslate nohighlight">\(x=0\)</span>, <span class="math notranslate nohighlight">\(x=-1\)</span>, and <span class="math notranslate nohighlight">\(x=2\)</span>. Calculating the value of the second derivative at the critical points:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{align}
+&amp;x=0 \implies f''(0)=-12 \text{ (local maximum)} \\
+&amp;x=-1 \implies f''(-1)=18 \text{ (local minimum)} \\
+&amp;x=2 \implies f''(2)=36 \text{ (local minimum)}\end{align}\end{split}\]</div>
+<p>Now we can check on the plot of <span class="math notranslate nohighlight">\(f(x)\)</span> that we have successfully determined the local extrema.</p>
+<img alt="../../_images/plot_extrema.png" src="../../_images/plot_extrema.png" />
+</div>
+</section>
+<section id="inflection-points">
+<h3>Inflection points<a class="headerlink" href="#inflection-points" title="Link to this heading">#</a></h3>
+<p>As we saw before, given a function <span class="math notranslate nohighlight">\(f(x)\)</span>, for some values of <span class="math notranslate nohighlight">\(x\)</span>, there might be a change in the concavity of the function:</p>
+<img alt="../../_images/inflection.png" src="../../_images/inflection.png" />
+<p>The points <span class="math notranslate nohighlight">\(x\)</span> for where the function changes its concavity are called <strong>inflection points</strong>. If <span class="math notranslate nohighlight">\(f\)</span> is twice differentiable and <span class="math notranslate nohighlight">\(c\)</span> is an inflection point, then <span class="math notranslate nohighlight">\(f''(c)=0\)</span>.</p>
+<p>Inflection points are the points where the function is (locally) steepest. It therefore helps us answer questions like when the growth/decline was strongest.</p>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>Note again that the points <span class="math notranslate nohighlight">\(x=c\)</span> where <span class="math notranslate nohighlight">\(f''(c)=0\)</span> are only candidates for inflection points. See that if <span class="math notranslate nohighlight">\(f(x)=x^{4}\)</span>, <span class="math notranslate nohighlight">\(f''(x)=12x^{2}\)</span>. Then, <span class="math notranslate nohighlight">\(f''(0)=0\)</span>, but <span class="math notranslate nohighlight">\(x=0\)</span> is not an inflection point of <span class="math notranslate nohighlight">\(f\)</span>.</p>
+<p>After calculating the candidate points <span class="math notranslate nohighlight">\(x=c\)</span>, if the concavities of the function (given by the sign of the second derivative) to the left and to the right of <span class="math notranslate nohighlight">\(c\)</span> change, then <span class="math notranslate nohighlight">\(c\)</span> is an inflection point.</p>
+</div>
+<div class="admonition-example admonition">
+<p class="admonition-title">Example</p>
+<p>Consider the Holling Type III functional response</p>
+<div class="math notranslate nohighlight">
+\[f(N)=\frac{aN^{2}}{1+ahN^{2}},\]</div>
+<p>where <span class="math notranslate nohighlight">\(f(N)\)</span> is prey consumption rate per predator, <span class="math notranslate nohighlight">\(N\)</span> is the prey population density, <span class="math notranslate nohighlight">\(a\)</span> is the attack rate, and <span class="math notranslate nohighlight">\(h\)</span> is handling time. Let us check if this function presents any inflection point. We have:</p>
+<p><span class="math notranslate nohighlight">\(\begin{align}f'(N) 
+&amp;= \frac{2aN(1+ahN^{2})-aN^{2}(2ahN)}{(1+ahN^{2})^{2}} \\
+&amp;= \frac{2aN}{(1+ahN^{2})^{2}}\end{align}\)</span></p>
+<br />
+<p><span class="math notranslate nohighlight">\(\begin{align}f''(N)
+&amp;=\frac{2a(1+ahN^{2})^{2}-2aN[2(1+ahN^{2})\cdot2ahN]}{(1+ahN^{2})^{4}} \\
+&amp;=\frac{2a(1+2ahN^{2}+a^{2}h^{2}N^{4})-2a[4ahN^{2}+4a^{2}h^{2}N^{4}]}{(1+ahN^{2})^{4}} \\
+&amp;=\frac{2a(1-3ahN^{2})}{(1+ahN^{2})^{3}}\end{align}\)</span></p>
+<p>See that the function <span class="math notranslate nohighlight">\(f''(N)\)</span> has two roots: <span class="math notranslate nohighlight">\(N=-\sqrt{\displaystyle\frac{1}{3ah}}\)</span> and <span class="math notranslate nohighlight">\(N=\sqrt{\displaystyle\frac{1}{3ah}}\)</span>. Since <span class="math notranslate nohighlight">\(N\)</span> is a population density, it only makes sense to analyse <span class="math notranslate nohighlight">\(N \ge 0\)</span>. It is also possible to show that:</p>
+<div class="math notranslate nohighlight">
+\[N&lt;\sqrt{\displaystyle\frac{1}{3ah}} \implies f''(N) &gt; 0\]</div>
+<div class="math notranslate nohighlight">
+\[N&gt;\sqrt{\displaystyle\frac{1}{3ah}} \implies f''(N) &lt; 0\]</div>
+<p>This means that we have a change of concavity at the point <span class="math notranslate nohighlight">\(N=\sqrt{\displaystyle\frac{1}{3ah}}\)</span> and thus this is and inflection point of this curve.</p>
+<p>Analyse how the function <span class="math notranslate nohighlight">\(f(N)\)</span> changes with <span class="math notranslate nohighlight">\(N\)</span>.</p>
+<img alt="../../_images/hollingIII.png" src="../../_images/hollingIII.png" />
+<p>How does this function compares with the Holling type II functional response?</p>
+</div>
+<div class="admonition tip">
+<p class="admonition-title">Tip</p>
+<p>For the intervals where the function is concave up you can also think of it as representing “accelerated change” of that function (since the first derivative, the “velocity” is increasing), while the intervals where it is concave down represent “deccelerated change” of the function. Understanding this can be particularly important if you function has a specific biological meaning, as in the previous example.</p>
+</div>
+</section>
+<section id="series-expansions">
+<h3>Series expansions<a class="headerlink" href="#series-expansions" title="Link to this heading">#</a></h3>
+<p>From the formal definition of a derivative</p>
+<div class="math notranslate nohighlight">
+\[f'(a)= \lim_{x \to a} \frac{f(x)-f(a)}{(x-a)} \xrightarrow{\text{$x$ very close to $a$}} f'(a) \approx \frac{f(x)-f(a)}{(x-a)},\]</div>
+<p>which means that if <span class="math notranslate nohighlight">\(x\)</span> is sufficiently close to <span class="math notranslate nohighlight">\(a\)</span>, the average rate of change is sufficiently close to the value of the derivative at that point. From the previous approximation, we have:</p>
+<div class="math notranslate nohighlight">
+\[f(x) = f(a)+f'(a)(x-a),\]</div>
+<p>which provides a linear approximation of <span class="math notranslate nohighlight">\(f(x)\)</span> around <span class="math notranslate nohighlight">\(a\)</span> (see that if <span class="math notranslate nohighlight">\(\alpha= f(a)\)</span> and <span class="math notranslate nohighlight">\(\beta=f'(a)\)</span>, both constants, then <span class="math notranslate nohighlight">\(f(x) = \alpha+\beta(x-a)\)</span>, which can easily be arranged on the form <span class="math notranslate nohighlight">\(y=mx+b\)</span>).</p>
+<div class="admonition-example admonition">
+<p class="admonition-title">Example</p>
+</div>
+<p>One way of trying to make the approximation better is to add terms of higher order of <span class="math notranslate nohighlight">\(x\)</span>. Suppose we want to approximate the function <span class="math notranslate nohighlight">\(f(x)\)</span> at point <span class="math notranslate nohighlight">\(a\)</span> by a given polynomial of degree <span class="math notranslate nohighlight">\(n\)</span></p>
+<div class="math notranslate nohighlight">
+\[P(x)=a_{0}+a_{1}(x-a)+a_{2}(x-a)^{2}+\cdots+a_n(x-a)^{n}\]</div>
+<p>so that the derivatives of <span class="math notranslate nohighlight">\(f(x)\)</span> and <span class="math notranslate nohighlight">\(P(x)\)</span> are the same at <span class="math notranslate nohighlight">\(x=a\)</span>:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{align}f(a)&amp;=P(a) \\
+f'(a)&amp;=P'(a)\\
+&amp;\vdots \\
+f^{(n)}(a)&amp;=P^{(n)}(a)\end{align}.\end{split}\]</div>
+<p>But for the polynomial, we have:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{align}  
+P(a)&amp;=a_{0} \\
+P'(a)&amp;=1 \cdot a_{1} \\
+P''(a)&amp;=2 \cdot 1 \cdot a_{2} \\
+P'''(a)&amp;=3 \cdot 2 \cdot 1 \cdot a_{3} \\  
+P''''(a)&amp;=4 \cdot 3 \cdot 2 \cdot 1 \cdot a_{4} \\
+&amp;\vdots \\
+P^{(n)}(a)&amp;= \underbrace{n(n-1)(n-2) \cdots 3 \cdot 2 \cdot 1}_{\displaystyle\text{$$n!$$}}\cdot a_{n} \end{align}\end{split}\]</div>
+<p>where <span class="math notranslate nohighlight">\(n! = n\cdot(n-1)\cdot(n-2) \cdots 3 \cdot 2 \cdot 1\)</span> is the factorial of <span class="math notranslate nohighlight">\(n\)</span>.</p>
+<p>Thus, by making derivatives of <span class="math notranslate nohighlight">\(f(x)\)</span> and <span class="math notranslate nohighlight">\(P(x)\)</span> equal at <span class="math notranslate nohighlight">\(x=a\)</span>, we obtain:</p>
+<div class="math notranslate nohighlight">
+\[f^{(n)}(a) = P^{(n)}(a) \implies f^{(n)}(a) = n!\, a_n \implies a_{n} = \frac{f^{(n)}(a)}{n!}\]</div>
+<p>The polynomial with these coefficients is called <strong>Taylor polynomial</strong> of degree <span class="math notranslate nohighlight">\(n\)</span> at <span class="math notranslate nohighlight">\(x = a\)</span>:</p>
+<div class="math notranslate nohighlight">
+\[P(x)=f(a)+f'(a)(x-a)+\frac{f''(a)}{2!}(x-a)^{2}+\cdots+\frac{f^{(n)}(a)}{n!}(x-a)^{n}.\]</div>
+<div class="admonition-example admonition">
+<p class="admonition-title">Example</p>
+<p>To find approximation of degree 3 for <span class="math notranslate nohighlight">\(f(x)=e^{x}\)</span>, at <span class="math notranslate nohighlight">\(x=0\)</span>, we have:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}f(0)=1 \\
+f'(0)=1 \\
+f''(0)=1 \\
+f'''(0)=1\end{split}\]</div>
+<p>Thus,</p>
+<div class="math notranslate nohighlight">
+\[P_{3}(x)=1+(x-0)+\frac{1}{2!}(x-0)^2+\frac{1}{3!}(x-0)^3=1+x+\frac{x^2}{2}+\frac{x^3}{6}\]</div>
+<p>See how the approximation to the function <span class="math notranslate nohighlight">\(f(x)=e^{x}\)</span>  at <span class="math notranslate nohighlight">\(x=0\)</span> gets increasingly better when more terms are added to the polynomial.</p>
+<img alt="../../_images/exp_series.gif" src="../../_images/exp_series.gif" />
+</div>
+<div class="admonition-example admonition">
+<p class="admonition-title">Example</p>
+<p>For the function <span class="math notranslate nohighlight">\(f(x)=\displaystyle\frac{1}{1-x}\ (x\neq1)\)</span> at <span class="math notranslate nohighlight">\(x=0\)</span>, we have:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}f(x)=\frac{1}{1-x} \implies f(0)=1 \\
+f'(x)=\frac{1}{(1-x)^{2}} \implies f'(0)=1 \\
+f''(x)=\frac{2}{(1-x)^{3}} \implies f''(0)=2 \\
+f'''(x)=\frac{3\cdot2}{(1-x)^{4}} \implies f'''(0)=3! \\
+f^{(n)}(x)=\frac{n!}{(1-x)^{n+1}} \implies f^{(n)}(0)= n!\end{split}\]</div>
+<p>Thus the Taylor polynomial of degree <span class="math notranslate nohighlight">\(n\)</span> will be:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{align}P(x)
+&amp;=1+1(x-0)+\frac{2}{2}(x-0)^{2}+\frac{3!}{3!}(x-0)^{3}+\cdots+\frac{n!}{n!}(x-0)^{n} \\
+&amp;=1+x+x^{2}+x^{3}+\cdots+x^{n}\end{align}\end{split}\]</div>
+</div>
+<p>For some functions, the approximation <span class="math notranslate nohighlight">\(f(x)=P_n(x)\)</span> at a given reference point <span class="math notranslate nohighlight">\(x=a\)</span> can be made increasingly better by adding more terms to <span class="math notranslate nohighlight">\(P(x)\)</span>, so that in the limit <span class="math notranslate nohighlight">\(n\rightarrow \infty\)</span> the Taylor polynomials (called <strong>Taylor series</strong> in this limit) for the functions <span class="math notranslate nohighlight">\(e^{x}\)</span>, <span class="math notranslate nohighlight">\(\mbox{sin}(x)\)</span>, <span class="math notranslate nohighlight">\(\mbox{cos}(x)\)</span> converge exactly for all values of <span class="math notranslate nohighlight">\(x\)</span>. For other functions, such as <span class="math notranslate nohighlight">\(\mbox{ln}(1+x)\)</span> and <span class="math notranslate nohighlight">\(\displaystyle\frac{1}{1-x}\)</span>, there is a restricted range of values of <span class="math notranslate nohighlight">\(x\)</span> for which the approximation can be made better by adding terms.</p>
+<p>We have the following approximations by Taylor series, with ranges in <span class="math notranslate nohighlight">\(x\)</span>, at <span class="math notranslate nohighlight">\(x = 0\)</span>:</p>
+<div class="math notranslate nohighlight">
+\[e^x=\sum_{n=0}^\infty \frac{x^n}{n!} =1+x+\frac{x^2}{2}+\frac{x^3}{3!}+\cdots,\ -\infty&lt;x&lt;\infty\]</div>
+<div class="math notranslate nohighlight">
+\[\mbox{sin}(x)=\sum_{n=0}^\infty \frac{(-1)^nx^{1+2n}}{(1+2n)!} = x-\frac{x^3}{3}+\frac{x^5}{5!}-\frac{x^7}{7!}+\cdots,\ -\infty&lt;x&lt;\infty\]</div>
+<div class="math notranslate nohighlight">
+\[\mbox{cos}(x)=\sum_{n=0}^\infty \frac{(-1)^nx^{2n}}{(2n)!} = 1-\frac{x^2}{2}+\frac{x^4}{4!}-\frac{x^6}{6!}+\cdots,\ -\infty&lt;x&lt;\infty\]</div>
+<div class="math notranslate nohighlight">
+\[\mbox{ln}(1+x)=-\sum_{n=1}^\infty \frac{(-1)^n x^n}{n} = \frac{x^2}{2!}+\frac{x^3}{3!}-\frac{x^4}{4!}+\frac{x^5}{5!}+\cdots,\ -1 &lt; x \leq 1\]</div>
+<div class="math notranslate nohighlight">
+\[\frac{1}{1-x} = \sum_{n=0}^\infty x^n = 1+x+x^2+x^3+\cdots,\ -1 \leq x &lt; 1\]</div>
+<p>Knowing the Taylor series often gives insight into the properties of a function.</p>
+</section>
+</section>
+<section id="calculating-accumulated-change">
+<h2>Calculating accumulated change<a class="headerlink" href="#calculating-accumulated-change" title="Link to this heading">#</a></h2>
+<section id="series">
+<h3>Series<a class="headerlink" href="#series" title="Link to this heading">#</a></h3>
+<p>For every sequence <span class="math notranslate nohighlight">\(\{a_n\}\)</span> of real numbers, it is possible to define another sequence <span class="math notranslate nohighlight">\(\{S_n\}\)</span> for which the elements are given by the sum of the first <span class="math notranslate nohighlight">\(n\)</span> terms of the sequence <span class="math notranslate nohighlight">\(\{a_n\}\)</span>:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{align}
+S_0 &amp;= a_0 \\
+S_1 &amp;= a_0 + a_1 \\
+S_2 &amp;= a_0 + a_1 + a_2 \\
+&amp;\vdots \\
+S_n &amp;= a_0 + a_1 + \cdots + a_{n-1} + a_n \\
+S_n &amp;= \sum_{i = 0}^n a_i \end{align}\end{split}\]</div>
+<p>The sequence <span class="math notranslate nohighlight">\(\{S_n\}\)</span> is called a <em>series</em> (or an infinite series, to emphasize the sum over increasing number of terms <span class="math notranslate nohighlight">\(\{a_n\}\)</span>).</p>
+<p>The same way as we asked about <span class="math notranslate nohighlight">\(\lim a_n\)</span>, we can ask if <span class="math notranslate nohighlight">\(\lim S_n\)</span> exists. If it exists, or, in other words, if the sum of infinite terms <span class="math notranslate nohighlight">\(\{a_n\}\)</span> is equal to a finite number, we say that the series <span class="math notranslate nohighlight">\(S_n\)</span> is convergent.</p>
+<div class="admonition-examples admonition">
+<p class="admonition-title">Examples</p>
+<p>The following expressions are example of convergent series:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{align}&amp;\sum_{n=0}^\infty \frac{x^n}{n!} \hspace{0.1cm}, \hspace{0.5cm} x \in \mathbb{R} \\
+&amp;\sum_{n=0}^\infty x^n \hspace{0.1cm}, \hspace{0.5cm} -1&lt;x&lt;1 \\
+&amp;\sum_{n=0}^\infty (-1)^n \frac{x^{2n+1}}{(2n+1)!} \hspace{0.1cm}, \hspace{0.5cm} x \in \mathbb{R}\end{align}\end{split}\]</div>
+</div>
+</section>
+<section id="calculating-areas">
+<h3>Calculating areas<a class="headerlink" href="#calculating-areas" title="Link to this heading">#</a></h3>
+<p>The idea of summing infinite quantities can be used to solve the problem of determining the area below a given curve. Let <span class="math notranslate nohighlight">\(f(x)\)</span> be a function defined on the interval <span class="math notranslate nohighlight">\(a \le x \le b\)</span> as:</p>
+<img alt="../../_images/sum_areas.png" src="../../_images/sum_areas.png" />
+<p>The area below the curve of <span class="math notranslate nohighlight">\(f(x)\)</span> can be found dividing the interval <span class="math notranslate nohighlight">\(a \le x \le b\)</span> into <span class="math notranslate nohighlight">\(n\)</span> subintervals:</p>
+<div class="math notranslate nohighlight">
+\[a=x_0&lt;x_1&lt;x_2&lt;\cdots&lt;x_{n-1}&lt;x_n=b\]</div>
+<p>where each interval defines a rectangle of height <span class="math notranslate nohighlight">\(f(x_i)\)</span> and width <span class="math notranslate nohighlight">\((x_{i+1}-x_i) = \Delta x_i = \displaystyle\frac{(b-a)}{n}\)</span>. The area below the curve will then be approximately the sum of the areas of the rectangles</p>
+<div class="math notranslate nohighlight">
+\[A \approx \sum_{i=0}^n f(x_i)\Delta x_i.\]</div>
+<p>As we increasethe number of intervals <span class="math notranslate nohighlight">\(n\)</span>, while decreasing the widths of each rectangle, the sum of the areas of the rectangles gives a better approximation of the area below the curve. The exact result is given by the limit <span class="math notranslate nohighlight">\(n \rightarrow \infty\)</span>. In this limit, the sum is represented by the following symbol:</p>
+<div class="math notranslate nohighlight">
+\[\lim_{n \to \infty} \sum_{i=0}^n f(x_i) \Delta x_i = \int_a^b f(x)dx\]</div>
+<p>If this limit exists, we say that the function <span class="math notranslate nohighlight">\(f\)</span> is <strong>integrable</strong> on the interval <span class="math notranslate nohighlight">\(a \lt x \lt b\)</span> and denote the limit by the <strong>definite integral</strong> <span class="math notranslate nohighlight">\(\int_a^b f(x)dx\)</span>.</p>
+<p>The interpretation of the integral as area can be generalized for when <span class="math notranslate nohighlight">\(f(x)\)</span> assumes negative values</p>
+<img alt="../../_images/defin_int.png" src="../../_images/defin_int.png" />
+<p>In the case of the previous figure, we will have:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{align} \int_a^b f(x)dx 
+&amp;= A_1 - A_2 + A_3 \hspace{0.1cm}, \hspace{0.5cm} (A_i \geq 0) \\
+&amp;= [\text{area above x-axis}] - [\text{area below x-axis}] \end{align}\end{split}\]</div>
+<p>Other intuitive results related to the calculation of the area are:</p>
+<div class="math notranslate nohighlight">
+\[\int_a^a f(x)dx = 0\quad \mbox{(size of the interval equal to zero)}\]</div>
+<div class="math notranslate nohighlight">
+\[\int_a^b f(x)dx = -\int_b^a f(x)dx\quad  \mbox{(orientation of the integral)}\]</div>
+<p>Furthermore, some of the properties of the integral follows directly from the properties of limits:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{align}
+\int_a^b kf(x)dx &amp;= k \int_a^b f(x)dx \\
+\int_a^b [f(x) + g(x)]dx &amp;= \int_a^b f(x)dx + \int_a^b g(x)dx \\
+\int_a^b f(x)dx &amp;= \int_a^c f(x)dx + \int_c^b f(x)dx ,\ \  (a \le c \le b) \end{align}\end{split}\]</div>
+</section>
+<section id="fundamental-theorem-of-calculus">
+<h3>Fundamental theorem of Calculus<a class="headerlink" href="#fundamental-theorem-of-calculus" title="Link to this heading">#</a></h3>
+<p>The problem of calculating integrals as limits of sums requires specific methods for each function and becomes quite impractical. Fortunately, this problem can be solved with the very important result that the area problem is intrinsically related to the tangent problem.</p>
+<p>Suppose we define a function <span class="math notranslate nohighlight">\(F(x)\)</span> that gives the area below the curve <span class="math notranslate nohighlight">\(f(u)\)</span> from <span class="math notranslate nohighlight">\(a\)</span> until a variable value <span class="math notranslate nohighlight">\(x\)</span>. By definition:</p>
+<div class="math notranslate nohighlight">
+\[F(x) = \int_a^x f(u)du\]</div>
+<img alt="../../_images/fund_calc.png" src="../../_images/fund_calc.png" />
+<p>If we consider a small increment <span class="math notranslate nohighlight">\(h\)</span> to <span class="math notranslate nohighlight">\(x\)</span>, the new area below the curve shall be given by:</p>
+<div class="math notranslate nohighlight">
+\[F(x+h) = \int_a^{x+h} f(u)du\]</div>
+<p>The difference between these two areas shall be approximately given by the area of the retangle with width <span class="math notranslate nohighlight">\(h\)</span> and height <span class="math notranslate nohighlight">\(f(x)\)</span>:</p>
+<div class="math notranslate nohighlight">
+\[F(x+h) - F(x) \approx hf(x) \implies \displaystyle\frac{F(x+h) - F(x)}{h} \approx f(x)\]</div>
+<p>But see that in the limit where the increment <span class="math notranslate nohighlight">\(h\)</span> gets very small, the approximation gets exactly equal to the increment in area, and we obtain the definition of the derivative of <span class="math notranslate nohighlight">\(F(x)\)</span>:</p>
+<div class="math notranslate nohighlight">
+\[\lim_{h\rightarrow 0} \displaystyle\frac{F(x+h) - F(x)}{h} = \displaystyle\frac{dF}{dx} = f(x).\]</div>
+<p>This results is known as the <em>Fundamental theorem of Calculus</em>, and it say that the instantaneous rate of change of the area below a given curve <span class="math notranslate nohighlight">\(f\)</span> at a point <span class="math notranslate nohighlight">\(x\)</span> is given by the value of the function at that point. In other words, the area <span class="math notranslate nohighlight">\(F(x)\)</span> is a function that, if differentiated, gives the function <span class="math notranslate nohighlight">\(f(x)\)</span>.</p>
+<p>The function <span class="math notranslate nohighlight">\(F(x)\)</span> is then called an <strong>antiderivative</strong> of <span class="math notranslate nohighlight">\(f(x)\)</span>. In general, there is a <em>family</em> of antiderivates of <span class="math notranslate nohighlight">\(f(x)\)</span>, since <span class="math notranslate nohighlight">\(\displaystyle\frac{d(F(x) + C)}{dx}=\displaystyle\frac{dF}{dx}=f(x)\)</span>, independent of the value of the constant <span class="math notranslate nohighlight">\(C\)</span>. We therefore call <strong>indefinite integral</strong>:</p>
+<div class="math notranslate nohighlight">
+\[\int f(x)dx = F(x) + C\]</div>
+<p>It also directly follows that</p>
+<div class="math notranslate nohighlight">
+\[\int_b^c f(x)dx = F(c) - F(b)\]</div>
+<p>Thus the problem of calculating the area below the curve <span class="math notranslate nohighlight">\(f(x)\)</span> from <span class="math notranslate nohighlight">\(b\)</span> to <span class="math notranslate nohighlight">\(c\)</span>, which, by definition, is given by the infinite sum of terms <span class="math notranslate nohighlight">\(f(x_i)\Delta x_i\)</span> (with <span class="math notranslate nohighlight">\(\Delta x_i \rightarrow 0\)</span>), can be reduced to obtaining the antiderivative of <span class="math notranslate nohighlight">\(f\)</span>, <span class="math notranslate nohighlight">\(F(x)\)</span>, and subtracting the values of this function at points <span class="math notranslate nohighlight">\(c\)</span> and <span class="math notranslate nohighlight">\(b\)</span>. The main problem then becomes how to calculate the antiderivates of a given function <span class="math notranslate nohighlight">\(f(x)\)</span>, for which different methods can be used.</p>
+</section>
+<section id="calculating-integrals">
+<h3>Calculating integrals<a class="headerlink" href="#calculating-integrals" title="Link to this heading">#</a></h3>
+<p>The simplest integrals can be calculated by inverting the differentiation of fundamental functions, as</p>
+<div class="math notranslate nohighlight">
+\[\int x^n \,dx = \frac{x^{n+1}}{n+1} + C\]</div>
+<div class="math notranslate nohighlight">
+\[\int e^x \,dx = e^x + C\]</div>
+<div class="math notranslate nohighlight">
+\[\int \frac{1}{x}\,dx = \ln |x| + C\]</div>
+<div class="math notranslate nohighlight">
+\[\int \mbox{cos}(x)\, dx = \mbox{sin}(x) + C\]</div>
+<div class="math notranslate nohighlight">
+\[\int \mbox{sin}(x)\, dx = -\mbox{cos}(x) + C\]</div>
+<p>This are obtained simply because we know that the derivative of the functions on the righ-hand sides of the equations give exactly the function that is being integrated. When the integrals are not so simple, we must apply other methods to turn them into integrals we know.</p>
+<section id="integration-by-substitution">
+<h4>Integration by substitution<a class="headerlink" href="#integration-by-substitution" title="Link to this heading">#</a></h4>
+<p>This method consists in finding an appropriate substitution of the variable of integration so that we end up with a simpler integral to calculate.</p>
+<div class="admonition-examples admonition">
+<p class="admonition-title">Examples</p>
+<ul>
+<li><p><span class="math notranslate nohighlight">\(\int\frac{1}{\sqrt{x+4}}dx\)</span></p>
+<p>Note that if <span class="math notranslate nohighlight">\(u=x+4 \implies \displaystyle\frac{du}{dx}=1 \implies du=dx\)</span>. Thus, the integral can be given by:</p>
+<div class="math notranslate nohighlight">
+\[\int u^{-\frac{1}{2}}du = \frac{1}{\left(-\frac{1}{2}+1\right)} \cdot u^{\frac{1}{2}} + C = 2\sqrt{x+4} + C.\]</div>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>Note that three mathematicians die everytime we operate with <span class="math notranslate nohighlight">\(du\)</span> and <span class="math notranslate nohighlight">\(dx\)</span> as if they were numbers on a fraction. Formally, the correct substitution on the integral would be given by a function <span class="math notranslate nohighlight">\(u=u(x)\)</span>, so that <span class="math notranslate nohighlight">\( \int f(u(x)) u'(x)dx = \int f(u)du\)</span>. But we can procede with the heresy above as long as we have this in mind.</p>
+</div>
+  <br />
+</li>
+<li><p><span class="math notranslate nohighlight">\(\int xe^{(x^2-1)}dx\)</span></p>
+<p>See that <span class="math notranslate nohighlight">\(u=x^2-1 \implies \displaystyle\frac{du}{dx}=2x \implies  dx=\displaystyle\frac{du}{2x}\)</span>. Then:</p>
+<div class="math notranslate nohighlight">
+\[\int \frac{1}{2}e^{u}du = \frac{1}{2}e^u + C = \frac{e^{(x^2-1)}}{2} + C \]</div>
+</li>
+</ul>
+</div>
+</section>
+<section id="integration-by-parts">
+<h4>Integration by parts<a class="headerlink" href="#integration-by-parts" title="Link to this heading">#</a></h4>
+<p>When integrating by parts, we have to find an appropriate association <span class="math notranslate nohighlight">\((u=u(x), v=v(x))\)</span>, so that:</p>
+<div class="math notranslate nohighlight">
+\[\int uv'dx=uv-\int u'vdx\]</div>
+<div class="admonition-example admonition">
+<p class="admonition-title">Example</p>
+<ul>
+<li><p><span class="math notranslate nohighlight">\(\int x\ln x\, dx \)</span></p>
+<p>Note that</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{align}
+    u=\ln x \implies u' = \displaystyle\frac{1}{x} \\
+    v'=x \implies v=\displaystyle\frac{x^2}{2} \end{align}\end{split}\]</div>
+<p>Thus:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{align}\int x\ln x\, dx 
+&amp;= \frac{x^2}{2}\ln x - \int \frac{x^2}{2} \cdot \frac{1}{x}dx \\
+&amp;= \frac{x^2 \ln x}{2} - \frac{1}{2}\int xdx \\
+&amp;= \frac{x^2 \ln x}{2} - \frac{1}{4}x^2 + C \\
+&amp;= \frac{1}{4} x^2 (2\ln x - 1) + C \end{align}\end{split}\]</div>
+</li>
+<li><p><span class="math notranslate nohighlight">\(\int xe^x \,dx\)</span></p>
+<p>Note that</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{align}
+    u=x \implies u'=1 \\
+    v'=e^x \implies v=e^x \end{align}\end{split}\]</div>
+<p>Thus:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{align}\int xe^xdx  
+    &amp;= xe^x - \int e^x dx \hspace{0.1cm} \\
+    &amp;= xe^x-e^x + C \\
+    &amp;= e^x (x-1) + C \end{align}\end{split}\]</div>
+</li>
+</ul>
+</div>
+<div class="admonition tip">
+<p class="admonition-title">Tip</p>
+<p>For complicated integrals, computer algebra programs like Mathematica or Sympy can be helpful. An especially user-friendly option is the website WolframAlpha.</p>
+</div>
+</section>
+<section id="integration-by-partial-fractions">
+<h4>Integration by partial fractions<a class="headerlink" href="#integration-by-partial-fractions" title="Link to this heading">#</a></h4>
+<p>If the integral has the form</p>
+<div class="math notranslate nohighlight">
+\[\int \frac{P(x)}{Q(x)}dx\]</div>
+<p>where the degree of <span class="math notranslate nohighlight">\(P(x)\)</span> is smaller than the degree of <span class="math notranslate nohighlight">\(Q(x)\)</span>, and <span class="math notranslate nohighlight">\(Q(x)=a(x-x_1)(x-x_2)\cdots(x-x_n)\)</span>, we can write</p>
+<div class="math notranslate nohighlight">
+\[\frac{P(x)}{Q(x)} = \frac{1}{a}\left[\frac{A_1}{x-x_1}+\frac{A_2}{x-x_2}+\cdots+\frac{A_n}{x-x_n}\right]\]</div>
+<p>with <span class="math notranslate nohighlight">\(\{A_n\}\)</span> to be determined.</p>
+<div class="admonition-example admonition">
+<p class="admonition-title">Example</p>
+<ul>
+<li><p><span class="math notranslate nohighlight">\(\int \frac{1}{x(x-1)}\, dx\)</span></p>
+<p>First, note that <span class="math notranslate nohighlight">\(\displaystyle\frac{1}{x(x-1)} =  \displaystyle\frac{A}{x} + \displaystyle\frac{B}{(x-1)} \implies 1 = A(x-1) + Bx \implies 1 = x(A+B) - A\)</span>. By comparing the coeffients of the terms in <span class="math notranslate nohighlight">\(x\)</span> and the terms independent of <span class="math notranslate nohighlight">\(x\)</span>, we obtain:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{cases}
+    A + B &amp;= 0\\
+       -A &amp;= 1
+    \end{cases} \implies  
+    \begin{cases}
+    A &amp;= -1 \\
+    B &amp;= 1\end{cases}\end{split}\]</div>
+<p>Thus:</p>
+<div class="math notranslate nohighlight">
+\[\frac{1}{x(x-1)} = -\frac{1}{x} + \frac{1}{x-1}\]</div>
+<p>Then we will have for the integral:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{align} \int\frac{1}{x(x-1)}dx  
+    &amp;= \int\left(-\frac{1}{x}+\frac{1}{x-1}\right)dx \\
+    &amp;= -\int \frac{1}{x}dx + \int\frac{1}{x-1}dx \\
+    &amp;= - \ln|x| + \ln|x-1| + C \\
+    &amp;= \ln \left|\frac{x-1}{x}\right| + C\end{align}\end{split}\]</div>
+</li>
+</ul>
+</div>
+</section>
+</section>
+</section>
+<section id="exercise-set-1">
+<h2>Exercise Set 1<a class="headerlink" href="#exercise-set-1" title="Link to this heading">#</a></h2>
+<ol class="arabic">
+<li><p>An autocatalytic reaction uses its resulting product for the formation of a new product, as in the reaction</p>
+<div class="math notranslate nohighlight">
+\[A + X \rightarrow X\]</div>
+<p>If we assume that this reaction occurs in a closed vessel, then the reaction rate is given by</p>
+<div class="math notranslate nohighlight">
+\[
+       R(x) = kx(a - x)
+    \]</div>
+<p>for <span class="math notranslate nohighlight">\(0 \le x \le a\)</span>, where <span class="math notranslate nohighlight">\(a\)</span> is the initial concentration of <span class="math notranslate nohighlight">\(A\)</span> and <span class="math notranslate nohighlight">\(x\)</span> is the concentration of <span class="math notranslate nohighlight">\(X\)</span>.</p>
+<br/>
+<p>(a) Show that <span class="math notranslate nohighlight">\(R(x)\)</span> is a polynomial and determine its degree.
+<br/>
+(b) Without using any graphic visualiser, graph <span class="math notranslate nohighlight">\(R(x)\)</span> for <span class="math notranslate nohighlight">\(k = 2\)</span> and <span class="math notranslate nohighlight">\(a = 6\)</span>. Find the value of <span class="math notranslate nohighlight">\(x\)</span> at which the reaction rate is maximal.</p>
+<br/>
+</li>
+<li><p>Find the maximum/minimum value of the following polynomials</p>
+ <br/>
+<p>(a) <span class="math notranslate nohighlight">\(p(x) = x^2 - 2x - 3\)</span></p>
+<p>(b) <span class="math notranslate nohighlight">\(p(x) = -x^2 - 6x + 8\)</span></p>
+<p>(<em>Hint:</em> Calculate the roots and consider the symmetry of the parabola)</p>
+ <br/>
+</li>
+<li><p>There is a function that is frequently used to describe the per capita growth rate of organisms when the rate depends on the concentration of some nutrient and becomes saturated for large enough nutrient concentrations. If we denote the concentration of the nutrient by <span class="math notranslate nohighlight">\(N\)</span>, then the per capita growth rate <span class="math notranslate nohighlight">\(r(N)\)</span> is given by the <em>Monod growth function</em></p>
+<div class="math notranslate nohighlight">
+\[
+        r(N)=\frac{aN}{k+N},\ \ \ N \ge 0
+    \]</div>
+<p>where <span class="math notranslate nohighlight">\(a\)</span> and <span class="math notranslate nohighlight">\(k\)</span> are positive constants.</p>
+ <br/>
+<p>(a) Use Python or R to investigate the Monod growth function. Graph <span class="math notranslate nohighlight">\(r(N)\)</span> for (i) <span class="math notranslate nohighlight">\(a = 5\)</span> and <span class="math notranslate nohighlight">\(k = 1\)</span>, (ii) <span class="math notranslate nohighlight">\(a = 5\)</span> and <span class="math notranslate nohighlight">\(k = 3\)</span>, and (iii) <span class="math notranslate nohighlight">\(a = 8\)</span> and k = 1. Place all three graphs in the same coordinate system.</p>
+<p>(b) On the basis of the graphs in (a), describe in words what happens when you change <span class="math notranslate nohighlight">\(a\)</span>.</p>
+<p>(c) On the basis of the graphs in (a), describe in words what happens when you change <span class="math notranslate nohighlight">\(k\)</span>.</p>
+ <br/>
+</li>
+<li><p>The function</p>
+<div class="math notranslate nohighlight">
+\[
+        f(x) = \frac{x^n}{b^n + x^n},\ \ \ x \ge 0
+    \]</div>
+<p>where <span class="math notranslate nohighlight">\(n\)</span> is a positive integer and <span class="math notranslate nohighlight">\(b\)</span> is a positive real number, is used in biochemistry to model reaction rates as a function of the concentration of some reactants</p>
+ <br/>
+<p>(a) Use Python or R to graph <span class="math notranslate nohighlight">\(f(x)\)</span> for <span class="math notranslate nohighlight">\(n = 1, 2,\ \mbox{and}\ 3\)</span> in the same coordinate system when <span class="math notranslate nohighlight">\(b = 2\)</span>.</p>
+<p>(b) Where do the three graphs in (a) intersect?</p>
+<p>(c) What happens to <span class="math notranslate nohighlight">\(f(x)\)</span> as <span class="math notranslate nohighlight">\(x\)</span> gets larger?</p>
+<p>(d) For an arbitrary positive value of <span class="math notranslate nohighlight">\(b\)</span>, show that <span class="math notranslate nohighlight">\(f (b) = 1/2\)</span>. On the basis of this demonstration and your answer in C. ,  explain why <span class="math notranslate nohighlight">\(b\)</span> is called the half-saturation constant.</p>
+ <br/>
+</li>
+<li><p>The file <a class="reference external" href="https://imperialcollegelondon.box.com/s/940y521dq1owarx8pt4d4mu2d0f2cz85">words_moby_dick.csv</a> contains data on the cumulative distribution of the number of times specific words occur in the text of the novel <em>Moby Dick</em>, by Herman Melville. Assume that this distribution is a power law of the form</p>
+<div class="math notranslate nohighlight">
+\[
+        y = x^D
+    \]</div>
+<p>(a) Using the provided data, and R/Python, estimate the exponent of this power law.
+(<em>Hint:</em> If you plot this data in log-log scale, what should correspond to the exponent <span class="math notranslate nohighlight">\(D\)</span>?)</p>
+<p>(b) Can you think of a better way to estimate D?</p>
+ <br/>
+</li>
+<li><p>Assume that a population size at time <span class="math notranslate nohighlight">\(t\)</span> is <span class="math notranslate nohighlight">\(N(t)\)</span> and that</p>
+<div class="math notranslate nohighlight">
+\[
+        N(t) = 40 \cdot 2^t,\ \ \ t \ge 0
+    \]</div>
+<p>(a) Find the population size at time t = 0.</p>
+<p>(b) Show that</p>
+<div class="math notranslate nohighlight">
+\[
+        N(t) = 40\, \mbox{e}^{t\, ln 2},\ \ \ t \ge 0
+    \]</div>
+<p>(c) How long will it take until the population size reaches 1000?</p>
+<p>(<em>Hint</em>: Find <span class="math notranslate nohighlight">\(t\)</span> so that <span class="math notranslate nohighlight">\(N(t) = 1000\)</span>.)</p>
+<br/>
+</li>
+<li><p>(<em>Adapted from Moss, 1980</em>) Hall (1964) investigated the change in population size of the zooplankton species <em>Daphnia galeata mendota</em> in Base Line Lake, Michigan. The population size <span class="math notranslate nohighlight">\(N(t)\)</span> at time <span class="math notranslate nohighlight">\(t\)</span> was modeled by the equation</p>
+<div class="math notranslate nohighlight">
+\[ 
+        N(t) = N_0\mbox{e}^{rt}
+    \]</div>
+<p>where <span class="math notranslate nohighlight">\(N_0\)</span> denotes the population size at time <span class="math notranslate nohighlight">\(0\)</span>. The constant <span class="math notranslate nohighlight">\(r\)</span> is called the <strong>intrinsic rate of growth</strong>.</p>
+ <br/>
+<p>(a) Plot <span class="math notranslate nohighlight">\(N(t)\)</span> as a function of <span class="math notranslate nohighlight">\(t\)</span> if <span class="math notranslate nohighlight">\(N_0 = 100\)</span> and <span class="math notranslate nohighlight">\(r = 2\)</span>. Compare your graph against the graph of <span class="math notranslate nohighlight">\(N(t)\)</span> when <span class="math notranslate nohighlight">\(N_0 = 100\)</span> and <span class="math notranslate nohighlight">\(r = 3\)</span>. Which population grows faster?</p>
+<p>(b) The constant <span class="math notranslate nohighlight">\(r\)</span> is an important quantity because it describes how quickly the population changes. Suppose that you determine the size of the population at the beginning and at the end of a period of length <span class="math notranslate nohighlight">\(1\)</span>, and you find that at the beginning there were <span class="math notranslate nohighlight">\(200\)</span> individuals and after one unit of time there were <span class="math notranslate nohighlight">\(250\)</span> individuals. Determine <span class="math notranslate nohighlight">\(r\)</span>. (<em>Hint</em>: Consider the ratio <span class="math notranslate nohighlight">\(N(t+1)/N(t)\)</span>.)</p>
+ <br/>
+</li>
+<li><p>Fish are indeterminate growers; that is, they grow throughout their lifetime. The growth of fish can be described by the von Bertalanffy function</p>
+<div class="math notranslate nohighlight">
+\[
+        L(x) = L_{\infty}(1-\textrm{e}^{-kx} )
+    \]</div>
+<p>for <span class="math notranslate nohighlight">\(x \ge 0\)</span>, where <span class="math notranslate nohighlight">\(L(x)\)</span> is the length of the fish at age <span class="math notranslate nohighlight">\(x\)</span> and <span class="math notranslate nohighlight">\(k\)</span> and <span class="math notranslate nohighlight">\(L_{\infty}\)</span> are positive</p>
+ <br/>
+<p>(a) Using Python or R, graph <span class="math notranslate nohighlight">\(L(x)\)</span> for <span class="math notranslate nohighlight">\(L_{\infty} = 20\)</span>, for (i) <span class="math notranslate nohighlight">\(k = 1\)</span> and (ii) <span class="math notranslate nohighlight">\(k = 0.1\)</span>.</p>
+<p>(b) For <span class="math notranslate nohighlight">\(k = 1\)</span>, find <span class="math notranslate nohighlight">\(x\)</span> so that the length is 90% of <span class="math notranslate nohighlight">\(L_{\infty}\)</span>. Repeat for 99% of <span class="math notranslate nohighlight">\(L_{\infty}\)</span>. Can the fish ever attain length <span class="math notranslate nohighlight">\(L_{\infty}\)</span>? Interpret the meaning of <span class="math notranslate nohighlight">\(L_{\infty}\)</span>.</p>
+<p>(c) Compare the graphs obtained in A. Which growth curve reaches 90% of <span class="math notranslate nohighlight">\(L_{\infty}\)</span> faster? Can you explain what happens to the curve of <span class="math notranslate nohighlight">\(L(x)\)</span> when you vary <span class="math notranslate nohighlight">\(k\)</span> (for fixed <span class="math notranslate nohighlight">\(L_{\infty}\)</span>)?</p>
+ <br/>
+</li>
+<li><p>For the following pairs of functions, plot the ratio (quotient) between the two using R or Python. Based on the behaviour of the ratio when <span class="math notranslate nohighlight">\(x \rightarrow \infty\)</span> (very large values of <span class="math notranslate nohighlight">\(x\)</span>), how does each of these functions compare to the other in the velocity that they grow?</p>
+ <br/>  
+<p>(a) <span class="math notranslate nohighlight">\( f(x) = e^x \quad \text{ and } \quad g(x) = x^2 \)</span></p>
+<p>(b) <span class="math notranslate nohighlight">\( f(x) = e^x \quad \text{ and } \quad g(x) = x^{20} \)</span></p>
+<p>(c) <span class="math notranslate nohighlight">\( f(x) = x \quad \text{ and } \quad g(x) = \log(x) \)</span></p>
+ <br/>
+</li>
+<li><p>A community measure that takes both species abundance and species richness into account is the Shannon diversity index <span class="math notranslate nohighlight">\(H\)</span>. To calculate <span class="math notranslate nohighlight">\(H\)</span>, the proportion <span class="math notranslate nohighlight">\(p_i\)</span> of species <span class="math notranslate nohighlight">\(i\)</span> in the community is used. Assume that the community consists of <span class="math notranslate nohighlight">\(S\)</span> species. Then</p>
+<div class="math notranslate nohighlight">
+\[
+        H = -\sum_{i=1}^S p_i\, \textrm{ln}\, p_i = -( p_1 \textrm{ln}\, p_1 + p_2 \textrm{ln}\, p_2 + \cdots + p_S \textrm{ln}\, p_S )
+    \]</div>
+<br/>
+<p>(a) Assume that <span class="math notranslate nohighlight">\(S = 5\)</span> and that all species are equally abundant; that is, <span class="math notranslate nohighlight">\(p_1 = p_2 = \cdots = p_5\)</span>. Compute <span class="math notranslate nohighlight">\(H\)</span>.</p>
+<p>(b) Assume that <span class="math notranslate nohighlight">\(S = 10\)</span> and that all species are equally abundant; that is, <span class="math notranslate nohighlight">\(p_1 = p_2 = \cdots = p_{10}\)</span>. Compute <span class="math notranslate nohighlight">\(H\)</span>.</p>
+<p>(c) A measure of equitability (or evenness) of the species distribution can be measured by dividing the diversity index <span class="math notranslate nohighlight">\(H\)</span> by <span class="math notranslate nohighlight">\(\textrm{ln}\, S\)</span>. Compute <span class="math notranslate nohighlight">\(H/\textrm{ln}\, S\)</span> for <span class="math notranslate nohighlight">\(S = 5\)</span> and <span class="math notranslate nohighlight">\(S = 10\)</span>.</p>
+</li>
+</ol>
+</section>
+<section id="exercise-set-2">
+<h2>Exercise Set 2<a class="headerlink" href="#exercise-set-2" title="Link to this heading">#</a></h2>
+<ol class="arabic">
+<li><p>Solve the following systems of linear equations
+<br/></p>
+<div class="math notranslate nohighlight">
+\[\begin{split}
+        \mbox{(a)}  \ 
+        \begin{cases}
+            5x - y + 2z = 6 \\
+            x + 2y - z = -1 \\
+            3x + 2y - 2z = 1\\
+        \end{cases}
+    \end{split}\]</div>
+ <br/>
+<div class="math notranslate nohighlight">
+\[\begin{split}
+        \mbox{(b)}  \ 
+        \begin{cases}
+            2x - y + 3z = 3 \\
+            2x + y + 4z = 4 \\
+            2x - 3y + 2z = 2\\
+        \end{cases}
+    \end{split}\]</div>
+ <br/>
+</li>
+<li><p>Three different species of insects are reared together in a laboratory cage. They are supplied with two different types of food each day. Each individual of species 1 consumes 3 units of food <span class="math notranslate nohighlight">\(A\)</span> and 5 units of food <span class="math notranslate nohighlight">\(B\)</span>, each individual of species 2 consumes 2 units of food <span class="math notranslate nohighlight">\(A\)</span> and 3 units of food <span class="math notranslate nohighlight">\(B\)</span>, and individual of species 3 consumes 1 unit of food <span class="math notranslate nohighlight">\(A\)</span> and 2 units of food <span class="math notranslate nohighlight">\(B\)</span>. Each day, 500 units of food <span class="math notranslate nohighlight">\(A\)</span> and 900 units of food <span class="math notranslate nohighlight">\(B\)</span> are supplied. How many individuals of each species can be reared together? Is there more than one solution? What happens if we add 550 units of a third type of food, called <span class="math notranslate nohighlight">\(C\)</span>, and each individual of species 1 consumes 2 units of food <span class="math notranslate nohighlight">\(C\)</span>, each individual of species 2 consumes 4 units of food <span class="math notranslate nohighlight">\(C\)</span>, and each individual of species 3 consumes 1 unit of food <span class="math notranslate nohighlight">\(C\)</span>?</p>
+ <br/>
+</li>
+<li><p>Networks are abstractions of real systems used to identify and study patterns of connectivity between the individual units that compose those systems. Networks (often referred to as <em>graphs</em>) are mathematically defined as a set of <em>nodes</em>, or vertices, (representing the individual units) and a set of <em>links</em>, or connections, between these nodes, which account for the interaction patterns within the studied systems. In Ecology, graphs have been widely used in studies investigating food-webs, plant-pollinator interactions, and disease transmission in populations.</p>
+ <br />
+<p>If the nodes of a network of size <span class="math notranslate nohighlight">\(N\)</span> are indexed with positive integers such as <span class="math notranslate nohighlight">\(i = 1, 2, \ldots, N\)</span>, an useful method for representing the structure of a network is to construct a matrix <span class="math notranslate nohighlight">\(A\)</span> for which the elements <span class="math notranslate nohighlight">\(a_{ij}\)</span> are given by:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}a_{ij} = \begin{cases}1,\ \mbox{if nodes}\ i \ \mbox{and}\ j \ \mbox{are linked} \\ 0,\ \mbox{otherwise}  \end{cases}\end{split}\]</div>
+<p>Matrix <span class="math notranslate nohighlight">\(A\)</span> is called the <em>adjacency matrix</em> of the network. One example of a network with <span class="math notranslate nohighlight">\(6\)</span> nodes is shown below.</p>
+ <br/>  
+<img alt="../../_images/graph_example.png" src="../../_images/graph_example.png" />
+<p>(a) Based on the graphical representation of this network, write down its adjacency matrix.</p>
+ <br/>
+<p>(b) By the definition of the adjacency matrix, the sum of the matrix elements on row <span class="math notranslate nohighlight">\(i\)</span>, <span class="math notranslate nohighlight">\(k_i = \sum_{j=1}^N a_{ij}\)</span>, corresponds to the number of links of node <span class="math notranslate nohighlight">\(i\)</span>. The quantity <span class="math notranslate nohighlight">\(k_i\)</span> is called the <em>degree</em> of node <span class="math notranslate nohighlight">\(i\)</span>. Calculate the corresponding degrees for each node on the previous network.</p>
+ <br/>
+<p>(c) The Laplacian matrix is a mathematical object that can be used to find many useful properties of a network. By definition, <span class="math notranslate nohighlight">\(L\)</span> is constructed as</p>
+ <br/>  
+<div class="math notranslate nohighlight">
+\[ L = D - A \]</div>
+ <br/>  
+<p>Where <span class="math notranslate nohighlight">\(A\)</span> is the adjacency matrix, and <span class="math notranslate nohighlight">\(D\)</span> is the degree matrix. The degree matrix <span class="math notranslate nohighlight">\(D\)</span> is a diagonal matrix which contains information about the degree of each node. With your answer for itens (a) and (b), calculate <span class="math notranslate nohighlight">\(L\)</span>.</p>
+ <br/>
+<p>(d) Use R/Python to numerically calculate the eigenvalues of the Laplacian matrix.</p>
+ <br/>
+</li>
+<li><p>Write each system in matrix form
+<br/></p>
+<div class="math notranslate nohighlight">
+\[\begin{split}
+        \mbox{(a)}  \ 
+        \begin{cases}
+            2x_2-x_1 = x_3 \\
+            4x_1 + x_3 = 7x_2 \\
+            x_2 - x_1 = x_3\\
+        \end{cases}
+    \end{split}\]</div>
+ <br/>
+<div class="math notranslate nohighlight">
+\[\begin{split}
+        \mbox{(b)}  \ 
+        \begin{cases}
+            2x_1 - 3x_2 = 4 \\
+            -x_1 + x_2 = 3 \\
+            3x_1 = 4\\
+        \end{cases}
+    \end{split}\]</div>
+ <br/>
+</li>
+<li><p>Suppose that</p>
+ <br/>
+<div class="math notranslate nohighlight">
+\[\begin{split}A = \begin{pmatrix}
+            2 &amp; 4 \\
+            3 &amp; 6 
+            \end{pmatrix}
+        \ \ \ \text{,}\ \ \
+        X = \begin{pmatrix}
+            x \\
+            y 
+            \end{pmatrix} \ \ \ \text{and}\ \ \
+            B = \begin{pmatrix}
+            b_1 \\
+            b_2 \\
+            \end{pmatrix}\end{split}\]</div>
+<p>(a) Write <span class="math notranslate nohighlight">\(AX = B\)</span> as a system of linear equation.</p>
+ <br/>
+<p>(b) Show that if</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}
+        B = \begin{pmatrix}
+            3 \\
+            \frac92 \\
+            \end{pmatrix}
+    \end{split}\]</div>
+<p>then <span class="math notranslate nohighlight">\(AX = B\)</span> has infinitely many solutions. Graph the two straight lines associated with the corresponding system of linear equations, and explain why the system has infinitely many solutions.</p>
+ <br/>
+<p>(c) Find a column vector</p>
+ <br/>
+<div class="math notranslate nohighlight">
+\[\begin{split}B = \begin{pmatrix}
+            b_1 \\
+            b_2 \\
+            \end{pmatrix}\end{split}\]</div>
+<p>so that <span class="math notranslate nohighlight">\(AX = B\)</span> has no solutions.</p>
+ <br/>
+</li>
+<li><p>Assume that a population is divided into three age classes and that 80% of the individuals age 0 and 10% of the individuals age 1 survive until the end of the next breeding season. Assume further that individuals age 1 have an average of 1.6 offspring and individuals age 2 have an average of 3.9 offspring. If, at time 0, the population consists of 1000 individuals age 0, 100 individuals age 1, and 20 individuals age 2, find the projection matrix and the age distribution at time 3.</p>
+ <br/>
+</li>
+<li><p>Consider the following projection matrix representing a population with five age classes:</p>
+ <br/>  
+<div class="math notranslate nohighlight">
+\[\begin{split}
+        P = \begin{pmatrix}
+        0 &amp; 5 &amp; 3 &amp; 2 &amp; 1 \\
+        0.9 &amp; 0 &amp; 0 &amp; 0 &amp; 0 \\
+        0 &amp; 0.3 &amp; 0 &amp; 0 &amp; 0 \\
+        0 &amp; 0 &amp; 0.1 &amp; 0 &amp; 0 \\
+        0 &amp; 0 &amp; 0 &amp; 0.05 &amp; 0 \\
+    \end{pmatrix}
+    \end{split}\]</div>
+ <br/>
+<p>(a) Making use of Python or R code, begin with a population distributed as <span class="math notranslate nohighlight">\((0,0,0,0,10)\)</span> individuals at time <span class="math notranslate nohighlight">\(t=0\)</span>, project the population at time <span class="math notranslate nohighlight">\(t=20\)</span>, and plot the total number over time. Plot the logarithm of the total population over time.</p>
+ <br/>
+<p>(b) Repeat the above but begin with <span class="math notranslate nohighlight">\((80,16,5,1,1)\)</span> individuals. How do these results compare to the results in (a) ?</p>
+ <br/>
+</li>
+<li><p>In the problems below, find the eigenvalues <span class="math notranslate nohighlight">\(\lambda_1\)</span> and <span class="math notranslate nohighlight">\(\lambda_2\)</span> and corresponding eigenvectors <span class="math notranslate nohighlight">\(\mathbf{v_1}\)</span> and <span class="math notranslate nohighlight">\(\mathbf{v_2}\)</span> for each matrix <span class="math notranslate nohighlight">\(B\)</span>. Determine the equations of the lines through the origin in the direction of the eigenvectors <span class="math notranslate nohighlight">\(\mathbf{v_1}\)</span> and <span class="math notranslate nohighlight">\(\mathbf{v_2}\)</span>, and graph the lines together with the eigenvectors <span class="math notranslate nohighlight">\(\mathbf{v_1}\)</span> and <span class="math notranslate nohighlight">\(\mathbf{v_2}\)</span> and the vectors <span class="math notranslate nohighlight">\(A\mathbf{v_1}\)</span> and <span class="math notranslate nohighlight">\(A\mathbf{v_2}\)</span>.</p>
+ <br/>
+<div class="math notranslate nohighlight">
+\[\begin{split}
+    \text{(a)} \ \ B = \begin{pmatrix}
+    2 &amp; 3\\
+    0 &amp; -1
+    \end{pmatrix}
+    \end{split}\]</div>
+ <br/>
+<div class="math notranslate nohighlight">
+\[\begin{split}
+    \text{(b)} \ \ B = \begin{pmatrix}
+    3 &amp; 6\\
+    -1 &amp; -4
+    \end{pmatrix}  
+    \end{split}\]</div>
+ <br/>
+</li>
+<li><p>Suppse that</p>
+ <br/>
+<div class="math notranslate nohighlight">
+\[\begin{split}
+    P = \begin{pmatrix}
+    0.5 &amp; 2.3\\
+    a &amp; 0
+    \end{pmatrix}
+    \end{split}\]</div>
+ <br/>
+<p>is the projection matrix of a population with two age classes. For which values of <span class="math notranslate nohighlight">\(a\)</span> does this population grow?</p>
+ <br/>
+</li>
+<li><p>Suppose that</p>
+<br/>
+<div class="math notranslate nohighlight">
+\[\begin{split}
+    P = \begin{pmatrix}
+    0.5 &amp; 2.0\\
+    0.1 &amp; 0
+    \end{pmatrix}
+    \end{split}\]</div>
+<br/>
+<p>is the projection matrix of a population with two age classes.</p>
+<br/>
+<p>(a) If you were to manage this population, would you need to be concerned about its long-term survival?</p>
+<br/>
+<p>(b) Suppose that you can improve either the fecundity or the survival of the zero-year-olds, but due to physiological and environmental constraints, the fecundity of zero-year-olds will not exceed 1.5 and the survival of zero-year-olds will not exceed 0.4. Investigate how the growth rate of the population is affected by changing either the survival or the fecundity of zero-year-olds, or both. What would be the maximum achievable growth rate?</p>
+<br/>
+<p>(c) In real situations, what other factors might you need to consider when you decide on management strategies?</p>
+</li>
+</ol>
+</section>
+<section id="exercise-set-3">
+<h2>Exercise Set 3<a class="headerlink" href="#exercise-set-3" title="Link to this heading">#</a></h2>
+<ol class="arabic">
+<li><p><strong>Island Model</strong> Assume that a species lives in a habitat that consists of many islands close to a mainland. The species occupies both the mainland and the islands, but, although it is present on the mainland at all times, it frequently goes extinct on the islands. Islands can be recolonized by migrants from the mainland. The following model keeps track of the fraction of islands occupied: Denote the fraction of islands occupied at time <span class="math notranslate nohighlight">\(t\)</span> by <span class="math notranslate nohighlight">\(p(t)\)</span>. Assume that each island experiences a constant risk of extinction and that vacant islands (the fraction <span class="math notranslate nohighlight">\(1-p\)</span>) are colonized from the mainland at a constant rate. Then</p>
+<div class="math notranslate nohighlight">
+\[
+        \frac{dp}{dt}=c(1-p)-ep
+    \]</div>
+<p>where <span class="math notranslate nohighlight">\(c\)</span> and <span class="math notranslate nohighlight">\(e\)</span> are positive constants.</p>
+ <br/>
+<p>(a) The gain from colonization is <span class="math notranslate nohighlight">\(f (p) = c(1 - p)\)</span> and the loss from extinction is <span class="math notranslate nohighlight">\(g(p) = ep\)</span>. Graph (without graph visualisers) <span class="math notranslate nohighlight">\(f(p)\)</span> and <span class="math notranslate nohighlight">\(g(p)\)</span> for <span class="math notranslate nohighlight">\(0 \le p \le 1\)</span> in the same coordinate system. Explain why the two graphs intersect whenever <span class="math notranslate nohighlight">\(e\)</span> and <span class="math notranslate nohighlight">\(c\)</span> are both positive. Compute the point of intersection and interpret its biological meaning.</p>
+  <br/>
+<p>(b) The parameter <span class="math notranslate nohighlight">\(c\)</span> measures how quickly a vacant island becomes colonized from the mainland. The closer the islands, the larger is the value of <span class="math notranslate nohighlight">\(c\)</span>. Use your graph in (a) to explain what happens to the point of intersection of the two lines as c increases. Interpret your result in biological terms.</p>
+ <br />  
+</li>
+<li><p><strong>Biotic Diversity</strong> (Adapted from Valentine, 1985.) Walker and Valentine (1984) suggested a model for species diversity which assumes that species extinction rates are independent of diversity but speciation rates are regulated by competition. Denoting the number of species at time <span class="math notranslate nohighlight">\(t\)</span> by <span class="math notranslate nohighlight">\(S(t)\)</span>, the speciation rate by <span class="math notranslate nohighlight">\(b\)</span>, and the extinction rate by <span class="math notranslate nohighlight">\(a\)</span>, they used the model</p>
+<div class="math notranslate nohighlight">
+\[
+        \frac{dS}{dt} = S\left[b\left(1-\frac{S}{K}\right)-a \right]
+    \]</div>
+<p>where <span class="math notranslate nohighlight">\(K\)</span> denotes the number of “niches”, or potential places for species in the ecosystem.</p>
+ <br/>
+<p>(a) Find possible equilibria (<span class="math notranslate nohighlight">\(dS/dt = 0\)</span>) under the condition <span class="math notranslate nohighlight">\(a &lt; b\)</span>.</p>
+ <br/>
+<p>(b) Use your result in (a) to explain the following statement by Valentine (1985):</p>
+<div class="admonition-quotation admonition">
+<p class="admonition-title">Quotation</p>
+<p>“In this situation, ecosystems are never ‘full’, with all potential niches occupied by species so long as the extinction rate is above zero.”</p>
+</div>
+ <br/>
+<p>(c) What happens when <span class="math notranslate nohighlight">\(a \ge b\)</span>?</p>
+ <br />  
+</li>
+<li><p>Find a point on the curve</p>
+<div class="math notranslate nohighlight">
+\[y = 1 - 3x^3\]</div>
+<p>whose tangent line is parallel to the line <span class="math notranslate nohighlight">\(y = -x\)</span>. Is there more than one such point? If so, find all other points with this property.</p>
+ <br />  
+</li>
+<li><p>Find a point on the curve</p>
+<div class="math notranslate nohighlight">
+\[y = 2x^3 - 4x + 1\]</div>
+<p>whose tangent line is parallel to the line <span class="math notranslate nohighlight">\(y-2x = 1\)</span>. Is there more than one such point? If so, find all other points with this property.</p>
+ <br />  
+</li>
+<li><p>In the following, use the product and quotient rules to find the derivative with respect to the independent variable.</p>
+ <br/>
+<p>(a) <span class="math notranslate nohighlight">\(f(x) = (2x^3 - 1)(3 + 2x^2)\)</span></p>
+ <br/>
+<p>(b) <span class="math notranslate nohighlight">\(h(s) = (4 - 3s^2 + 4s^3)^2\)</span></p>
+ <br/>
+<p>(c) <span class="math notranslate nohighlight">\(f(x) =\sqrt{3x}(x^2 - 1)\)</span></p>
+ <br/>
+<p>(d) <span class="math notranslate nohighlight">\(g(t) = 4(3t^2 - 1)(2t + 1)\)</span></p>
+ <br/>
+<p>(e) <span class="math notranslate nohighlight">\(f(x) = 3(x^2 + 2)(4x^2 - 5x^4) - 3\)</span></p>
+ <br/>
+<p>(f) <span class="math notranslate nohighlight">\(g(x) = \displaystyle\frac{1 - 4x^3}{1 - x}\)</span></p>
+ <br/>
+<p>(g) <span class="math notranslate nohighlight">\(h(x) = \displaystyle\frac{1 + 2x^2 - 4x^4}{3x^3 - 5x^5}\)</span></p>
+ <br/>
+<p>(h) <span class="math notranslate nohighlight">\(f(t) =\displaystyle\frac{3 - t^2}{(t + 1)^2}\)</span></p>
+ <br/>
+<p>(i) <span class="math notranslate nohighlight">\(h(s) = \displaystyle\frac{2s^3 - 4s^2 + 5s - 7}{(s^2 - 3)^2}\)</span></p>
+ <br/>
+<p>(j) <span class="math notranslate nohighlight">\(g(s) = \displaystyle\frac{s^{1/7} - s^{2/7}}{s^{3/7} + s^{4/7}}\)</span></p>
+ <br />  
+</li>
+<li><p><strong>Logistic Growth</strong></p>
+ <br/>
+<p>(a) Find the derivative of the logistic growth curve</p>
+<div class="math notranslate nohighlight">
+\[N(t)=\frac{K}{1+\left(\frac{K}{N(0)}-1\right)e^{-rt}}\]</div>
+<p>where <span class="math notranslate nohighlight">\(r\)</span> and <span class="math notranslate nohighlight">\(K\)</span> are positive constants and <span class="math notranslate nohighlight">\(N(0)\)</span> is the population size at time <span class="math notranslate nohighlight">\(0\)</span>.</p>
+ <br/>
+<p>(b) Show that <span class="math notranslate nohighlight">\(N(t)\)</span> satisfies the equation</p>
+<div class="math notranslate nohighlight">
+\[\frac{dN}{dt}=rN\left(1-\frac{N}{K}\right)\]</div>
+<p>[<em>Hint:</em> Use the function <span class="math notranslate nohighlight">\(N(t)\)</span> given in (a) for the right-hand side, and simplify until you obtain the derivative of N(t) that you computed in A.]</p>
+ <br/>
+<p>(c) What happens to the <em>per-capita</em> rate of growth <span class="math notranslate nohighlight">\(\frac{1}{N}\frac{dN}{dt}\)</span> as you increase N? What biological principle and what type of interaction is your model describing?</p>
+ <br />  
+</li>
+<li><p>In the following, find the derivative with respect to the independent variable (<em>Hint</em>: use the chain rule).</p>
+ <br/>
+<p>(a) <span class="math notranslate nohighlight">\(f(x) = \mbox{sin}(e^{2x} + x)\)</span></p>
+ <br/>
+<p>(b) <span class="math notranslate nohighlight">\(h(x) = \mbox{exp}[x^2 - 2 \mbox{cos}\, x]\)</span></p>
+ <br/>
+<p>(c) <span class="math notranslate nohighlight">\(f(x) =\displaystyle\frac{x-2^{−x}}
+{1+x2^{−x}}\)</span></p>
+ <br/>
+<p>(d) <span class="math notranslate nohighlight">\(g(t) = \mbox{log}\left(\displaystyle\frac{1 - t}{1 + 2t}\right)\)</span></p>
+ <br/>
+<p>(e) <span class="math notranslate nohighlight">\(f(t) = \mbox{sin}(\mbox{ln}(3t))\)</span></p>
+ <br/>
+<p>(f) <span class="math notranslate nohighlight">\(g(x) = \mbox{ln}(\mbox{cos}(1 - x))\)</span></p>
+ <br/>
+<p>(g) <span class="math notranslate nohighlight">\(h(x) = \displaystyle\frac{\sqrt{x^2 - 1}}{2+\sqrt{x^2+1}}\)</span></p>
+ <br/>
+<p>(h) <span class="math notranslate nohighlight">\(f(t) =\sqrt[3]{t^2+\sqrt{1-t}}\)</span></p>
+ <br/>
+</li>
+<li><p>The functional responses of some predators are sigmoidal; that is, the number of prey attacked per predator as a function of prey density has a sigmoidal shape. If we denote the prey density by <span class="math notranslate nohighlight">\(N\)</span>, the predator density by <span class="math notranslate nohighlight">\(P\)</span>, the time available for searching for prey by <span class="math notranslate nohighlight">\(T\)</span>, and the handling time of each prey item per predator by <span class="math notranslate nohighlight">\(T_h\)</span>, then the number of prey encounters per predator as a function of <span class="math notranslate nohighlight">\(N\)</span>, <span class="math notranslate nohighlight">\(T\)</span>, and <span class="math notranslate nohighlight">\(T_h\)</span> can be expressed as</p>
+<div class="math notranslate nohighlight">
+\[
+        f (N, T, T_h) = \frac{b^2N^2T}{1 + cN + bT_hN^2}
+    \]</div>
+<p>where <span class="math notranslate nohighlight">\(b\)</span> and <span class="math notranslate nohighlight">\(c\)</span> are positive constants.</p>
+ <br/>
+<p>(a) Calculate the first-order partial derivative of <span class="math notranslate nohighlight">\(f(N, T, T_h)\)</span> with respect to the prey density <span class="math notranslate nohighlight">\(N\)</span>. If <span class="math notranslate nohighlight">\(N\)</span> increases, what happens to <span class="math notranslate nohighlight">\(f(N, T, T_h)\)</span>?</p>
+ <br/>
+<p>(b) Calculate the first-order partial derivative of <span class="math notranslate nohighlight">\(f(N, T, T_h)\)</span> with respect to the the time <span class="math notranslate nohighlight">\(T\)</span> available for search. If <span class="math notranslate nohighlight">\(T\)</span> increases, what happens to <span class="math notranslate nohighlight">\(f(N, T, T_h)\)</span>?</p>
+ <br/>
+<p>(c) Calculate the first-order partial derivative of <span class="math notranslate nohighlight">\(f(N, T, T_h)\)</span> with respect to the handling time <span class="math notranslate nohighlight">\(T_h\)</span>. If <span class="math notranslate nohighlight">\(T_h\)</span> increases, what happens to <span class="math notranslate nohighlight">\(f(N, T, T_h)\)</span>?</p>
+<p>[<em>Hint:</em> The function <span class="math notranslate nohighlight">\(f(N, T, T_h)\)</span> will be an increasing (decreasing) function of the independent variable if the first-order partial derivative of <span class="math notranslate nohighlight">\(f\)</span> with respect to this variable is positive (negative).]</p>
+</li>
+</ol>
+</section>
+</section>
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+    <i class="fa-solid fa-list"></i> Contents
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+  <nav class="bd-toc-nav page-toc">
+    <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#making-mathematical-statements">Making mathematical statements</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#sets-and-relations">Sets and relations</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#polynomial-functions">Polynomial functions</a><ul class="nav section-nav flex-column">
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#degree-0">Degree 0</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#degree-1">Degree 1</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#degree-2">Degree 2</a><ul class="nav section-nav flex-column">
+<li class="toc-h5 nav-item toc-entry"><a class="reference internal nav-link" href="#concavity">Concavity</a></li>
+<li class="toc-h5 nav-item toc-entry"><a class="reference internal nav-link" href="#intercept-roots">Intercept &amp; Roots</a></li>
+<li class="toc-h5 nav-item toc-entry"><a class="reference internal nav-link" href="#minimum-maximum">Minimum / Maximum</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#rational-functions">Rational functions</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#power-functions">Power functions</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#exponential-functions">Exponential functions</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#logarithmic-functions">Logarithmic functions</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#describing-shapes-and-patterns">Describing shapes and patterns</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#measuring-angles">Measuring angles</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#trigonometric-circle">Trigonometric circle</a><ul class="nav section-nav flex-column">
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#trigonometric-identities">Trigonometric identities</a></li>
+</ul>
+</li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#trigonometric-functions">Trigonometric functions</a><ul class="nav section-nav flex-column">
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#amplitude-and-period">Amplitude and Period</a></li>
+</ul>
+</li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#modelling-complex-periodic-phenomena">Modelling complex periodic phenomena</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#transformation-on-graphs">Transformation on graphs</a><ul class="nav section-nav flex-column">
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#translations">Translations</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#stretching-compression">Stretching / Compression</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#reflection">Reflection</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#analysing-change-one-step-at-a-time">Analysing change, one step at a time</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#sequences">Sequences</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#limits">Limits</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#discrete-time-population-models">Discrete-time population models</a><ul class="nav section-nav flex-column">
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#density-dependent-population-growth">Density-dependent population growth</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#annual-plants">Annual plants</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#fibonacci-sequence">Fibonacci sequence</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#from-table-arrangements-to-powerful-tools">From table arrangements to powerful tools</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#systems-of-linear-equations">Systems of linear equations</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#matrix-operations">Matrix operations</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#inverse-matrix">Inverse matrix</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#determinant">Determinant</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#structured-models">Structured models</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#linear-transformations">Linear transformations</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#eigenvalues-and-eigenvectors">Eigenvalues and eigenvectors</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#vector-decomposition">Vector decomposition</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#structured-models-revisited">Structured models revisited</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#analysing-change-continuously">Analysing change, continuously</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#average-rate-of-change">Average rate of change</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#instantaneous-rate-of-change">Instantaneous rate of change</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#calculating-derivatives">Calculating derivatives</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#chain-rule">Chain rule</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#higher-order-derivatives">Higher order derivatives</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#functions-of-several-variables">Functions of several variables</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#analysing-change-continuously-part-2">Analysing change, continuously (Part 2)</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#extreme-values-of-functions">Extreme values of functions</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#monotonicity-and-concavity">Monotonicity and concavity</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#finding-extrema">Finding extrema</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#inflection-points">Inflection points</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#series-expansions">Series expansions</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#calculating-accumulated-change">Calculating accumulated change</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#series">Series</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#calculating-areas">Calculating areas</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#fundamental-theorem-of-calculus">Fundamental theorem of Calculus</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#calculating-integrals">Calculating integrals</a><ul class="nav section-nav flex-column">
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#integration-by-substitution">Integration by substitution</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#integration-by-parts">Integration by parts</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#integration-by-partial-fractions">Integration by partial fractions</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#exercise-set-1">Exercise Set 1</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#exercise-set-2">Exercise Set 2</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#exercise-set-3">Exercise Set 3</a></li>
+</ul>
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diff --git a/notebooks/Appendix-MiniProj.html b/notebooks/Appendix-MiniProj.html
index c295069e..8eb5191d 100644
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     <meta charset="utf-8" />
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-    <title>The Computing Miniproject &#8212; TheMulQuaBio</title>
+    <title>The Computing Miniproject &#8212; MulQuaBio</title>
   
   
   
@@ -153,8 +153,8 @@
       
     
     
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+    <img src="../_static/logo.png" class="logo__image only-light" alt="MulQuaBio - Home"/>
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 </a></div>
@@ -188,23 +188,9 @@
 <li class="toctree-l1"><a class="reference internal" href="Python.html">Biological Computing in Python</a></li>
 <li class="toctree-l1"><a class="reference internal" href="R.html">Biological Computing in R</a></li>
 </ul>
-<p aria-level="2" class="caption" role="heading"><span class="caption-text">Maths for Biologists</span></p>
-<ul class="nav bd-sidenav">
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture01/01_Making_mathematical_statements.html">01 - Making mathematical statements</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture02/02_Describing_shapes_and_patterns.html">02   - Describing shapes and patterns</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture03/03_Analysing_change%2C_one_step_at_a_time.html">03   - Analysing change, one step at a time</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture04/04_From_table_arrangements_to_powerful_tools.html">04 - From table arrangements to powerful tools</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture05/05_From_table_arrangements_to_powerful_tools_part2.html">05 - From table arrangements to powerful tools - part 2</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture06/06_Analysing_change_continuously.html">06 - Analysing change, continuously</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture07/07_Analysing_change_continuously_part2.html">07 - Analysing change, continuously (Part 2)</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture08/08_Calculating_accumulated_change.html">08 - Calculating accumulated change</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/tutorials/lecture01/Tutorial_01.html">Tutorial 01</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/tutorials/lecture05/Tutorial_05.html">Tutorial 05</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/tutorials/lecture06/Tutorial_06.html">Tutorial 06</a></li>
-</ul>
 <p aria-level="2" class="caption" role="heading"><span class="caption-text">Basic Data Analyses and Statistics</span></p>
 <ul class="nav bd-sidenav">
-<li class="toctree-l1"><a class="reference internal" href="../Stats-Intro.html">Basic Data Science and Statistics: Introduction</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../intro-Stats.html">Introduction</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Data_R.html">Data Management and Visualization</a></li>
 <li class="toctree-l1"><a class="reference internal" href="ExpDesign.html">Experimental design and Data exploration</a></li>
 <li class="toctree-l1"><a class="reference internal" href="t_F_tests.html">Basic hypothesis testing: <span class="math notranslate nohighlight">\(t\)</span> and <span class="math notranslate nohighlight">\(F\)</span> tests</a></li>
@@ -224,6 +210,7 @@
 <li class="toctree-l1"><a class="reference internal" href="Appendix-JupyIntro.html">Introduction to Jupyter</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-Data-Python.html">Data analyses  with Python &amp; Jupyter</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-Maths.html">Mathematical models in Jupyter</a></li>
+<li class="toctree-l1"><a class="reference internal" href="Appendix-MathsForBiologists/MathsForBiologists.html">Maths for Biologists</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-Databases.html">Databases <span class="tocSkip"></span></a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-NLLS-Python.html">Model fitting in Python</a></li>
 <li class="toctree-l1 current active"><a class="current reference internal" href="#">The Computing Miniproject</a></li>
@@ -283,7 +270,7 @@
       
       
       
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+      <li><a href="https://github.com/MulQuaBio/MQB/issues/new?title=Issue%20on%20page%20%2Fnotebooks/Appendix-MiniProj.html&body=Your%20issue%20content%20here." target="_blank"
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@@ -2822,7 +2809,7 @@ <h3>Thermal Performance Curves<a class="headerlink" href="#id9" title="Link to t
   <div class="footer-item">
     
 <p class="component-author">
-By TheMulQuaBio collective!
+By MulQuaBio collective!
 </p>
 
   </div>
diff --git a/notebooks/Appendix-NLLS-Python.html b/notebooks/Appendix-NLLS-Python.html
index feb7ad78..4810b8df 100644
--- a/notebooks/Appendix-NLLS-Python.html
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@@ -8,7 +8,7 @@
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     <meta name="viewport" content="width=device-width, initial-scale=1.0" /><meta name="viewport" content="width=device-width, initial-scale=1" />
 
-    <title>Model fitting in Python &#8212; TheMulQuaBio</title>
+    <title>Model fitting in Python &#8212; MulQuaBio</title>
   
   
   
@@ -153,8 +153,8 @@
       
     
     
-    <img src="../_static/logo.png" class="logo__image only-light" alt="TheMulQuaBio - Home"/>
-    <script>document.write(`<img src="../_static/logo.png" class="logo__image only-dark" alt="TheMulQuaBio - Home"/>`);</script>
+    <img src="../_static/logo.png" class="logo__image only-light" alt="MulQuaBio - Home"/>
+    <script>document.write(`<img src="../_static/logo.png" class="logo__image only-dark" alt="MulQuaBio - Home"/>`);</script>
   
   
 </a></div>
@@ -188,23 +188,9 @@
 <li class="toctree-l1"><a class="reference internal" href="Python.html">Biological Computing in Python</a></li>
 <li class="toctree-l1"><a class="reference internal" href="R.html">Biological Computing in R</a></li>
 </ul>
-<p aria-level="2" class="caption" role="heading"><span class="caption-text">Maths for Biologists</span></p>
-<ul class="nav bd-sidenav">
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture01/01_Making_mathematical_statements.html">01 - Making mathematical statements</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture02/02_Describing_shapes_and_patterns.html">02   - Describing shapes and patterns</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture03/03_Analysing_change%2C_one_step_at_a_time.html">03   - Analysing change, one step at a time</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture04/04_From_table_arrangements_to_powerful_tools.html">04 - From table arrangements to powerful tools</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture05/05_From_table_arrangements_to_powerful_tools_part2.html">05 - From table arrangements to powerful tools - part 2</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture06/06_Analysing_change_continuously.html">06 - Analysing change, continuously</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture07/07_Analysing_change_continuously_part2.html">07 - Analysing change, continuously (Part 2)</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture08/08_Calculating_accumulated_change.html">08 - Calculating accumulated change</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/tutorials/lecture01/Tutorial_01.html">Tutorial 01</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/tutorials/lecture05/Tutorial_05.html">Tutorial 05</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/tutorials/lecture06/Tutorial_06.html">Tutorial 06</a></li>
-</ul>
 <p aria-level="2" class="caption" role="heading"><span class="caption-text">Basic Data Analyses and Statistics</span></p>
 <ul class="nav bd-sidenav">
-<li class="toctree-l1"><a class="reference internal" href="../Stats-Intro.html">Basic Data Science and Statistics: Introduction</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../intro-Stats.html">Introduction</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Data_R.html">Data Management and Visualization</a></li>
 <li class="toctree-l1"><a class="reference internal" href="ExpDesign.html">Experimental design and Data exploration</a></li>
 <li class="toctree-l1"><a class="reference internal" href="t_F_tests.html">Basic hypothesis testing: <span class="math notranslate nohighlight">\(t\)</span> and <span class="math notranslate nohighlight">\(F\)</span> tests</a></li>
@@ -224,6 +210,7 @@
 <li class="toctree-l1"><a class="reference internal" href="Appendix-JupyIntro.html">Introduction to Jupyter</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-Data-Python.html">Data analyses  with Python &amp; Jupyter</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-Maths.html">Mathematical models in Jupyter</a></li>
+<li class="toctree-l1"><a class="reference internal" href="Appendix-MathsForBiologists/MathsForBiologists.html">Maths for Biologists</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-Databases.html">Databases <span class="tocSkip"></span></a></li>
 <li class="toctree-l1 current active"><a class="current reference internal" href="#">Model fitting in Python</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-MiniProj.html">The Computing Miniproject</a></li>
@@ -283,7 +270,7 @@
       
       
       
-      <li><a href="https://github.com/mhasoba/TheMulQuaBio" target="_blank"
+      <li><a href="https://github.com/MulQuaBio/MQB" target="_blank"
    class="btn btn-sm btn-source-repository-button dropdown-item"
    title="Source repository"
    data-bs-placement="left" data-bs-toggle="tooltip"
@@ -300,7 +287,7 @@
       
       
       
-      <li><a href="https://github.com/mhasoba/TheMulQuaBio/issues/new?title=Issue%20on%20page%20%2Fnotebooks/Appendix-NLLS-Python.html&body=Your%20issue%20content%20here." target="_blank"
+      <li><a href="https://github.com/MulQuaBio/MQB/issues/new?title=Issue%20on%20page%20%2Fnotebooks/Appendix-NLLS-Python.html&body=Your%20issue%20content%20here." target="_blank"
    class="btn btn-sm btn-source-issues-button dropdown-item"
    title="Open an issue"
    data-bs-placement="left" data-bs-toggle="tooltip"
@@ -1124,7 +1111,7 @@ <h2>Readings &amp; Resources<a class="headerlink" href="#readings-resources" tit
   <div class="footer-item">
     
 <p class="component-author">
-By TheMulQuaBio collective!
+By MulQuaBio collective!
 </p>
 
   </div>
diff --git a/notebooks/Bayesian.html b/notebooks/Bayesian.html
index ec87fd35..99c776e5 100644
--- a/notebooks/Bayesian.html
+++ b/notebooks/Bayesian.html
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     <meta charset="utf-8" />
     <meta name="viewport" content="width=device-width, initial-scale=1.0" /><meta name="viewport" content="width=device-width, initial-scale=1" />
 
-    <title>Model Fitting the Bayesian Way &#8212; TheMulQuaBio</title>
+    <title>Model Fitting the Bayesian Way &#8212; MulQuaBio</title>
   
   
   
@@ -153,8 +153,8 @@
       
     
     
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-    <script>document.write(`<img src="../_static/logo.png" class="logo__image only-dark" alt="TheMulQuaBio - Home"/>`);</script>
+    <img src="../_static/logo.png" class="logo__image only-light" alt="MulQuaBio - Home"/>
+    <script>document.write(`<img src="../_static/logo.png" class="logo__image only-dark" alt="MulQuaBio - Home"/>`);</script>
   
   
 </a></div>
@@ -188,23 +188,9 @@
 <li class="toctree-l1"><a class="reference internal" href="Python.html">Biological Computing in Python</a></li>
 <li class="toctree-l1"><a class="reference internal" href="R.html">Biological Computing in R</a></li>
 </ul>
-<p aria-level="2" class="caption" role="heading"><span class="caption-text">Maths for Biologists</span></p>
-<ul class="nav bd-sidenav">
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture01/01_Making_mathematical_statements.html">01 - Making mathematical statements</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture02/02_Describing_shapes_and_patterns.html">02   - Describing shapes and patterns</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture03/03_Analysing_change%2C_one_step_at_a_time.html">03   - Analysing change, one step at a time</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture04/04_From_table_arrangements_to_powerful_tools.html">04 - From table arrangements to powerful tools</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture05/05_From_table_arrangements_to_powerful_tools_part2.html">05 - From table arrangements to powerful tools - part 2</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture06/06_Analysing_change_continuously.html">06 - Analysing change, continuously</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture07/07_Analysing_change_continuously_part2.html">07 - Analysing change, continuously (Part 2)</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture08/08_Calculating_accumulated_change.html">08 - Calculating accumulated change</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/tutorials/lecture01/Tutorial_01.html">Tutorial 01</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/tutorials/lecture05/Tutorial_05.html">Tutorial 05</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/tutorials/lecture06/Tutorial_06.html">Tutorial 06</a></li>
-</ul>
 <p aria-level="2" class="caption" role="heading"><span class="caption-text">Basic Data Analyses and Statistics</span></p>
 <ul class="nav bd-sidenav">
-<li class="toctree-l1"><a class="reference internal" href="../Stats-Intro.html">Basic Data Science and Statistics: Introduction</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../intro-Stats.html">Introduction</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Data_R.html">Data Management and Visualization</a></li>
 <li class="toctree-l1"><a class="reference internal" href="ExpDesign.html">Experimental design and Data exploration</a></li>
 <li class="toctree-l1"><a class="reference internal" href="t_F_tests.html">Basic hypothesis testing: <span class="math notranslate nohighlight">\(t\)</span> and <span class="math notranslate nohighlight">\(F\)</span> tests</a></li>
@@ -224,6 +210,7 @@
 <li class="toctree-l1"><a class="reference internal" href="Appendix-JupyIntro.html">Introduction to Jupyter</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-Data-Python.html">Data analyses  with Python &amp; Jupyter</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-Maths.html">Mathematical models in Jupyter</a></li>
+<li class="toctree-l1"><a class="reference internal" href="Appendix-MathsForBiologists/MathsForBiologists.html">Maths for Biologists</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-Databases.html">Databases <span class="tocSkip"></span></a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-NLLS-Python.html">Model fitting in Python</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-MiniProj.html">The Computing Miniproject</a></li>
@@ -283,7 +270,7 @@
       
       
       
-      <li><a href="https://github.com/mhasoba/TheMulQuaBio" target="_blank"
+      <li><a href="https://github.com/MulQuaBio/MQB" target="_blank"
    class="btn btn-sm btn-source-repository-button dropdown-item"
    title="Source repository"
    data-bs-placement="left" data-bs-toggle="tooltip"
@@ -300,7 +287,7 @@
       
       
       
-      <li><a href="https://github.com/mhasoba/TheMulQuaBio/issues/new?title=Issue%20on%20page%20%2Fnotebooks/Bayesian.html&body=Your%20issue%20content%20here." target="_blank"
+      <li><a href="https://github.com/MulQuaBio/MQB/issues/new?title=Issue%20on%20page%20%2Fnotebooks/Bayesian.html&body=Your%20issue%20content%20here." target="_blank"
    class="btn btn-sm btn-source-issues-button dropdown-item"
    title="Open an issue"
    data-bs-placement="left" data-bs-toggle="tooltip"
@@ -2212,7 +2199,7 @@ <h2>Readings and Resources <a id='Readings'></a><a class="headerlink" href="#rea
   <div class="footer-item">
     
 <p class="component-author">
-By TheMulQuaBio collective!
+By MulQuaBio collective!
 </p>
 
   </div>
diff --git a/notebooks/Data_R.html b/notebooks/Data_R.html
index 117d2300..117c6721 100644
--- a/notebooks/Data_R.html
+++ b/notebooks/Data_R.html
@@ -8,7 +8,7 @@
     <meta charset="utf-8" />
     <meta name="viewport" content="width=device-width, initial-scale=1.0" /><meta name="viewport" content="width=device-width, initial-scale=1" />
 
-    <title>Data Management and Visualization &#8212; TheMulQuaBio</title>
+    <title>Data Management and Visualization &#8212; MulQuaBio</title>
   
   
   
@@ -61,7 +61,7 @@
     <link rel="index" title="Index" href="../genindex.html" />
     <link rel="search" title="Search" href="../search.html" />
     <link rel="next" title="Experimental design and Data exploration" href="ExpDesign.html" />
-    <link rel="prev" title="Basic Data Science and Statistics: Introduction" href="../Stats-Intro.html" />
+    <link rel="prev" title="Introduction" href="../intro-Stats.html" />
   <meta name="viewport" content="width=device-width, initial-scale=1"/>
   <meta name="docsearch:language" content="en"/>
   </head>
@@ -153,8 +153,8 @@
       
     
     
-    <img src="../_static/logo.png" class="logo__image only-light" alt="TheMulQuaBio - Home"/>
-    <script>document.write(`<img src="../_static/logo.png" class="logo__image only-dark" alt="TheMulQuaBio - Home"/>`);</script>
+    <img src="../_static/logo.png" class="logo__image only-light" alt="MulQuaBio - Home"/>
+    <script>document.write(`<img src="../_static/logo.png" class="logo__image only-dark" alt="MulQuaBio - Home"/>`);</script>
   
   
 </a></div>
@@ -188,23 +188,9 @@
 <li class="toctree-l1"><a class="reference internal" href="Python.html">Biological Computing in Python</a></li>
 <li class="toctree-l1"><a class="reference internal" href="R.html">Biological Computing in R</a></li>
 </ul>
-<p aria-level="2" class="caption" role="heading"><span class="caption-text">Maths for Biologists</span></p>
-<ul class="nav bd-sidenav">
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture01/01_Making_mathematical_statements.html">01 - Making mathematical statements</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture02/02_Describing_shapes_and_patterns.html">02   - Describing shapes and patterns</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture03/03_Analysing_change%2C_one_step_at_a_time.html">03   - Analysing change, one step at a time</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture04/04_From_table_arrangements_to_powerful_tools.html">04 - From table arrangements to powerful tools</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture05/05_From_table_arrangements_to_powerful_tools_part2.html">05 - From table arrangements to powerful tools - part 2</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture06/06_Analysing_change_continuously.html">06 - Analysing change, continuously</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture07/07_Analysing_change_continuously_part2.html">07 - Analysing change, continuously (Part 2)</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture08/08_Calculating_accumulated_change.html">08 - Calculating accumulated change</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/tutorials/lecture01/Tutorial_01.html">Tutorial 01</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/tutorials/lecture05/Tutorial_05.html">Tutorial 05</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/tutorials/lecture06/Tutorial_06.html">Tutorial 06</a></li>
-</ul>
 <p aria-level="2" class="caption" role="heading"><span class="caption-text">Basic Data Analyses and Statistics</span></p>
 <ul class="current nav bd-sidenav">
-<li class="toctree-l1"><a class="reference internal" href="../Stats-Intro.html">Basic Data Science and Statistics: Introduction</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../intro-Stats.html">Introduction</a></li>
 <li class="toctree-l1 current active"><a class="current reference internal" href="#">Data Management and Visualization</a></li>
 <li class="toctree-l1"><a class="reference internal" href="ExpDesign.html">Experimental design and Data exploration</a></li>
 <li class="toctree-l1"><a class="reference internal" href="t_F_tests.html">Basic hypothesis testing: <span class="math notranslate nohighlight">\(t\)</span> and <span class="math notranslate nohighlight">\(F\)</span> tests</a></li>
@@ -224,6 +210,7 @@
 <li class="toctree-l1"><a class="reference internal" href="Appendix-JupyIntro.html">Introduction to Jupyter</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-Data-Python.html">Data analyses  with Python &amp; Jupyter</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-Maths.html">Mathematical models in Jupyter</a></li>
+<li class="toctree-l1"><a class="reference internal" href="Appendix-MathsForBiologists/MathsForBiologists.html">Maths for Biologists</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-Databases.html">Databases <span class="tocSkip"></span></a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-NLLS-Python.html">Model fitting in Python</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-MiniProj.html">The Computing Miniproject</a></li>
@@ -283,7 +270,7 @@
       
       
       
-      <li><a href="https://github.com/mhasoba/TheMulQuaBio" target="_blank"
+      <li><a href="https://github.com/MulQuaBio/MQB" target="_blank"
    class="btn btn-sm btn-source-repository-button dropdown-item"
    title="Source repository"
    data-bs-placement="left" data-bs-toggle="tooltip"
@@ -300,7 +287,7 @@
       
       
       
-      <li><a href="https://github.com/mhasoba/TheMulQuaBio/issues/new?title=Issue%20on%20page%20%2Fnotebooks/Data_R.html&body=Your%20issue%20content%20here." target="_blank"
+      <li><a href="https://github.com/MulQuaBio/MQB/issues/new?title=Issue%20on%20page%20%2Fnotebooks/Data_R.html&body=Your%20issue%20content%20here." target="_blank"
    class="btn btn-sm btn-source-issues-button dropdown-item"
    title="Open an issue"
    data-bs-placement="left" data-bs-toggle="tooltip"
@@ -1962,7 +1949,7 @@ <h4>Adding smoothers and regression lines<a class="headerlink" href="#adding-smo
 <a class="reference internal image-reference" href="../_images/5b16aaf6baa4c48819128352bb6f781bc5cd11c9ff3579d4c5fcbee98acf6a0c.png"><img alt="../_images/5b16aaf6baa4c48819128352bb6f781bc5cd11c9ff3579d4c5fcbee98acf6a0c.png" src="../_images/5b16aaf6baa4c48819128352bb6f781bc5cd11c9ff3579d4c5fcbee98acf6a0c.png" style="width: 420px; height: 360px;" /></a>
 </div>
 </div>
-<p><code class="docutils literal notranslate"><span class="pre">lm</span></code> stands for <code class="docutils literal notranslate"><span class="pre">l</span></code>inear <code class="docutils literal notranslate"><span class="pre">m</span></code>odels. Linear regression is a type of linear model. More on this in the <a class="reference internal" href="../Stats-Intro.html"><span class="std std-doc">BASIC DATA ANALYSES AND STATISTICS</span></a> section of these notes.</p>
+<p><code class="docutils literal notranslate"><span class="pre">lm</span></code> stands for <code class="docutils literal notranslate"><span class="pre">l</span></code>inear <code class="docutils literal notranslate"><span class="pre">m</span></code>odels. Linear regression is a type of linear model. More on this in the <a class="reference internal" href="#../Stats-Intro.md"><span class="xref myst">BASIC DATA ANALYSES AND STATISTICS</span></a> section of these notes.</p>
 <p>We can also add a “smoother” for each type of interaction:</p>
 <div class="cell docutils container">
 <div class="cell_input docutils container">
@@ -2997,12 +2984,12 @@ <h3>Data visualization<a class="headerlink" href="#id4" title="Link to this head
                   
 <div class="prev-next-area">
     <a class="left-prev"
-       href="../Stats-Intro.html"
+       href="../intro-Stats.html"
        title="previous page">
       <i class="fa-solid fa-angle-left"></i>
       <div class="prev-next-info">
         <p class="prev-next-subtitle">previous</p>
-        <p class="prev-next-title">Basic Data Science and Statistics: Introduction</p>
+        <p class="prev-next-title">Introduction</p>
       </div>
     </a>
     <a class="right-next"
@@ -3121,7 +3108,7 @@ <h3>Data visualization<a class="headerlink" href="#id4" title="Link to this head
   <div class="footer-item">
     
 <p class="component-author">
-By TheMulQuaBio collective!
+By MulQuaBio collective!
 </p>
 
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diff --git a/notebooks/Ecosystems.html b/notebooks/Ecosystems.html
new file mode 100644
index 00000000..d6abe1c8
--- /dev/null
+++ b/notebooks/Ecosystems.html
@@ -0,0 +1,688 @@
+
+<!DOCTYPE html>
+
+
+<html lang="en" data-content_root="../" >
+
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+    <meta name="viewport" content="width=device-width, initial-scale=1.0" /><meta name="viewport" content="width=device-width, initial-scale=1" />
+
+    <title>Ecosystems &#8212; MulQuaBio</title>
+  
+  
+  
+  <script data-cfasync="false">
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+    document.documentElement.dataset.theme = localStorage.getItem("theme") || "";
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+    <h1>Ecosystems</h1>
+    <!-- Table of contents -->
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+            
+            <div>
+                <h2> Contents </h2>
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+            <nav aria-label="Page">
+                <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#introduction">Introduction</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#ecological-networks">Ecological Networks</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#trophic-networks">Trophic Networks</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#mutualistic-networks">Mutualistic Networks</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#competitive-networks">Competitive Networks</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#behavioral-networks">Behavioral Networks</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#structure-and-stability">Structure and Stability</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#id1">Trophic Networks</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#id2">Mutualistic Networks</a></li>
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+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#id3">Trophic Networks</a></li>
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+                  
+  <section class="tex2jax_ignore mathjax_ignore" id="ecosystems">
+<h1>Ecosystems<a class="headerlink" href="#ecosystems" title="Link to this heading">#</a></h1>
+<ul class="simple">
+<li><p>Communities, Food-webs and Ecosystems</p>
+<ul>
+<li><p>Community vs Ecosystem</p>
+<ul>
+<li><p>Microbial community vs microbiome</p></li>
+</ul>
+</li>
+<li><p>Food-webs</p></li>
+<li><p>Random matrix theory</p></li>
+<li><p>Evolutionary dynamics (!!)</p>
+<ul>
+<li><p>Microbial systems as an example</p></li>
+</ul>
+</li>
+<li><p>Applications/Examples</p></li>
+</ul>
+</li>
+</ul>
+<section id="introduction">
+<h2>Introduction<a class="headerlink" href="#introduction" title="Link to this heading">#</a></h2>
+<p>Ecological and mutualistic networks are fundamental to understanding how species interact and coexist within ecosystems. These networks, encompassing trophic (consumer-resource) and mutualistic (e.g., plant-pollinator) interactions, provide insights into biodiversity, ecosystem functioning, and resilience to disturbances.</p>
+<blockquote>
+<div><p>“The presence of a feline animal in large numbers in a district might determine, through the intervention first of mice and then of bees, the frequency of certain flowers in that district.”<br />
+– Charles Darwin, 1859</p>
+</div></blockquote>
+<p>This quote illustrates the intricate interdependence within ecological systems.</p>
+</section>
+<section id="ecological-networks">
+<h2>Ecological Networks<a class="headerlink" href="#ecological-networks" title="Link to this heading">#</a></h2>
+<p>Ecological networks represent interactions where nodes correspond to individuals or species populations and links denote interactions such as predation, competition, or mutualism. A classic example is the Silwood Park food web, which illustrates trophic relationships among species.</p>
+<section id="trophic-networks">
+<h3>Trophic Networks<a class="headerlink" href="#trophic-networks" title="Link to this heading">#</a></h3>
+<p>Trophic networks describe feeding relationships among species. These are often represented as directed graphs, with arrows indicating the flow of energy from prey to predator. For example, a primary producer (e.g., a plant) might feed a herbivore, which in turn feeds a carnivore. Trophic networks often exhibit hierarchical structures, with top predators occupying the apex of the food web.</p>
+</section>
+<section id="mutualistic-networks">
+<h3>Mutualistic Networks<a class="headerlink" href="#mutualistic-networks" title="Link to this heading">#</a></h3>
+<p>Mutualistic networks describe interactions where species mutually benefit, such as:</p>
+<ul class="simple">
+<li><p><strong>Plant-fungal networks:</strong> Mycorrhizal fungi exchange nutrients with plants.</p></li>
+<li><p><strong>Bacterial networks:</strong> Exchange of metabolites, such as in yogurt and cheese production.</p></li>
+<li><p><strong>Plant-pollinator networks:</strong> Pollinators gain nectar, while plants achieve reproduction.</p></li>
+<li><p><strong>Seed dispersal networks:</strong> Animals disperse seeds in exchange for fruit.</p></li>
+</ul>
+<p>These networks contribute to critical ecosystem services, including pollination and nutrient cycling. They are often characterized by nested structures, where specialists interact with subsets of generalist species.</p>
+</section>
+<section id="competitive-networks">
+<h3>Competitive Networks<a class="headerlink" href="#competitive-networks" title="Link to this heading">#</a></h3>
+<p>Competitive networks involve interactions where species compete for the same limited resources, such as nutrients, light, or space. For example, plants in dense forests compete for sunlight, while microbes in soil compete for nutrients. These networks are often depicted as undirected graphs where the links indicate negative interactions.</p>
+</section>
+<section id="behavioral-networks">
+<h3>Behavioral Networks<a class="headerlink" href="#behavioral-networks" title="Link to this heading">#</a></h3>
+<p>Behavioral networks describe interactions within social or cooperative groups, such as ant colonies or primate troops. These interactions can include cooperation, altruism, or dominance hierarchies. Behavioral networks are critical for understanding group dynamics and evolutionary strategies.</p>
+</section>
+</section>
+<section id="structure-and-stability">
+<h2>Structure and Stability<a class="headerlink" href="#structure-and-stability" title="Link to this heading">#</a></h2>
+<p>The structure of ecological networks is shaped by modules or motifs, small groups of interconnected nodes that enhance stability. Stability in ecological networks can be formally defined as:</p>
+<ul class="simple">
+<li><p><strong>Local Stability:</strong> The ability of a system to return to equilibrium after a small perturbation.</p></li>
+<li><p><strong>Global Stability:</strong> The capacity of a system to remain functional across a wide range of disturbances.</p></li>
+</ul>
+<section id="id1">
+<h3>Trophic Networks<a class="headerlink" href="#id1" title="Link to this heading">#</a></h3>
+<p>In trophic networks, stability is often influenced by modularity (the division of the network into relatively independent clusters) and connectivity (the number of interactions between species). Highly connected but modular networks tend to be more resilient to species loss.</p>
+</section>
+<section id="id2">
+<h3>Mutualistic Networks<a class="headerlink" href="#id2" title="Link to this heading">#</a></h3>
+<p>Mutualistic networks often exhibit nestedness, a structural pattern where specialist species interact with subsets of generalist species. This nestedness enhances stability by reducing competition and fostering redundancy. Modularity in mutualistic networks can also increase resilience by containing disturbances within specific clusters.</p>
+</section>
+</section>
+<section id="mathematical-modeling">
+<h2>Mathematical Modeling<a class="headerlink" href="#mathematical-modeling" title="Link to this heading">#</a></h2>
+<section id="id3">
+<h3>Trophic Networks<a class="headerlink" href="#id3" title="Link to this heading">#</a></h3>
+<p>Trophic interactions are often modeled using Lotka-Volterra equations:</p>
+<div class="math notranslate nohighlight">
+\[
+\frac{dN}{dt} = r_m N \left(1 - \frac{N}{K}\right) - aNC,
+\]</div>
+<div class="math notranslate nohighlight">
+\[
+\frac{dC}{dt} = eaNC - zC,
+\]</div>
+<p>where <span class="math notranslate nohighlight">\(N\)</span> and <span class="math notranslate nohighlight">\(C\)</span> represent resource and consumer populations, respectively, and parameters include growth rates (<span class="math notranslate nohighlight">\(r_m\)</span>), carrying capacity (<span class="math notranslate nohighlight">\(K\)</span>), attack rate (<span class="math notranslate nohighlight">\(a\)</span>), conversion efficiency (<span class="math notranslate nohighlight">\(e\)</span>), and mortality rate (<span class="math notranslate nohighlight">\(z\)</span>). These models help predict population dynamics and the effects of perturbations.</p>
+</section>
+<section id="id4">
+<h3>Mutualistic Networks<a class="headerlink" href="#id4" title="Link to this heading">#</a></h3>
+<p>The dynamics of mutualistic networks can be described using interaction matrices that quantify the strength and symmetry of benefits between species. For example, the dynamics of plant-pollinator networks can be modeled as:</p>
+<div class="math notranslate nohighlight">
+\[
+\frac{dP}{dt} = r_P P \left(1 - \frac{P}{K_P}\right) + \beta_{PM} M,
+\]</div>
+<div class="math notranslate nohighlight">
+\[
+\frac{dM}{dt} = r_M M \left(1 - \frac{M}{K_M}\right) + \beta_{MP} P,
+\]</div>
+<p>where <span class="math notranslate nohighlight">\(P\)</span> and <span class="math notranslate nohighlight">\(M\)</span> represent plant and pollinator populations, and <span class="math notranslate nohighlight">\(\beta_{PM}\)</span> and <span class="math notranslate nohighlight">\(\beta_{MP}\)</span> are mutualistic interaction coefficients.</p>
+</section>
+</section>
+<section id="conclusion">
+<h2>Conclusion<a class="headerlink" href="#conclusion" title="Link to this heading">#</a></h2>
+<p>Ecological and mutualistic networks are integral to understanding biodiversity and ecosystem functioning. By studying their structure and stability, ecologists can predict responses to environmental changes and design conservation strategies. A comprehensive understanding of these networks is crucial for maintaining ecosystem resilience in the face of anthropogenic pressures.</p>
+</section>
+<section id="discussion-questions">
+<h2>Discussion Questions<a class="headerlink" href="#discussion-questions" title="Link to this heading">#</a></h2>
+<ol class="arabic simple">
+<li><p>What sorts of interactions might ancient microbes have had? What type(s) of Ecological Network(s)?</p></li>
+<li><p>What are the differences and similarities between microbial networks (who-eats-what type) and food webs (who-eats-whom type)?</p></li>
+<li><p>What role do indirect interactions play in trophic network / food web dynamics?</p></li>
+<li><p>How are trophic/biomass/ecological pyramids related to body size structure in food webs?</p></li>
+<li><p>How do you think climatic temperature (and warming) affects trophic networks / food webs?</p></li>
+<li><p>What are the differences between trophic (e.g., food web) and mutualistic networks?</p></li>
+<li><p>Can we truly study Mutualistic and Trophic networks separately?</p></li>
+<li><p>What role might body size play in Mutualistic (e.g., plant-pollinator) networks?</p></li>
+<li><p>How do you think global climate change might affect plant-pollinator networks and pollination services?</p></li>
+<li><p>How do you think increasing mutualism increases the rate of microbial community functioning (e.g., decomposition of leaf litter)? What role can temperature play in this? How does this relate to the global carbon cycle?</p></li>
+</ol>
+</section>
+</section>
+
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+    <i class="fa-solid fa-list"></i> Contents
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+    <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#introduction">Introduction</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#ecological-networks">Ecological Networks</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#trophic-networks">Trophic Networks</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#mutualistic-networks">Mutualistic Networks</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#competitive-networks">Competitive Networks</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#behavioral-networks">Behavioral Networks</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#structure-and-stability">Structure and Stability</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#id1">Trophic Networks</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#id2">Mutualistic Networks</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#mathematical-modeling">Mathematical Modeling</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#id3">Trophic Networks</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#id4">Mutualistic Networks</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#conclusion">Conclusion</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#discussion-questions">Discussion Questions</a></li>
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diff --git a/notebooks/ExpDesign.html b/notebooks/ExpDesign.html
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-    <title>Experimental design and Data exploration &#8212; TheMulQuaBio</title>
+    <title>Experimental design and Data exploration &#8212; MulQuaBio</title>
   
   
   
@@ -153,8 +153,8 @@
       
     
     
-    <img src="../_static/logo.png" class="logo__image only-light" alt="TheMulQuaBio - Home"/>
-    <script>document.write(`<img src="../_static/logo.png" class="logo__image only-dark" alt="TheMulQuaBio - Home"/>`);</script>
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+    <script>document.write(`<img src="../_static/logo.png" class="logo__image only-dark" alt="MulQuaBio - Home"/>`);</script>
   
   
 </a></div>
@@ -188,23 +188,9 @@
 <li class="toctree-l1"><a class="reference internal" href="Python.html">Biological Computing in Python</a></li>
 <li class="toctree-l1"><a class="reference internal" href="R.html">Biological Computing in R</a></li>
 </ul>
-<p aria-level="2" class="caption" role="heading"><span class="caption-text">Maths for Biologists</span></p>
-<ul class="nav bd-sidenav">
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture01/01_Making_mathematical_statements.html">01 - Making mathematical statements</a></li>
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-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture04/04_From_table_arrangements_to_powerful_tools.html">04 - From table arrangements to powerful tools</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture05/05_From_table_arrangements_to_powerful_tools_part2.html">05 - From table arrangements to powerful tools - part 2</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture06/06_Analysing_change_continuously.html">06 - Analysing change, continuously</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture07/07_Analysing_change_continuously_part2.html">07 - Analysing change, continuously (Part 2)</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture08/08_Calculating_accumulated_change.html">08 - Calculating accumulated change</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/tutorials/lecture01/Tutorial_01.html">Tutorial 01</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/tutorials/lecture05/Tutorial_05.html">Tutorial 05</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/tutorials/lecture06/Tutorial_06.html">Tutorial 06</a></li>
-</ul>
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 <ul class="current nav bd-sidenav">
-<li class="toctree-l1"><a class="reference internal" href="../Stats-Intro.html">Basic Data Science and Statistics: Introduction</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../intro-Stats.html">Introduction</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Data_R.html">Data Management and Visualization</a></li>
 <li class="toctree-l1 current active"><a class="current reference internal" href="#">Experimental design and Data exploration</a></li>
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@@ -224,6 +210,7 @@
 <li class="toctree-l1"><a class="reference internal" href="Appendix-JupyIntro.html">Introduction to Jupyter</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-Data-Python.html">Data analyses  with Python &amp; Jupyter</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-Maths.html">Mathematical models in Jupyter</a></li>
+<li class="toctree-l1"><a class="reference internal" href="Appendix-MathsForBiologists/MathsForBiologists.html">Maths for Biologists</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-Databases.html">Databases <span class="tocSkip"></span></a></li>
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@@ -283,7 +270,7 @@
       
       
       
-      <li><a href="https://github.com/mhasoba/TheMulQuaBio" target="_blank"
+      <li><a href="https://github.com/MulQuaBio/MQB" target="_blank"
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    title="Source repository"
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@@ -300,7 +287,7 @@
       
       
       
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+      <li><a href="https://github.com/MulQuaBio/MQB/issues/new?title=Issue%20on%20page%20%2Fnotebooks/ExpDesign.html&body=Your%20issue%20content%20here." target="_blank"
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@@ -1177,7 +1164,7 @@ <h2>From data exploration to statistical analysis<a class="headerlink" href="#fr
   <div class="footer-item">
     
 <p class="component-author">
-By TheMulQuaBio collective!
+By MulQuaBio collective!
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index 9be64841..f904aa3f 100644
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-    <title>Version control with Git &#8212; TheMulQuaBio</title>
+    <title>Version control with Git &#8212; MulQuaBio</title>
   
   
   
@@ -153,8 +153,8 @@
       
     
     
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+    <img src="../_static/logo.png" class="logo__image only-light" alt="MulQuaBio - Home"/>
+    <script>document.write(`<img src="../_static/logo.png" class="logo__image only-dark" alt="MulQuaBio - Home"/>`);</script>
   
   
 </a></div>
@@ -188,23 +188,9 @@
 <li class="toctree-l1"><a class="reference internal" href="Python.html">Biological Computing in Python</a></li>
 <li class="toctree-l1"><a class="reference internal" href="R.html">Biological Computing in R</a></li>
 </ul>
-<p aria-level="2" class="caption" role="heading"><span class="caption-text">Maths for Biologists</span></p>
-<ul class="nav bd-sidenav">
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture01/01_Making_mathematical_statements.html">01 - Making mathematical statements</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture02/02_Describing_shapes_and_patterns.html">02   - Describing shapes and patterns</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture03/03_Analysing_change%2C_one_step_at_a_time.html">03   - Analysing change, one step at a time</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture04/04_From_table_arrangements_to_powerful_tools.html">04 - From table arrangements to powerful tools</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture05/05_From_table_arrangements_to_powerful_tools_part2.html">05 - From table arrangements to powerful tools - part 2</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture06/06_Analysing_change_continuously.html">06 - Analysing change, continuously</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture07/07_Analysing_change_continuously_part2.html">07 - Analysing change, continuously (Part 2)</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture08/08_Calculating_accumulated_change.html">08 - Calculating accumulated change</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/tutorials/lecture01/Tutorial_01.html">Tutorial 01</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/tutorials/lecture05/Tutorial_05.html">Tutorial 05</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/tutorials/lecture06/Tutorial_06.html">Tutorial 06</a></li>
-</ul>
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 <ul class="nav bd-sidenav">
-<li class="toctree-l1"><a class="reference internal" href="../Stats-Intro.html">Basic Data Science and Statistics: Introduction</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../intro-Stats.html">Introduction</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Data_R.html">Data Management and Visualization</a></li>
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@@ -224,6 +210,7 @@
 <li class="toctree-l1"><a class="reference internal" href="Appendix-JupyIntro.html">Introduction to Jupyter</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-Data-Python.html">Data analyses  with Python &amp; Jupyter</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-Maths.html">Mathematical models in Jupyter</a></li>
+<li class="toctree-l1"><a class="reference internal" href="Appendix-MathsForBiologists/MathsForBiologists.html">Maths for Biologists</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-Databases.html">Databases <span class="tocSkip"></span></a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-NLLS-Python.html">Model fitting in Python</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-MiniProj.html">The Computing Miniproject</a></li>
@@ -283,7 +270,7 @@
       
       
       
-      <li><a href="https://github.com/mhasoba/TheMulQuaBio" target="_blank"
+      <li><a href="https://github.com/MulQuaBio/MQB" target="_blank"
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@@ -300,7 +287,7 @@
       
       
       
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+      <li><a href="https://github.com/MulQuaBio/MQB/issues/new?title=Issue%20on%20page%20%2Fnotebooks/Git.html&body=Your%20issue%20content%20here." target="_blank"
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new file mode 100644
index 00000000..9d676a60
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+++ b/notebooks/Interactions.html
@@ -0,0 +1,678 @@
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+    <h1>Species Interactions</h1>
+    <!-- Table of contents -->
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+            
+            <div>
+                <h2> Contents </h2>
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+            <nav aria-label="Page">
+                <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#introduction">Introduction</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#energy-metabolism-and-consumption">Energy, Metabolism, and Consumption</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#size-and-metabolic-needs">Size and Metabolic Needs</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#biomechanical-implications-of-size">Biomechanical Implications of Size</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#dissecting-consumer-resource-interactions">Dissecting Consumer-Resource Interactions</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#predicting-consumer-resource-interactions">Predicting Consumer-Resource Interactions</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#temperature-and-consumer-resource-interactions">Temperature and Consumer-Resource Interactions</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#consumer-resource-dynamics">Consumer-Resource Dynamics</a></li>
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+                  
+  <section class="tex2jax_ignore mathjax_ignore" id="species-interactions">
+<h1>Species Interactions<a class="headerlink" href="#species-interactions" title="Link to this heading">#</a></h1>
+<ul class="simple">
+<li><p>Species Interactions</p>
+<ul>
+<li><p>Consumer-resource interactions</p>
+<ul>
+<li><p>Predator-prey (VJ)</p></li>
+<li><p>Bacterial-substrate</p></li>
+<li><p>MacArthur</p></li>
+</ul>
+</li>
+<li><p>The Lotka-Volterra equations</p>
+<ul>
+<li><p>Every LV system is “effective”</p>
+<ul>
+<li><p>Normal form?</p></li>
+</ul>
+</li>
+<li><p>Bifurcations (VJ)</p></li>
+</ul>
+</li>
+<li><p>Evolutionary dynamics</p></li>
+<li><p>Stochasticity</p></li>
+<li><p>Applications/Examples</p></li>
+</ul>
+</li>
+</ul>
+<section id="introduction">
+<h2>Introduction<a class="headerlink" href="#introduction" title="Link to this heading">#</a></h2>
+<p>Consumers “live to eat” and “eat to live.” They are heterotrophs that harvest energy from other organisms, such as tigers, or from organic sources, such as bacteria. As Ludwig Boltzmann noted in 1886:</p>
+<blockquote>
+<div><p>“The struggle for existence of living beings is not for the fundamental constituents of food, but for the possession of the free energy obtained, chiefly by means of the green plant, from the transfer of radiant energy from the hot sun to the cold earth.”</p>
+</div></blockquote>
+<p>Consumer-resource interactions play a crucial role in the flow of energy through ecosystems. The metabolic theory of ecology provides a foundation for understanding these interactions, linking individual metabolic processes to population and ecosystem dynamics.</p>
+</section>
+<section id="energy-metabolism-and-consumption">
+<h2>Energy, Metabolism, and Consumption<a class="headerlink" href="#energy-metabolism-and-consumption" title="Link to this heading">#</a></h2>
+<p>All living organisms must maintain an energy balance:</p>
+<div class="math notranslate nohighlight">
+\[
+\text{Energy Consumed} = \text{Energy Used (Maintenance + Growth + Movement)}.
+\]</div>
+<p>Different organisms allocate energy to maintenance, growth, and movement in varying proportions. To do more than just maintain themselves, organisms must consume energy at a rate faster than their metabolic rate. This requires investment in activities such as foraging, which links energy acquisition to body size. For instance, autotrophs utilize photosynthesis, while heterotrophs depend on consuming organic material.</p>
+</section>
+<section id="size-and-metabolic-needs">
+<h2>Size and Metabolic Needs<a class="headerlink" href="#size-and-metabolic-needs" title="Link to this heading">#</a></h2>
+<p>Resting metabolic rate (<span class="math notranslate nohighlight">\(B\)</span>) increases with body size (<span class="math notranslate nohighlight">\(M\)</span>) following a power-law relationship:</p>
+<div class="math notranslate nohighlight">
+\[
+B = B_0 M^b,
+\]</div>
+<p>where <span class="math notranslate nohighlight">\(B_0\)</span> is a normalization constant and <span class="math notranslate nohighlight">\(b\)</span> is the scaling exponent, typically around <span class="math notranslate nohighlight">\(3/4\)</span>. Larger organisms have higher absolute metabolic rates but lower mass-specific rates:</p>
+<div class="math notranslate nohighlight">
+\[
+\frac{B}{M} = B_0 M^{b-1}.
+\]</div>
+<p>This relationship implies that larger organisms need to consume more energy overall but are more efficient per unit of body mass. This efficiency has significant implications for energy flow in ecosystems, as larger consumers typically occupy higher trophic levels and have lower population densities.</p>
+<section id="biomechanical-implications-of-size">
+<h3>Biomechanical Implications of Size<a class="headerlink" href="#biomechanical-implications-of-size" title="Link to this heading">#</a></h3>
+<p>Size also affects biomechanics and interactions with the physical environment. As Haldane (1926) observed:</p>
+<blockquote>
+<div><p>“You can drop a mouse down a thousand-yard mine shaft; and, on arriving at the bottom, it gets a slight shock and walks away, provided that the ground is fairly soft. A rat is killed, a man is broken, a horse splashes.”</p>
+</div></blockquote>
+<p>Size influences how organisms move, forage, and interact with resources, ultimately affecting their consumption rates. Smaller organisms, for example, often forage faster but consume smaller amounts per event, while larger organisms are slower but consume larger quantities.</p>
+</section>
+</section>
+<section id="dissecting-consumer-resource-interactions">
+<h2>Dissecting Consumer-Resource Interactions<a class="headerlink" href="#dissecting-consumer-resource-interactions" title="Link to this heading">#</a></h2>
+<p>Components of consumer-resource interactions determine consumption rate (<span class="math notranslate nohighlight">\(c\)</span>):</p>
+<div class="math notranslate nohighlight">
+\[
+ c = \frac{aR}{1 + ahR},
+\]</div>
+<p>where <span class="math notranslate nohighlight">\(a\)</span> is the search rate, <span class="math notranslate nohighlight">\(R\)</span> is resource density, and <span class="math notranslate nohighlight">\(h\)</span> is handling time. This is known as the Type II functional response, where resource handling dominates at high resource densities, while searching dominates at low densities.</p>
+<p>These interactions are further influenced by environmental factors, such as temperature and resource availability, which alter the rates of searching and handling.</p>
+</section>
+<section id="predicting-consumer-resource-interactions">
+<h2>Predicting Consumer-Resource Interactions<a class="headerlink" href="#predicting-consumer-resource-interactions" title="Link to this heading">#</a></h2>
+<p>Key components of consumer-resource interactions scale with size:</p>
+<ul class="simple">
+<li><p>Velocity and detection distance increase with consumer size.</p></li>
+<li><p>Handling time decreases with size.</p></li>
+<li><p>Consumption rate scales positively with size and interaction dimensionality (e.g., 2D vs. 3D).</p></li>
+</ul>
+<p>For example, in 3D environments like oceans, consumption rates scale more steeply with size compared to 2D environments. Studies have shown that aquatic predators exhibit 10x higher consumption rates at a given size compared to terrestrial predators due to the higher dimensionality of their interaction space.</p>
+</section>
+<section id="temperature-and-consumer-resource-interactions">
+<h2>Temperature and Consumer-Resource Interactions<a class="headerlink" href="#temperature-and-consumer-resource-interactions" title="Link to this heading">#</a></h2>
+<p>Temperature affects interaction components via the Boltzmann-Arrhenius equation:</p>
+<div class="math notranslate nohighlight">
+\[
+k = A e^{-\frac{E_a}{k_B T}},
+\]</div>
+<p>where <span class="math notranslate nohighlight">\(k\)</span> is the reaction rate, <span class="math notranslate nohighlight">\(A\)</span> is a constant, <span class="math notranslate nohighlight">\(E_a\)</span> is activation energy, <span class="math notranslate nohighlight">\(k_B\)</span> is Boltzmann’s constant, and <span class="math notranslate nohighlight">\(T\)</span> is temperature. Higher temperatures generally increase foraging velocity and reduce handling time, leading to higher consumption rates. For example, ectothermic organisms show a strong temperature dependence in their foraging activities, while endotherms have a more stable consumption rate due to thermal regulation.</p>
+</section>
+<section id="consumer-resource-dynamics">
+<h2>Consumer-Resource Dynamics<a class="headerlink" href="#consumer-resource-dynamics" title="Link to this heading">#</a></h2>
+<p>The metabolic basis of consumer-resource interactions can predict population dynamics using models like the Lotka-Volterra equations:</p>
+<div class="math notranslate nohighlight">
+\[
+\frac{dN}{dt} = r_m N \left(1 - \frac{N}{K}\right) - aNC,
+\]</div>
+<div class="math notranslate nohighlight">
+\[
+\frac{dC}{dt} = eaNC - zC,
+\]</div>
+<p>where <span class="math notranslate nohighlight">\(r_m\)</span> is the resource’s maximum growth rate, <span class="math notranslate nohighlight">\(K\)</span> is carrying capacity, <span class="math notranslate nohighlight">\(a\)</span> is search rate, <span class="math notranslate nohighlight">\(e\)</span> is conversion efficiency, and <span class="math notranslate nohighlight">\(z\)</span> is mortality rate. These parameters depend on size and temperature, linking individual metabolism to population-level outcomes. Additionally, such models help explain phenomena like population cycles and the effects of environmental changes on consumer-resource interactions.</p>
+</section>
+<section id="summary">
+<h2>Summary<a class="headerlink" href="#summary" title="Link to this heading">#</a></h2>
+<ul class="simple">
+<li><p>Metabolic rate and biomechanics shape species’ consumer-resource interactions and consumption rates.</p></li>
+<li><p>Size, temperature, and dimensionality influence these interactions.</p></li>
+<li><p>Models incorporating metabolic constraints can predict population dynamics and ecological patterns.</p></li>
+<li><p>Understanding these principles provides insights into energy flow and stability in ecosystems.</p></li>
+</ul>
+</section>
+<section id="discussion-questions">
+<h2>Discussion Questions<a class="headerlink" href="#discussion-questions" title="Link to this heading">#</a></h2>
+<ol class="arabic simple">
+<li><p>What are the main advantages of a consumer (heterotrophic) lifestyle compared to an producer (autotrophic) one? What are the disadvantages?</p></li>
+<li><p>Which is the most common and diverse organisms on planet earth ? Why? Think in terms of what type of consumer they are, and what their size is.</p></li>
+<li><p>What is the largest consumer on Earth? What is the smallest? In what respects is their “struggle for existence” similar? In what respects is it different?</p></li>
+<li><p>Which consumer(s) on earth operate(s) at the hottest temperatures? Which operate at the coldest? How is their “struggle for existence” similar or different?</p></li>
+</ol>
+</section>
+</section>
+
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+    <i class="fa-solid fa-list"></i> Contents
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+  <nav class="bd-toc-nav page-toc">
+    <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#introduction">Introduction</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#energy-metabolism-and-consumption">Energy, Metabolism, and Consumption</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#size-and-metabolic-needs">Size and Metabolic Needs</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#biomechanical-implications-of-size">Biomechanical Implications of Size</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#dissecting-consumer-resource-interactions">Dissecting Consumer-Resource Interactions</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#predicting-consumer-resource-interactions">Predicting Consumer-Resource Interactions</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#temperature-and-consumer-resource-interactions">Temperature and Consumer-Resource Interactions</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#consumer-resource-dynamics">Consumer-Resource Dynamics</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#summary">Summary</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#discussion-questions">Discussion Questions</a></li>
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diff --git a/notebooks/LaTeX.html b/notebooks/LaTeX.html
index 1b6db5c7..ac35135c 100644
--- a/notebooks/LaTeX.html
+++ b/notebooks/LaTeX.html
@@ -8,7 +8,7 @@
     <meta charset="utf-8" />
     <meta name="viewport" content="width=device-width, initial-scale=1.0" /><meta name="viewport" content="width=device-width, initial-scale=1" />
 
-    <title>Scientific documents with \(\LaTeX\) &#8212; TheMulQuaBio</title>
+    <title>Scientific documents with \(\LaTeX\) &#8212; MulQuaBio</title>
   
   
   
@@ -153,8 +153,8 @@
       
     
     
-    <img src="../_static/logo.png" class="logo__image only-light" alt="TheMulQuaBio - Home"/>
-    <script>document.write(`<img src="../_static/logo.png" class="logo__image only-dark" alt="TheMulQuaBio - Home"/>`);</script>
+    <img src="../_static/logo.png" class="logo__image only-light" alt="MulQuaBio - Home"/>
+    <script>document.write(`<img src="../_static/logo.png" class="logo__image only-dark" alt="MulQuaBio - Home"/>`);</script>
   
   
 </a></div>
@@ -188,23 +188,9 @@
 <li class="toctree-l1"><a class="reference internal" href="Python.html">Biological Computing in Python</a></li>
 <li class="toctree-l1"><a class="reference internal" href="R.html">Biological Computing in R</a></li>
 </ul>
-<p aria-level="2" class="caption" role="heading"><span class="caption-text">Maths for Biologists</span></p>
-<ul class="nav bd-sidenav">
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture01/01_Making_mathematical_statements.html">01 - Making mathematical statements</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture02/02_Describing_shapes_and_patterns.html">02   - Describing shapes and patterns</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture03/03_Analysing_change%2C_one_step_at_a_time.html">03   - Analysing change, one step at a time</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture04/04_From_table_arrangements_to_powerful_tools.html">04 - From table arrangements to powerful tools</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture05/05_From_table_arrangements_to_powerful_tools_part2.html">05 - From table arrangements to powerful tools - part 2</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture06/06_Analysing_change_continuously.html">06 - Analysing change, continuously</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture07/07_Analysing_change_continuously_part2.html">07 - Analysing change, continuously (Part 2)</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture08/08_Calculating_accumulated_change.html">08 - Calculating accumulated change</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/tutorials/lecture01/Tutorial_01.html">Tutorial 01</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/tutorials/lecture05/Tutorial_05.html">Tutorial 05</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/tutorials/lecture06/Tutorial_06.html">Tutorial 06</a></li>
-</ul>
 <p aria-level="2" class="caption" role="heading"><span class="caption-text">Basic Data Analyses and Statistics</span></p>
 <ul class="nav bd-sidenav">
-<li class="toctree-l1"><a class="reference internal" href="../Stats-Intro.html">Basic Data Science and Statistics: Introduction</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../intro-Stats.html">Introduction</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Data_R.html">Data Management and Visualization</a></li>
 <li class="toctree-l1"><a class="reference internal" href="ExpDesign.html">Experimental design and Data exploration</a></li>
 <li class="toctree-l1"><a class="reference internal" href="t_F_tests.html">Basic hypothesis testing: <span class="math notranslate nohighlight">\(t\)</span> and <span class="math notranslate nohighlight">\(F\)</span> tests</a></li>
@@ -224,6 +210,7 @@
 <li class="toctree-l1"><a class="reference internal" href="Appendix-JupyIntro.html">Introduction to Jupyter</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-Data-Python.html">Data analyses  with Python &amp; Jupyter</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-Maths.html">Mathematical models in Jupyter</a></li>
+<li class="toctree-l1"><a class="reference internal" href="Appendix-MathsForBiologists/MathsForBiologists.html">Maths for Biologists</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-Databases.html">Databases <span class="tocSkip"></span></a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-NLLS-Python.html">Model fitting in Python</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-MiniProj.html">The Computing Miniproject</a></li>
@@ -283,7 +270,7 @@
       
       
       
-      <li><a href="https://github.com/mhasoba/TheMulQuaBio" target="_blank"
+      <li><a href="https://github.com/MulQuaBio/MQB" target="_blank"
    class="btn btn-sm btn-source-repository-button dropdown-item"
    title="Source repository"
    data-bs-placement="left" data-bs-toggle="tooltip"
@@ -300,7 +287,7 @@
       
       
       
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+      <li><a href="https://github.com/MulQuaBio/MQB/issues/new?title=Issue%20on%20page%20%2Fnotebooks/LaTeX.html&body=Your%20issue%20content%20here." target="_blank"
    class="btn btn-sm btn-source-issues-button dropdown-item"
    title="Open an issue"
    data-bs-placement="left" data-bs-toggle="tooltip"
@@ -673,8 +660,8 @@ <h3>Typesetting math<a class="headerlink" href="#typesetting-math" title="Link t
 </pre></div>
 </div>
 <p>becomes</p>
-<div class="amsmath math notranslate nohighlight" id="equation-a1fe610c-78fe-4df8-ba04-3eeb2dc044ce">
-<span class="eqno">(1)<a class="headerlink" href="#equation-a1fe610c-78fe-4df8-ba04-3eeb2dc044ce" title="Permalink to this equation">#</a></span>\[\begin{equation}
+<div class="amsmath math notranslate nohighlight" id="equation-4484992c-b5a1-4b02-a8eb-d3e9306390b8">
+<span class="eqno">(1)<a class="headerlink" href="#equation-4484992c-b5a1-4b02-a8eb-d3e9306390b8" title="Permalink to this equation">#</a></span>\[\begin{equation}
     \int_0^1 \left(\ln \left( \frac{1}{x} \right) 
     \right)^y dx = y!
 \end{equation}\]</div>
@@ -1056,7 +1043,7 @@ <h3>Word Count in <span class="math notranslate nohighlight">\(\LaTeX\)</span><a
   <div class="footer-item">
     
 <p class="component-author">
-By TheMulQuaBio collective!
+By MulQuaBio collective!
 </p>
 
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diff --git a/notebooks/MLE.html b/notebooks/MLE.html
index 06af5f9d..34f7d9db 100644
--- a/notebooks/MLE.html
+++ b/notebooks/MLE.html
@@ -8,7 +8,7 @@
     <meta charset="utf-8" />
     <meta name="viewport" content="width=device-width, initial-scale=1.0" /><meta name="viewport" content="width=device-width, initial-scale=1" />
 
-    <title>Model Fitting using Maximum Likelihood &#8212; TheMulQuaBio</title>
+    <title>Model Fitting using Maximum Likelihood &#8212; MulQuaBio</title>
   
   
   
@@ -153,8 +153,8 @@
       
     
     
-    <img src="../_static/logo.png" class="logo__image only-light" alt="TheMulQuaBio - Home"/>
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+    <img src="../_static/logo.png" class="logo__image only-light" alt="MulQuaBio - Home"/>
+    <script>document.write(`<img src="../_static/logo.png" class="logo__image only-dark" alt="MulQuaBio - Home"/>`);</script>
   
   
 </a></div>
@@ -188,23 +188,9 @@
 <li class="toctree-l1"><a class="reference internal" href="Python.html">Biological Computing in Python</a></li>
 <li class="toctree-l1"><a class="reference internal" href="R.html">Biological Computing in R</a></li>
 </ul>
-<p aria-level="2" class="caption" role="heading"><span class="caption-text">Maths for Biologists</span></p>
-<ul class="nav bd-sidenav">
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture01/01_Making_mathematical_statements.html">01 - Making mathematical statements</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture02/02_Describing_shapes_and_patterns.html">02   - Describing shapes and patterns</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture03/03_Analysing_change%2C_one_step_at_a_time.html">03   - Analysing change, one step at a time</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture04/04_From_table_arrangements_to_powerful_tools.html">04 - From table arrangements to powerful tools</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture05/05_From_table_arrangements_to_powerful_tools_part2.html">05 - From table arrangements to powerful tools - part 2</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture06/06_Analysing_change_continuously.html">06 - Analysing change, continuously</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture07/07_Analysing_change_continuously_part2.html">07 - Analysing change, continuously (Part 2)</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture08/08_Calculating_accumulated_change.html">08 - Calculating accumulated change</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/tutorials/lecture01/Tutorial_01.html">Tutorial 01</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/tutorials/lecture05/Tutorial_05.html">Tutorial 05</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/tutorials/lecture06/Tutorial_06.html">Tutorial 06</a></li>
-</ul>
 <p aria-level="2" class="caption" role="heading"><span class="caption-text">Basic Data Analyses and Statistics</span></p>
 <ul class="nav bd-sidenav">
-<li class="toctree-l1"><a class="reference internal" href="../Stats-Intro.html">Basic Data Science and Statistics: Introduction</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../intro-Stats.html">Introduction</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Data_R.html">Data Management and Visualization</a></li>
 <li class="toctree-l1"><a class="reference internal" href="ExpDesign.html">Experimental design and Data exploration</a></li>
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@@ -224,6 +210,7 @@
 <li class="toctree-l1"><a class="reference internal" href="Appendix-JupyIntro.html">Introduction to Jupyter</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-Data-Python.html">Data analyses  with Python &amp; Jupyter</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-Maths.html">Mathematical models in Jupyter</a></li>
+<li class="toctree-l1"><a class="reference internal" href="Appendix-MathsForBiologists/MathsForBiologists.html">Maths for Biologists</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-Databases.html">Databases <span class="tocSkip"></span></a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-NLLS-Python.html">Model fitting in Python</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-MiniProj.html">The Computing Miniproject</a></li>
@@ -283,7 +270,7 @@
       
       
       
-      <li><a href="https://github.com/mhasoba/TheMulQuaBio" target="_blank"
+      <li><a href="https://github.com/MulQuaBio/MQB" target="_blank"
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    title="Source repository"
    data-bs-placement="left" data-bs-toggle="tooltip"
@@ -300,7 +287,7 @@
       
       
       
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+      <li><a href="https://github.com/MulQuaBio/MQB/issues/new?title=Issue%20on%20page%20%2Fnotebooks/MLE.html&body=Your%20issue%20content%20here." target="_blank"
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@@ -864,7 +851,7 @@ <h2>Readings and Resources <a id='Readings'></a><a class="headerlink" href="#rea
   <div class="footer-item">
     
 <p class="component-author">
-By TheMulQuaBio collective!
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new file mode 100644
index 00000000..f5fc5b20
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+++ b/notebooks/MathsForBiologists/MathsForBiologists.html
@@ -0,0 +1,3002 @@
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+    <h1>Maths for Biologists</h1>
+    <!-- Table of contents -->
+    <div id="print-main-content">
+        <div id="jb-print-toc">
+            
+            <div>
+                <h2> Contents </h2>
+            </div>
+            <nav aria-label="Page">
+                <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#making-mathematical-statements">Making mathematical statements</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#sets-and-relations">Sets and relations</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#polynomial-functions">Polynomial functions</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#degree-0">Degree 0</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#degree-1">Degree 1</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#degree-2">Degree 2</a><ul class="nav section-nav flex-column">
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#concavity">Concavity</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#intercept-roots">Intercept &amp; Roots</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#minimum-maximum">Minimum/ maximum</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#rational-functions">Rational functions</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#power-functions">Power functions</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#exponential-functions">Exponential functions</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#logarithmic-function">Logarithmic function</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#describing-shapes-and-patterns">Describing shapes and patterns</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#measuring-angles">Measuring angles</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#trigonometric-circle">Trigonometric circle</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#trigonometric-identities">Trigonometric identities</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#trigonometric-functions">Trigonometric functions</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#amplitude-and-period">Amplitude and Period</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#modelling-complex-periodic-phenomena">Modelling complex periodic phenomena</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#transformation-on-graphs">Transformation on graphs</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#translations">Translations</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#stretching-compression">Stretching/ Compression</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#reflection">Reflection</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#analysing-change-one-step-at-a-time">Analysing change, one step at a time</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#sequences">Sequences</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#limits">Limits</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#discrete-time-population-models">Discrete-time population models</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#density-dependent-population-growth">Density-dependent population growth</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#annual-plants">Annual plants</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#fibonacci-sequence">Fibonacci sequence</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#from-table-arrangements-to-powerful-tools">From table arrangements to powerful tools</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#systems-of-linear-equations">Systems of linear equations</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#matrix-operations">Matrix operations</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#inverse-matrix">Inverse matrix</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#determinant">Determinant</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#structured-models">Structured models</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#from-table-arrangements-to-powerful-tools-part-2">From table arrangements to powerful tools - part 2</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#linear-transformations">Linear transformations</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#eigenvalues-and-eigenvectors">Eigenvalues and eigenvectors</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#vector-decomposition">Vector decomposition</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#structured-models-revisited">Structured models revisited</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#analysing-change-continuously">Analysing change, continuously</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#average-rate-of-change">Average rate of change</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#instantaneous-rate-of-change">Instantaneous rate of change</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#calculating-derivatives">Calculating derivatives</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#chain-rule">Chain rule</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#higher-order-derivatives">Higher order derivatives</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#functions-of-several-variables">Functions of several variables</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#analysing-change-continuously-part-2">Analysing change, continuously (Part 2)</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#extreme-values-of-functions">Extreme values of functions</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#monotonicity-and-concavity">Monotonicity and concavity</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#finding-extrema">Finding extrema</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#inflection-points">Inflection points</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#series-expansions">Series expansions</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#calculating-accumulated-change">Calculating accumulated change</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#series">Series</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#calculating-areas">Calculating areas</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#fundamental-theorem-of-calculus">Fundamental theorem of Calculus</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#calculating-integrals">Calculating integrals</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#integration-by-substitution">Integration by substitution</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#integration-by-parts">Integration by parts</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#integration-by-partial-fractions">Integration by partial fractions</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#tutorial-01">Tutorial 01</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#tutorial-05">Tutorial 05</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#tutorial-06">Tutorial 06</a></li>
+</ul>
+            </nav>
+        </div>
+    </div>
+</div>
+
+              
+                
+<div id="searchbox"></div>
+                <article class="bd-article">
+                  
+  <section class="tex2jax_ignore mathjax_ignore" id="maths-for-biologists">
+<h1>Maths for Biologists<a class="headerlink" href="#maths-for-biologists" title="Link to this heading">#</a></h1>
+<section id="making-mathematical-statements">
+<h2>Making mathematical statements<a class="headerlink" href="#making-mathematical-statements" title="Link to this heading">#</a></h2>
+</section>
+<section id="sets-and-relations">
+<h2>Sets and relations<a class="headerlink" href="#sets-and-relations" title="Link to this heading">#</a></h2>
+<p>Start by watching the <a class="reference external" href="https://web.microsoftstream.com/video/a9ad1a8c-3d80-4910-b9f2-fab98c01b40b">video</a>.</p>
+</section>
+<section id="polynomial-functions">
+<h2>Polynomial functions<a class="headerlink" href="#polynomial-functions" title="Link to this heading">#</a></h2>
+<p>One of the simplest elemntary functions is given by:</p>
+<div class="math notranslate nohighlight">
+\[ f(x) = a_n x^n + a_{n-1} x^{n-1}+ \cdots + a_1 x^1 + a_0, \]</div>
+<p>where <span class="math notranslate nohighlight">\(n\)</span> is a <em>non-negative integer</em>, and the coefficients <span class="math notranslate nohighlight">\(a_n\)</span>, <span class="math notranslate nohighlight">\(a_{n-1}\)</span>, …, <span class="math notranslate nohighlight">\(a_0\)</span>. The highest value of <span class="math notranslate nohighlight">\(n\)</span> (for which the coefficient <span class="math notranslate nohighlight">\(a_n\)</span> is different from <span class="math notranslate nohighlight">\(0\)</span>) defines the <em>degree</em> of the polynomial.</p>
+<div class="admonition-examples admonition">
+<p class="admonition-title">Examples</p>
+<p><span class="math notranslate nohighlight">\(f(x) = 3x^5 + 12x^4 - 6x^3 - x^2 + x - 16\)</span> (Polynomial function of degree 5)<br />
+<span class="math notranslate nohighlight">\(g(x) = \frac{3}{4}x^2 - \frac{5}{2} \)</span> (Polynomial function of degree 2)<br />
+<span class="math notranslate nohighlight">\(h(x) = 1.6x^8 - 0.576x^4 + 1.89x^2 - 0.7\)</span> (Polynomial function of degree 8)</p>
+</div>
+<div class="admonition tip">
+<p class="admonition-title">Tip</p>
+<p>When the domain of the polynomial function is not explicitly given, it is frequently assumed to be all real numbers, <span class="math notranslate nohighlight">\({\rm I\!R}\)</span>.</p>
+</div>
+<div class="admonition tip">
+<p class="admonition-title">Tip</p>
+<p>When no application is specified, the function name ‘<span class="math notranslate nohighlight">\(f\)</span>’ and variable name ‘<span class="math notranslate nohighlight">\(x\)</span>’ are usually chosen. For application purposes, the function and variable are often named for what they represent; for example, <span class="math notranslate nohighlight">\(A(r)=\pi r^2\)</span> for the area of a circle dependent on its radius.</p>
+</div>
+<p>Let us discuss three of the most common forms of polynomial functions with degrees <span class="math notranslate nohighlight">\(0\)</span>, <span class="math notranslate nohighlight">\(1\)</span>, and <span class="math notranslate nohighlight">\(2\)</span>.</p>
+<section id="degree-0">
+<h3>Degree 0<a class="headerlink" href="#degree-0" title="Link to this heading">#</a></h3>
+<p>Polynomial functions of degree 0 assume the form:</p>
+<div class="math notranslate nohighlight">
+\[ f(x) = c, \]</div>
+<p>where <span class="math notranslate nohighlight">\(c\)</span> is a real number. This function is graphically represented by:</p>
+<img alt="../../_images/poly_deg0.png" src="../../_images/poly_deg0.png" />
+<div class="admonition-example admonition">
+<p class="admonition-title">Example</p>
+<p>For most mobile phone contracts nowadays a fixed fee is paid for the month with unlimited usage. So the cost as a function of used minutes would be a polynomial of degree zero, since the cost does not depend on the usage.</p>
+</div>
+</section>
+<section id="degree-1">
+<h3>Degree 1<a class="headerlink" href="#degree-1" title="Link to this heading">#</a></h3>
+<p>Polynomial functions of degree 1 are also called <em>linear functions</em> and are given by:</p>
+<div class="math notranslate nohighlight">
+\[ f(x) = mx + b,\]</div>
+<p>where again <span class="math notranslate nohighlight">\(m\)</span> and <span class="math notranslate nohighlight">\(b\)</span> are real numbers (with <span class="math notranslate nohighlight">\(m \ne 0\)</span>).</p>
+<p>The constants <span class="math notranslate nohighlight">\(m\)</span> and <span class="math notranslate nohighlight">\(b\)</span> are frequently referred to as <em>slope</em> and <em>intercept</em>. The intercept <span class="math notranslate nohighlight">\(b\)</span> shows where the function crosses the <span class="math notranslate nohighlight">\(y\)</span>-axis, while the slope <span class="math notranslate nohighlight">\(m\)</span> is related to the inclination of the function.</p>
+<p>The sign of the slope <span class="math notranslate nohighlight">\(m\)</span> also determines if the function increases or decreases as a function of <span class="math notranslate nohighlight">\(x\)</span>. We have:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split} \begin{align}
+  m&gt;0 &amp;\rightarrow \mbox{Increasing function of } x \\
+  m&lt;0 &amp;\rightarrow \mbox{Decreasing function of } x
+  \end{align} \end{split}\]</div>
+<p>One can also ask what is the value of <span class="math notranslate nohighlight">\(x\)</span> where the function <span class="math notranslate nohighlight">\(f(x)\)</span> crosses de <span class="math notranslate nohighlight">\(x\)</span>-axis, or, in mathematical notation, what is <span class="math notranslate nohighlight">\(x\ \mbox{so that}\ f(x) = 0\)</span>. This value of x is called a <strong>root of <span class="math notranslate nohighlight">\(f\)</span></strong> and it can be obtained by:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{align}
+    f(x) &amp;= 0 \\
+    mx + b &amp;= 0 \\
+    mx &amp;= -b \\
+    x &amp;= -\frac{b}{m}
+    \end{align} \end{split}\]</div>
+<div class="admonition-example admonition">
+<p class="admonition-title">Example</p>
+</div>
+<div class="admonition-examples admonition">
+<p class="admonition-title">Examples</p>
+<p>The metabolic rate of some organisms depends (approximately) linearly on the body’s temperature, with higher rates at higher temperatures.</p>
+</div>
+</section>
+<section id="degree-2">
+<h3>Degree 2<a class="headerlink" href="#degree-2" title="Link to this heading">#</a></h3>
+<p>Now we look at polynomial functions of degree 2, given by the general form:</p>
+<div class="math notranslate nohighlight">
+\[f(x) = ax^2 + bx +c,\]</div>
+<p>the coefficients <span class="math notranslate nohighlight">\(a\)</span>, <span class="math notranslate nohighlight">\(b\)</span>, and <span class="math notranslate nohighlight">\(c\)</span> are real numbers (with <span class="math notranslate nohighlight">\(a \ne 0\)</span>). The graph of a polynomial function of degree 2 is a <em>parabola</em> and many of its properties can be readily known by analysing the coefficients.</p>
+<section id="concavity">
+<h4>Concavity<a class="headerlink" href="#concavity" title="Link to this heading">#</a></h4>
+</section>
+<section id="intercept-roots">
+<h4>Intercept &amp; Roots<a class="headerlink" href="#intercept-roots" title="Link to this heading">#</a></h4>
+<p>The expression inside the square root distinguish between different properties of this function and therefore receives the name of <em>discriminant</em>:</p>
+<div class="math notranslate nohighlight">
+\[ \Delta = b^2 -4ac.\]</div>
+<p>Depending on the value of the discriminant a quick conclusion can be reached regarding the roots of the function.</p>
+<div class="admonition-example admonition">
+<p class="admonition-title">Example</p>
+<p>If we throw a ball that is heavy enough so that air resistance is negligible (alternatively we can travel to the moon) the height of the ball as a function of time follows a quadratic equation. Knowing the roots we can calculate when and where the ball will land.</p>
+</div>
+</section>
+<section id="minimum-maximum">
+<h4>Minimum/ maximum<a class="headerlink" href="#minimum-maximum" title="Link to this heading">#</a></h4>
+<p>Depending on the concavity of the parabola (up or down), there will be a point where the function assumes a minimum or a maximum value. This point is called the <em>vertex</em> of the parabola. This extreme point can be easily determined by the symmetry properties of the function curve.</p>
+<div class="admonition-example admonition">
+<p class="admonition-title">Example</p>
+<p>Imagine the population of a specific region having a negative quadratic relationship - it undergoes initial growth, reaches its peak, and then declines. To determine both the maximum population size and the time when it peaked, we need to identify the extrema of this parabolic curve.</p>
+</div>
+<p>Now watch this <a class="reference external" href="https://web.microsoftstream.com/video/f64abd8a-9997-4b5e-b11f-0ca09b7603fe">video</a> for a short introduction to complex numbers.</p>
+</section>
+</section>
+</section>
+<section id="rational-functions">
+<h2>Rational functions<a class="headerlink" href="#rational-functions" title="Link to this heading">#</a></h2>
+<p>Rational functions are defined as quotients of polynomial functions:</p>
+<div class="math notranslate nohighlight">
+\[f(x) = \frac{p(x)}{q(x)},\ \ \ q(x) \ne 0\]</div>
+<div class="admonition-example admonition">
+<p class="admonition-title">Example</p>
+<p>Rational functions are often used to express ratios, for example of chemicals. Lets assume chemical A increases quadratically <span class="math notranslate nohighlight">\(A(t) = 2 t^2\)</span> and chemical B increases linearly <span class="math notranslate nohighlight">\(B(t)=t+5\)</span>, the ratio will be:</p>
+<div class="math notranslate nohighlight">
+\[r(t) = \frac{A(t)}{B(t)} = 2\frac{t^2}{t+5}\]</div>
+</div>
+</section>
+<section id="power-functions">
+<h2>Power functions<a class="headerlink" href="#power-functions" title="Link to this heading">#</a></h2>
+<p>Another type of function used in many applications are power functions (sometimes also called power laws). These functions have the functional form:</p>
+<div class="math notranslate nohighlight">
+\[f(x) = Cx^{\alpha}\]</div>
+<p>where <span class="math notranslate nohighlight">\(C\)</span> and <span class="math notranslate nohighlight">\(\alpha\)</span> are real constants (with <span class="math notranslate nohighlight">\(C \ne 0\)</span>).</p>
+<p>Many commonly used functions are examples of power functions, e.g. the square root <span class="math notranslate nohighlight">\(\sqrt{x}=x^{\frac 12}\)</span> or the inverse function <span class="math notranslate nohighlight">\(x^{-1}\)</span>.</p>
+<p>Watch this <a class="reference external" href="https://web.microsoftstream.com/video/bf7415e1-823f-449e-ace1-e379fac8fe0a">video</a> about additional properties of power law functions.</p>
+</section>
+<section id="exponential-functions">
+<h2>Exponential functions<a class="headerlink" href="#exponential-functions" title="Link to this heading">#</a></h2>
+<p>Similar to power functions, but with very different properties are the exponential functions:</p>
+<div class="math notranslate nohighlight">
+\[f(x) = a^x, \]</div>
+<p>where <span class="math notranslate nohighlight">\(a\)</span> is a positive constant. The parameter <span class="math notranslate nohighlight">\(a\)</span> is called <em>base</em> and the variable <span class="math notranslate nohighlight">\(x\)</span> is the exponent (remember that for power functions, the variable <span class="math notranslate nohighlight">\(x\)</span> is the base, raised to the power of the parameter <span class="math notranslate nohighlight">\(\alpha\)</span>).</p>
+<p>Exponential functions are used to model many different real-world applications, from bacterial growth to radioactive decay.</p>
+<div class="admonition-example admonition">
+<p class="admonition-title">Example</p>
+<p>Suppose we start a bacterial colony with a single individual on a Petri dish. At each time step, counted as an average reproduction interval, each individual bacterium duplicates, originating two other individuals. The number of bacteria in the colony as a function of time, <span class="math notranslate nohighlight">\(N(t)\)</span>, can be counted as:</p>
+<div class="pst-scrollable-table-container"><table class="table">
+<thead>
+<tr class="row-odd"><th class="head"><p><span class="math notranslate nohighlight">\(t\)</span></p></th>
+<th class="head text-center"><p><span class="math notranslate nohighlight">\(N(t)\)</span></p></th>
+</tr>
+</thead>
+<tbody>
+<tr class="row-even"><td><p><span class="math notranslate nohighlight">\(0\)</span></p></td>
+<td class="text-center"><p><span class="math notranslate nohighlight">\(1\ (= 2^0)\)</span></p></td>
+</tr>
+<tr class="row-odd"><td><p><span class="math notranslate nohighlight">\(1\)</span></p></td>
+<td class="text-center"><p><span class="math notranslate nohighlight">\(2\ (= 2^1)\)</span></p></td>
+</tr>
+<tr class="row-even"><td><p><span class="math notranslate nohighlight">\(2\)</span></p></td>
+<td class="text-center"><p><span class="math notranslate nohighlight">\(4\ (= 2^2)\)</span></p></td>
+</tr>
+<tr class="row-odd"><td><p><span class="math notranslate nohighlight">\(3\)</span></p></td>
+<td class="text-center"><p><span class="math notranslate nohighlight">\(8\ (= 2^3)\)</span></p></td>
+</tr>
+<tr class="row-even"><td><p><span class="math notranslate nohighlight">\(4\)</span></p></td>
+<td class="text-center"><p><span class="math notranslate nohighlight">\(16\ (= 2^4)\)</span></p></td>
+</tr>
+<tr class="row-odd"><td><p><span class="math notranslate nohighlight">\(5\)</span></p></td>
+<td class="text-center"><p><span class="math notranslate nohighlight">\(32\ (= 2^5)\)</span></p></td>
+</tr>
+</tbody>
+</table>
+</div>
+<p>If no limitation (on space or resources) is imposed to the colony, the number of bacteria at each time step <span class="math notranslate nohighlight">\(t\)</span> can be well modelled by an exponential function of the form:</p>
+<div class="math notranslate nohighlight">
+\[ N(t) = 2^t. \]</div>
+</div>
+<p>Exponential functions of the form <span class="math notranslate nohighlight">\(f(x) = a^x\)</span> (with <span class="math notranslate nohighlight">\(a&gt;0\)</span>) can never assume negative values and their behaviour depends on the value of <span class="math notranslate nohighlight">\(a\)</span>.</p>
+<img alt="../../_images/exponential_01.png" class="align-center" src="../../_images/exponential_01.png" />
+<div class="admonition tip">
+<p class="admonition-title">Tip</p>
+<p>Note that the function <em>increases</em> with <span class="math notranslate nohighlight">\(x\)</span> if <span class="math notranslate nohighlight">\(a&gt;1\)</span> (growth) and <em>decreases</em> with <span class="math notranslate nohighlight">\(x\)</span> if <span class="math notranslate nohighlight">\(0&lt;a&lt;1\)</span> (decay). To remember this, one can directly apply the properties of exponentiation. If <span class="math notranslate nohighlight">\(f(x) = (1/3)^x\)</span> (thus <span class="math notranslate nohighlight">\(a = (1/3) &lt; 1\)</span>), we have:</p>
+<div class="math notranslate nohighlight">
+\[f(x) = \left(\frac{1}{3}\right)^x = \frac{1}{3^x},\]</div>
+<p>and since <span class="math notranslate nohighlight">\(3^x\)</span> is an increasing function of <span class="math notranslate nohighlight">\(x\)</span>, then <span class="math notranslate nohighlight">\(1/3^x\)</span> should be a decreasing function of <span class="math notranslate nohighlight">\(x\)</span> (as the denominator increases, the fraction decreases).</p>
+</div>
+<div class="admonition-example admonition">
+<p class="admonition-title">Example</p>
+<p>In the early stages of a new desease, we can approximate the spread as exponential. In this case <span class="math notranslate nohighlight">\(a\)</span> is the number of people every infected person infects themselves. For <span class="math notranslate nohighlight">\(a&lt;1\)</span> every person infects less than one other person on average and the desease will slow down, for <span class="math notranslate nohighlight">\(a&gt;1\)</span> the desease will spread. At the beginning of the COVID-19 pandemic, an estimate for <span class="math notranslate nohighlight">\(a\)</span> (or in this case <span class="math notranslate nohighlight">\(R\)</span> for reproductive value) was <span class="math notranslate nohighlight">\(a=4\)</span>. If we start with 1 infected person, how many do we have after 10 infection cycles?</p>
+</div>
+<p>Two of the most used bases are <span class="math notranslate nohighlight">\(a=10\)</span> and <span class="math notranslate nohighlight">\(a=e\)</span>, where <span class="math notranslate nohighlight">\(e=2.71828\)</span> (called <em>Euler’s number</em> or <em>Napier’s constant</em>). Exponential functions with <span class="math notranslate nohighlight">\(a=e\)</span> are commonly written as:</p>
+<div class="math notranslate nohighlight">
+\[f(x) = e^x = \mbox{exp}(x) \]</div>
+<p>The unique charachteristic of the exponential of base <span class="math notranslate nohighlight">\(e\)</span> is, that it is its own derivative</p>
+<div class="math notranslate nohighlight">
+\[\frac{\text{d}}{\text{d}x} e^x = e^x \]</div>
+<p>The both formulations can be changed into one another via</p>
+<div class="math notranslate nohighlight">
+\[f(x) = e^{\lambda x} = ({\underbrace{e^\lambda}_a})^x.\]</div>
+<p>We have exponential growth when <span class="math notranslate nohighlight">\(a&gt;1\)</span> and therefore when <span class="math notranslate nohighlight">\(\lambda&gt;0\)</span> and decay when <span class="math notranslate nohighlight">\(0&lt;a&lt;1\)</span> and therefore <span class="math notranslate nohighlight">\(\lambda&lt;0\)</span>.</p>
+</section>
+<section id="logarithmic-function">
+<h2>Logarithmic function<a class="headerlink" href="#logarithmic-function" title="Link to this heading">#</a></h2>
+<p>Let us start this section by understanding what the expression</p>
+<div class="math notranslate nohighlight">
+\[ \mbox{log}_a b = x \]</div>
+<p>actually means. One possible way to read it is: “<span class="math notranslate nohighlight">\(\mbox{log}_a b\)</span> is the number with which I have to exponentiate the base <span class="math notranslate nohighlight">\(a\)</span> so that I find the number <span class="math notranslate nohighlight">\(b\)</span>. Since “<span class="math notranslate nohighlight">\(\mbox{log}_a b = x\)</span>, the above expression is equivalent to:</p>
+<div class="math notranslate nohighlight">
+\[ a^x = b.\]</div>
+<p>The expression <span class="math notranslate nohighlight">\(\mbox{log}_a b\)</span> is called the <em>logarithm of b in the base a</em> and it is also possible to define a logarithmic function given by:</p>
+<div class="math notranslate nohighlight">
+\[f(x) = \mbox{log}_a x,\]</div>
+<p>where <span class="math notranslate nohighlight">\(a\)</span> is the base of the logarithm and it is assumed <span class="math notranslate nohighlight">\(a&gt;0\)</span> and <span class="math notranslate nohighlight">\(a \ne 1\)</span>.</p>
+<p>By defining a function like this, for each <span class="math notranslate nohighlight">\(x\)</span> we will have a number <span class="math notranslate nohighlight">\(y = f(x)\)</span> for which <span class="math notranslate nohighlight">\(a^y = x\)</span>. Therefore, the logarithmic function is only defined for <span class="math notranslate nohighlight">\(x&gt;0\)</span>. Like the exponential function, the behaviour of the logarithmic function for increasing <span class="math notranslate nohighlight">\(x\)</span> will depend on the value of the base <span class="math notranslate nohighlight">\(a\)</span>.</p>
+<div class="admonition tip">
+<p class="admonition-title">Tip</p>
+<p>The similarities between the graphs for exponential and logarithmic functions are not coincidental. In fact, the graph for a logarithmic function with base <span class="math notranslate nohighlight">\(a\)</span> can be obtained by drawing the graph for an exponential function with base <span class="math notranslate nohighlight">\(a\)</span> and reflecting it in relation to the line <span class="math notranslate nohighlight">\(y = x\)</span>.  [MAKE COMMENT ON INVERSE FUNCTIONS]</p>
+<img alt="../../_images/log_base_02.png" src="../../_images/log_base_02.png" />
+<p>(here we use the common nomenclature <span class="math notranslate nohighlight">\(\mbox{log}_e x = \mbox{ln}\ x\)</span>).</p>
+</div>
+</section>
+<section id="describing-shapes-and-patterns">
+<h2>Describing shapes and patterns<a class="headerlink" href="#describing-shapes-and-patterns" title="Link to this heading">#</a></h2>
+</section>
+<section id="measuring-angles">
+<h2>Measuring angles<a class="headerlink" href="#measuring-angles" title="Link to this heading">#</a></h2>
+<p>An <strong>angle</strong> is a primitive geometrical element, such as the <em>point</em>, the <em>curve</em> or the <em>plane</em>. It is defined as the figure formed by two line segments that extend from a common point.</p>
+<br/>
+<img alt="../../_images/angle.png" src="../../_images/angle.png" />
+<br/>
+<br/>
+<p>The two most common units for measuring angles are the <em>degree</em> and the <em>radian</em>.</p>
+<p>For a given angle, transforming between units of degrees and radians can be simply done with:</p>
+<div class="math notranslate nohighlight">
+\[\frac{360^{\circ}}{\theta} = \frac{2\pi}{\phi},\]</div>
+<p>where <span class="math notranslate nohighlight">\(\theta\)</span> is the angle measured in degrees and <span class="math notranslate nohighlight">\(\phi\)</span>, measured in radians. In this lecture, we will consider the angles measured in radians (independent of the symbol used for the variable), unless stated otherwise.</p>
+<div class="admonition-example admonition">
+<p class="admonition-title">Example</p>
+<p>If <span class="math notranslate nohighlight">\(\theta = 90^{\circ}\)</span>, then using the previous expression we have <span class="math notranslate nohighlight">\(\phi = \displaystyle\frac{\pi}{2}\)</span>. Since <span class="math notranslate nohighlight">\(\theta\)</span> and $\phi are directly proportional, we also have:</p>
+<div class="math notranslate nohighlight">
+\[\theta = 180^{\circ} \longrightarrow \phi = 2 \cdot \frac{\pi}{2} = \pi \]</div>
+<div class="math notranslate nohighlight">
+\[\theta = 270^{\circ} \longrightarrow \phi = 3 \cdot \frac{\pi}{2} = \frac{3\pi}{2} \]</div>
+<div class="math notranslate nohighlight">
+\[\theta = 360^{\circ} \longrightarrow \phi = 4 \cdot \frac{\pi}{2} = 2\pi \]</div>
+</div>
+</section>
+<section id="trigonometric-circle">
+<h2>Trigonometric circle<a class="headerlink" href="#trigonometric-circle" title="Link to this heading">#</a></h2>
+<p>We obtain the trigonometric circle by superimposing a circle of radius <span class="math notranslate nohighlight">\(1\)</span> to the cartesian system of coordinates:</p>
+<img alt="../../_images/trig_circ.png" src="../../_images/trig_circ.png" />
+<p>The projections on the <span class="math notranslate nohighlight">\(x\)</span> and <span class="math notranslate nohighlight">\(y\)</span>-axes have special meanings, if we consider the angle <span class="math notranslate nohighlight">\(\theta\)</span> between the line segment in the radial direction and the <span class="math notranslate nohighlight">\(x\)</span>-axis.</p>
+<p>From the geometric relations on the right triangle (triangle in which one internal angle is a right angle):</p>
+<p>Back to the trigonometric circle, it is possible to construct a right triangle where the hypothenuse corresponds to the radius of the superimposed circle. In this case, since this radius equals to <span class="math notranslate nohighlight">\(1\)</span>, we have:</p>
+<div class="math notranslate nohighlight">
+\[\mbox{sin}(\theta) = y\]</div>
+<div class="math notranslate nohighlight">
+\[\mbox{cos}(\theta) = x.\]</div>
+<p>In other words, the projections of the end of the radial line segment on the <span class="math notranslate nohighlight">\(y\)</span> and <span class="math notranslate nohighlight">\(x\)</span>-axes correspond to the sine and cosine of the angle <span class="math notranslate nohighlight">\(\theta\)</span>. Sine and cosine for this angle can be obtained by the projections even if <span class="math notranslate nohighlight">\(\theta &gt; \displaystyle\frac{\pi}{2}\)</span>.</p>
+<section id="trigonometric-identities">
+<h3>Trigonometric identities<a class="headerlink" href="#trigonometric-identities" title="Link to this heading">#</a></h3>
+<p>With the aid of the projections on the trigonometric circle, some trigonometric identities become imediately clear. Consider <span class="math notranslate nohighlight">\(\theta &gt; 0\)</span> for counterclockwise angle measurements starting from the <span class="math notranslate nohighlight">\(x\)</span>-axis and <span class="math notranslate nohighlight">\(\theta &lt; 0\)</span> for clockwise measurements. We have:</p>
+<div class="math notranslate nohighlight">
+\[\mbox{sin}(0) = 0,\ \ \ \mbox{sin}\left(\frac{\pi}{2}\right) = 1,\ \ \ \mbox{sin}(\pi) = 0,\ \ \ \mbox{sin}\left(\frac{3\pi}{2}\right) = -1,\ \ \ \mbox{sin}(2\pi) = 0\]</div>
+<div class="math notranslate nohighlight">
+\[\mbox{cos}(0) = 1,\ \ \ \mbox{cos}\left(\frac{\pi}{2}\right) = 0,\ \ \ \mbox{cos}(\pi) = -1,\ \ \ \mbox{cos}\left(\frac{3\pi}{2}\right) = 0,\ \ \ \mbox{cos}(2\pi) = 1\]</div>
+<p>From the fact that a full rotation brings you back to the same point:</p>
+<div class="math notranslate nohighlight">
+\[\mbox{sin}(\theta) = \mbox{sin}(\theta + 2\pi)\]</div>
+<div class="math notranslate nohighlight">
+\[\mbox{cos}(\theta) = \mbox{cos}(\theta + 2\pi)\]</div>
+<p>or from Pythagoras’ theorem on the right triangle in the trigonometric circle:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{align}
+[\mbox{sin}(\theta)]^2 + [\mbox{cos}(\theta)]^2 =&amp; 1^2 \\
+\mbox{sin}^2(\theta) + \mbox{cos}^2(\theta) =&amp; 1
+\end{align}\end{split}\]</div>
+<div class="admonition tip">
+<p class="admonition-title">Tip</p>
+<p>The notation <span class="math notranslate nohighlight">\(\mbox{sin}^2(\theta)\)</span> is frequently used to express the square of <span class="math notranslate nohighlight">\(\mbox{sin}(\theta)\)</span>, or:</p>
+<div class="math notranslate nohighlight">
+\[ \mbox{sin}^2(\theta) = [\mbox{sin}(\theta)]^2 = [\mbox{sin}(\theta)]\cdot[\mbox{sin}(\theta)],\]</div>
+<p>which is different from <span class="math notranslate nohighlight">\(\mbox{sin}(\theta^2)\)</span>.</p>
+</div>
+<p>Now watch this <a class="reference external" href="https://web.microsoftstream.com/video/b3023026-7d45-4378-a8ee-c166a43eb2e7">video</a> for a different application of the same trigonometric construct.</p>
+</section>
+</section>
+<section id="trigonometric-functions">
+<h2>Trigonometric functions<a class="headerlink" href="#trigonometric-functions" title="Link to this heading">#</a></h2>
+<p>Since sine and cosine are defined for any real value of a given angle <span class="math notranslate nohighlight">\(x\)</span> (<span class="math notranslate nohighlight">\(x \in {\rm I\!R}\)</span>), we can define the functions:</p>
+<div class="math notranslate nohighlight">
+\[f(x) = \mbox{sin}(x)\]</div>
+<div class="math notranslate nohighlight">
+\[f(x) = \mbox{cos}(x).\]</div>
+<p>Thus, the association of sine and cosine with pure geometric ratios gives place for a more generic definition of functions of real variables. The graphs of these functions are given by:</p>
+<img alt="../../_images/sin_cos.png" src="../../_images/sin_cos.png" />
+<p>All the other known trigonometric relations can also be defined as functions in a similar way. Take for example the tangent of an angle <span class="math notranslate nohighlight">\(\theta\)</span>, which is:</p>
+<div class="math notranslate nohighlight">
+\[tan(\theta) = \displaystyle\frac{\mbox{sin}(\theta)}{\mbox{cos}(\theta)}.\]</div>
+<p>If <span class="math notranslate nohighlight">\(x \in {\rm I\!R}\)</span>, than we can define a function</p>
+<div class="math notranslate nohighlight">
+\[ f(x) = \mbox{tan}(x) = \displaystyle\frac{\mbox{sin}(x)}{cos(x)}, \ \ \ (\mbox{for }\mbox{cos}(x) \ne 0).\]</div>
+<p>The graph from the tangent function will be given by:</p>
+<img alt="../../_images/tan.png" src="../../_images/tan.png" />
+<p>Note by the above graphs that the functions sine and cosine have patterns that repeat themselves every interval of <span class="math notranslate nohighlight">\(2\pi\)</span>, while the tangent repeats itself every interval of <span class="math notranslate nohighlight">\(\pi\)</span>. We say that these three functions are <strong>periodic</strong>.</p>
+<p>A function <span class="math notranslate nohighlight">\(f(x)\)</span> is periodic if there is a positive constant <span class="math notranslate nohighlight">\(a\)</span>, so that:</p>
+<div class="math notranslate nohighlight">
+\[f(x + a) = f(x),\]</div>
+<p>for all values of <span class="math notranslate nohighlight">\(x\)</span> for which the function f(x) is defined.</p>
+<p>If <span class="math notranslate nohighlight">\(a\)</span> is the smallest number with this property, we call it the <em>period</em> of f(x).</p>
+<div class="admonition-example admonition">
+<p class="admonition-title">Example</p>
+<p>Since <span class="math notranslate nohighlight">\(\mbox{sin}(x + 2\pi) = \mbox{sin}(x)\)</span> for any value of <span class="math notranslate nohighlight">\(x\)</span> and since <span class="math notranslate nohighlight">\(2\pi\)</span> is the smallest number for which this property holds, then sine is a periodic function with period <span class="math notranslate nohighlight">\(2\pi\)</span>. With the same reasoning, it can be shown that the tangent is periodic with period <span class="math notranslate nohighlight">\(\pi\)</span>.</p>
+</div>
+<div class="admonition-example admonition">
+<p class="admonition-title">Example</p>
+<p>Many natural processes can be modelled as periodic. Think for example of a pendulum, the tides, the seasons. Often periodicity can be a good first order approximation, even if it does not strictly hold. For example when describimg the shape of the cloud cover in the picture below.</p>
+</div>
+<img alt="../../_images/cloud.png" src="../../_images/cloud.png" />
+<section id="amplitude-and-period">
+<h3>Amplitude and Period<a class="headerlink" href="#amplitude-and-period" title="Link to this heading">#</a></h3>
+<p>From the basic functions <span class="math notranslate nohighlight">\(\mbox{sin}(x)\)</span> or <span class="math notranslate nohighlight">\(\mbox{cos}(x)\)</span> we can write a more general function <span class="math notranslate nohighlight">\(f(x)\)</span> given by</p>
+<div class="math notranslate nohighlight">
+\[f(x) = A\mbox{sin}(\omega x),\ \ \ \ \ \ A, \omega \in {\rm I\!R}\ \ (A, \omega \ne 0)\]</div>
+<p>Since the sine ranges from <span class="math notranslate nohighlight">\(-1\)</span> to <span class="math notranslate nohighlight">\(1\)</span> (see the graph for <span class="math notranslate nohighlight">\(sin(x)\)</span> for reference), by its definition, <span class="math notranslate nohighlight">\(f(x)\)</span> will range from <span class="math notranslate nohighlight">\(-A\)</span> to <span class="math notranslate nohighlight">\(A\)</span>. We call <span class="math notranslate nohighlight">\(A\)</span> the <strong>amplitude</strong> of <span class="math notranslate nohighlight">\(f(x)\)</span> and it gives the extent of the variation of the oscillation of the function.</p>
+<p>Being sine a periodic function, <span class="math notranslate nohighlight">\(f(x)\)</span> will also be, possibly with a different period. The period <span class="math notranslate nohighlight">\(T\)</span> of <span class="math notranslate nohighlight">\(f(x)\)</span>, as discussed before, will be determined by the equation <span class="math notranslate nohighlight">\(f(x + T) = f(x)\)</span>, that should be valid for every <span class="math notranslate nohighlight">\(x\)</span>. Thus:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{align}
+    f(x + T) &amp;= f(x) \\
+    A\mbox{sin}(\omega(x+T)) &amp;= A\mbox{sin}(\omega x) \\
+    \mbox{sin}(\omega(x+T)) &amp;= \mbox{sin}(\omega x) \\
+    \mbox{sin}(\omega x+\omega T)) &amp;= \mbox{sin}(\omega x) \\
+    \mbox{sin}(z+\omega T)) &amp;= \mbox{sin}(z)\ \ \ \ \ \ (\mbox{where we change the variable to } z=\omega x)
+  \end{align}\end{split}\]</div>
+<p>But since we know that sine is periodic with period <span class="math notranslate nohighlight">\(2\pi\)</span>, we should have <span class="math notranslate nohighlight">\(|\omega|T = 2\pi\)</span>. Thus, the <strong>period</strong> of the function <span class="math notranslate nohighlight">\(f(x)\)</span> will be:</p>
+<div class="math notranslate nohighlight">
+\[T = \frac{2\pi}{\omega}.\]</div>
+<p>The constant <span class="math notranslate nohighlight">\(\omega = \displaystyle\frac{2\pi}{T}\)</span> is called <strong>angular frequency</strong> and measures the number of full rotations the function <span class="math notranslate nohighlight">\(f(x)\)</span> oscillates in one interval of <span class="math notranslate nohighlight">\(T\)</span>.</p>
+<div class="admonition-example admonition">
+<p class="admonition-title">Example</p>
+<p>The function <span class="math notranslate nohighlight">\(f(x) = 3.6\, \mbox{cos}\left(\displaystyle\frac{\pi}{3}x\right)\)</span> has an amplitude <span class="math notranslate nohighlight">\(A=3.6\)</span>, an angular frequency <span class="math notranslate nohighlight">\(\omega = \displaystyle\frac{\pi}{3}\)</span>, and a period <span class="math notranslate nohighlight">\(T = \displaystyle\frac{2\pi}{\omega} = \displaystyle\frac{2 \pi}{\frac{\pi}{3}} = 6\)</span>.</p>
+</div>
+</section>
+</section>
+<section id="modelling-complex-periodic-phenomena">
+<h2>Modelling complex periodic phenomena<a class="headerlink" href="#modelling-complex-periodic-phenomena" title="Link to this heading">#</a></h2>
+<p>Watch this <a class="reference external" href="https://web.microsoftstream.com/video/02f373a2-621b-4313-a18a-265f5fb65656">video</a> about periodic functions decomposition.</p>
+</section>
+<section id="transformation-on-graphs">
+<h2>Transformation on graphs<a class="headerlink" href="#transformation-on-graphs" title="Link to this heading">#</a></h2>
+<p>Starting from the elementary functions we have discussed</p>
+<div class="math notranslate nohighlight">
+\[ax+b,\ \ ax^2+bx+c,\ \ \sqrt{x},\ \ \mbox{e}^x,\ \ \mbox{log}(x),\ \ \mbox{sin}(x),\ \ \mbox{cos}(x)\]</div>
+<p>we can introduce several transformations to obtain new functions, by adding or multiplying constants to the argument (function variable) or to the whole function. These are the simplest cases of what is know as <em>function composition</em> with the resulting function called <em>composite function</em>.</p>
+<p>When working with mathematical models, this will help you to understand how the shape of a given function might change by changing one of the parameters, or how to construct a set of functions (that might be useful for your particular model) with small modifications of basic functions.</p>
+<section id="translations">
+<h3>Translations<a class="headerlink" href="#translations" title="Link to this heading">#</a></h3>
+<p>A given function can be dislocated, either vertically or horiontally, by <em>adding</em> a constant <span class="math notranslate nohighlight">\(c\)</span> to the whole function or to its argument. In general, starting from a function <span class="math notranslate nohighlight">\(f(x)\)</span>, translations are built following the rules:</p>
+<ul>
+<li><p><strong>Horizontal</strong> (Constant <span class="math notranslate nohighlight">\(c\)</span> summed to the argument of the function)</p>
+<p><span class="math notranslate nohighlight">\(y = f(x + c)\ \ \ \  -  \left\{
+\begin{array}{ll}
+    c &gt; 0 &amp; \rightarrow \ \  \mbox{function moves to the left} \\
+    c &lt; 0 &amp; \rightarrow \ \  \mbox{function moves to the right} \\
+\end{array} 
+\right. \)</span></p>
+</li>
+<li><p><strong>Vertical</strong> (Constant <span class="math notranslate nohighlight">\(c\)</span> summed to the whole function)</p>
+<p><span class="math notranslate nohighlight">\(y = f(x) + c\ \ \ \  -  \left\{
+\begin{array}{ll}
+    c &gt; 0 &amp; \rightarrow \ \  \mbox{function moves up} \\
+    c &lt; 0 &amp; \rightarrow \ \  \mbox{function moves down} \\
+\end{array} 
+\right. \)</span></p>
+</li>
+</ul>
+<div class="admonition-example admonition">
+<p class="admonition-title">Example</p>
+<p>If we start with the function <span class="math notranslate nohighlight">\(f(x) = x^3\)</span>, the function <span class="math notranslate nohighlight">\(y = f(x + 2) = (x+2)^3\)</span> will represent a horizontal translation, while the function <span class="math notranslate nohighlight">\(y = f(x) + 2 = x^3+2\)</span> will represent a vertical translation.</p>
+</div>
+<p>To see how this works, try to implement the following code in your own jupyter notebook. Try to use different functions <span class="math notranslate nohighlight">\(f(x)\)</span> (defined in ‘func(x)’) to see how different graphs behave to horizontal and vertical translations.</p>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">ipywidgets</span> <span class="kn">import</span> <span class="n">interact</span>
+<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
+<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+
+<span class="k">def</span> <span class="nf">func</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
+    <span class="k">return</span> <span class="n">x</span><span class="o">**</span><span class="mi">3</span>
+
+<span class="k">def</span> <span class="nf">htrans</span><span class="p">(</span><span class="n">c</span><span class="p">):</span>
+    <span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span><span class="mi">3</span><span class="p">)</span>
+    
+    <span class="n">y</span> <span class="o">=</span> <span class="n">func</span><span class="p">(</span><span class="n">x</span><span class="o">+</span><span class="n">c</span><span class="p">)</span> <span class="c1">#constant c in the argument of the function</span>
+    
+    <span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;$f(x+c)$&#39;</span><span class="p">)</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s1">&#39;Horizontal translation&#39;</span><span class="p">)</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">12</span><span class="p">)</span>
+    
+    <span class="n">plt</span><span class="o">.</span><span class="n">ylim</span><span class="p">(</span><span class="o">-</span><span class="mi">8</span><span class="p">,</span> <span class="mi">8</span><span class="p">)</span> <span class="c1">#remember to change the limits for better visualisation if necessary</span>
+    
+    <span class="n">plt</span><span class="o">.</span><span class="n">axhline</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;black&#39;</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">axvline</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;black&#39;</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
+    
+    
+
+    <span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
+
+<span class="k">def</span> <span class="nf">vtrans</span><span class="p">(</span><span class="n">c</span><span class="p">):</span>
+    <span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span><span class="mi">3</span><span class="p">)</span>
+    
+    <span class="n">y</span> <span class="o">=</span> <span class="n">func</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">+</span><span class="n">c</span> <span class="c1">#constant c summing the whole function</span>
+    
+    <span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">,</span>  <span class="n">label</span><span class="o">=</span><span class="s1">&#39;$f(x)+c$&#39;</span><span class="p">)</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s1">&#39;Vertical translation&#39;</span><span class="p">)</span>
+    
+    <span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s1">&#39;x&#39;</span><span class="p">)</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s1">&#39;y&#39;</span><span class="p">)</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">12</span><span class="p">)</span>
+    
+    <span class="n">plt</span><span class="o">.</span><span class="n">ylim</span><span class="p">(</span><span class="o">-</span><span class="mi">8</span><span class="p">,</span> <span class="mi">8</span><span class="p">)</span> <span class="c1">#remember to change the limits for better visualisation if necessary</span>
+        
+    <span class="n">plt</span><span class="o">.</span><span class="n">axhline</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;black&#39;</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">axvline</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;black&#39;</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
+    
+    <span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
+    
+<span class="c1"># create a slider</span>
+
+<span class="n">interact</span><span class="p">(</span><span class="n">htrans</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="p">(</span><span class="o">-</span><span class="mf">2.0</span><span class="p">,</span><span class="mf">2.0</span><span class="p">))</span>
+<span class="n">interact</span><span class="p">(</span><span class="n">vtrans</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="p">(</span><span class="o">-</span><span class="mf">2.0</span><span class="p">,</span><span class="mf">2.0</span><span class="p">))</span>
+</pre></div>
+</div>
+</section>
+<section id="stretching-compression">
+<h3>Stretching/ Compression<a class="headerlink" href="#stretching-compression" title="Link to this heading">#</a></h3>
+<p>A given function can also be stretched or compressed, either vertically or horiontally, by <em>multiplying</em> a constant <span class="math notranslate nohighlight">\(c\)</span> to the whole function or to its argument. Starting from a function <span class="math notranslate nohighlight">\(f(x)\)</span>, stretchings/compressions are built following the rules:</p>
+<ul>
+<li><p><strong>Horizontal</strong> (Constant <span class="math notranslate nohighlight">\(c\)</span> multiplied to the argument of the function)</p>
+<p><span class="math notranslate nohighlight">\(y = f(cx)\ \ \ \  -  \left\{
+\begin{array}{ll}
+    c &gt; 1 &amp; \rightarrow \ \  \mbox{function is horizontally compressed} \\
+    0 &lt; c &lt; 1 &amp; \rightarrow \ \  \mbox{function is horizontally stretched} \\
+\end{array} 
+\right. \)</span></p>
+</li>
+<li><p><strong>Vertical</strong> (Constant <span class="math notranslate nohighlight">\(c\)</span> multiplied to the whole function)</p>
+<p><span class="math notranslate nohighlight">\(y = cf(x)\ \ \ \  -  \left\{
+\begin{array}{ll}
+    c &gt; 1 &amp; \rightarrow \ \  \mbox{function is vertically stretched} \\
+    0 &lt; c &lt; 1 &amp; \rightarrow \ \  \mbox{function is vertically compressed} \\
+\end{array} 
+\right. \)</span></p>
+</li>
+</ul>
+<div class="admonition-example admonition">
+<p class="admonition-title">Example</p>
+<p>For the function <span class="math notranslate nohighlight">\(f(x) = x^3-2x\)</span>, the function <span class="math notranslate nohighlight">\(y = f(2x) = (2x)^3-2(2x) = 8x^3-4x\)</span> will represent a horizontal compression, while the function <span class="math notranslate nohighlight">\(y = 2f(x) = 2x^3 - 4x\)</span> will represent a vertical stretching.</p>
+</div>
+<div class="admonition-example admonition">
+<p class="admonition-title">Example</p>
+<p>One example where we stretch or compress functions comes from chemical processes. Many chemical processes happen faster when the temperature is higher. If we describe the concentration of a certain chemical as a function of time <span class="math notranslate nohighlight">\(c(t)\)</span>, increasing the temperature will lead to a horizontal compression of the function.</p>
+</div>
+<p>Now play with the code below to see how different graphs behave to horizontal and vertical stretching/compression.</p>
+<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">ipywidgets</span> <span class="kn">import</span> <span class="n">interact</span>
+<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
+<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+
+<span class="k">def</span> <span class="nf">func</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
+    <span class="k">return</span> <span class="n">x</span><span class="o">**</span><span class="mi">3</span><span class="o">-</span><span class="mi">2</span><span class="o">*</span><span class="n">x</span>
+
+<span class="k">def</span> <span class="nf">hstetcom</span><span class="p">(</span><span class="n">c</span><span class="p">):</span>
+    <span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span><span class="mi">3</span><span class="p">)</span>
+    
+    <span class="n">y</span> <span class="o">=</span> <span class="n">func</span><span class="p">(</span><span class="n">c</span><span class="o">*</span><span class="n">x</span><span class="p">)</span> <span class="c1">#constant c multiplying the argument of the function</span>
+    
+    <span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;$f\, (cx)$&#39;</span><span class="p">)</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s1">&#39;Horizontal stretching/compression&#39;</span><span class="p">)</span>
+        
+    <span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s1">&#39;x&#39;</span><span class="p">)</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s1">&#39;y&#39;</span><span class="p">)</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">12</span><span class="p">)</span>
+    
+    <span class="n">plt</span><span class="o">.</span><span class="n">ylim</span><span class="p">(</span><span class="o">-</span><span class="mi">4</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span> <span class="c1">#remember to change the limits for better visualisation if necessary</span>
+    
+    <span class="n">plt</span><span class="o">.</span><span class="n">axhline</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;black&#39;</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">axvline</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;black&#39;</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
+    
+    
+
+    <span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
+
+<span class="k">def</span> <span class="nf">vstetcom</span><span class="p">(</span><span class="n">c</span><span class="p">):</span>
+    <span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span><span class="mi">3</span><span class="p">)</span>
+    
+    <span class="n">y</span> <span class="o">=</span> <span class="n">c</span><span class="o">*</span><span class="n">func</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="c1">#constant c multiplying the whole function</span>
+    
+    <span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">,</span>  <span class="n">label</span><span class="o">=</span><span class="s1">&#39;$c\, f\, (x)$&#39;</span><span class="p">)</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s1">&#39;Vertical stretching/compression&#39;</span><span class="p">)</span>
+    
+    <span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s1">&#39;x&#39;</span><span class="p">)</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s1">&#39;y&#39;</span><span class="p">)</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">fontsize</span><span class="o">=</span><span class="mi">12</span><span class="p">)</span>
+    
+    <span class="n">plt</span><span class="o">.</span><span class="n">ylim</span><span class="p">(</span><span class="o">-</span><span class="mi">4</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span> <span class="c1">#remember to change the limits for better visualisation if necessary</span>
+        
+    <span class="n">plt</span><span class="o">.</span><span class="n">axhline</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;black&#39;</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
+    <span class="n">plt</span><span class="o">.</span><span class="n">axvline</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s1">&#39;black&#39;</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
+    
+    <span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
+    
+<span class="c1"># create a slider</span>
+
+<span class="n">interact</span><span class="p">(</span><span class="n">hstetcom</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="p">(</span><span class="mf">0.1</span><span class="p">,</span><span class="mf">3.0</span><span class="p">))</span>
+<span class="n">interact</span><span class="p">(</span><span class="n">vstetcom</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="p">(</span><span class="mf">0.1</span><span class="p">,</span><span class="mf">3.0</span><span class="p">))</span>
+</pre></div>
+</div>
+</section>
+<section id="reflection">
+<h3>Reflection<a class="headerlink" href="#reflection" title="Link to this heading">#</a></h3>
+<p>If the constant <span class="math notranslate nohighlight">\(c\)</span> multiplying the whole function or its argument has the particular value <span class="math notranslate nohighlight">\(c = -1\)</span>, we will have a <em>reflection</em>. Reflections can be done in relation to the <span class="math notranslate nohighlight">\(x\)</span> or <span class="math notranslate nohighlight">\(y\)</span>-axes and they are built, starting from a function <span class="math notranslate nohighlight">\(f(x)\)</span>, following the rules:</p>
+<ul class="simple">
+<li><p><strong>About the x-axis</strong> - <span class="math notranslate nohighlight">\(y = -f(x)\)</span></p></li>
+<li><p><strong>About the y-axis</strong> - <span class="math notranslate nohighlight">\(y = f(-x)\)</span></p></li>
+</ul>
+<p>The figure below shows these two reflections using the function <span class="math notranslate nohighlight">\(f(x) = (x-1)^2\)</span>.</p>
+<img alt="../../_images/reflection.png" src="../../_images/reflection.png" />
+</section>
+</section>
+<section id="analysing-change-one-step-at-a-time">
+<h2>Analysing change, one step at a time<a class="headerlink" href="#analysing-change-one-step-at-a-time" title="Link to this heading">#</a></h2>
+<p>Imagine a colony of wasps is founded by a queen with the following pattern of eggs laid:</p>
+<br />  
+<div class="pst-scrollable-table-container"><table class="table">
+<thead>
+<tr class="row-odd"><th class="head"><p>Days</p></th>
+<th class="head text-center"><p>Eggs</p></th>
+</tr>
+</thead>
+<tbody>
+<tr class="row-even"><td><p><span class="math notranslate nohighlight">\(0\)</span></p></td>
+<td class="text-center"><p><span class="math notranslate nohighlight">\(100\)</span></p></td>
+</tr>
+<tr class="row-odd"><td><p><span class="math notranslate nohighlight">\(1\)</span></p></td>
+<td class="text-center"><p><span class="math notranslate nohighlight">\(135\)</span></p></td>
+</tr>
+<tr class="row-even"><td><p><span class="math notranslate nohighlight">\(2\)</span></p></td>
+<td class="text-center"><p><span class="math notranslate nohighlight">\(170\)</span></p></td>
+</tr>
+<tr class="row-odd"><td><p><span class="math notranslate nohighlight">\(3\)</span></p></td>
+<td class="text-center"><p><span class="math notranslate nohighlight">\(205\)</span></p></td>
+</tr>
+<tr class="row-even"><td><p><span class="math notranslate nohighlight">\(4\)</span></p></td>
+<td class="text-center"><p><span class="math notranslate nohighlight">\(240\)</span></p></td>
+</tr>
+</tbody>
+</table>
+</div>
+<p>Since there is an additive pattern for the increase in the total number of eggs laid, with the number of eggs in any given day, we can determine the number of eggs the colony will have in the following day. If we index the days by a sequence of natural numbers <em>n</em>, we have for the above colony:</p>
+<div class="math notranslate nohighlight">
+\[a_{n+1} = a_{n} + 35,\]</div>
+<p>where <span class="math notranslate nohighlight">\(a_n\)</span> represents the total amount of eggs in a given day <span class="math notranslate nohighlight">\(n\)</span> and <span class="math notranslate nohighlight">\(a_{n+1}\)</span> the same variable on the day following day <span class="math notranslate nohighlight">\(n\)</span>. The constant <span class="math notranslate nohighlight">\(35\)</span> represents the daily increment (or decrement) on this total number. A sequence of number built with constant additive increments receives the name of <em>arithmetic progression</em>.</p>
+<p>The above expression defines an iterative process (also known as a <em>recursion</em>) with which we can construct the whole sequence starting from a single initial value. If we want to know how many eggs will have been laid on a given day <span class="math notranslate nohighlight">\(n\)</span>, this shall be given by the expression:</p>
+<div class="math notranslate nohighlight">
+\[a_n = 100 + 35n.\]</div>
+<p>Thus, the total amount of eggs laid on day <span class="math notranslate nohighlight">\(n=100\)</span> is <span class="math notranslate nohighlight">\(a_{100} = 100 + 35 \cdot 100 = 3600\)</span>.</p>
+<br />
+<br />
+Now imagine a single *Escherichia coli* cell put on a Petri dish with a favorable growth medium. If given the right conditions, *E. coli* cells divide, generating two other cells, in a time of approximately $30$ minutes. For our experiment, we would have:  
+<br />  
+<div class="pst-scrollable-table-container"><table class="table">
+<thead>
+<tr class="row-odd"><th class="head"><p>Minutes</p></th>
+<th class="head text-center"><p># cells</p></th>
+</tr>
+</thead>
+<tbody>
+<tr class="row-even"><td><p><span class="math notranslate nohighlight">\(0\)</span></p></td>
+<td class="text-center"><p><span class="math notranslate nohighlight">\(1\)</span></p></td>
+</tr>
+<tr class="row-odd"><td><p><span class="math notranslate nohighlight">\(30\)</span></p></td>
+<td class="text-center"><p><span class="math notranslate nohighlight">\(2\)</span></p></td>
+</tr>
+<tr class="row-even"><td><p><span class="math notranslate nohighlight">\(60\)</span></p></td>
+<td class="text-center"><p><span class="math notranslate nohighlight">\(4\)</span></p></td>
+</tr>
+<tr class="row-odd"><td><p><span class="math notranslate nohighlight">\(90\)</span></p></td>
+<td class="text-center"><p><span class="math notranslate nohighlight">\(8\)</span></p></td>
+</tr>
+<tr class="row-even"><td><p><span class="math notranslate nohighlight">\(120\)</span></p></td>
+<td class="text-center"><p><span class="math notranslate nohighlight">\(16\)</span></p></td>
+</tr>
+</tbody>
+</table>
+</div>
+<p>If <span class="math notranslate nohighlight">\(t\)</span> represents the number of reproduction intervals since the start of the experiment (assuming values <span class="math notranslate nohighlight">\(t=0, 1, 2, \ldots\)</span>), then we can construct the recursion:</p>
+<div class="math notranslate nohighlight">
+\[N_{t+1} = 2N_t,\]</div>
+<p>where <span class="math notranslate nohighlight">\(N_t\)</span> represents the number of <em>E. coli</em> cells after <span class="math notranslate nohighlight">\(t\)</span> intervals of reproduction (each interval corresponding to <span class="math notranslate nohighlight">\(30\)</span> minutes) and <span class="math notranslate nohighlight">\(N_{t+1}\)</span> represents the number of cells on the interval following <span class="math notranslate nohighlight">\(t\)</span>. See that this time the constant <span class="math notranslate nohighlight">\(2\)</span> provides a multiplicative increment, so that the sequence formed by this multiplicative iteration is called <em>geometric progression</em>.</p>
+<p>If we want <span class="math notranslate nohighlight">\(N_t\)</span> for a given interval <span class="math notranslate nohighlight">\(t\)</span>, we then have:</p>
+<div class="math notranslate nohighlight">
+\[N_t = 2^t,\]</div>
+<p>leading to an exponential growth in discrete steps.</p>
+<p>In general, if we have a population growth described by a geometric progression, starting with a number of individuals <span class="math notranslate nohighlight">\(N_0\)</span>, the number of individuals on the subsequent generations can be described by the recursion:</p>
+<div class="math notranslate nohighlight">
+\[N_{t+1} = RN_t,\]</div>
+<p>where <span class="math notranslate nohighlight">\(t\)</span> indicates the number of generations and <span class="math notranslate nohighlight">\(R\)</span> is a positive constant (<span class="math notranslate nohighlight">\(R&gt;0\)</span>).</p>
+<p>With this recursion, starting from <span class="math notranslate nohighlight">\(N_0\)</span>, we can obtain the full sequence by construction:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{align}  
+N_1 &amp;= RN_0 \\
+N_2 &amp;= RN_1 = R(RN_0) = R^2N_0 \\
+N_3 &amp;= RN_2 = R(RN_1) = R[R(RN_0)] = R^3N_0 \\
+    &amp;\vdots \end{align}\end{split}\]</div>
+<p>From this process of construction, we can obtain what we call the <em>solution</em> of the above recursion, given by:</p>
+<div class="math notranslate nohighlight">
+\[N_t = N_0R^t.\]</div>
+<p>In problems of population dynamics, <span class="math notranslate nohighlight">\(R\)</span> is called <strong>growth constant</strong> and its value determines the long-term behaviour of the function:</p>
+<img alt="../../_images/expgrowth.png" src="../../_images/expgrowth.png" />
+</section>
+<section id="sequences">
+<h2>Sequences<a class="headerlink" href="#sequences" title="Link to this heading">#</a></h2>
+<p>The two last problems are examples of <em>sequences</em> for which a formal definition is given as:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{align}
+    f\!:\ &amp;{\rm I\!N} \rightarrow {\rm I\!R} \\
+       &amp; n \rightarrow f(n),\end{align}\end{split}\]</div>
+<p>in other words, a sequence is a function <span class="math notranslate nohighlight">\(f\)</span> from the set of natural numbers to the set of real numbers. For each value of the independent variable <span class="math notranslate nohighlight">\(n\)</span> (a natural number), the sequence determines its image <span class="math notranslate nohighlight">\(f(n)\)</span>.</p>
+<div class="admonition-example admonition">
+<p class="admonition-title">Example</p>
+<p>The sequence defined by the rule <span class="math notranslate nohighlight">\(f(n) = \displaystyle\frac{1}{n+1}\)</span> defines the sequence of numbers (starting from <span class="math notranslate nohighlight">\(n=0\)</span>):</p>
+<div class="math notranslate nohighlight">
+\[1,\frac{1}{2},\frac{1}{3},\frac{1}{4},\frac{1}{5},\ldots,\]</div>
+<p>corresponding to the set of values <span class="math notranslate nohighlight">\(n=0,1,2,3,4,\ldots\)</span>.</p>
+</div>
+<p>Sequences are usually written as ordered lists of numbers <span class="math notranslate nohighlight">\(a_0, a_1, a_2, a_3, \ldots\)</span> where <span class="math notranslate nohighlight">\(a_n=f(n)\)</span> and referred to with the notation <span class="math notranslate nohighlight">\(\{a_n\}\)</span> (which is short for <span class="math notranslate nohighlight">\(\{a_n: n \in {\rm I\!N}\}\)</span>, making it explicit that <span class="math notranslate nohighlight">\(n\)</span> is a natural number).</p>
+<p>As we have seen before, two forms of representation are possible:</p>
+</section>
+<section id="limits">
+<h2>Limits<a class="headerlink" href="#limits" title="Link to this heading">#</a></h2>
+<p>In many applications, we are interested to know what is the <em>long-term behaviour</em> of a given sequence <span class="math notranslate nohighlight">\(\{a_n\}\)</span>. For example, is a given population tending towards a stable number of individuals as time passes? In other words, we want to know if there is a number <span class="math notranslate nohighlight">\(L\)</span> for which the sequence tends to as <span class="math notranslate nohighlight">\(n\)</span> increases. To represent this we use the following notation:</p>
+<div class="math notranslate nohighlight">
+\[\lim_{n\rightarrow \infty} a_n = L\ \ (\mbox{or simply}\ \lim a_n =L)\]</div>
+<p>If such a number exists, we say that the sequence <span class="math notranslate nohighlight">\(\{a_n\}\)</span> is <strong>convergent</strong>.</p>
+<div class="admonition-examples admonition">
+<p class="admonition-title">Examples</p>
+<ol class="arabic">
+<li><p><span class="math notranslate nohighlight">\(a_n = \displaystyle\frac{1}{n+1}: \ \ \ \left\{1,\displaystyle\frac{1}{2},\displaystyle\frac{1}{3},\displaystyle\frac{1}{4},\displaystyle\frac{1}{5},\ldots\right\}\)</span><br />
+<br />
+Note that each term is smaller than the previous and they are all positive. We can conclude that:</p>
+<div class="math notranslate nohighlight">
+\[\lim a_n = 0.\]</div>
+</li>
+<li><p><span class="math notranslate nohighlight">\(b_n = (-1)^n: \ \ \ \left\{1,-1,1,-1,1,-1,\ldots\right\}\)</span><br />
+<br />
+The terms in this sequence are alternating between <span class="math notranslate nohighlight">\(1\)</span> and <span class="math notranslate nohighlight">\(-1\)</span>, never convergeing for a specific value. Thus, the limit of sequence <span class="math notranslate nohighlight">\(\{b_n\}\)</span> does not exist.<br />
+<br /></p></li>
+<li><p><span class="math notranslate nohighlight">\(c_n = 2^n: \ \ \ \left\{1,2,4,8,16,32,\ldots\right\}\)</span><br />
+<br />
+For this sequence, the terms are ever increasing. We frequently say that the sequence tends to infinity (or tends to <span class="math notranslate nohighlight">\(\infty\)</span>). However, <span class="math notranslate nohighlight">\(\infty\)</span> represents an abstraction (“ever increasing”), not a specific number. Therefore, the limit for this sequence also does not exist.<br />
+<br /></p></li>
+<li><p><span class="math notranslate nohighlight">\(d_n = \displaystyle\frac{n+2}{n+1}: \ \ \ \left\{2,\displaystyle\frac{3}{2},\displaystyle\frac{4}{3},\displaystyle\frac{5}{4},\displaystyle\frac{6}{5},\ldots\right\}\)</span><br />
+<br />
+For this sequence, each term is smaller than the previous (look at the decimal forms of the fractions: <span class="math notranslate nohighlight">\(2, 1.5, 1.333\cdots, 1.25, 1.2 \ldots\)</span>) and they are greater than <span class="math notranslate nohighlight">\(1\)</span>. We can conclude that:</p>
+<div class="math notranslate nohighlight">
+\[\lim a_n = 1.\]</div>
+</li>
+</ol>
+</div>
+<p>Although there is a formal way to <em>prove</em> that a sequence <span class="math notranslate nohighlight">\(\{a_n\}\)</span> converges to a certain limit, in practice we use a set of rules to apply limits to sequences and composition of sequences.</p>
+<div class="admonition tip">
+<p class="admonition-title">Tip</p>
+<p>Consider two sequences <span class="math notranslate nohighlight">\(\{a_n\}\)</span> and <span class="math notranslate nohighlight">\(\{b_n\}\)</span>, and <span class="math notranslate nohighlight">\(C\)</span> as a real constant. If these two sequences are convergent with  <span class="math notranslate nohighlight">\(\lim a_n = L\)</span> and <span class="math notranslate nohighlight">\(\lim b_n = M\)</span>, we have:</p>
+<div class="math notranslate nohighlight">
+\[\lim (a_n + b_n) = \lim a_n + \lim b_n = L + M\]</div>
+<div class="math notranslate nohighlight">
+\[\lim (Ca_n) = C\lim a_n = CL\]</div>
+<div class="math notranslate nohighlight">
+\[\lim (a_nb_n) = (\lim a_n)(\lim b_n) = LM\]</div>
+<div class="math notranslate nohighlight">
+\[\lim \left(\frac{a_n}{b_n}\right) = \frac{\lim a_n}{\lim b_n} = \frac{L}{M},\ \ (\mbox{if}\ M \ne 0)\]</div>
+<p>In other other words, if you can decompose a given sequence into the sum, product or division of two other convergent sequences, the limit of this sequence will be the composition of the limits of the two others.</p>
+</div>
+<div class="admonition-examples admonition">
+<p class="admonition-title">Examples</p>
+<p>If <span class="math notranslate nohighlight">\(a_n = \displaystyle\frac{4n^2-1}{n^2}:\)</span></p>
+<div class="math notranslate nohighlight">
+\[\lim\left(\displaystyle\frac{4n^2-1}{n^2}\right) = \lim\left(4-\displaystyle\frac{1}{n^2}\right) = \lim 4 - \left(\lim\displaystyle\frac{1}{n}\right)\left(\lim\displaystyle\frac{1}{n}\right) = 4,\]</div>
+<p>since as we know: <span class="math notranslate nohighlight">\(\ \lim 4 = 4\ \)</span> and <span class="math notranslate nohighlight">\(\ \lim \displaystyle\frac{1}{n} = 0\)</span>.</p>
+</div>
+<p>For recursions, one way to find the limit is to find the solution (corresponding explicit description) and then calculate the limit of the sequence with the limit rules. However, finding the explicit form of a given sequence is not always possible.</p>
+<p>There is a procedure to find <em>candidates</em> for the long-term behaviour of a given sequence. We do this by calculating <strong>fixed points</strong>.</p>
+<p>Fixed points are specific values on a given sequence for which all subsequent values equal the first. That is, if <span class="math notranslate nohighlight">\(a\)</span> is a fixed point of the sequence <span class="math notranslate nohighlight">\(\{a_n\}\)</span>, and if <span class="math notranslate nohighlight">\(a_0 = a\)</span>, then <span class="math notranslate nohighlight">\(a_1 = a\)</span>, <span class="math notranslate nohighlight">\(a_2 = a\)</span>, and all the other values will be <span class="math notranslate nohighlight">\(a\)</span>. Thus, given the recursion <span class="math notranslate nohighlight">\(a_{n+1} = g(a_n)\)</span>, if <span class="math notranslate nohighlight">\(a\)</span> is a fixed point we will have:</p>
+<div class="math notranslate nohighlight">
+\[a = g(a)\]</div>
+<p>and the last equation provides a way of find possible values of <span class="math notranslate nohighlight">\(a\)</span>.</p>
+<div class="admonition-example admonition">
+<p class="admonition-title">Example</p>
+<p>Consider the following recursion</p>
+<div class="math notranslate nohighlight">
+\[a_{n+1} = \sqrt{3a_n},\ \ \ a_0 = 2.\]</div>
+<p>To obtain the fixed points we calculate:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{align}
+  a &amp;= \sqrt{3a} \\
+  a^2 &amp;= 3a \ \ \longrightarrow a = 0\ \mbox{ or }\ a = 3 \end{align}\end{split}\]</div>
+<p>Starting with <span class="math notranslate nohighlight">\(a_0 = 2\)</span>, we can use the recursion to construct the sequence as:</p>
+<div class="math notranslate nohighlight">
+\[\left\{\sqrt{6},\sqrt{3\sqrt{6}},\sqrt{3\sqrt{3\sqrt{6}}},\sqrt{3\sqrt{3\sqrt{3\sqrt{6}}}},\ldots \right\}\]</div>
+<p>or also: <span class="math notranslate nohighlight">\(\{2.449\cdots,2.711\cdots,2.852\cdots,2.925\cdots, \ldots\}\)</span>. We see that starting with <span class="math notranslate nohighlight">\(a_0=2\)</span>, we have <span class="math notranslate nohighlight">\(a_n&gt;2\)</span> for all <span class="math notranslate nohighlight">\(n\)</span>, with increasing values. Since <span class="math notranslate nohighlight">\(a=3\)</span> is a fixed point, we conclude that <span class="math notranslate nohighlight">\(\lim a_n = 3\)</span>.</p>
+</div>
+<div class="admonition-example admonition">
+<p class="admonition-title">Example</p>
+<p>Whenever we are asking questions like ‘where does this process stabilise’ or ‘how high/much will it be in the end,’ we are interested in the limit of some series. This could be a variety of things, like population size, ground water levels, global mean temperature, number of trees, etc.</p>
+</div>
+<div class="admonition warning">
+<p class="admonition-title">Warning</p>
+<p>Remember that fixed points are only <strong>candidates</strong> for long-term behaviour. See for example the recursion:</p>
+<div class="math notranslate nohighlight">
+\[a_{n+1} = \displaystyle\frac{3}{a_n}\]</div>
+<p>For the fixed points, we calculate:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{align}
+  a &amp;= \displaystyle\frac{3}{a} \\
+  a^2 &amp;= 3 \\
+  &amp;\longrightarrow a = -\sqrt{3}\ \mbox{ or }\ a = \sqrt{3}. \end{align}\end{split}\]</div>
+<p>Indeed, if we construct the sequence from the recursion starting with any of these two values, all the subsequent elements of the sequence will have the same value. However:</p>
+<div class="math notranslate nohighlight">
+\[\mbox{if } a_0 = 2 \ \mbox{the sequence will be given by: }\ \left\{2,\displaystyle\frac{3}{2},2,\displaystyle\frac{3}{2},\ldots \right\}\]</div>
+<p>or</p>
+<div class="math notranslate nohighlight">
+\[\mbox{If } a_0 = -3 \ \mbox{the sequence will be given by: }\ \left\{-3,-1,-3,-1,\ldots \right\}\]</div>
+<p>showing that for any initial value chosen (different from the fixed points), the sequence will oscillate between two values, without a clear tendency to any limit.</p>
+</div>
+</section>
+<section id="discrete-time-population-models">
+<h2>Discrete-time population models<a class="headerlink" href="#discrete-time-population-models" title="Link to this heading">#</a></h2>
+<p>Sequences are generally used in many applications in Biology. Important examples are models for <strong>seasonally breeding</strong> populations.</p>
+<section id="density-dependent-population-growth">
+<h3>Density-dependent population growth<a class="headerlink" href="#density-dependent-population-growth" title="Link to this heading">#</a></h3>
+<p>As we have seen before, the recursion relation for the exponential growth model is given by <span class="math notranslate nohighlight">\(N_{t+1} = RN_t\)</span>. Rearranging the terms of this equation we have:</p>
+<div class="math notranslate nohighlight">
+\[\displaystyle\frac{N_t}{N_{t+1}}=\frac{1}{R}.\]</div>
+<p>The left-hand term of the last equation is known as <em>parent-offspring ratio</em> as it denotes the ratio between the population size at time <span class="math notranslate nohighlight">\(t\)</span> (parents) and the population size at the following time <span class="math notranslate nohighlight">\(t+1\)</span> (offspring). Here we are considering a population that breeds without overlapping generations. Each generation breeds, dies, and is replaced by their offspring on the following generation.</p>
+<p>On the equation we also see that the parent-offspring ratio is equal to <span class="math notranslate nohighlight">\(\displaystyle\frac{1}{R}\)</span>, which is a constant. Therefore, we say that this growth is <em>density-independent</em> as the parent-offspring ratio does not depend on the density of individuals at any time step. Relating the parent-offspring ratio to the value of <span class="math notranslate nohighlight">\(R\)</span> allows us to predict the behaviour of this population in terms of growth or decline:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{align}
+R&gt;1 \ &amp;- \ \mbox{there will be more offspring than parents (population grows)} \\
+R&lt;1 \ &amp;- \ \mbox{there will be more parents than offspring (population declines)} \\
+R=1 \ &amp;- \ \mbox{there will be no change in population size} \end{align}\end{split}\]</div>
+<p>We know, however, than in real scenarios, that the growth factor of a given population should depend on the current size of that population, due to space or resources limitation. These are called <em>density-dependent</em> effects. One of the simplest methods to include density-dependent effects in this population model is to consider the parent-offspring ratio as a linear function of <span class="math notranslate nohighlight">\(N_t\)</span>, instead of a constant. Let us assume that the parent-offspring ratio show the following linear behaviour:</p>
+<img alt="../../_images/parent-offspring.png" src="../../_images/parent-offspring.png" />
+<p>This means that the parent-offspring ratio is smaller than <span class="math notranslate nohighlight">\(1\)</span> (population growth) for values of <span class="math notranslate nohighlight">\(N_t\)</span> smaller than a given population threshold <span class="math notranslate nohighlight">\(K\)</span>, and greater than <span class="math notranslate nohighlight">\(1\)</span> (population decline) if the population size is greater than this threshold. The graph above leads to the following expression:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{align}
+\displaystyle\frac{N_t}{N_{t+1}} = \frac{1}{R} + \frac{1-\frac{1}{R}}{K}N_t \\
+N_{t+1} = \frac{N_t}{\frac{1}{R} + \frac{1-\frac{1}{R}}{K}N_t}, \end{align}\end{split}\]</div>
+<p>which leads, after rearranging the terms, to:</p>
+<div class="math notranslate nohighlight">
+\[N_{t+1} = \frac{RN_t}{1 + \frac{(R-1)}{K}N_t}.\]</div>
+<p>The previous expression is known as <strong>Beverton-Holt recruitment curve</strong> (or Beverton-Holt model) and it describes a type of population growth called <em>logistic growth</em>.</p>
+<p>The fixed points <span class="math notranslate nohighlight">\(\bar{N}\)</span> for this recursion can be calculated as:</p>
+<div class="math notranslate nohighlight">
+\[\bar{N} = \frac{R\bar{N}}{1 + \frac{(R-1)}{K}\bar{N}}\ \longrightarrow \ \bar{N}\left(1 + \frac{(R-1)}{K}\bar{N}\right) = R\bar{N}\]</div>
+<p>From the last term we can show (try it as an exercise) that <span class="math notranslate nohighlight">\(\bar{N}=0\)</span> and <span class="math notranslate nohighlight">\(\bar{N}=K\)</span> are the two fixed points. Starting from different values for the initial population size <span class="math notranslate nohighlight">\(N_0\)</span> it can be shown that the population tends to <span class="math notranslate nohighlight">\(K\)</span> as time increases as:</p>
+<img alt="../../_images/logistic.png" src="../../_images/logistic.png" />
+<p>The constant <span class="math notranslate nohighlight">\(K\)</span> is know as <em>carrying capacity</em> and represents the maximum population size a given population can reach at a given place, due to intra-specific competition for resources or space.</p>
+<p>Watch this <a class="reference external" href="https://www.youtube.com/watch?v=Kas0tIxDvrg">video</a> about application of growth models to epidemics.</p>
+</section>
+<section id="annual-plants">
+<h3>Annual plants<a class="headerlink" href="#annual-plants" title="Link to this heading">#</a></h3>
+<p>Suppose a given population of plants produces, on average <span class="math notranslate nohighlight">\(f\)</span> seeds per individual in the reproductive period. Seeds survive winter with a probability <span class="math notranslate nohighlight">\(\sigma\)</span> and germinate on the next season with probability <span class="math notranslate nohighlight">\(\alpha\)</span> (these seeds reach reproductive maturity on the same season). If <span class="math notranslate nohighlight">\(p_n\)</span> is the plant population size on season <span class="math notranslate nohighlight">\(n\)</span> and <span class="math notranslate nohighlight">\(p_{n+1}\)</span> the population size on the season following <span class="math notranslate nohighlight">\(n\)</span>, we shall have:</p>
+<div class="math notranslate nohighlight">
+\[p_{n+1} = (\alpha\sigma f)p_n\]</div>
+<p>which correspond to the density-independent model since <span class="math notranslate nohighlight">\(R=\alpha\sigma f\)</span> is a constant.</p>
+<p>Now suppose that some of the seeds that were produced on the previous season and survived the previous winter, but did not germinate on the current season, might have a second chance of germinating. They might do so on the following season, provided they survive the winter and germinate on the following season. From the average number of seeds produced last season <span class="math notranslate nohighlight">\(fp_{n-1}\)</span>, a number <span class="math notranslate nohighlight">\(\sigma fp_{n-1}\)</span> of them survived last winter. From this number, a fraction <span class="math notranslate nohighlight">\((1-\alpha)\)</span> did not germinate this season (1 minus the probability of germinating). These seeds will have a second chance provided they survive next winter (with probability <span class="math notranslate nohighlight">\(\sigma\)</span>) and germinate on the next season (with probability <span class="math notranslate nohighlight">\(\alpha\)</span>). Thus, the number plant population size on the season following <span class="math notranslate nohighlight">\(n\)</span> will be:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{align}
+p_{n+1} = \alpha\sigma fp_n + \alpha\sigma(1-\alpha)\sigma fp_{n-1} \\  
+p_{n+1} = (\alpha\sigma f)p_n + [\alpha(1-\alpha)\sigma^2 f]p_{n-1} \\  
+p_{n+1} = \beta p_n + \gamma p_{n-1}\end{align}\end{split}\]</div>
+<p>where <span class="math notranslate nohighlight">\(\beta=\alpha\sigma f\)</span> and <span class="math notranslate nohighlight">\(\gamma = \alpha(1-\alpha)\sigma^2 f\)</span>.</p>
+<p>The previous model is an example of a discrete population model with overlapping populations. Since the term <span class="math notranslate nohighlight">\(n+1\)</span> depends not only on the term <span class="math notranslate nohighlight">\(n\)</span> but also on <span class="math notranslate nohighlight">\(n-1\)</span> this is called a <em>second-order difference equation</em> or <em>delay (or lagged) difference equation</em>.</p>
+</section>
+<section id="fibonacci-sequence">
+<h3>Fibonacci sequence<a class="headerlink" href="#fibonacci-sequence" title="Link to this heading">#</a></h3>
+<p>The Fibonacci sequence is a sequence defined by the following rule: each number is equal to the sum of the previous two numbers (starting with <span class="math notranslate nohighlight">\(0\)</span> and <span class="math notranslate nohighlight">\(1\)</span>). Thus the following sequence might be formed:</p>
+<div class="math notranslate nohighlight">
+\[\left\{ 0,1,1,2,3,5,8,13,21,\ldots \right\}. \]</div>
+<p>It can also be defined by the second-order difference equation:</p>
+<div class="math notranslate nohighlight">
+\[F_{n+1} = F_{n} + F_{n-1}.\]</div>
+<p>Although this is clearly not a convergent sequence (with sustained growth for each element), if we define a new sequence given by the ratio between successive elements of the Fibonacci sequence as <span class="math notranslate nohighlight">\(b_n = \displaystyle\frac{F_{n+1}}{F_n}\)</span>, it can be show that this sequence is convergent.</p>
+<p>Assuming that <span class="math notranslate nohighlight">\(\lim \displaystyle\frac{F_{n+1}}{F_n} = \lambda\)</span>, we have:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{align}
+F_{n+1} &amp;= F_{n} + F_{n-1}\quad \mbox{(Fibonacci seq.)} \\  
+\displaystyle\frac{F_{n+1}}{F_{n}} &amp;= 1 + \frac{F_{n-1}}{F_{n}}\quad \mbox{(divide both sides by}\ F_{n}\mbox{)} \\
+\lim \left(\displaystyle\frac{F_{n+1}}{F_{n}}\right) &amp;= \lim\left(1 + \frac{F_{n-1}}{F_{n}}\right) \\
+\lambda &amp;= 1 + \frac{1}{\lambda} \\
+\lambda^2 &amp;- \lambda -1 = 0 \end{align}\end{split}\]</div>
+<p>The previous equation is solved finding the roots of the second-degree polynomial, which gives <span class="math notranslate nohighlight">\(\lambda = \displaystyle\frac{1+\sqrt{5}}{2}\)</span> or <span class="math notranslate nohighlight">\(\lambda = \displaystyle\frac{1-\sqrt{5}}{2}\)</span>. Since the solution <span class="math notranslate nohighlight">\(\displaystyle\frac{1-\sqrt{5}}{2}\)</span> is negative, it can be the limit of the ratio of Fibonacci terms (all of them positive). Hence we conclude:</p>
+<div class="math notranslate nohighlight">
+\[\lim \left(\displaystyle\frac{F_{n+1}}{F_{n}}\right) = \displaystyle\frac{1+\sqrt{5}}{2} \approx 1.618.\]</div>
+<p>The previous number is known as the <em>Golden ratio</em> and it has fascinated mathematicians, phylosophers, and artists since ancient times. Fibonacci numbers and the Golden ratio can be found in several interesting natural patterns, as you can see in this <a class="reference external" href="https://www.youtube.com/watch?v=ahXIMUkSXX0">video</a>.</p>
+</section>
+</section>
+<section id="from-table-arrangements-to-powerful-tools">
+<h2>From table arrangements to powerful tools<a class="headerlink" href="#from-table-arrangements-to-powerful-tools" title="Link to this heading">#</a></h2>
+</section>
+<section id="systems-of-linear-equations">
+<h2>Systems of linear equations<a class="headerlink" href="#systems-of-linear-equations" title="Link to this heading">#</a></h2>
+<p>As we saw previously, the equation <span class="math notranslate nohighlight">\(ax + by = c\)</span>, with <span class="math notranslate nohighlight">\(a\)</span>, <span class="math notranslate nohighlight">\(b\)</span>, and <span class="math notranslate nohighlight">\(c\)</span> constants, defines a straight line on the <span class="math notranslate nohighlight">\(xy\)</span>-plane (note that it can also be written as <span class="math notranslate nohighlight">\(y = mx + k\)</span>, <span class="math notranslate nohighlight">\(m\)</span> and <span class="math notranslate nohighlight">\(k\)</span> constants, by a suitable change of the coeffients). Since the coefficients do not depend on the variables <span class="math notranslate nohighlight">\(x\)</span> and <span class="math notranslate nohighlight">\(y\)</span>, we call it a <em>linear equation</em>. A possible plot for this equation would be:</p>
+<img alt="../../_images/lineq.png" src="../../_images/lineq.png" />
+<p>Note that all the points <span class="math notranslate nohighlight">\((x,y)\)</span> on the blue line <em>satisfy</em> or <em>solve</em> the lienar equation <span class="math notranslate nohighlight">\(ax + by = c\)</span>.</p>
+<p>Now consider that we have the following:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{cases} ax + by = c \\ dx +ey = f \end{cases}\end{split}\]</div>
+<p>where <span class="math notranslate nohighlight">\(a, b, c, d, e,\ \mbox{and}\ f\)</span> are all <a class="reference external" href="http://constants.In">constants.In</a> this case we have a <em>system of linear equations</em>. Solving this system means to find all possible pairs <span class="math notranslate nohighlight">\((x,y)\)</span> that solve both equations <strong>simultaneously</strong>. For the previous system, we would have 3 options:</p>
+<img alt="../../_images/syslineq.png" src="../../_images/syslineq.png" />
+<p>On the left panel, the two lines intersect at a <em>single point</em>. This point is the only one that solves both linear equations at the same time, and, therefore, constitutes the <strong>only solution</strong> for the system of two equations. On the central panel, the two lines are <em>parallel</em>, so there is no point of intersection. In this case there is <strong>no solution</strong> and the system of equations is said to be <em>inconsistent</em>. On the right panel, the two lines coincide. Thus, every point in those lines correspond to a possible solution for the system, so that it has <strong>infinite solutions</strong>.</p>
+<div class="admonition-example admonition">
+<p class="admonition-title">Example</p>
+<p>Given the system of linear equations</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{cases} 2x + 3y = 6 \\ 2x + y = 4 \end{cases}\end{split}\]</div>
+<p>we can obtain the solution by isolating one of the variables in one of the equations and substituting on the other equation. Isolating <span class="math notranslate nohighlight">\(x\)</span> on the first equation gives:</p>
+<div class="math notranslate nohighlight">
+\[x = 3 - \displaystyle\frac{3}{2}y\]</div>
+<p>Then by substituting on the second we have:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{align} 
+2\left(3 - \displaystyle\frac{3}{2}y \right) + y &amp;= 4 \\
+6 - 3y + y &amp;= 4 \\
+-2y = -2 \\
+y = 1 \end{align}\end{split}\]</div>
+<p>The value of <span class="math notranslate nohighlight">\(x\)</span> shall then be given by:</p>
+<div class="math notranslate nohighlight">
+\[x = 3 - \displaystyle\frac{3}{2}\cdot 1 = 3 - \frac{3}{2} = \frac{3}{2}\]</div>
+<p>We see that the system has a single solution, given by <span class="math notranslate nohighlight">\(\left(\displaystyle\frac{3}{2},1\right)\)</span>.</p>
+</div>
+<div class="admonition-example admonition">
+<p class="admonition-title">Example</p>
+<p>Now consider the sytem</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{cases} 2x + 3y &amp;= 6 \\ -4x + -6y &amp;= -12 \end{cases}\end{split}\]</div>
+<p>Isolating <span class="math notranslate nohighlight">\(x\)</span> on the first equation gives the same as before:</p>
+<div class="math notranslate nohighlight">
+\[x = 3 - \displaystyle\frac{3}{2}y\]</div>
+<p>Then by substituting on the second we have:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{align} 
+-4\left(3 - \displaystyle\frac{3}{2}y \right) - 6y &amp;= -12 \\
+-12 + 6y - 6y &amp;= -12 \\
+0 = 0 \end{align}\end{split}\]</div>
+<p>which is just a trivial equation.</p>
+<p>When we look at the equations in our system we see that the second equation is simply a constant multiplied by the first <span class="math notranslate nohighlight">\((\text{Eqn.} 2) = -2\cdot(\text{Eqn.} 1)\)</span>, and thus they should represent the same straight line.</p>
+<p>This system has then infinite solutions which are represented by the set of points <span class="math notranslate nohighlight">\(\left(3 - \displaystyle\frac{3}{2}y,y\right)\)</span></p>
+<div class="admonition tip">
+<p class="admonition-title">Tip</p>
+<p>Note that writing the solution as <span class="math notranslate nohighlight">\(\left(3 - \displaystyle\frac{3}{2}y,y\right)\)</span> represents an infinite set of points, since for every real value of <span class="math notranslate nohighlight">\(y\)</span> we would have a different ordered pair. This set of points forms the straight line corresponding to <span class="math notranslate nohighlight">\(2x+3y = 6\)</span>.</p>
+</div>
+</div>
+<p>Solving the system by this method of substitutions, however, can get much more complicated if we have more than two variables. For these cases, we need a simpler method.</p>
+<p>One strategy is to find equivalent systems, which have the same solution as the original one, but are much easier to solve. We can apply the following set of operations without changing the solutions of a given system:</p>
+<ul class="simple">
+<li><p>Multiplication of an entire row by a non-zero constant</p></li>
+<li><p>Adding/subtracting one row to another</p></li>
+<li><p>Rearranging the order of the rows</p></li>
+</ul>
+<p>The examples below show this procedure applied to two systems.</p>
+<div class="admonition-examples admonition">
+<p class="admonition-title">Examples</p>
+<p>(i) <span class="math notranslate nohighlight">\(\begin{cases} 3x + 5y - z = 10 \\ 2x –  y + 3z = 9 \\ 4x + 2y - 3z = -1 \end{cases}\)</span></p>
+<p>See the solution on this <a class="reference external" href="https://web.microsoftstream.com/video/194b0152-903a-4614-b608-306ce71d9a39">video</a></p>
+<p>(ii) <span class="math notranslate nohighlight">\(\begin{cases} x - 3y + z = 4 \\ x – 2y + 3z = 6 \\ 2x - 6y + 2z = 8 \end{cases}\)</span></p>
+<p>See the solution on this <a class="reference external" href="https://web.microsoftstream.com/video/067c8938-5e45-4f08-bf90-783cb125c6f5">video</a></p>
+</div>
+<p>Note that by applying the rules of matrix multiplication, the system <em>(ii)</em> can also be written in the form:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{pmatrix} 1 &amp; -3 &amp; 1\\ 1 &amp; -2 &amp; 3 \\ 2 &amp; -6 &amp; 2  \end{pmatrix} \begin{pmatrix} x \\ y \\ z   \end{pmatrix} = \begin{pmatrix} 4 \\ 6 \\ 8   \end{pmatrix} \quad \longrightarrow \quad A\mathbf{x}=\mathbf{b}\end{split}\]</div>
+<p>where <span class="math notranslate nohighlight">\(A\)</span> is the matrix of coefficients and <span class="math notranslate nohighlight">\(\mathbf{x}\)</span> is the vector of variables.</p>
+<div class="admonition tip">
+<p class="admonition-title">Tip</p>
+<p>We will be using the common notation of uppercase letters (<span class="math notranslate nohighlight">\(A\)</span>) to represent matrices and lowercase bold letters (<span class="math notranslate nohighlight">\(\mathbf{x}\)</span>) to represent vectors.</p>
+</div>
+<p>In general, the system with <span class="math notranslate nohighlight">\(n\)</span> variables <span class="math notranslate nohighlight">\(\{x_1, x_2, \ldots, x_n \}\)</span> and <span class="math notranslate nohighlight">\(m\)</span> equations can be written as</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{cases} a_{11}x_1 + a_{12}x_2 + \ldots + a_{1n}x_n = b_1 \\ a_{21}x_1 + a_{22}x_2 + \ldots + a_{2n}x_n = b_2 \\ \dots \\ a_{m1}x_1 + a_{m2}x_2 + \ldots + a_{mn}x_n = b_m \end{cases}\end{split}\]</div>
+<p>can be written as <span class="math notranslate nohighlight">\(A\mathbf{x}=\mathbf{b}\)</span>, where</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}A=\begin{pmatrix} a_{11} &amp; a_{12} &amp; \cdots &amp; a_{1n} \\ a_{21} &amp; a_{22} &amp; \cdots &amp; a_{2n} \\ \vdots &amp; &amp; \ddots &amp; \vdots \\ a_{m1} &amp; a_{m2} &amp; \cdots &amp; a_{mn} \end{pmatrix}\end{split}\]</div>
+<p>and</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\mathbf{x} = \begin{pmatrix} x_1 \\ x_2 \\ \vdots \\ x_n   \end{pmatrix}\quad \mbox{and} \quad \mathbf{b} = \begin{pmatrix} b_1 \\ b_2 \\ \vdots \\ b_m   \end{pmatrix}\end{split}\]</div>
+<p>The matrix <span class="math notranslate nohighlight">\(A\)</span> has size <span class="math notranslate nohighlight">\(m \times n\)</span> (since the number of equations can, in principle, be different from the number of variables), while the vectors <span class="math notranslate nohighlight">\(\mathbf{x}\)</span> and <span class="math notranslate nohighlight">\(\mathbf{b}\)</span> have sizes <span class="math notranslate nohighlight">\(n \times 1\)</span> and <span class="math notranslate nohighlight">\(m \times 1\)</span>, respectively.</p>
+</section>
+<section id="matrix-operations">
+<h2>Matrix operations<a class="headerlink" href="#matrix-operations" title="Link to this heading">#</a></h2>
+<p>Let us quickly remind the basic operations that can be performed with matrices. If <span class="math notranslate nohighlight">\(A\)</span> and <span class="math notranslate nohighlight">\(B\)</span> are both matrices of size <span class="math notranslate nohighlight">\(m \times n\)</span> (<span class="math notranslate nohighlight">\(m\)</span> rows and <span class="math notranslate nohighlight">\(n\)</span> columns), we have:</p>
+<ul>
+<li><p><span class="math notranslate nohighlight">\(A=B\ \)</span> if and only if <span class="math notranslate nohighlight">\(a_{ij} = b_{ij}\)</span> for every <span class="math notranslate nohighlight">\(1 \le i \le m,  1 \le j \le n\)</span> (elements at the same position have to be equal).</p></li>
+<li><p><span class="math notranslate nohighlight">\(C = A + B\)</span> is the matrix for which <span class="math notranslate nohighlight">\(c_{ij} = a_{ij} + b_{ij}\)</span> (sum is performed summing the elements at the same position)</p></li>
+<li><p>If <span class="math notranslate nohighlight">\(k \in \rm I\!R\)</span> than <span class="math notranslate nohighlight">\(kA\)</span> is the <span class="math notranslate nohighlight">\(m \times n\)</span> matrix with elements <span class="math notranslate nohighlight">\(ka_{ij}\)</span></p></li>
+<li><p>The following properties are valid:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{align}
+    A+B &amp;= B+A \\
+    (A+B)+C &amp;= A+(B+C) \\
+    A + 0 &amp;= A\ (0\ \mbox{here represents the matrix with null elements}) \end{align}\end{split}\]</div>
+<div class="admonition-example admonition">
+<p class="admonition-title">Example</p>
+<p>If <span class="math notranslate nohighlight">\(A=\begin{pmatrix} 1 &amp; 0  \\ -1 &amp; 2  \end{pmatrix}\)</span> and <span class="math notranslate nohighlight">\(B=\begin{pmatrix} 2 &amp; 3  \\ -1 &amp; 1  \end{pmatrix}\)</span>, then:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}C = 2A+B = \begin{pmatrix} 2 &amp; 0  \\ -2 &amp; 4  \end{pmatrix} + \begin{pmatrix} 2 &amp; 3  \\ -1 &amp; 1  \end{pmatrix} = \begin{pmatrix} 4 &amp; 3  \\ -3 &amp; 5  \end{pmatrix}\end{split}\]</div>
+</div>
+</li>
+<li><p>If <span class="math notranslate nohighlight">\(A\)</span> is the <span class="math notranslate nohighlight">\(m \times n\)</span> matrix with elements <span class="math notranslate nohighlight">\(a_{ij}\)</span>, the transpose of <span class="math notranslate nohighlight">\(A\)</span> (denoted by <span class="math notranslate nohighlight">\(A'\)</span>) is the <span class="math notranslate nohighlight">\(n \times m\)</span> matrix with elements <span class="math notranslate nohighlight">\(a_{ij}' = a_{ji}\)</span>.</p>
+<div class="admonition-example admonition">
+<p class="admonition-title">Example</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}A=\begin{pmatrix} 1 &amp; 2 &amp; 3 \\ 4 &amp; 5 &amp; 6  \end{pmatrix} \longrightarrow A'=\begin{pmatrix} 1 &amp; 4 \\ 2 &amp; 5 \\ 3 &amp; 6   \end{pmatrix}\end{split}\]</div>
+<div class="math notranslate nohighlight">
+\[\begin{split}B=\begin{pmatrix} 1 &amp; 2 \end{pmatrix} \longrightarrow B' = \begin{pmatrix} 1 \\ 2 \end{pmatrix}\end{split}\]</div>
+</div>
+</li>
+<li><p><strong>(Matrix multiplication)</strong> Consider now that <span class="math notranslate nohighlight">\(A\)</span> and <span class="math notranslate nohighlight">\(B\)</span> are matrices with sizes <span class="math notranslate nohighlight">\(m \times l\)</span> and <span class="math notranslate nohighlight">\(l \times n\)</span> (number of columns of <span class="math notranslate nohighlight">\(A\)</span> is equal to the number of rows of <span class="math notranslate nohighlight">\(B\)</span>). Then we can define <span class="math notranslate nohighlight">\(C = AB\)</span>  as the <span class="math notranslate nohighlight">\(m \times n\)</span> matrix with elements <span class="math notranslate nohighlight">\(c_{ij}\)</span> formed by multiplying every element of the <span class="math notranslate nohighlight">\(i\)</span>-th row of <span class="math notranslate nohighlight">\(A\)</span> with the corresponding element of the <span class="math notranslate nohighlight">\(j\)</span>-th column of <span class="math notranslate nohighlight">\(B\)</span>.</p>
+<div class="admonition-example admonition">
+<p class="admonition-title">Example</p>
+<p>If <span class="math notranslate nohighlight">\(A=\begin{pmatrix} 1 &amp; 2  \\ -1 &amp; 0  \end{pmatrix}\)</span> and <span class="math notranslate nohighlight">\(B=\begin{pmatrix} 1 &amp; 2 &amp; 3  \\ 0 &amp; -1 &amp; 4  \end{pmatrix}\)</span>, then:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}C = AB = \begin{pmatrix} 1 &amp; 0 &amp; 11  \\ -1 &amp; -2 &amp; -3  \end{pmatrix}.\end{split}\]</div>
+<p>Note that the matrix <span class="math notranslate nohighlight">\(BA\)</span> cannot be defined here since the number of columns of <span class="math notranslate nohighlight">\(B\)</span> is different from the number of rows of <span class="math notranslate nohighlight">\(A\)</span>.</p>
+</div>
+<div class="admonition warning">
+<p class="admonition-title">Warning</p>
+<p>If matrices <span class="math notranslate nohighlight">\(A\)</span> and <span class="math notranslate nohighlight">\(B\)</span> are <em>square matrices</em> (same number of rows and columns) then we can define both <span class="math notranslate nohighlight">\(AB\)</span> and <span class="math notranslate nohighlight">\(BA\)</span>. But, in general, we <strong>cannot assume</strong> that <span class="math notranslate nohighlight">\(AB=BA\)</span>. For example, if  <span class="math notranslate nohighlight">\(A=\begin{pmatrix} 2 &amp; -1  \\ 4 &amp; -2  \end{pmatrix}\)</span> and <span class="math notranslate nohighlight">\(B=\begin{pmatrix} 1 &amp; -1  \\ 2 &amp; -2  \end{pmatrix}\)</span>, then:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}AB = \begin{pmatrix} 0 &amp; 0  \\ 0 &amp; 0  \end{pmatrix} \quad and \quad BA = \begin{pmatrix} -2 &amp; 1  \\ -4 &amp; 2  \end{pmatrix}\end{split}\]</div>
+</div>
+</li>
+<li><p>For matrix multiplication, the following properties are valid:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{align}
+    (A+B)C&amp;=AC+BC \\
+    A(B+C)&amp;=AB+AC \\
+    (AB)C &amp;= A(BC) \\
+    A0 &amp;= 0A = 0 \end{align}\end{split}\]</div>
+</li>
+<li><p>We can also define powers of <span class="math notranslate nohighlight">\(A\)</span>. If <span class="math notranslate nohighlight">\(k\)</span> is a positive integer:</p>
+<div class="math notranslate nohighlight">
+\[A^k = A^{k-1}A = AA^{k-1} = A\cdot A\cdot A \cdots A\ (\mbox{product of}\ k\ \mbox{matrices})\]</div>
+</li>
+</ul>
+</section>
+<section id="inverse-matrix">
+<h2>Inverse matrix<a class="headerlink" href="#inverse-matrix" title="Link to this heading">#</a></h2>
+<p>Suppose that <span class="math notranslate nohighlight">\(A\)</span> is a <span class="math notranslate nohighlight">\(n \times n\)</span> matrix. If there exists a <span class="math notranslate nohighlight">\(n \times n\)</span> matrix <span class="math notranslate nohighlight">\(B\)</span> so that</p>
+<div class="math notranslate nohighlight">
+\[ AB = BA = I_n,\]</div>
+<p>where <span class="math notranslate nohighlight">\(I=\begin{pmatrix} 1 &amp; 0 &amp; \cdots &amp; 0 \\ 0 &amp; 1 &amp; \cdots &amp; 0 \\ \vdots &amp; &amp; \ddots &amp; \vdots \\ 0 &amp; 0 &amp; \cdots &amp; 1 \end{pmatrix}\)</span> is the <span class="math notranslate nohighlight">\(n \times n\)</span> <em>identity matrix</em> (with elements <span class="math notranslate nohighlight">\(1\)</span> on the main diagonal and <span class="math notranslate nohighlight">\(0\)</span> otherwise).</p>
+<p>In this case we call the matrix <span class="math notranslate nohighlight">\(B\)</span> the <em>inverse</em> of <span class="math notranslate nohighlight">\(A\)</span> and denote <span class="math notranslate nohighlight">\(B=A^{-1}\)</span>. If <span class="math notranslate nohighlight">\(A\)</span> has an inverse than it is called <em>invertible</em> or <em>non-singular</em>. If it does not have an inverse, <span class="math notranslate nohighlight">\(A\)</span> is <em>singular</em>.</p>
+<p>Finding the inverse of <span class="math notranslate nohighlight">\(A\)</span> automatically gives us the solution to the system of linear equations <span class="math notranslate nohighlight">\(A\mathbf x = \mathbf x\)</span></p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{align}  
+A\mathbf{x}&amp;=\mathbf{b} \\
+A^{-1} A\mathbf{x} &amp;= A^{-1}\mathbf{b} \\
+\mathbf{x} &amp;= A^{-1}\mathbf{b} \end{align}\end{split}\]</div>
+<p>Finding ways of inverting matrices is therefore needed for a wide range of applications. A whole research field is dealing with optimising matrix inversions.</p>
+<div class="admonition-example admonition">
+<p class="admonition-title">Example</p>
+<p>To find the inverse of matrix <span class="math notranslate nohighlight">\(A= \begin{pmatrix} 2 &amp; 5  \\ 1 &amp; 3  \end{pmatrix}\)</span>, we have to find the matrix <span class="math notranslate nohighlight">\(B = \begin{pmatrix} b_{11} &amp; b_{12}  \\ b_{21} &amp; b_{22}  \end{pmatrix}\)</span> so that</p>
+<div class="math notranslate nohighlight">
+\[ AB = I \quad (\mbox{then check that }\ BA = I.\]</div>
+<p>Thus</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{align}  
+  \begin{pmatrix} 2 &amp; 5  \\ 1 &amp; 3  \end{pmatrix}\begin{pmatrix} b_{11} &amp; b_{12}  \\ b_{21} &amp; b_{22}  \end{pmatrix} = \begin{pmatrix} 1 &amp; 0  \\ 0 &amp; 1  \end{pmatrix} \\  
+  \begin{pmatrix} 2b_{11} + 5b_{21}  &amp; 2b_{12} + 5b_{22} \\ b_{11} + 3b_{21} &amp; b_{12} + 3b_{22}  \end{pmatrix} = \begin{pmatrix} 1 &amp; 0  \\ 0 &amp; 1  \end{pmatrix}, \\
+\end{align}\end{split}\]</div>
+<p>which leads to two systems of equations:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{cases} 2b_{11} + 5b_{21} &amp;= 1 \\ b_{11} + 3b_{21} &amp;= 0 \end{cases}\quad \mbox{and}\quad \begin{cases} 2b_{12} + 5b_{22} &amp;= 0 \\ b_{12} + 3b_{22} &amp;= 1 \end{cases}\end{split}\]</div>
+<p>The solutions of these two systems, leads to <span class="math notranslate nohighlight">\(B= \begin{pmatrix} 3 &amp; -5  \\ -1 &amp; 2  \end{pmatrix}\)</span></p>
+</div>
+</section>
+<section id="determinant">
+<h2>Determinant<a class="headerlink" href="#determinant" title="Link to this heading">#</a></h2>
+<p>If <span class="math notranslate nohighlight">\(A = \begin{pmatrix} a_{11} &amp; a_{12}  \\ a_{21} &amp; a_{22}  \end{pmatrix}\)</span> then the expression:</p>
+<div class="math notranslate nohighlight">
+\[\mbox{det}A = a_{11}a_{22} - a_{21}a_{12}\]</div>
+<p>given by the product of the elements on the main diagonal minus the product of the elements on secondary diagonal. The term <span class="math notranslate nohighlight">\(\mbox{det}A\)</span> is called the <em>determinant</em> of <span class="math notranslate nohighlight">\(A\)</span>. The determinant can be calculated for matrices of any size, but the expression gets increasingly more complicated for larger matrices.</p>
+<p>The determinant establishes a clear criterium for the existence of <span class="math notranslate nohighlight">\(A^{-1}\)</span>. The matrix <span class="math notranslate nohighlight">\(A\)</span> will be invertible if and only if <span class="math notranslate nohighlight">\(\mbox{det}A \ne 0\)</span>.</p>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>The previous criterium has an important implication. Consider the system of linear equations defined by <span class="math notranslate nohighlight">\(A\mathbf{x}=\mathbf{b}\)</span> with equal number of equations and variables. If <span class="math notranslate nohighlight">\(\mbox{det}A \ne 0\)</span> then the inverse of <span class="math notranslate nohighlight">\(A\)</span> exists. Thus, if we multiply <span class="math notranslate nohighlight">\(A^{-1}\)</span> to the left of each side of the equation defining the linear system, we have:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{align}  
+A\mathbf{x}&amp;=\mathbf{b} \\
+A^{-1}(A\mathbf{x}) &amp;= A^{-1}\mathbf{b} \\
+(A^{-1}A)\mathbf{x} &amp;= A^{-1}\mathbf{b} \\
+\mathbf{x} &amp;= A^{-1}\mathbf{b} \end{align}\end{split}\]</div>
+<p>which means that if <span class="math notranslate nohighlight">\(\mbox{det}A \ne 0\)</span> the system will have a <strong>single solution</strong> with the variables assuming the values <span class="math notranslate nohighlight">\(\mathbf{x} = A^{-1}\mathbf{b}\)</span>.</p>
+</div>
+</section>
+<section id="structured-models">
+<h2>Structured models<a class="headerlink" href="#structured-models" title="Link to this heading">#</a></h2>
+<p>In the models we have seen until now, all the individuals in a given population are identical. However, real populations have importance differences between individuals according to their life stages. For example, some insects have stages of eggs, larvae, pupae, and adults, while many plants pass through stages of seeds, seedlings, saplings, and mature individuals. Thus, understanding the differences in the dynamics of each life stage is important to add realism to our models (especially in applied models, such as pest control or harvesting management).</p>
+<p>Suppose our population is composed by two classes: <em>immature</em> and <em>mature</em> individuals. Let us assume non-overlapping breeding seasons and that the timescales for reproduction and maturation are similar. Thus, time can be counted in discrete steps corresponding to this reproduction/maturation interval. At time <span class="math notranslate nohighlight">\(t\)</span> we have <span class="math notranslate nohighlight">\(X_t\)</span> immature individuals and <span class="math notranslate nohighlight">\(Y_t\)</span> mature individuals. Immature individuals survive until maturity with a probability <span class="math notranslate nohighlight">\(p\)</span>, while mature individuals reproduce and generate, on average, <span class="math notranslate nohighlight">\(m\)</span> immature individuals for the next generation. Mature individuals die after reproduction. With these assumptions, we can calculate the sizes of each population class on the step <span class="math notranslate nohighlight">\(t+1\)</span> as:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{align}
+X_{t+1} = m Y_t \\
+Y_{t+1} = p X_t \end{align}\end{split}\]</div>
+<p>See that now we have a model with two equations describing the variable class sizes in our population. Moreover, the dynamics of each class depends on the other, so that we have a system of <em>coupled</em> difference equations.</p>
+<p>Interestingly, we can also write this system in matrix form, as:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{pmatrix} X_{t+1}  \\ Y_{t+1} \end{pmatrix} = \begin{pmatrix} 0 &amp; m  \\ p &amp; 0  \end{pmatrix} \begin{pmatrix} X_{t}  \\ Y_{t} \end{pmatrix}, \quad \mbox{or}\end{split}\]</div>
+<div class="math notranslate nohighlight">
+\[\mathbf{n_{t+1}} = P\mathbf{n_{t}}\]</div>
+<p>where the sum of the components of the population vector <span class="math notranslate nohighlight">\(\mathbf{n_{t}}\)</span>, gives the total population (immature plus mature individuals) at time <span class="math notranslate nohighlight">\(t\)</span>: <span class="math notranslate nohighlight">\(N_t = X_t + Y_t\)</span>. The matrix <span class="math notranslate nohighlight">\(P=\begin{pmatrix} 0 &amp; m  \\ p &amp; 0  \end{pmatrix}\)</span> is called <em>projection matrix</em> or <em>Leslie matrix</em>. Multiplying matrix <span class="math notranslate nohighlight">\(P\)</span> by the population vector at time <span class="math notranslate nohighlight">\(t\)</span>, <span class="math notranslate nohighlight">\(\mathbf{n_{t}}\)</span>, we can project the next population size at time <span class="math notranslate nohighlight">\(t+1\)</span>. Thus, from an initial population state, say <span class="math notranslate nohighlight">\(\mathbf{n_{0}}\)</span>, the projection matrix governs the dynamics of the model.</p>
+<br />
+<p>This model can be easily generalised for a larger number of age classes. Suppose we now have <span class="math notranslate nohighlight">\(q+1\)</span> classes, <span class="math notranslate nohighlight">\(X_0, X_1, X_2, \ldots, X_q\)</span>, from the youngest (meaningful) age <span class="math notranslate nohighlight">\(X_0\)</span> until the oldest age <span class="math notranslate nohighlight">\(X_q\)</span>. At each time step <span class="math notranslate nohighlight">\(t\)</span>, individuals from class <span class="math notranslate nohighlight">\(X_i\)</span> have a chance of surviving to the next age class <span class="math notranslate nohighlight">\(X_{i+1}\)</span>, with probability <span class="math notranslate nohighlight">\(p_{i+1\ i}\)</span> (imagine an arrow for direction of age class progession <span class="math notranslate nohighlight">\(p_{i+1\leftarrow i}\)</span>. We are going to omit the arrow for simplicity). Additionally, suppose that each age class <span class="math notranslate nohighlight">\(X_i\)</span> excluding the youngest, will, in principle, be able to reproduce, with average fecundity <span class="math notranslate nohighlight">\(m_i\)</span> generating new individuals for class <span class="math notranslate nohighlight">\(X_0\)</span>. Thus, with the age class sizes at time <span class="math notranslate nohighlight">\(t\)</span>, and the parameters for survival probabilities and average fecundities, we have:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{align}
+X_0(t+1) &amp;= m_1X_1(t) +  m_2X_2(t) + m_3X_3(t) + \ldots + m_qX_q(t) \\
+X_1(t+1) &amp;= p_{10}X_0(t) \\
+X_2(t+1) &amp;= p_{21}X_1(t) \\
+X_3(t+1) &amp;= p_{32}X_2(t) \\
+         &amp;\vdots \\
+X_q(t+1) &amp;= p_{q\ q-1}X_{q-1}(t)
+    \end{align}\end{split}\]</div>
+<p>This can also be written in matrix form as:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{pmatrix} X_0(t+1)  \\ X_1(t+1) \\X_2(t+1)  \\X_3(t+1) \\ \vdots \\X_q(t+1)  \\ \end{pmatrix} = \begin{pmatrix} 0 &amp; m_1 &amp; m_2 &amp; m_3 &amp; \cdots &amp; m_q \\ p_{10} &amp; 0 &amp; 0 &amp; 0 &amp; \cdots &amp;0 \\ 0 &amp; p_{21} &amp; 0 &amp; 0 &amp; \cdots &amp; 0 \\ 0 &amp; 0 &amp; p_{32} &amp; 0 &amp; \cdots &amp; 0 \\ \vdots &amp; \vdots &amp; \vdots &amp; \vdots &amp; \ddots &amp; \vdots \\ 0 &amp; 0 &amp; 0 &amp; \cdots &amp; p_{q\ q-1} &amp; 0\end{pmatrix} \begin{pmatrix} X_0(t)  \\ X_1(t) \\X_2(t) \\X_3(t) \\ \vdots \\X_q(t)  \\ \end{pmatrix}\end{split}\]</div>
+<div class="math notranslate nohighlight">
+\[\mathbf{n_{t+1}} = P\mathbf{n_{t}}\]</div>
+<p>The generalised projection matrix <span class="math notranslate nohighlight">\(P\)</span> has then all the surviving probabilities just below the main diagonal, and the fecundities for each class in its <span class="math notranslate nohighlight">\(0\)</span>-th row.</p>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>Two important differences here. First, the rows and columns for the matrix are indexed starting from <span class="math notranslate nohighlight">\(0\)</span> until <span class="math notranslate nohighlight">\(q\)</span>, not starting from <span class="math notranslate nohighlight">\(1\)</span> as usual. Second, the time steps are written in parentheses for each age class, instead of a subscript as we have seen before for other population models, so that it does not conflict with the index for the age classes (going from <span class="math notranslate nohighlight">\(0\)</span> to <span class="math notranslate nohighlight">\(q\)</span>).</p>
+</div>
+<div class="admonition-example admonition">
+<p class="admonition-title">Example</p>
+<p>Consider a population with three age classes <span class="math notranslate nohighlight">\(X_0, X_1,\mbox{ and}\ X_2\)</span>, and the following projection matrix</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}P=\begin{pmatrix} 0 &amp; 5 &amp; 10\\ 0.5 &amp; 0 &amp; 0  \\ 0 &amp; 0.2 &amp; 0 \end{pmatrix}\end{split}\]</div>
+<p>If the population starts with <span class="math notranslate nohighlight">\(10\)</span> individuals in the oldest category at time <span class="math notranslate nohighlight">\(t=0\)</span>, the population vector at time <span class="math notranslate nohighlight">\(t=1\)</span> will be:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\mathbf{n_{1}} = P\mathbf{n_{0}} \longrightarrow \begin{pmatrix} X_0(1)  \\ X_1(1) \\X_2(1) \end{pmatrix}=\begin{pmatrix} 0 &amp; 5 &amp; 10\\ 0.5 &amp; 0 &amp; 0  \\ 0 &amp; 0.2 &amp; 0 \end{pmatrix}\begin{pmatrix} 0  \\ 0 \\ 10 \end{pmatrix} \longrightarrow \begin{pmatrix} 100  \\ 0 \\ 0 \end{pmatrix}\end{split}\]</div>
+<p>The population age distribution at each time step will then be given be the successive application of the matrix <span class="math notranslate nohighlight">\(P\)</span> to the previous population distribution. For the first 10 time steps we have:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{pmatrix}0 \\ 0 \\ 10 \end{pmatrix} \rightarrow \begin{pmatrix} 100  \\ 0 \\ 0 \end{pmatrix} \rightarrow \begin{pmatrix} 0  \\ 50 \\ 0 \end{pmatrix} \rightarrow \begin{pmatrix} 250  \\ 0 \\ 10 \end{pmatrix} \rightarrow \begin{pmatrix} 100  \\ 125 \\ 0 \end{pmatrix} \rightarrow \begin{pmatrix} 625  \\ 50 \\ 25 \end{pmatrix} \rightarrow \begin{pmatrix} 500  \\ 312 \\ 10 \end{pmatrix} \rightarrow \begin{pmatrix} 1662  \\ 250 \\ 62 \end{pmatrix} \rightarrow \begin{pmatrix} 1875  \\ 831 \\ 50 \end{pmatrix} \rightarrow \begin{pmatrix} 4656  \\ 937 \\ 166 \end{pmatrix}\end{split}\]</div>
+<p>Two important things are worth noting here. First, the total population size (sum accros all age classes), follows the sequence <span class="math notranslate nohighlight">\(N_t = \left\{ 10,100,50,260,225,700,822,1975,2756,5759,\ldots \right \}\)</span>. The reproduction process does not always lead to increasing population sizes in the beginning, but is subject to size fluctuations due to the demographic distribution of this population. After a few steps increases in population size become more consistent. Second, as the population size continues to increase, the population age distribution tends to a given proportion of the population classes given by: <span class="math notranslate nohighlight">\(X_0 \approx 76\%, X_1 \approx 22\%, \mbox{and}\ X_2 \approx 2\%\)</span>.</p>
+<p>Now consider that, instead of <span class="math notranslate nohighlight">\(10\)</span> individuals in the oldest class, we initiate the population with the following distribution <span class="math notranslate nohighlight">\(\mathbf{n_{0}} = \begin{pmatrix} 7.6  \\ 2.2 \\ 0.2 \end{pmatrix}\)</span>. The population progression by successive application of the projection matrix will be</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{pmatrix} 7.6  \\ 2.2 \\ 0.2 \end{pmatrix} \longrightarrow \begin{pmatrix} 13  \\ 4 \\ 0.4 \end{pmatrix} \longrightarrow \begin{pmatrix} 23  \\ 6 \\ 0.76 \end{pmatrix} \longrightarrow \begin{pmatrix} 40  \\ 12 \\ 1 \end{pmatrix} \longrightarrow \begin{pmatrix} 72  \\ 20 \\ 2 \end{pmatrix} \longrightarrow \begin{pmatrix} 124  \\ 36 \\ 4 \end{pmatrix} \longrightarrow \begin{pmatrix} 219  \\ 62 \\ 7 \end{pmatrix} \longrightarrow \begin{pmatrix} 381  \\ 109 \\ 12 \end{pmatrix} \end{split}\]</div>
+<p>You can check that although each age cathegory is increasing, the percentage contribution of each class to the total population remains nearly constant. To this distribution of classes we give the name of <strong>stable age distribution</strong>.</p>
+<p>Moreover, at each time step, each age class increases with approximately the same factor in relation to the previous step, around <span class="math notranslate nohighlight">\(\lambda = 1.75\)</span>. If for each age class <span class="math notranslate nohighlight">\(X_i(t+1) = \lambda X_i(t)\)</span> is valid,then we have:</p>
+<div class="math notranslate nohighlight">
+\[\mathbf{n_{t+1}}=\lambda\mathbf{n_{t}},\]</div>
+<p>so that we recover the usual model for exponential growth for the total population size <span class="math notranslate nohighlight">\(N_{t+1} = \lambda N_t\)</span>.</p>
+</div>
+</section>
+<section id="from-table-arrangements-to-powerful-tools-part-2">
+<h2>From table arrangements to powerful tools - part 2<a class="headerlink" href="#from-table-arrangements-to-powerful-tools-part-2" title="Link to this heading">#</a></h2>
+</section>
+<section id="linear-transformations">
+<h2>Linear transformations<a class="headerlink" href="#linear-transformations" title="Link to this heading">#</a></h2>
+<p>In general terms, a given matrix <span class="math notranslate nohighlight">\(A\)</span> is a representation of what we call a <em>linear transformation</em>, a function from a special kind of set called <em>vector space</em> into another vector space. Vector spaces are generalised sets with particular properties, but for the moment you can think of two-dimensional and three-dimensional cartesian spaces as usual vector spaces. Then vectors are gonna be represented as arrays of real numbers as before: <span class="math notranslate nohighlight">\(\mathbf{n_{0}} = \begin{pmatrix} 7.6  \\ 2.2  \end{pmatrix}\)</span> or <span class="math notranslate nohighlight">\(\mathbf{v} = \begin{pmatrix} x \\ y \\ z \end{pmatrix}\)</span>.</p>
+<p>Back to linear transformations, they are usually denoted as:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{align}
+A:V &amp;\longrightarrow W \\
+  \mathbf{v} &amp;\rightarrow A(\mathbf{v}) \end{align}\end{split}\]</div>
+<p>where <span class="math notranslate nohighlight">\(V\)</span> and <span class="math notranslate nohighlight">\(W\)</span> are the domain and codomain vector spaces, and the transformation <span class="math notranslate nohighlight">\(A\)</span> is a function that projects vectors <span class="math notranslate nohighlight">\(\mathbf{v}\)</span> into the images <span class="math notranslate nohighlight">\(A(\mathbf{v})\)</span>. The property that defines linear transformations is the following:</p>
+<div class="math notranslate nohighlight">
+\[A(a\mathbf{v}+b\mathbf{w})=aA(\mathbf{v})+bA(\mathbf{w}),\]</div>
+<p>where <span class="math notranslate nohighlight">\(a\)</span> and <span class="math notranslate nohighlight">\(b\)</span> are constants. In other words,for a linear transformation, the image of the transformation <span class="math notranslate nohighlight">\(A\)</span> of a linear combination of two vectors (<span class="math notranslate nohighlight">\(\mathbf{v}\)</span> and <span class="math notranslate nohighlight">\(\mathbf{w}\)</span>) is equal to the linear combination of the images of this vectors by <span class="math notranslate nohighlight">\(A\)</span>.</p>
+<p>If we represent vectors with their components in a cartesian system, the images of the transformation <span class="math notranslate nohighlight">\(A\)</span> will be given be the multiplication of the matrix representation of <span class="math notranslate nohighlight">\(A\)</span> on the vectors.</p>
+<div class="admonition-example admonition">
+<p class="admonition-title">Example</p>
+<p>Given the matrix <span class="math notranslate nohighlight">\(A = \begin{pmatrix} 1 &amp; 2 \\ 3 &amp; 2 \end{pmatrix}\)</span> and the vectors <span class="math notranslate nohighlight">\(\mathbf{v_1} = \begin{pmatrix} 3  \\ -1  \end{pmatrix}\)</span> and <span class="math notranslate nohighlight">\(\mathbf{v_2} = \begin{pmatrix} 2  \\ 3  \end{pmatrix}\)</span>, then we have:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{align} A\mathbf{v_1} &amp;= \begin{pmatrix} 1 &amp; 2 \\ 3 &amp; 2 \end{pmatrix}\begin{pmatrix} 3  \\ -1  \end{pmatrix} = \begin{pmatrix} 1  \\ 7  \end{pmatrix} \\  
+  A\mathbf{v_2} &amp;= \begin{pmatrix} 1 &amp; 2 \\ 3 &amp; 2 \end{pmatrix}\begin{pmatrix} 2  \\ 3  \end{pmatrix} = \begin{pmatrix} 8  \\ 12  \end{pmatrix} = 4\begin{pmatrix} 2  \\ 3  \end{pmatrix} \end{align}\end{split}\]</div>
+<p>The vectors and their respective transformations are given by:</p>
+<img alt="../../_images/lin_trans.png" src="../../_images/lin_trans.png" />
+<p>We see that under the transformation <span class="math notranslate nohighlight">\(A\)</span>, vector <span class="math notranslate nohighlight">\(\mathbf{v_1}\)</span> gets rotated (counterclockwise) and stretched, while vector <span class="math notranslate nohighlight">\(\mathbf{v_2}\)</span> only gets stretched without changing the direction.</p>
+</div>
+</section>
+<section id="eigenvalues-and-eigenvectors">
+<h2>Eigenvalues and eigenvectors<a class="headerlink" href="#eigenvalues-and-eigenvectors" title="Link to this heading">#</a></h2>
+<p>As seen on the previous example, there might be some vectors that upon application of <span class="math notranslate nohighlight">\(A\)</span> do not have their direction changed, only their magnitude (or <em>module</em> of the vector). For these vectors, we have, in general:</p>
+<div class="math notranslate nohighlight">
+\[A\mathbf{v} = \lambda\mathbf{v}, \quad \mbox{for some constant }\ \lambda\]</div>
+<p>But this is equivalent to:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{align}
+A\mathbf{v} - \lambda\mathbf{v} = 0 \\  
+A\mathbf{v} - \lambda I\mathbf{v} = 0 \\
+\left(A - \lambda I \right)\mathbf{v} = 0 \end{align},\end{split}\]</div>
+<p>where <span class="math notranslate nohighlight">\(I\)</span> is the identity matrix. The equation <span class="math notranslate nohighlight">\(\left(A - \lambda I \right)\mathbf{v} = 0\)</span> defines a system of linear equations where <span class="math notranslate nohighlight">\(A - \lambda I\)</span> is the coefficient matrix, <span class="math notranslate nohighlight">\(\mathbf{v}\)</span> is the vector of variables, and <span class="math notranslate nohighlight">\(\mathbf{b}=0\)</span>. See that since <span class="math notranslate nohighlight">\(\mathbf{b}=0\)</span> we already know one possible solution for this system, which is <span class="math notranslate nohighlight">\(\mathbf{v}=0\)</span> (all the variables equal to <span class="math notranslate nohighlight">\(0\)</span>). Therefore, either this system has an unique solution (the null solution) or it has infinite solutions, but definitely it is not inconsistent.  Thus, if we want to know all possible non-zero vectors <span class="math notranslate nohighlight">\(\mathbf{v}\)</span> that satisfy <span class="math notranslate nohighlight">\(A\mathbf{v} = \lambda\mathbf{v}\)</span>, we have to impose that the system assumes infinite solutions. This criterium should be defined by the determinant of the matrix of coefficients, as:</p>
+<div class="math notranslate nohighlight">
+\[\mbox{det}(A-\lambda I) = 0.\]</div>
+<p>The determinant then defines a polynomial in the variable <span class="math notranslate nohighlight">\(\lambda\)</span>, <span class="math notranslate nohighlight">\(\mbox{det}(A-\lambda I)=p(\lambda)\)</span>, called <em>characteristic polynomial</em>. Since we are interested in the condition <span class="math notranslate nohighlight">\(p(\lambda)=0\)</span>, what we want to know are the roots <span class="math notranslate nohighlight">\(\lambda\)</span> of this polynomial.</p>
+<p>The solutions <span class="math notranslate nohighlight">\(\lambda\)</span> for these equations are called <strong>eigenvalues</strong> and their corresponding vectors are <strong>eigenvectors</strong>. In summary, the eigenvectors are all the vectors that, upon application of <span class="math notranslate nohighlight">\(A\)</span>, remain on the same direction, only changing magnitude, with the eigenvalue being the stretching/compressing factor.</p>
+<div class="admonition-example admonition">
+<p class="admonition-title">Example</p>
+<p>Let us calculate the eigenvalues and eigenvectors of <span class="math notranslate nohighlight">\(A = \begin{pmatrix} 1 &amp; 2 \\ 3 &amp; 2 \end{pmatrix}\)</span>. First we need to obtain matrix <span class="math notranslate nohighlight">\((A-\lambda I)\)</span> and to calculate its determinant. We have:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}A-\lambda I = \begin{pmatrix} 1 &amp; 2 \\ 3 &amp; 2 \end{pmatrix} - \lambda \begin{pmatrix} 1 &amp; 0 \\ 0 &amp; 1 \end{pmatrix} = \begin{pmatrix} 1 -\lambda &amp; 2 \\ 3 &amp; 2 -\lambda  \end{pmatrix}\end{split}\]</div>
+<p>The characteristic polynomial will be given by <span class="math notranslate nohighlight">\(p(\lambda)=\mbox{det}(A-\lambda I)\)</span>, and applying the rule for the calculus of determinants of matrices size 2, we have:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}p(\lambda)=\mbox{det}(A-\lambda I) = \mbox{det}\begin{pmatrix} 1 -\lambda &amp; 2 \\ 3 &amp; 2 -\lambda  \end{pmatrix} = (1 -\lambda)(2 -\lambda) - 3\cdot 2 \end{split}\]</div>
+<div class="math notranslate nohighlight">
+\[p(\lambda) = 2 -\lambda -2\lambda +\lambda^2 - 6 = \lambda^2 - 3\lambda -4.\]</div>
+<p>Thus, calculating the roots of this 2nd-degree polynomial, we find <span class="math notranslate nohighlight">\(\lambda_1=-1\)</span> and <span class="math notranslate nohighlight">\(\lambda_2=4\)</span>. For each of the eigenvalues, to obtain the eigenvectors we have to solve the linear system  <span class="math notranslate nohighlight">\(A\mathbf{v} = \lambda\mathbf{v}\)</span> for some general vector <span class="math notranslate nohighlight">\(\mathbf{v} = \begin{pmatrix} x  \\ y  \end{pmatrix}\)</span> (with coordinates <span class="math notranslate nohighlight">\(x\)</span> and <span class="math notranslate nohighlight">\(y\)</span> to be specified). Thus:</p>
+<ul>
+<li><p><span class="math notranslate nohighlight">\(\mathbf{\lambda_1 = -1}\)</span>:</p>
+<p><span class="math notranslate nohighlight">\(\begin{align}
+A\mathbf{v} = \lambda\mathbf{v} \longrightarrow \begin{pmatrix} 1 &amp; 2 \\ 3 &amp; 2 \end{pmatrix}\begin{pmatrix} x  \\ y \end{pmatrix}= -1\begin{pmatrix} x  \\ y \end{pmatrix}\longrightarrow \cases{x + 2y = -x \\ 3x + 2y = -y} \longrightarrow \cases{2x + 2y = 0 \\ 3x + 3y = 0}
+\end{align}\)</span></p>
+<p>Note that both equations correspond to the same linear equation <span class="math notranslate nohighlight">\(x + y = 0\)</span>, which is exactly what we would expect since we built the system to be underdetermined when we did <span class="math notranslate nohighlight">\(\mbox{det}(A-\lambda I) = 0\)</span>. The eigenvector corresponding to <span class="math notranslate nohighlight">\(\lambda_1\)</span> will be such that <span class="math notranslate nohighlight">\(x + y = 0\)</span> or <span class="math notranslate nohighlight">\(x = -y\)</span>. Thus, we can choose <span class="math notranslate nohighlight">\(\mathbf{v_1} = \begin{pmatrix} 1  \\ -1  \end{pmatrix}\)</span>.</p>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>Remember that infinite solutions are possible for the same system. The important is that the condition <span class="math notranslate nohighlight">\(x = -y\)</span> holds. In fact, multiplying a non-zero constant for the vector <span class="math notranslate nohighlight">\(\begin{pmatrix} 1  \\ -1  \end{pmatrix}\)</span> would still give a vector in the same direction. So if <span class="math notranslate nohighlight">\(\mathbf{v_1}\)</span> is an eigenvector, <span class="math notranslate nohighlight">\(k\mathbf{v_1}\ (k \in {\rm I\!R})\)</span> also is.</p>
+</div>
+  <br />  
+</li>
+<li><p><span class="math notranslate nohighlight">\(\mathbf{\lambda_1 = 4}\)</span>:</p>
+<p><span class="math notranslate nohighlight">\(\begin{align}
+A\mathbf{v} = \lambda\mathbf{v} \longrightarrow \begin{pmatrix} 1 &amp; 2 \\ 3 &amp; 2 \end{pmatrix}\begin{pmatrix} x  \\ y \end{pmatrix}= 4\begin{pmatrix} x  \\ y \end{pmatrix}\longrightarrow \cases{x + 2y = 4x \\ 3x + 2y = 4y} \longrightarrow \cases{-3x + 2y = 0 \\ 3x - 2y = 0}
+\end{align}\)</span></p>
+<p>Since the eigenvector needs to satisfy <span class="math notranslate nohighlight">\(3x=2y\)</span>, we choose <span class="math notranslate nohighlight">\(\mathbf{v_2} = \begin{pmatrix} 2/3  \\ 1  \end{pmatrix}\)</span>.</p>
+<p>The representation of these vectors on the cartesian plane will be:</p>
+</li>
+</ul>
+<img alt="../../_images/eigenvec_01.png" src="../../_images/eigenvec_01.png" />
+</div>
+<div class="admonition-example admonition">
+<p class="admonition-title">Example</p>
+<p>Let us calculate the eigenvalues and eigenvectors of matrix <span class="math notranslate nohighlight">\(A = \begin{pmatrix} -2 &amp; 1 \\ 0 &amp; -1 \end{pmatrix}\)</span>. We have:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}p(\lambda)=\mbox{det}(A-\lambda I) = \mbox{det}\begin{pmatrix} -2 -\lambda &amp; 1 \\ 0 &amp; -1 -\lambda  \end{pmatrix} = (-2 -\lambda)(-1 -\lambda) - 1\cdot 0 \end{split}\]</div>
+<div class="math notranslate nohighlight">
+\[p(\lambda)=(-2 -\lambda)(-1 -\lambda)\]</div>
+<p>Since we need <span class="math notranslate nohighlight">\(p(\lambda)=0\)</span> either one of the two factors has to be zero. So, <span class="math notranslate nohighlight">\(\lambda_1=-2\)</span> and <span class="math notranslate nohighlight">\(\lambda_2=-1\)</span>.<br />
+For the eigenvectors:</p>
+<ul>
+<li><p><span class="math notranslate nohighlight">\(\mathbf{\lambda_1 = -2}\)</span>:</p>
+<p><span class="math notranslate nohighlight">\(\begin{align}
+A\mathbf{v} = \lambda\mathbf{v} \longrightarrow \begin{pmatrix} -2 &amp; 1 \\ 0 &amp; -1 \end{pmatrix}\begin{pmatrix} x  \\ y \end{pmatrix}= -2\begin{pmatrix} x  \\ y \end{pmatrix}\longrightarrow \cases{-2x + y = -2x \\ -y = -2y} \longrightarrow \cases{y = 0 \\ y = 0}
+\end{align}\)</span></p>
+<p>Since <span class="math notranslate nohighlight">\(y=0\)</span>, we choose <span class="math notranslate nohighlight">\(\mathbf{v_1} = \begin{pmatrix} 1  \\ 0  \end{pmatrix}\)</span>.</p>
+</li>
+<li><p><span class="math notranslate nohighlight">\(\mathbf{\lambda_2 = -1}\)</span>:</p>
+<p><span class="math notranslate nohighlight">\(\begin{align}
+A\mathbf{v} = \lambda\mathbf{v} \longrightarrow \begin{pmatrix} -2 &amp; 1 \\ 0 &amp; -1 \end{pmatrix}\begin{pmatrix} x  \\ y \end{pmatrix}= -1\begin{pmatrix} x  \\ y \end{pmatrix}\longrightarrow \cases{-2x + y = -x \\ -y = -y} \longrightarrow \cases{-x + y = 0 \\ y = y}
+\end{align}\)</span></p>
+<p>The second equation is trivial and provides no information. From the first equation, since <span class="math notranslate nohighlight">\(x=y\)</span>, we choose <span class="math notranslate nohighlight">\(\mathbf{v_2} = \begin{pmatrix} 1  \\ 1  \end{pmatrix}\)</span>.</p>
+</li>
+</ul>
+<div class="admonition tip">
+<p class="admonition-title">Tip</p>
+<p>Note that the two eigenvalues <span class="math notranslate nohighlight">\(\lambda_1 = -2\)</span> and <span class="math notranslate nohighlight">\(\lambda_2 = -1\)</span> correspond to the elements of the main diagonal of the matrix <span class="math notranslate nohighlight">\(A\)</span>. Whenever the matrix <span class="math notranslate nohighlight">\(A\)</span> is <em>triangular</em>, meaning all the elements <em>below</em> the main diagonal are <span class="math notranslate nohighlight">\(0\)</span>, the eigenvalues will correspond to the elements of the main diagonal.</p>
+</div>
+</div>
+</section>
+<section id="vector-decomposition">
+<h2>Vector decomposition<a class="headerlink" href="#vector-decomposition" title="Link to this heading">#</a></h2>
+<p>Now suppose matrix <span class="math notranslate nohighlight">\(A\)</span> has eigenvalues <span class="math notranslate nohighlight">\(\lambda_1\)</span> and <span class="math notranslate nohighlight">\(\lambda_2\)</span>, with corresponding eigenvectors <span class="math notranslate nohighlight">\(\mathbf{u_1}\)</span> and <span class="math notranslate nohighlight">\(\mathbf{u_2}\)</span>. If <span class="math notranslate nohighlight">\(\lambda_1 \ne \lambda_2\)</span>, then the vectors <span class="math notranslate nohighlight">\(\mathbf{u_1}\)</span> and <span class="math notranslate nohighlight">\(\mathbf{u_2}\)</span> are <em>linearly independent</em> meaning one can not be written as a multiple (multiplied by a real constant) of the other. Geometrically, this means that the directions corresponding to this two eigenvectors are not on the same line.</p>
+<p>In two dimensions, this is enough to say that any given vector <span class="math notranslate nohighlight">\(\mathbf{v}\)</span> can be written as a linear combination of <span class="math notranslate nohighlight">\(\mathbf{u_1}\)</span> and <span class="math notranslate nohighlight">\(\mathbf{u_2}\)</span> (or that <span class="math notranslate nohighlight">\(\mathbf{u_1}\)</span> and <span class="math notranslate nohighlight">\(\mathbf{u_2}\)</span> consitute a <em>basis</em> of this space). We have:</p>
+<div class="math notranslate nohighlight">
+\[\mathbf{v}= a_1\mathbf{u_1} + a_2\mathbf{u_2},\]</div>
+<p>for constants <span class="math notranslate nohighlight">\(a_1\)</span> and <span class="math notranslate nohighlight">\(a_2\)</span> to be determined.</p>
+<p>But given that <span class="math notranslate nohighlight">\(A\)</span> is a linear transformation, it should hold that:</p>
+<div class="math notranslate nohighlight">
+\[A\mathbf{v} = A(a_1\mathbf{u_1} + a_2\mathbf{u_2}) = a_1A(\mathbf{u_1}) + a_2A(\mathbf{u_2}) = a_1\lambda_1\mathbf{u_1} + a_2\lambda_2\mathbf{u_2},\]</div>
+<p>with the last step coming from the fact that <span class="math notranslate nohighlight">\(\mathbf{u_1}\)</span> and <span class="math notranslate nohighlight">\(\mathbf{u_2}\)</span> are eigenvectors of <span class="math notranslate nohighlight">\(A\)</span>. Since <span class="math notranslate nohighlight">\(A\mathbf{v}\)</span> is also a vector in this space, we can go ahead and apply the transformation <span class="math notranslate nohighlight">\(A\)</span> again, and we will have:</p>
+<div class="math notranslate nohighlight">
+\[A(A\mathbf{v}) = A(a_1\lambda_1\mathbf{u_1} + a_2\lambda_2\mathbf{u_2})=a_1\lambda_1 A(\mathbf{u_1}) + a_2\lambda_2 A(\mathbf{u_2}) = a_1\lambda_1^2\mathbf{u_1} + a_2\lambda_2^2\mathbf{u_2}.\]</div>
+<p>So, in general:</p>
+<div class="math notranslate nohighlight">
+\[A(A(A \cdots A\mathbf{v}) \cdots )) = A^n\mathbf{v} = a_1\lambda_1^n\mathbf{u_1} + a_2\lambda_2^n\mathbf{u_2},\]</div>
+<p>or, in other words, if we know the eigenvectors and eigenvalues of <span class="math notranslate nohighlight">\(A\)</span> (with <span class="math notranslate nohighlight">\(\lambda_1 \ne \lambda_2\)</span>), and we know the coeffiencients <span class="math notranslate nohighlight">\(a_1\)</span> and <span class="math notranslate nohighlight">\(a_2\)</span> of the expansion of <span class="math notranslate nohighlight">\(\mathbf{v}\)</span> into the eigenvectors, then it is easy to calculate what is going to be the image ov vector <span class="math notranslate nohighlight">\(\mathbf{v}\)</span> by any number of successive applications of <span class="math notranslate nohighlight">\(A\)</span>. This is also equivalent of applying the <span class="math notranslate nohighlight">\(n\)</span>-th power of matrix <span class="math notranslate nohighlight">\(A\)</span> to <span class="math notranslate nohighlight">\(\mathbf{v}\)</span>.</p>
+<p>This fact will have many important applications, as we will se next.</p>
+</section>
+<section id="structured-models-revisited">
+<h2>Structured models revisited<a class="headerlink" href="#structured-models-revisited" title="Link to this heading">#</a></h2>
+<p>Let us considered a structured population model defined by the projection matrix <span class="math notranslate nohighlight">\(P=\begin{pmatrix} 1.5 &amp; 2  \\ 0.08 &amp; 0  \end{pmatrix}\)</span> and an initial population vector given by <span class="math notranslate nohighlight">\(\mathbf{n_0} = \begin{pmatrix} 105  \\ 1 \end{pmatrix}\)</span>. If we calculate the eigenvalues and eigenvectors, we will have:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}p(\lambda)=\mbox{det}(P-\lambda I) = \mbox{det}\begin{pmatrix} 1.5 -\lambda &amp; 2 \\ 0.08 &amp; -\lambda  \end{pmatrix} = (1.5 -\lambda)(-\lambda) - 0.08\cdot 2 \end{split}\]</div>
+<div class="math notranslate nohighlight">
+\[p(\lambda)=\lambda^2-1.5\lambda-0.16=(\lambda-1.6)(\lambda+0.1)\]</div>
+<p>Thus, we have <span class="math notranslate nohighlight">\(\lambda_1=1.6\)</span> and <span class="math notranslate nohighlight">\(\lambda_2=-0.1\)</span>.</p>
+<ul>
+<li><p><span class="math notranslate nohighlight">\(\mathbf{\lambda_1 = 1.6}\)</span>:</p>
+<p><span class="math notranslate nohighlight">\(\begin{align}
+A\mathbf{v} = \lambda\mathbf{v} \longrightarrow \begin{pmatrix} 1.5 &amp; 2  \\ 0.08 &amp; 0 \end{pmatrix}\begin{pmatrix} x  \\ y \end{pmatrix}= 1.6\begin{pmatrix} x  \\ y \end{pmatrix}\longrightarrow \cases{1.5x + 2y = 1.6x \\ 0.08x = 1.6y} \longrightarrow \cases{x - 20y = 0 \\ x - 20y = 0}
+\end{align}\)</span></p>
+<p>Since <span class="math notranslate nohighlight">\(x=20y\)</span>, we choose <span class="math notranslate nohighlight">\(\mathbf{u_1} = \begin{pmatrix} 20  \\ 1  \end{pmatrix}\)</span>.</p>
+<br />
+</li>
+<li><p><span class="math notranslate nohighlight">\(\mathbf{\lambda_2 = -0.1}\)</span>:</p>
+<p><span class="math notranslate nohighlight">\(\begin{align}
+A\mathbf{v} = \lambda\mathbf{v} \longrightarrow \begin{pmatrix} 1.5 &amp; 2  \\ 0.08 &amp; 0 \end{pmatrix}\begin{pmatrix} x  \\ y \end{pmatrix}= -0.1\begin{pmatrix} x  \\ y \end{pmatrix}\longrightarrow \cases{1.5x + 2y = -0.1x \\ 0.08x = -0.1y} \longrightarrow \cases{0.8x + y = 0 \\ 0.08x + 0.1y = 0}
+\end{align}\)</span></p>
+<p>Since <span class="math notranslate nohighlight">\(-\displaystyle\frac{4}{5}x=y\)</span>, we choose <span class="math notranslate nohighlight">\(\mathbf{u_2} = \begin{pmatrix} 5  \\ -4  \end{pmatrix}\)</span>.</p>
+</li>
+</ul>
+<br />
+<p>We can also note that the initial population vector can be decomposed into the eigenvectors <span class="math notranslate nohighlight">\(\mathbf{u_1}\)</span> and <span class="math notranslate nohighlight">\(\mathbf{u_2}\)</span> of the matrix <span class="math notranslate nohighlight">\(P\)</span> according to:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\mathbf{n_0}=\begin{pmatrix} 105  \\ 1 \end{pmatrix} = 5\begin{pmatrix} 20  \\ 1 \end{pmatrix} + \begin{pmatrix} 5  \\ -4 \end{pmatrix} = 5\mathbf{u_1} + \mathbf{u_2}\end{split}\]</div>
+<p>We know that to find the population vector for the following step, we have to apply the projection matrix on the current population vector. Thus:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\mathbf{n_1} = P\mathbf{n_0}= P(5\mathbf{u_1} + \mathbf{u_2})=5P(\mathbf{u_1}) + P(\mathbf{u_2})=5(1.6)\begin{pmatrix} 20  \\ 1 \end{pmatrix} + (-0.1)\begin{pmatrix} 5  \\ -4 \end{pmatrix} = \begin{pmatrix} 159.5  \\ 8.4 \end{pmatrix}\end{split}\]</div>
+<p>If we perform successive applications of <span class="math notranslate nohighlight">\(P\)</span>, we will finally have:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\mathbf{n_t} = P^t\mathbf{n_0} = P^t(5\mathbf{u_1} + \mathbf{u_2})=5(1.6)^t\begin{pmatrix} 20  \\ 1 \end{pmatrix} + (-0.1)^t\begin{pmatrix} 5  \\ -4 \end{pmatrix}.\end{split}\]</div>
+<br />  
+<p>By decomposing into eigenvalues and eigenvectors, we know the behaviour of our population for any time <span class="math notranslate nohighlight">\(t\)</span>, which is much easier than applying <span class="math notranslate nohighlight">\(P\)</span> a number <span class="math notranslate nohighlight">\(t\)</span> of times, or than calculating <span class="math notranslate nohighlight">\(P^t\)</span>. Furthermore, as <span class="math notranslate nohighlight">\(t\rightarrow\infty\)</span> the first term will dominate (with the second term dying out) and the proportions for each age class in this population will be dictated by the vector <span class="math notranslate nohighlight">\(\begin{pmatrix} 20  \\ 1 \end{pmatrix}\)</span> (or, if we divide by the sum of the components: <span class="math notranslate nohighlight">\(\displaystyle\frac{1}{21}\begin{pmatrix} 20  \\ 1 \end{pmatrix} = \begin{pmatrix} 95\%  \\ 5\% \end{pmatrix}\)</span>.</p>
+<p>In general the eigenvector corresponding to the <strong>largest eigenvalue</strong> (also called <em>dominant eigenvalue</em>) of the projection matrix, which is always <em>real</em> and <em>positive</em>, will correspond to the stable age distribution, for which our population vector converges after we wait enough time.</p>
+<p>As we have seen, after enough time, the dominant eigenvalue will also correspond to the growth factor for each of the age classes. Thus, if <span class="math notranslate nohighlight">\(\lambda_1&gt;1\)</span> the population size will increase, whereas if <span class="math notranslate nohighlight">\(0&lt;\lambda_1&lt;1\)</span> the population size will decrease.</p>
+</section>
+<section id="analysing-change-continuously">
+<h2>Analysing change, continuously<a class="headerlink" href="#analysing-change-continuously" title="Link to this heading">#</a></h2>
+</section>
+<section id="average-rate-of-change">
+<h2>Average rate of change<a class="headerlink" href="#average-rate-of-change" title="Link to this heading">#</a></h2>
+<p>Watch this <a class="reference external" href="https://web.microsoftstream.com/video/2953e661-e1c1-43ff-ba78-ca52099ed1ec">video</a>
+for a short introduction to this lecture.</p>
+</section>
+<section id="instantaneous-rate-of-change">
+<h2>Instantaneous rate of change<a class="headerlink" href="#instantaneous-rate-of-change" title="Link to this heading">#</a></h2>
+<p>If we have a continuous function <span class="math notranslate nohighlight">\(f(x)\)</span>, for each two points <span class="math notranslate nohighlight">\(x_{0}\)</span> and <span class="math notranslate nohighlight">\(x_{1}\)</span>, we can define the average rate of change between these points by the ratio between how much <span class="math notranslate nohighlight">\(f\)</span> has changed and how much <span class="math notranslate nohighlight">\(x\)</span> has changed:</p>
+<div class="math notranslate nohighlight">
+\[\frac{\Delta f(x)}{\Delta x} = \frac{f(x_{1}) - f(x_{0})}{x_{1} - x_{0}}\]</div>
+<p>For points <span class="math notranslate nohighlight">\(x_{0}\)</span> and <span class="math notranslate nohighlight">\(x_{1}\)</span> this can be graphically represented by:</p>
+<img alt="../../_images/sec_tan.png" src="../../_images/sec_tan.png" />
+<p>The line that passes through the points <span class="math notranslate nohighlight">\((x_0, f(x_0))\)</span> and <span class="math notranslate nohighlight">\((x_1, f(x_1))\)</span> (yellow line) is named <em>secant line</em>. As we have seen before in our treatment of linear functions, the slope of the secant line will be given by the average rate of change between this points (remember that the slope for linear functions is defined as <span class="math notranslate nohighlight">\(m=\displaystyle\frac{\Delta y}{\Delta x})\)</span>.</p>
+<p>Now consider that we change the value of <span class="math notranslate nohighlight">\(x_1\)</span> so that it slowly approaches <span class="math notranslate nohighlight">\(x_0\)</span> from the right. In this case, the secant lines also change (from yellow to red), as we change the image of <span class="math notranslate nohighlight">\(x_1\)</span> the function, <span class="math notranslate nohighlight">\(f(x_1)\)</span>. In the limit where <span class="math notranslate nohighlight">\(x_1\)</span> is very close to <span class="math notranslate nohighlight">\(x_0\)</span>, the secant line approaches the <em>tangent</em> (sark red line) of the function <span class="math notranslate nohighlight">\(f(x)\)</span> at the point <span class="math notranslate nohighlight">\(x_0\)</span> (the tangent is the straight line that “just touches” the curve at that point).</p>
+<p>If we represent the slope of the tangent line at the point <span class="math notranslate nohighlight">\(x_0\)</span> by the term <span class="math notranslate nohighlight">\(f'(x_0)\)</span>, we will have:</p>
+<div class="math notranslate nohighlight">
+\[\lim_{x_{1} \to x_{0}} \frac{\Delta f(x)}{\Delta x} = \frac{f(x_{1}) - f(x_{0})}{x_{1} - x_{0}} = f'(x_{0})\]</div>
+<p>Since we can do the same process as we change the value <span class="math notranslate nohighlight">\(x_0\)</span>, we define a new function <span class="math notranslate nohighlight">\(f'(x_{0})\)</span> which we call the <strong>derivative</strong> of the function <span class="math notranslate nohighlight">\(f\)</span> at the point <span class="math notranslate nohighlight">\(x_0\)</span>, and its value tells us about the <em>instantaneous</em> rate of change of <span class="math notranslate nohighlight">\(f\)</span> at the point <span class="math notranslate nohighlight">\(x_0\)</span>.</p>
+<p>If we look instead at the interval between <span class="math notranslate nohighlight">\(x_{0}\)</span> and <span class="math notranslate nohighlight">\(x_{1}\)</span> and define <span class="math notranslate nohighlight">\(h = x_1 - x_0\)</span>, if <span class="math notranslate nohighlight">\(x_1\)</span> approaches <span class="math notranslate nohighlight">\(x_0\)</span>, then <span class="math notranslate nohighlight">\(h\)</span> approaches <span class="math notranslate nohighlight">\(0\)</span>. and we would have for the derivative of <span class="math notranslate nohighlight">\(f(x)\)</span>:</p>
+<div class="math notranslate nohighlight">
+\[\lim_{h \to 0} \frac{f(x + h) - f(x)}{h} = f'(x)\]</div>
+<p>Visit this <a class="reference external" href="https://www.demonstrations.wolfram.com/SecantAndTangentLines/">link</a> for an interactive demonstration of the limit of the tangent curve.</p>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>Note that in the previous expression the index <span class="math notranslate nohighlight">\(0\)</span> was omitted from <span class="math notranslate nohighlight">\(x_0\)</span>. We will assume that the derivative is defined for all the points in the domain of the function <span class="math notranslate nohighlight">\(f(x)\)</span> for which the limit in the expression exists.</p>
+<p>It is good to have a graphical intuition for when the derivative is not well defined. Some cases are shown below.</p>
+</div>
+<div class="admonition-example admonition">
+<p class="admonition-title">Example</p>
+<p>By the definition of the derivative of <span class="math notranslate nohighlight">\(f(x)\)</span>:</p>
+<div class="math notranslate nohighlight">
+\[f'(x)=\lim_{h \to 0} \frac{f(x + h) - f(x)}{h}\]</div>
+<p>Thus, if <span class="math notranslate nohighlight">\(f(x)=x^2\)</span>, we will have:</p>
+<div class="math notranslate nohighlight">
+\[f'(x)=\lim_{h \to 0} \frac{(x + h)^{2} - x^{2}}{h} = \lim_{h \to 0} \frac{x^{2} + 2xh + h^{2} - x^{2}}{h} = \lim_{h \to 0} (h + 2x) = 2x\]</div>
+<p>So that the derivative of <span class="math notranslate nohighlight">\(f(x)=x^2\)</span> is the function <span class="math notranslate nohighlight">\(f'(x)=2x\)</span>.</p>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>Although limits of functions can be defined in a more rigorous way, in this course we will treat them more intuitively, and use, if required, the known rules for limits. Thus, if <span class="math notranslate nohighlight">\(C\)</span> is a constant, <span class="math notranslate nohighlight">\(\lim_{x \rightarrow a} f(x) = L\)</span>, and <span class="math notranslate nohighlight">\(\lim_{x \rightarrow a} g(x) = M\)</span>, it is generally valid that:</p>
+<div class="math notranslate nohighlight">
+\[\lim_{x \rightarrow a} [f(x) + g(x)] = \lim_{x \rightarrow a} f(x) + \lim_{x \rightarrow a} g(x) = L + M\]</div>
+<div class="math notranslate nohighlight">
+\[\lim_{x \rightarrow a} [Cf(x)] = C\lim_{x \rightarrow a} f(x) = CL\]</div>
+<div class="math notranslate nohighlight">
+\[\lim_{x \rightarrow a} [f(x)g(x)] = [\lim_{x \rightarrow a} f(x)][\lim_{x \rightarrow a} g(x)] = LM\]</div>
+<div class="math notranslate nohighlight">
+\[\lim_{x \rightarrow a} \left[\frac{f(x)}{g(x)}\right] = \frac{\lim_{x \rightarrow a} f(x)}{\lim_{x \rightarrow a} g(x)} = \frac{L}{M},\ \ (\mbox{if}\ M \ne 0)\]</div>
+<div class="math notranslate nohighlight">
+\[\lim_{x \rightarrow a} [C^{f(x)}] = C^{\lim_{x \rightarrow a} f(x)} = C^L\]</div>
+</div>
+</div>
+<p>Other common notations for the derivative are given by:</p>
+<div class="math notranslate nohighlight">
+\[f'(x) = \frac{df}{dx} = \frac{d}{dx}f(x) = \dot{f}(x)\]</div>
+<p>Visit this <a class="reference external" href="https://the-learning-machine.com/article/calculus/derivatives">link</a> for examples that provide demonstrations of the intuitive interpretation of the derivative.</p>
+</section>
+<section id="calculating-derivatives">
+<h2>Calculating derivatives<a class="headerlink" href="#calculating-derivatives" title="Link to this heading">#</a></h2>
+<p>It is possible to generalise the previous example for <span class="math notranslate nohighlight">\(f(x)=x^2\)</span> and obtain the derivative for any power function <span class="math notranslate nohighlight">\(f(x)=x^n\)</span> (<span class="math notranslate nohighlight">\(n \ne 0\)</span>):</p>
+<div class="math notranslate nohighlight">
+\[f(x) = x^{n} \implies f'(x) = nx^{n-1}\]</div>
+<div class="admonition-example admonition">
+<p class="admonition-title">Example</p>
+<div class="math notranslate nohighlight">
+\[x^{2} \implies 2x^{2-1} = 2x\]</div>
+<div class="math notranslate nohighlight">
+\[\ x^{3} \implies 3x^{3-1} = 3x^{2}\]</div>
+<div class="math notranslate nohighlight">
+\[\ x^{4.2} \implies 4.2x^{4.2-1} = 4.2x^{3.2}\]</div>
+</div>
+<p>It is possible to show, from the definition of derivative and the rules for limits, that there are simple rules for derivatives when calculating them for a sum of functions or a product by a constant. If <span class="math notranslate nohighlight">\(C\)</span> is a constant, we generally have:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{align}\frac{d}{dx}[f(x)+g(x)] &amp;= \frac{df}{dx} + \frac{dg}{dx} \\
+   \frac{d}{dx}[Cf(x)] &amp;= C\frac{df}{dx}\end{align}\end{split}\]</div>
+<p>Thus, differentiating a given polynomial p(x) stands for differentiating each term separately and summing the result.</p>
+<div class="admonition-example admonition">
+<p class="admonition-title">Example</p>
+<div class="math notranslate nohighlight">
+\[f(x) = 3x^{2} - 4x + 6 \implies f'(x) = 3(2x) - 4(x^{0}) + 0 = 6x - 4\]</div>
+<div class="math notranslate nohighlight">
+\[f(x) = x^{60} - 3x^{20} - 4x \implies f'(x) = 60x^{59} - 60x^{19} - 4\]</div>
+</div>
+<p>For products and quotients of functions, however, we have different rules:</p>
+<div class="math notranslate nohighlight">
+\[\frac{d}{dx}[f(x)g(x)] = \left(\frac{df}{dx}\right) g(x) + f(x)\left(\frac{dg}{dx}\right)\]</div>
+<div class="math notranslate nohighlight">
+\[\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{\left(\displaystyle\frac{df}{dx}\right)g(x) - f(x)\left(\displaystyle\frac{dg}{dx}\right)}{[g(x)]^{2}}\]</div>
+<div class="admonition tip">
+<p class="admonition-title">Tip</p>
+<p>If you do not want to remember the quotients rule, you can use the product rule on <span class="math notranslate nohighlight">\(f(x)j(x)\)</span>, with <span class="math notranslate nohighlight">\(j(x)=g(x)^{-1}\)</span> and use the chain rule to evaluate <span class="math notranslate nohighlight">\(\frac{d}{dx} j(x)\)</span>.</p>
+</div>
+<div class="admonition-examples admonition">
+<p class="admonition-title">Examples</p>
+<div class="math notranslate nohighlight">
+\[f(x) = (x + 2)(x - 4) \implies f'(x) = (1)(x-4) + (x+2)(1) = x - 4 + x + 2 = 2x - 2\]</div>
+<div class="math notranslate nohighlight">
+\[f(x) = \frac{\alpha x}{1 + \beta x} \implies f'(x) =  \frac{(\alpha)(1 + \beta x) - (\alpha x)(\beta)}{(1 + \beta x)^{2}} = \frac{\alpha + \alpha\beta x - \alpha \beta x}{(1 + \beta x)^{2}} =\frac{\alpha}{(1 + \beta x)^{2}}\]</div>
+</div>
+<p>Moreover, expressions for the derivatives of the other elementary functions can also be found:</p>
+<ul class="simple">
+<li><p><span class="math notranslate nohighlight">\(f(x) = a^{x} \implies f'(x) = a^{x} \cdot (\ln a) ,\ (a &gt; 0, a \neq 1)\)</span><br />
+<span class="math notranslate nohighlight">\(\left(\mbox{in particular: }\ (e^{x})' = e^{x}\right)\)</span></p></li>
+<li><p><span class="math notranslate nohighlight">\(f(x) = \log_{a}x \implies f'(x) = \displaystyle\frac{1}{(\ln a)x}\)</span><br />
+<span class="math notranslate nohighlight">\(\left(\mbox{in particular: }\ (\ln x)' = \displaystyle\frac{1}{x}\right)\)</span></p></li>
+<li><p><span class="math notranslate nohighlight">\(f(x) = \mbox{sin}(x) \implies f'(x) = \mbox{cos}(x)\)</span></p></li>
+<li><p><span class="math notranslate nohighlight">\(f(x) = \mbox{cos}(x) \implies f'(x) = -\mbox{sin}(x)\)</span></p></li>
+</ul>
+<div class="admonition-example admonition">
+<p class="admonition-title">Example</p>
+<p>If <span class="math notranslate nohighlight">\(f(x) = \mbox{tan}(x) = \displaystyle\frac{\mbox{sin}(x)}{\mbox{cos}(x)}\)</span>, then:</p>
+<div class="math notranslate nohighlight">
+\[f'(x) = \frac{(\mbox{cos}(x))\mbox{cos}(x) - \mbox{sin}(x)(-\mbox{sin}(x))}{(\mbox{cos}(x))^{2}} = \frac{\mbox{sin}^{2}(x) + \mbox{cos}^{2}(x)}{\mbox{cos}^{2}(x)} = \frac{1}{\mbox{cos}^{2}(x)} = \mbox{sec}^{2}(x)\]</div>
+</div>
+</section>
+<section id="chain-rule">
+<h2>Chain rule<a class="headerlink" href="#chain-rule" title="Link to this heading">#</a></h2>
+<p>The calculation of derivatives for more complicated functions can generally be made easier by breaking the function into components, for example:</p>
+<ul class="simple">
+<li><p><span class="math notranslate nohighlight">\(f(x) = (3x^{2} - 2x + 1)^{3} = (g(x))^{3}\)</span>, where <span class="math notranslate nohighlight">\(g(x) = 3x^{2} - 2x + 1\)</span></p></li>
+<li><p><span class="math notranslate nohighlight">\(f(x) = \displaystyle\frac{\sqrt{2x - 1}}{1 + \sqrt{2x - 1}} = \displaystyle\frac{g(x)}{1 + g(x)}\)</span>, where <span class="math notranslate nohighlight">\(g(x) = \sqrt{h(x)}\)</span>, and <span class="math notranslate nohighlight">\(h(x) = 2x - 1\)</span></p></li>
+</ul>
+<p>We have seen this before when discussing transformation of graphs.  When one function is nested within the other, we have a composite function, or as for the last example above:</p>
+<div class="math notranslate nohighlight">
+\[f(x) = f(g(h(x)))\]</div>
+<p>The derivative of composite functions in relation to the dependent variable is then given by the <em>chain rule</em>:</p>
+<ul class="simple">
+<li><p><span class="math notranslate nohighlight">\(\displaystyle\frac{d}{dx}f(x) = \displaystyle\frac{d}{dx}f(g(x)) = \left(\displaystyle\frac{df}{dg}\right)\left(\displaystyle\frac{dg}{dx}\right)\)</span></p></li>
+<li><p><span class="math notranslate nohighlight">\(\displaystyle\frac{d}{dx}f(x) = \displaystyle\frac{d}{dx}f(g(h(x))) = \left(\displaystyle\frac{df}{dg}\right) \left(\displaystyle\frac{dg}{dh}\right) \left(\displaystyle\frac{dh}{dx}\right)\)</span></p></li>
+</ul>
+<p>The strategy then consists in identifying the nested functions, calculating their derivatives and multiplying the results.</p>
+<div class="admonition-examples admonition">
+<p class="admonition-title">Examples</p>
+<ul>
+<li><p>If <span class="math notranslate nohighlight">\(f(x) = (3x^{2} - 2x + 1)^{3}\)</span>, we have:</p>
+<div class="math notranslate nohighlight">
+\[f(g) = g^{3} \implies \displaystyle\frac{df}{dg} = 3g^2\]</div>
+<div class="math notranslate nohighlight">
+\[g(x) = 3x^{2} - 2x + 1 \implies \displaystyle\frac{dg}{dx} = 6x - 2\]</div>
+<p>Thus:</p>
+<div class="math notranslate nohighlight">
+\[\displaystyle\frac{df}{dx} = \left(\displaystyle\frac{df}{dg}\right)\left(\displaystyle\frac{dg}{dx}\right) = 3g^{2} \cdot g' = 3(3x^{2} - 2x + 1)^2(6x - 2)\]</div>
+  <br />  
+</li>
+<li><p>If <span class="math notranslate nohighlight">\(f(x) = \displaystyle\frac{\sqrt{2x - 1}}{1 + \sqrt{2x - 1}}\)</span>, then:</p>
+<div class="math notranslate nohighlight">
+\[f(g) = \displaystyle\frac{g}{1+g} \implies \displaystyle\frac{df}{dg} = \frac{(1+g) - g}{(1+g)^2} = \frac{1}{(1+g)^2}\]</div>
+<div class="math notranslate nohighlight">
+\[g(h) = \sqrt{h} \implies \displaystyle\frac{dg}{dh} = \frac{1}{2}h^{-1/2}\]</div>
+<div class="math notranslate nohighlight">
+\[h(x) = 2x-1 \implies \displaystyle\frac{dh}{dx} = 2\]</div>
+<p>Thus:</p>
+<div class="math notranslate nohighlight">
+\[\displaystyle\frac{df}{dx} =  \left(\displaystyle\frac{df}{dg}\right) \left(\displaystyle\frac{dg}{dh}\right) \left(\displaystyle\frac{dh}{dx}\right) = \frac{1}{(1+\sqrt{2x - 1})^2}\cdot \frac{1}{2(\sqrt{2x - 1})}\cdot 2 = \frac{1}{(\sqrt{2x - 1})(1+\sqrt{2x - 1})^2}\]</div>
+  <br />
+</li>
+<li><p>If <span class="math notranslate nohighlight">\(f(t) = 5 + 5\mbox{cos}\left[\displaystyle\frac{\pi}{12}(t - 3)\right]\)</span>, we have:</p>
+<div class="math notranslate nohighlight">
+\[f(g) = 5 + 5g \implies \displaystyle\frac{df}{dg} = 5\]</div>
+<div class="math notranslate nohighlight">
+\[g(h) = \mbox{cos}(h) \implies \displaystyle\frac{dg}{dh} = -\mbox{sin}(h)\]</div>
+<div class="math notranslate nohighlight">
+\[h(t) = \frac{\pi}{12}(t - 3) \implies \displaystyle\frac{dh}{dt} = \frac{\pi}{12}\]</div>
+<p>Thus:</p>
+<div class="math notranslate nohighlight">
+\[\displaystyle\frac{df}{dx} =  \left(\displaystyle\frac{df}{dg}\right) \left(\displaystyle\frac{dg}{dh}\right) \left(\displaystyle\frac{dh}{dx}\right) = 5 \cdot \left(-\mbox{sin}(h)\right) \cdot \frac{\pi}{12} = -\frac{5\pi}{12}\mbox{sin}\left[\displaystyle\frac{\pi}{12}(t - 3)\right]\]</div>
+<p>For this last example if we plot the function <span class="math notranslate nohighlight">\(f(t)=5 + 5\mbox{cos}\left[\displaystyle\frac{\pi}{12}(t - 3)\right]\)</span> together with its derivative <span class="math notranslate nohighlight">\(f'(t)=-\displaystyle\frac{5\pi}{12}\mbox{sin}\left[\displaystyle\frac{\pi}{12}(t - 3)\right]\)</span> we will have:</p>
+<img alt="../../_images/func_diff.png" src="../../_images/func_diff.png" />
+<p>Try to compare the two graphs. What happens with the function <span class="math notranslate nohighlight">\(f(t)\)</span> on the intervals where the derivative <span class="math notranslate nohighlight">\(f'(t)\)</span> is positive or negative? What happens at the points where the derivative is zero?</p>
+</li>
+</ul>
+</div>
+</section>
+<section id="higher-order-derivatives">
+<h2>Higher order derivatives<a class="headerlink" href="#higher-order-derivatives" title="Link to this heading">#</a></h2>
+<p>If the derivative of a given function is also a “well-behaved” function (without kinks or discontinuities), then we can also calculate the derivative of the derivative. Thus:</p>
+<div class="math notranslate nohighlight">
+\[\frac{d}{dx}\left(\frac{df}{dx}\right) = \frac{d^{2}f}{dx^{2}} = f''(x)\]</div>
+<p>The function <span class="math notranslate nohighlight">\(f''(x)\)</span> is called the <em>second derivative</em> with respect to <span class="math notranslate nohighlight">\(x\)</span>.</p>
+<p>We can continue this process and get higher-order derivatives:</p>
+<div class="math notranslate nohighlight">
+\[\frac{d^{n}f}{dx^{n}} = f^{(n)}(x),\]</div>
+<p>assuming again that these functions are also well-behaved. Function <span class="math notranslate nohighlight">\(f^{(n)}(x)\)</span> is called the <span class="math notranslate nohighlight">\(n\)</span>-th derivative of <span class="math notranslate nohighlight">\(f\)</span> with respect to <span class="math notranslate nohighlight">\(x\)</span> (note the parenthesis on <span class="math notranslate nohighlight">\((n)\)</span> to distinguish from powers of <span class="math notranslate nohighlight">\(f\)</span>).</p>
+<div class="admonition-examples admonition">
+<p class="admonition-title">Examples</p>
+<p>The velocity is the derivative of the position, and the acceleration is the derivative of the velocity. Therefore, acceleration is the second-order derivative of the position</p>
+</div>
+<div class="admonition-examples admonition">
+<p class="admonition-title">Examples</p>
+<ul>
+<li><p><span class="math notranslate nohighlight">\(f(x) = 3x^{2} - 6x + 2\)</span>, then:</p>
+<div class="math notranslate nohighlight">
+\[f'(x) = 6x - 6\]</div>
+<div class="math notranslate nohighlight">
+\[f''(x) = 6\]</div>
+<p>Polynomials are always well-behaved and they are infinitely differentiable. However, if <span class="math notranslate nohighlight">\(n\)</span> is the degree of the polynomial (<em>i.e.</em> the largest exponent in the polynomial), we will have: <span class="math notranslate nohighlight">\(p^{(k)}(x) = 0,\ \mbox{for}\ k \ge n+1\)</span>.</p>
+  <br />
+</li>
+<li><p><span class="math notranslate nohighlight">\(f(x) = e^{-x^{2}}\)</span>, then:</p>
+<div class="math notranslate nohighlight">
+\[f'(x) = e^{-x^{2}} \cdot (-2x) = -2xe^{-x^{2}}\]</div>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{align}f''(x)
+&amp;= -2[e^{-x^{2}} + x(-2x)(e^{-x^{2}})] \\
+&amp;= -2e^{-x^{2}} + 4x^{2}e^{-x^{2}} \\
+&amp;= -2e^{-x^{2}}(1 - 2x^{2})\end{align}\end{split}\]</div>
+</li>
+</ul>
+</div>
+</section>
+<section id="functions-of-several-variables">
+<h2>Functions of several variables<a class="headerlink" href="#functions-of-several-variables" title="Link to this heading">#</a></h2>
+<p>A function <span class="math notranslate nohighlight">\(z = f(x,y)\)</span> of two dependent variables <span class="math notranslate nohighlight">\(x\)</span> and <span class="math notranslate nohighlight">\(y\)</span> defines a surface on the three dimensional <span class="math notranslate nohighlight">\(xyz\)</span>-space. If this function is well-behave it will be smooth, like if we applied different deformations on a rubber sheet. For smooth functions, we can calculate derivatives with respect to each of the dependent variables. This is written as:</p>
+<div class="math notranslate nohighlight">
+\[ \frac{\partial f}{\partial x} \text{ or } \frac{\partial f}{\partial x}(x, y) \text{ or } \frac{\partial}{\partial x}f(x,y)\]</div>
+<div class="math notranslate nohighlight">
+\[\frac{\partial f}{\partial y}, \frac{\partial f}{\partial{y}}(x,y), \frac{\partial}{\partial y}f(x,y)\]</div>
+<p>The partial derivative informs us how the function <span class="math notranslate nohighlight">\(f\)</span> varies along the <span class="math notranslate nohighlight">\(x\)</span> (or the <span class="math notranslate nohighlight">\(y\)</span>) direction at a specific point.</p>
+<div class="admonition-example admonition">
+<p class="admonition-title">Example</p>
+<p>Consider <span class="math notranslate nohighlight">\(f(x,y) = 4x^2y-2xy^3\)</span>. To compute the partial derivative with respect to <span class="math notranslate nohighlight">\(x\)</span> we consider all terms in <span class="math notranslate nohighlight">\(y\)</span> constants, and calculate the derivative as usual. For the partial derivative with respect to <span class="math notranslate nohighlight">\(y\)</span>, we consider the <span class="math notranslate nohighlight">\(x\)</span> terms constant. We have:</p>
+<div class="math notranslate nohighlight">
+\[\frac{\partial f}{\partial x} = (4y) \cdot (2x) - 2y^{3} \cdot (1) = 8xy - 2y^{3}\]</div>
+<div class="math notranslate nohighlight">
+\[\frac{\partial f}{\partial y} = 4x^{2} - 2x(3y^{2}) = 4x^{2} - 6xy^{2}\]</div>
+</div>
+<p>Higher order partial derivatives can also be defined, and represented with the common notations:</p>
+<div class="math notranslate nohighlight">
+\[\frac{\partial}{\partial x}\left(\frac{\partial f}{\partial x}\right) = \frac{\partial^{2}f}{\partial x^{2}} = \partial_{xx} f = f_{xx}(x,y)\]</div>
+<div class="math notranslate nohighlight">
+\[\frac{\partial}{\partial y}\left(\frac{\partial f}{\partial y}\right) = \frac{\partial^{2}f}{\partial y^{2}} = 
+\partial_{yy} f = f_{yy}(x,y)\]</div>
+<p>with subscripts representing the variable and the order of differentiation.</p>
+<p>For well-behaved functions, the order of the variables chosen to differentiate does not matter (a special case of what is called Schwarz’s theorem):</p>
+<div class="math notranslate nohighlight">
+\[f_{yx}(x,y) = \frac{\partial}{\partial y}\left(\frac{\partial f}{\partial x}\right) = \frac{\partial^{2}f}{\partial y \partial x} = \frac{\partial^{2} f}{\partial x \partial y}= \frac{\partial}{\partial x}\left(\frac{\partial f}{\partial y}\right) = f_{xy}(x,y)\]</div>
+<div class="admonition-example admonition">
+<p class="admonition-title">Example</p>
+<p>Often, a quantity of interest does not depend only on one variable but on many. The population size of a species depends on various aspects, including the availability of food, the number of predators, and other environmental factors.</p>
+</div>
+</section>
+<section id="analysing-change-continuously-part-2">
+<h2>Analysing change, continuously (Part 2)<a class="headerlink" href="#analysing-change-continuously-part-2" title="Link to this heading">#</a></h2>
+<p>In many different contexts we may be interested in finding the maximum or minimum values a given function may assume. This process is referred to as <em>optimization</em>.</p>
+<p>In Ecology, some of the most common uses for optimization methods are:</p>
+<p>1 - <em>Statistical estimation and inference</em>: optimizing the likelihood of a model and its parameters being true, for a given set of observations.</p>
+<p>2 - <em>Resources management</em>: finding a balance between different wildlife and economic priorities (market, legal, natural constraints)</p>
+<p>3 - <em>Optimality theory</em>: how can the behavioural or life-history trade-offs result in optimal strategies for different organisms</p>
+</section>
+<section id="extreme-values-of-functions">
+<h2>Extreme values of functions<a class="headerlink" href="#extreme-values-of-functions" title="Link to this heading">#</a></h2>
+<p>For a given function <span class="math notranslate nohighlight">\(f(x)\)</span> that is <strong>continuous</strong> in an interval <span class="math notranslate nohighlight">\(a \le x \le b\)</span>, there will always be a global maximum and a global minimum, points at which the function assumes its maximum and minimum values in the interval.</p>
+<img alt="../../_images/extrema.png" src="../../_images/extrema.png" />
+<p>A function <span class="math notranslate nohighlight">\(f(x)\)</span> presents a <em>local maximum</em> value if for a given vicinity of the point <span class="math notranslate nohighlight">\(x_0\)</span>, <span class="math notranslate nohighlight">\(f(x_0)\)</span> is greater than all the other <span class="math notranslate nohighlight">\(f(x)\)</span> for <span class="math notranslate nohighlight">\(x\)</span> in this vicinity. The analogous definition can be made for a <em>local minimum</em> value when <span class="math notranslate nohighlight">\(f(x_0)\)</span> is smaller than all the other <span class="math notranslate nohighlight">\(f(x)\)</span> in the vicinity. Local minima and local maxima constitue the <em>local extrema</em> for <span class="math notranslate nohighlight">\(f(x)\)</span>, and with the set of all extrema we can find the global maximum and the global minimum.</p>
+<p>An important result is that, if the function <span class="math notranslate nohighlight">\(f\)</span> has a local extremum at a given point <span class="math notranslate nohighlight">\(x=c\)</span> in that interval (<span class="math notranslate nohighlight">\(a&lt;c&lt;b\)</span>) and the derivative of the function exists at that point (in other words, if <span class="math notranslate nohighlight">\(f'(c)\)</span> is well defined), then we will have:</p>
+<p><span class="math notranslate nohighlight">\(f'(c)=0\)</span></p>
+<div class="admonition warning">
+<p class="admonition-title">Warning</p>
+</div>
+<p>Thus the points <span class="math notranslate nohighlight">\(x=c\)</span>, for which <span class="math notranslate nohighlight">\(f'(c)\)</span>, will constitute <strong>candidates</strong> for local extrema, and will have to be evaluated under other sets of conditions. Also important to have in mind that point for which the derivative is not definided (kinks or discontinuities) may also constitute local extrema (see on the first figure).</p>
+<p>In general, when searching for local extrema, we have to adopt the following rules:</p>
+<ul class="simple">
+<li><p>Do not assume points <span class="math notranslate nohighlight">\(x\)</span> for which <span class="math notranslate nohighlight">\(f'(x) = 0\)</span> are extrema. They are candidates.</p></li>
+<li><p>Check points where derivative is not defined.</p></li>
+<li><p>Check endpoints of domain.</p></li>
+</ul>
+<div class="admonition tip">
+<p class="admonition-title">Tip</p>
+<p>If possible, create a plot. Visuals are powerful tools and can help verify whether calculations have been done correctly.</p>
+</div>
+</section>
+<section id="monotonicity-and-concavity">
+<h2>Monotonicity and concavity<a class="headerlink" href="#monotonicity-and-concavity" title="Link to this heading">#</a></h2>
+<p>Imagine a function <span class="math notranslate nohighlight">\(f(x)\)</span> defined in a given interval <span class="math notranslate nohighlight">\(a \le x \le b\)</span>. For every <span class="math notranslate nohighlight">\(x_1\)</span> and <span class="math notranslate nohighlight">\(x_2\)</span> in this interval, if <span class="math notranslate nohighlight">\(x_{2} &gt; x_{1}\)</span> implies that <span class="math notranslate nohighlight">\(f(x_{2}) &gt; f(x_{1})\)</span> then the function is <strong>increasing</strong> in this interval. If on the other hand, <span class="math notranslate nohighlight">\(x_{2} &gt; x_{1}\)</span> implies that <span class="math notranslate nohighlight">\(f(x_{2}) &lt; f(x_{1})\)</span> then the function is <strong>decreasing</strong> in this interval.  This can be graphically shown as:</p>
+<img alt="../../_images/incr_decr.png" src="../../_images/incr_decr.png" />
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>Since we are using the stric inequalities (”<span class="math notranslate nohighlight">\(&gt;\)</span>” or “<span class="math notranslate nohighlight">\(&lt;\)</span>”) it can be added that the function is <strong>monotonic</strong>, or monotonically increasing or decreasing. If we had <span class="math notranslate nohighlight">\(x_{2} &gt; x_{1}\)</span> implying that <span class="math notranslate nohighlight">\(f(x_{2}) \ge f(x_{1})\)</span> (or <span class="math notranslate nohighlight">\(f(x_{2}) \le f(x_{1})\)</span>), the function would be <em>non-decreasing</em> (or <em>non-increasing</em>).</p>
+</div>
+<p>Now suppose that <span class="math notranslate nohighlight">\(f\)</span> is well-behaved in a given interval <span class="math notranslate nohighlight">\(a \le x \le b\)</span> (in mathematical terms, we say: <span class="math notranslate nohighlight">\(f\)</span> is continuous and differentiable in <span class="math notranslate nohighlight">\((a,b)\)</span>), then an important results is that</p>
+<div class="math notranslate nohighlight">
+\[\mbox{If }\ f'(x) &gt; 0\ \mbox{ for}\ a \le x \le b \implies f(x)\ \mbox{is increasing for}\ a \le x \le b\]</div>
+<div class="math notranslate nohighlight">
+\[\mbox{If }\ f'(x) &lt; 0\ \mbox{ for}\ a \le x \le b \implies f(x)\ \mbox{is decreasing for}\ a \le x \le b\]</div>
+<p>In other words, if the derivative of function <span class="math notranslate nohighlight">\(f\)</span> is positive (negative) for every point <span class="math notranslate nohighlight">\(x\)</span> in a given interval then the function is increasing (decreasing) in this interval.</p>
+<div class="admonition-example admonition">
+<p class="admonition-title">Example</p>
+<p>Let us calculate the intervals where the function <span class="math notranslate nohighlight">\(f(x) = x^{3} - \displaystyle\frac{3}{2}x^{2} - 6x + 3\)</span> is increasing and decreasing. Since <span class="math notranslate nohighlight">\(f(x)\)</span> is continuos and differentiable for all real values of <span class="math notranslate nohighlight">\(x\)</span> (remember that all polynomial functions are well-behved), we can use the first derivative test:</p>
+<div class="math notranslate nohighlight">
+\[f'(x) = 3x^{2}-3x-6 = 3(x^{2}-x-2) = 3(x+1)(x-2)\]</div>
+<p>and we see that <span class="math notranslate nohighlight">\(f'(x)\)</span> has roots in <span class="math notranslate nohighlight">\(x=-1\)</span> and <span class="math notranslate nohighlight">\(x=2\)</span>. Since <span class="math notranslate nohighlight">\(f'(x)\)</span> is a polynomial of degree 2 with the coefficient of the leading term positive, it is <em>concave up</em>. From this, we can conclude that:</p>
+<div class="math notranslate nohighlight">
+\[x&lt;-1\ \mbox{or}\ x&gt;2 \implies f'(x) &gt; 0 \implies f(x)\ \mbox{is}\ increasing\]</div>
+<div class="math notranslate nohighlight">
+\[-1&lt;x&lt;2 \implies f'(x) &lt; 0 \implies f(x)\ \mbox{is}\ decreasing\]</div>
+<p>The following plot shows the function <span class="math notranslate nohighlight">\(f(x)\)</span> and its derivative. Compare the intervals where the derivative is positive (negative) with the intervals where the function <span class="math notranslate nohighlight">\(f(x)\)</span> is increasing (decreasing).</p>
+<img alt="notebooks/MathsForBiologists/func_diff2.png" src="notebooks/MathsForBiologists/func_diff2.png" />
+</div>
+<p>When we discussed polynomials of degree 2, depending on the sign of the coefficient of the leading term, we had the two cases:</p>
+<img alt="../../_images/concav.png" src="../../_images/concav.png" />
+<p>Let us now generalise the notion of concavity for any arbitrary function. Suppose <span class="math notranslate nohighlight">\(f(x)\)</span> is differentiable in a given interval <span class="math notranslate nohighlight">\(a \le x \le b\)</span> (in other words, the derivative <span class="math notranslate nohighlight">\(f'(x)\)</span> is well-defined for every point in this interval). Then we have the following results:</p>
+<div class="math notranslate nohighlight">
+\[\mbox{If }\ f'(x)\ \mbox{is increasing for}\ a \le x \le b \implies f(x)\ \mbox{is}\ concave\ up\]</div>
+<div class="math notranslate nohighlight">
+\[\mbox{If }\ f'(x)\ \mbox{is decreasing for}\ a \le x \le b \implies f(x)\ \mbox{is}\ concave\ down\]</div>
+<p>This means that if for a given function <span class="math notranslate nohighlight">\(f(x)\)</span> the slopes of the tangent lines are increasing in a given interval of <span class="math notranslate nohighlight">\(x\)</span>, in this interval the function should resemble an “upwards U-shape”. If on the other hand, the slopes of the tangent lines are decreasing in this interval, the function should resemble a “downwards U-shape”. This can be visualised on the plots below.</p>
+<img alt="../../_images/slopes_inc_dec.png" src="../../_images/slopes_inc_dec.png" />
+<p>If the function <span class="math notranslate nohighlight">\(f\)</span> is twice differentiable in <span class="math notranslate nohighlight">\(a \le x \le b\)</span> (<span class="math notranslate nohighlight">\(f''(x)\)</span> is well-define for <span class="math notranslate nohighlight">\(a \le x \le b\)</span>), we can use the last two results to find a general criterium for concavity. This will be given by:</p>
+<div class="math notranslate nohighlight">
+\[\mbox{If }\ f''(x) &gt; 0\ \mbox{ for}\ a \le x \le b \implies f'(x)\ \mbox{is increasing for}\ a \le x \le b \implies f(x)\ \mbox{is}\ concave\ up\]</div>
+<div class="math notranslate nohighlight">
+\[\mbox{If }\ f''(x) &lt; 0\ \mbox{ for}\ a \le x \le b \implies f'(x)\ \mbox{is decreasing for}\ a \le x \le b \implies f(x)\ \mbox{is}\ concave\ down\]</div>
+<div class="admonition-example admonition">
+<p class="admonition-title">Example</p>
+<p>If you have a population curve, these tools will help you analyze questions such as ‘How fast is the population growing?’ ‘Is the growth accelerating?’ or ‘Are any of these factors changing over different periods?</p>
+</div>
+<div class="admonition-example admonition">
+<p class="admonition-title">Example</p>
+<p>Let us analyse the function <span class="math notranslate nohighlight">\(f(x) = x^{3}-\displaystyle\frac{3}{2}x^{2}-6x+3\)</span> again. We have:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{align} 
+f'(x) = 3x^{2}-3x-6 \\
+f''(x) = 6x -3 \end{align}\end{split}\]</div>
+<p>The function f’’(x) has a root in <span class="math notranslate nohighlight">\(x=\displaystyle\frac{1}{2}\)</span>. Since its slope is positive, we have:</p>
+<div class="math notranslate nohighlight">
+\[x&lt;\displaystyle\frac{1}{2} \implies f''(x) &lt; 0 \implies f(x)\ \mbox{is}\ concave\ down\]</div>
+<div class="math notranslate nohighlight">
+\[x&gt;\displaystyle\frac{1}{2} \implies f''(x) &gt; 0 \implies f(x)\ \mbox{is}\ concave\ up\]</div>
+<p>On the plots below, compare the intervals where the second derivative is positive (negative) with the intervals where the function <span class="math notranslate nohighlight">\(f(x)\)</span> is concave up (concave down).</p>
+<img alt="../../_images/plot_concav.png" src="../../_images/plot_concav.png" />
+</div>
+</section>
+<section id="finding-extrema">
+<h2>Finding extrema<a class="headerlink" href="#finding-extrema" title="Link to this heading">#</a></h2>
+<p>Back to our strategy for finding maxima or minima of functions, we first determine the set:</p>
+<ul class="simple">
+<li><p>find all points <span class="math notranslate nohighlight">\(c\)</span> for which <span class="math notranslate nohighlight">\(f'(c) = 0\)</span> or all point <span class="math notranslate nohighlight">\(c\)</span> where <span class="math notranslate nohighlight">\(f'(c)\)</span> does not exist. These are called <strong>critical points</strong>.</p></li>
+<li><p>find endpoints of domain of <span class="math notranslate nohighlight">\(f\)</span>.</p></li>
+</ul>
+<p>Additionally, if <span class="math notranslate nohighlight">\(f\)</span> is twice differentiable (in some interval containing <span class="math notranslate nohighlight">\(c\)</span>):</p>
+<ul class="simple">
+<li><p>If <span class="math notranslate nohighlight">\(f'(c) = 0\)</span> and <span class="math notranslate nohighlight">\(f''(c) &lt; 0 \implies c\)</span> is a local maximum (since the function is concave down)</p></li>
+<li><p>If <span class="math notranslate nohighlight">\(f'(c) = 0\)</span> and <span class="math notranslate nohighlight">\(f''(c) &gt; 0 \implies c\)</span> is a local minimum (since the function is concave up)</p></li>
+</ul>
+<div class="admonition-example admonition">
+<p class="admonition-title">Example</p>
+<p>Let us analyse the function <span class="math notranslate nohighlight">\(f(x) = \displaystyle\frac{3}{2}x^{4}-2x^{3}-6x^{2}+2\)</span>, and find its local extrema. We have:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{align}
+f'(x)&amp;= 6x^{3}-6x^{2}-12x=6x(x-2)(x+1) \\
+f''(x)&amp;=18x^{2}-12x-12=6(3x^{2}-2x-2)\end{align}\end{split}\]</div>
+<p>Since this is a polynomial function, the derivative is well-defined for all points. Also, for the critical points (roots of <span class="math notranslate nohighlight">\(f'(x)\)</span>) we have <span class="math notranslate nohighlight">\(x=0\)</span>, <span class="math notranslate nohighlight">\(x=-1\)</span>, and <span class="math notranslate nohighlight">\(x=2\)</span>. Calculating the value of the second derivative at the critical points:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{align}
+&amp;x=0 \implies f''(0)=-12 \text{ (local maximum)} \\
+&amp;x=-1 \implies f''(-1)=18 \text{ (local minimum)} \\
+&amp;x=2 \implies f''(2)=36 \text{ (local minimum)}\end{align}\end{split}\]</div>
+<p>Now we can check on the plot of <span class="math notranslate nohighlight">\(f(x)\)</span> that we have successfully determined the local extrema.</p>
+<img alt="../../_images/plot_extrema.png" src="../../_images/plot_extrema.png" />
+</div>
+</section>
+<section id="inflection-points">
+<h2>Inflection points<a class="headerlink" href="#inflection-points" title="Link to this heading">#</a></h2>
+<p>As we saw before, given a function <span class="math notranslate nohighlight">\(f(x)\)</span>, for some values of <span class="math notranslate nohighlight">\(x\)</span>, there might be a change in the concavity of the function:</p>
+<img alt="../../_images/inflection.png" src="../../_images/inflection.png" />
+<p>The points <span class="math notranslate nohighlight">\(x\)</span> for where the function changes its concavity are called <strong>inflection points</strong>. If <span class="math notranslate nohighlight">\(f\)</span> is twice differentiable and <span class="math notranslate nohighlight">\(c\)</span> is an inflection point, then <span class="math notranslate nohighlight">\(f''(c)=0\)</span>.</p>
+<p>Inflection points are the points where the function is (locally) steepest. It therefore helps us answer questions like when the growth/decline was strongest.</p>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>Note again that the points <span class="math notranslate nohighlight">\(x=c\)</span> where <span class="math notranslate nohighlight">\(f''(c)=0\)</span> are only candidates for inflection points. See that if <span class="math notranslate nohighlight">\(f(x)=x^{4}\)</span>, <span class="math notranslate nohighlight">\(f''(x)=12x^{2}\)</span>. Then, <span class="math notranslate nohighlight">\(f''(0)=0\)</span>, but <span class="math notranslate nohighlight">\(x=0\)</span> is not an inflection point of <span class="math notranslate nohighlight">\(f\)</span>.</p>
+<p>After calculating the candidate points <span class="math notranslate nohighlight">\(x=c\)</span>, if the concavities of the function (given by the sign of the second derivative) to the left and to the right of <span class="math notranslate nohighlight">\(c\)</span> change, then <span class="math notranslate nohighlight">\(c\)</span> is an inflection point.</p>
+</div>
+<div class="admonition-example admonition">
+<p class="admonition-title">Example</p>
+<p>Consider the Holling Type III functional response</p>
+<div class="math notranslate nohighlight">
+\[f(N)=\frac{aN^{2}}{1+ahN^{2}},\]</div>
+<p>where <span class="math notranslate nohighlight">\(f(N)\)</span> is prey consumption rate per predator, <span class="math notranslate nohighlight">\(N\)</span> is the prey population density, <span class="math notranslate nohighlight">\(a\)</span> is the attack rate, and <span class="math notranslate nohighlight">\(h\)</span> is handling time. Let us check if this function presents any inflection point. We have:</p>
+<p><span class="math notranslate nohighlight">\(\begin{align}f'(N) 
+&amp;= \frac{2aN(1+ahN^{2})-aN^{2}(2ahN)}{(1+ahN^{2})^{2}} \\
+&amp;= \frac{2aN}{(1+ahN^{2})^{2}}\end{align}\)</span></p>
+<br />
+<p><span class="math notranslate nohighlight">\(\begin{align}f''(N)
+&amp;=\frac{2a(1+ahN^{2})^{2}-2aN[2(1+ahN^{2})\cdot2ahN]}{(1+ahN^{2})^{4}} \\
+&amp;=\frac{2a(1+2ahN^{2}+a^{2}h^{2}N^{4})-2a[4ahN^{2}+4a^{2}h^{2}N^{4}]}{(1+ahN^{2})^{4}} \\
+&amp;=\frac{2a(1-3ahN^{2})}{(1+ahN^{2})^{3}}\end{align}\)</span></p>
+<p>See that the function <span class="math notranslate nohighlight">\(f''(N)\)</span> has two roots: <span class="math notranslate nohighlight">\(N=-\sqrt{\displaystyle\frac{1}{3ah}}\)</span> and <span class="math notranslate nohighlight">\(N=\sqrt{\displaystyle\frac{1}{3ah}}\)</span>. Since <span class="math notranslate nohighlight">\(N\)</span> is a population density, it only makes sense to analyse <span class="math notranslate nohighlight">\(N \ge 0\)</span>. It is also possible to show that:</p>
+<div class="math notranslate nohighlight">
+\[N&lt;\sqrt{\displaystyle\frac{1}{3ah}} \implies f''(N) &gt; 0\]</div>
+<div class="math notranslate nohighlight">
+\[N&gt;\sqrt{\displaystyle\frac{1}{3ah}} \implies f''(N) &lt; 0\]</div>
+<p>This means that we have a change of concavity at the point <span class="math notranslate nohighlight">\(N=\sqrt{\displaystyle\frac{1}{3ah}}\)</span> and thus this is and inflection point of this curve.</p>
+<p>Analyse how the function <span class="math notranslate nohighlight">\(f(N)\)</span> changes with <span class="math notranslate nohighlight">\(N\)</span>.</p>
+<img alt="../../_images/hollingIII.png" src="../../_images/hollingIII.png" />
+<p>How does this function compares with the Holling type II functional response?</p>
+</div>
+<div class="admonition tip">
+<p class="admonition-title">Tip</p>
+<p>For the intervals where the function is concave up you can also think of it as representing “accelerated change” of that function (since the first derivative, the “velocity” is increasing), while the intervals where it is concave down represent “deccelerated change” of the function. Understanding this can be particularly important if you function has a specific biological meaning, as in the previous example.</p>
+</div>
+</section>
+<section id="series-expansions">
+<h2>Series expansions<a class="headerlink" href="#series-expansions" title="Link to this heading">#</a></h2>
+<p>From the formal definition of a derivative</p>
+<div class="math notranslate nohighlight">
+\[f'(a)= \lim_{x \to a} \frac{f(x)-f(a)}{(x-a)} \xrightarrow{\text{$x$ very close to $a$}} f'(a) \approx \frac{f(x)-f(a)}{(x-a)},\]</div>
+<p>which means that if <span class="math notranslate nohighlight">\(x\)</span> is sufficiently close to <span class="math notranslate nohighlight">\(a\)</span>, the average rate of change is sufficiently close to the value of the derivative at that point. From the previous approximation, we have:</p>
+<div class="math notranslate nohighlight">
+\[f(x) = f(a)+f'(a)(x-a),\]</div>
+<p>which provides a linear approximation of <span class="math notranslate nohighlight">\(f(x)\)</span> around <span class="math notranslate nohighlight">\(a\)</span> (see that if <span class="math notranslate nohighlight">\(\alpha= f(a)\)</span> and <span class="math notranslate nohighlight">\(\beta=f'(a)\)</span>, both constants, then <span class="math notranslate nohighlight">\(f(x) = \alpha+\beta(x-a)\)</span>, which can easily be arranged on the form <span class="math notranslate nohighlight">\(y=mx+b\)</span>).</p>
+<div class="admonition-example admonition">
+<p class="admonition-title">Example</p>
+</div>
+<p>One way of trying to make the approximation better is to add terms of higher order of <span class="math notranslate nohighlight">\(x\)</span>. Suppose we want to approximate the function <span class="math notranslate nohighlight">\(f(x)\)</span> at point <span class="math notranslate nohighlight">\(a\)</span> by a given polynomial of degree <span class="math notranslate nohighlight">\(n\)</span></p>
+<div class="math notranslate nohighlight">
+\[P(x)=a_{0}+a_{1}(x-a)+a_{2}(x-a)^{2}+\cdots+a_n(x-a)^{n}\]</div>
+<p>so that the derivatives of <span class="math notranslate nohighlight">\(f(x)\)</span> and <span class="math notranslate nohighlight">\(P(x)\)</span> are the same at <span class="math notranslate nohighlight">\(x=a\)</span>:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{align}f(a)&amp;=P(a) \\
+f'(a)&amp;=P'(a)\\
+&amp;\vdots \\
+f^{(n)}(a)&amp;=P^{(n)}(a)\end{align}.\end{split}\]</div>
+<p>But for the polynomial, we have:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{align}  
+P(a)&amp;=a_{0} \\
+P'(a)&amp;=1 \cdot a_{1} \\
+P''(a)&amp;=2 \cdot 1 \cdot a_{2} \\
+P'''(a)&amp;=3 \cdot 2 \cdot 1 \cdot a_{3} \\  
+P''''(a)&amp;=4 \cdot 3 \cdot 2 \cdot 1 \cdot a_{4} \\
+&amp;\vdots \\
+P^{(n)}(a)&amp;= \underbrace{n(n-1)(n-2) \cdots 3 \cdot 2 \cdot 1}_{\displaystyle\text{$$n!$$}}\cdot a_{n} \end{align}\end{split}\]</div>
+<p>where <span class="math notranslate nohighlight">\(n! = n\cdot(n-1)\cdot(n-2) \cdots 3 \cdot 2 \cdot 1\)</span> is the factorial of <span class="math notranslate nohighlight">\(n\)</span>.</p>
+<p>Thus, by making derivatives of <span class="math notranslate nohighlight">\(f(x)\)</span> and <span class="math notranslate nohighlight">\(P(x)\)</span> equal at <span class="math notranslate nohighlight">\(x=a\)</span>, we obtain:</p>
+<div class="math notranslate nohighlight">
+\[f^{(n)}(a) = P^{(n)}(a) \implies f^{(n)}(a) = n!\, a_n \implies a_{n} = \frac{f^{(n)}(a)}{n!}\]</div>
+<p>The polynomial with these coefficients is called <strong>Taylor polynomial</strong> of degree <span class="math notranslate nohighlight">\(n\)</span> at <span class="math notranslate nohighlight">\(x = a\)</span>:</p>
+<div class="math notranslate nohighlight">
+\[P(x)=f(a)+f'(a)(x-a)+\frac{f''(a)}{2!}(x-a)^{2}+\cdots+\frac{f^{(n)}(a)}{n!}(x-a)^{n}.\]</div>
+<div class="admonition-example admonition">
+<p class="admonition-title">Example</p>
+<p>To find approximation of degree 3 for <span class="math notranslate nohighlight">\(f(x)=e^{x}\)</span>, at <span class="math notranslate nohighlight">\(x=0\)</span>, we have:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}f(0)=1 \\
+f'(0)=1 \\
+f''(0)=1 \\
+f'''(0)=1\end{split}\]</div>
+<p>Thus,</p>
+<div class="math notranslate nohighlight">
+\[P_{3}(x)=1+(x-0)+\frac{1}{2!}(x-0)^2+\frac{1}{3!}(x-0)^3=1+x+\frac{x^2}{2}+\frac{x^3}{6}\]</div>
+<p>See how the approximation to the function <span class="math notranslate nohighlight">\(f(x)=e^{x}\)</span>  at <span class="math notranslate nohighlight">\(x=0\)</span> gets increasingly better when more terms are added to the polynomial.</p>
+<img alt="../../_images/exp_series.gif" src="../../_images/exp_series.gif" />
+</div>
+<div class="admonition-example admonition">
+<p class="admonition-title">Example</p>
+<p>For the function <span class="math notranslate nohighlight">\(f(x)=\displaystyle\frac{1}{1-x}\ (x\neq1)\)</span> at <span class="math notranslate nohighlight">\(x=0\)</span>, we have:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}f(x)=\frac{1}{1-x} \implies f(0)=1 \\
+f'(x)=\frac{1}{(1-x)^{2}} \implies f'(0)=1 \\
+f''(x)=\frac{2}{(1-x)^{3}} \implies f''(0)=2 \\
+f'''(x)=\frac{3\cdot2}{(1-x)^{4}} \implies f'''(0)=3! \\
+f^{(n)}(x)=\frac{n!}{(1-x)^{n+1}} \implies f^{(n)}(0)= n!\end{split}\]</div>
+<p>Thus the Taylor polynomial of degree <span class="math notranslate nohighlight">\(n\)</span> will be:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{align}P(x)
+&amp;=1+1(x-0)+\frac{2}{2}(x-0)^{2}+\frac{3!}{3!}(x-0)^{3}+\cdots+\frac{n!}{n!}(x-0)^{n} \\
+&amp;=1+x+x^{2}+x^{3}+\cdots+x^{n}\end{align}\end{split}\]</div>
+</div>
+<p>For some functions, the approximation <span class="math notranslate nohighlight">\(f(x)=P_n(x)\)</span> at a given reference point <span class="math notranslate nohighlight">\(x=a\)</span> can be made increasingly better by adding more terms to <span class="math notranslate nohighlight">\(P(x)\)</span>, so that in the limit <span class="math notranslate nohighlight">\(n\rightarrow \infty\)</span> the Taylor polynomials (called <strong>Taylor series</strong> in this limit) for the functions <span class="math notranslate nohighlight">\(e^{x}\)</span>, <span class="math notranslate nohighlight">\(\mbox{sin}(x)\)</span>, <span class="math notranslate nohighlight">\(\mbox{cos}(x)\)</span> converge exactly for all values of <span class="math notranslate nohighlight">\(x\)</span>. For other functions, such as <span class="math notranslate nohighlight">\(\mbox{ln}(1+x)\)</span> and <span class="math notranslate nohighlight">\(\displaystyle\frac{1}{1-x}\)</span>, there is a restricted range of values of <span class="math notranslate nohighlight">\(x\)</span> for which the approximation can be made better by adding terms.</p>
+<p>We have the following approximations by Taylor series, with ranges in <span class="math notranslate nohighlight">\(x\)</span>, at <span class="math notranslate nohighlight">\(x = 0\)</span>:</p>
+<div class="math notranslate nohighlight">
+\[e^x=\sum_{n=0}^\infty \frac{x^n}{n!} =1+x+\frac{x^2}{2}+\frac{x^3}{3!}+\cdots,\ -\infty&lt;x&lt;\infty\]</div>
+<div class="math notranslate nohighlight">
+\[\mbox{sin}(x)=\sum_{n=0}^\infty \frac{(-1)^nx^{1+2n}}{(1+2n)!} = x-\frac{x^3}{3}+\frac{x^5}{5!}-\frac{x^7}{7!}+\cdots,\ -\infty&lt;x&lt;\infty\]</div>
+<div class="math notranslate nohighlight">
+\[\mbox{cos}(x)=\sum_{n=0}^\infty \frac{(-1)^nx^{2n}}{(2n)!} = 1-\frac{x^2}{2}+\frac{x^4}{4!}-\frac{x^6}{6!}+\cdots,\ -\infty&lt;x&lt;\infty\]</div>
+<div class="math notranslate nohighlight">
+\[\mbox{ln}(1+x)=-\sum_{n=1}^\infty \frac{(-1)^n x^n}{n} = \frac{x^2}{2!}+\frac{x^3}{3!}-\frac{x^4}{4!}+\frac{x^5}{5!}+\cdots,\ -1 &lt; x \leq 1\]</div>
+<div class="math notranslate nohighlight">
+\[\frac{1}{1-x} = \sum_{n=0}^\infty x^n = 1+x+x^2+x^3+\cdots,\ -1 \leq x &lt; 1\]</div>
+<p>Knowing the Taylor series often gives insight into the properties of a function.</p>
+</section>
+<section id="calculating-accumulated-change">
+<h2>Calculating accumulated change<a class="headerlink" href="#calculating-accumulated-change" title="Link to this heading">#</a></h2>
+</section>
+<section id="series">
+<h2>Series<a class="headerlink" href="#series" title="Link to this heading">#</a></h2>
+<p>For every sequence <span class="math notranslate nohighlight">\(\{a_n\}\)</span> of real numbers, it is possible to define another sequence <span class="math notranslate nohighlight">\(\{S_n\}\)</span> for which the elements are given by the sum of the first <span class="math notranslate nohighlight">\(n\)</span> terms of the sequence <span class="math notranslate nohighlight">\(\{a_n\}\)</span>:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{align}
+S_0 &amp;= a_0 \\
+S_1 &amp;= a_0 + a_1 \\
+S_2 &amp;= a_0 + a_1 + a_2 \\
+&amp;\vdots \\
+S_n &amp;= a_0 + a_1 + \cdots + a_{n-1} + a_n \\
+S_n &amp;= \sum_{i = 0}^n a_i \end{align}\end{split}\]</div>
+<p>The sequence <span class="math notranslate nohighlight">\(\{S_n\}\)</span> is called a <em>series</em> (or an infinite series, to emphasize the sum over increasing number of terms <span class="math notranslate nohighlight">\(\{a_n\}\)</span>).</p>
+<p>The same way as we asked about <span class="math notranslate nohighlight">\(\lim a_n\)</span>, we can ask if <span class="math notranslate nohighlight">\(\lim S_n\)</span> exists. If it exists, or, in other words, if the sum of infinite terms <span class="math notranslate nohighlight">\(\{a_n\}\)</span> is equal to a finite number, we say that the series <span class="math notranslate nohighlight">\(S_n\)</span> is convergent.</p>
+<div class="admonition-examples admonition">
+<p class="admonition-title">Examples</p>
+<p>The following expressions are example of convergent series:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{align}&amp;\sum_{n=0}^\infty \frac{x^n}{n!} \hspace{0.1cm}, \hspace{0.5cm} x \in \mathbb{R} \\
+&amp;\sum_{n=0}^\infty x^n \hspace{0.1cm}, \hspace{0.5cm} -1&lt;x&lt;1 \\
+&amp;\sum_{n=0}^\infty (-1)^n \frac{x^{2n+1}}{(2n+1)!} \hspace{0.1cm}, \hspace{0.5cm} x \in \mathbb{R}\end{align}\end{split}\]</div>
+</div>
+</section>
+<section id="calculating-areas">
+<h2>Calculating areas<a class="headerlink" href="#calculating-areas" title="Link to this heading">#</a></h2>
+<p>The idea of summing infinite quantities can be used to solve the problem of determining the area below a given curve. Let <span class="math notranslate nohighlight">\(f(x)\)</span> be a function defined on the interval <span class="math notranslate nohighlight">\(a \le x \le b\)</span> as:</p>
+<img alt="../../_images/sum_areas.png" src="../../_images/sum_areas.png" />
+<p>The area below the curve of <span class="math notranslate nohighlight">\(f(x)\)</span> can be found dividing the interval <span class="math notranslate nohighlight">\(a \le x \le b\)</span> into <span class="math notranslate nohighlight">\(n\)</span> subintervals:</p>
+<div class="math notranslate nohighlight">
+\[a=x_0&lt;x_1&lt;x_2&lt;\cdots&lt;x_{n-1}&lt;x_n=b\]</div>
+<p>where each interval defines a rectangle of height <span class="math notranslate nohighlight">\(f(x_i)\)</span> and width <span class="math notranslate nohighlight">\((x_{i+1}-x_i) = \Delta x_i = \displaystyle\frac{(b-a)}{n}\)</span>. The area below the curve will then be approximately the sum of the areas of the rectangles</p>
+<div class="math notranslate nohighlight">
+\[A \approx \sum_{i=0}^n f(x_i)\Delta x_i.\]</div>
+<p>As we increasethe number of intervals <span class="math notranslate nohighlight">\(n\)</span>, while decreasing the widths of each rectangle, the sum of the areas of the rectangles gives a better approximation of the area below the curve. The exact result is given by the limit <span class="math notranslate nohighlight">\(n \rightarrow \infty\)</span>. In this limit, the sum is represented by the following symbol:</p>
+<div class="math notranslate nohighlight">
+\[\lim_{n \to \infty} \sum_{i=0}^n f(x_i) \Delta x_i = \int_a^b f(x)dx\]</div>
+<p>If this limit exists, we say that the function <span class="math notranslate nohighlight">\(f\)</span> is <strong>integrable</strong> on the interval <span class="math notranslate nohighlight">\(a \lt x \lt b\)</span> and denote the limit by the <strong>definite integral</strong> <span class="math notranslate nohighlight">\(\int_a^b f(x)dx\)</span>.</p>
+<p>The interpretation of the integral as area can be generalized for when <span class="math notranslate nohighlight">\(f(x)\)</span> assumes negative values</p>
+<img alt="../../_images/defin_int.png" src="../../_images/defin_int.png" />
+<p>In the case of the previous figure, we will have:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{align} \int_a^b f(x)dx 
+&amp;= A_1 - A_2 + A_3 \hspace{0.1cm}, \hspace{0.5cm} (A_i \geq 0) \\
+&amp;= [\text{area above x-axis}] - [\text{area below x-axis}] \end{align}\end{split}\]</div>
+<p>Other intuitive results related to the calculation of the area are:</p>
+<div class="math notranslate nohighlight">
+\[\int_a^a f(x)dx = 0\quad \mbox{(size of the interval equal to zero)}\]</div>
+<div class="math notranslate nohighlight">
+\[\int_a^b f(x)dx = -\int_b^a f(x)dx\quad  \mbox{(orientation of the integral)}\]</div>
+<p>Furthermore, some of the properties of the integral follows directly from the properties of limits:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{align}
+\int_a^b kf(x)dx &amp;= k \int_a^b f(x)dx \\
+\int_a^b [f(x) + g(x)]dx &amp;= \int_a^b f(x)dx + \int_a^b g(x)dx \\
+\int_a^b f(x)dx &amp;= \int_a^c f(x)dx + \int_c^b f(x)dx ,\ \  (a \le c \le b) \end{align}\end{split}\]</div>
+</section>
+<section id="fundamental-theorem-of-calculus">
+<h2>Fundamental theorem of Calculus<a class="headerlink" href="#fundamental-theorem-of-calculus" title="Link to this heading">#</a></h2>
+<p>The problem of calculating integrals as limits of sums requires specific methods for each function and becomes quite impractical. Fortunately, this problem can be solved with the very important result that the area problem is intrinsically related to the tangent problem.</p>
+<p>Suppose we define a function <span class="math notranslate nohighlight">\(F(x)\)</span> that gives the area below the curve <span class="math notranslate nohighlight">\(f(u)\)</span> from <span class="math notranslate nohighlight">\(a\)</span> until a variable value <span class="math notranslate nohighlight">\(x\)</span>. By definition:</p>
+<div class="math notranslate nohighlight">
+\[F(x) = \int_a^x f(u)du\]</div>
+<img alt="../../_images/fund_calc.png" src="../../_images/fund_calc.png" />
+<p>If we consider a small increment <span class="math notranslate nohighlight">\(h\)</span> to <span class="math notranslate nohighlight">\(x\)</span>, the new area below the curve shall be given by:</p>
+<div class="math notranslate nohighlight">
+\[F(x+h) = \int_a^{x+h} f(u)du\]</div>
+<p>The difference between these two areas shall be approximately given by the area of the retangle with width <span class="math notranslate nohighlight">\(h\)</span> and height <span class="math notranslate nohighlight">\(f(x)\)</span>:</p>
+<div class="math notranslate nohighlight">
+\[F(x+h) - F(x) \approx hf(x) \implies \displaystyle\frac{F(x+h) - F(x)}{h} \approx f(x)\]</div>
+<p>But see that in the limit where the increment <span class="math notranslate nohighlight">\(h\)</span> gets very small, the approximation gets exactly equal to the increment in area, and we obtain the definition of the derivative of <span class="math notranslate nohighlight">\(F(x)\)</span>:</p>
+<div class="math notranslate nohighlight">
+\[\lim_{h\rightarrow 0} \displaystyle\frac{F(x+h) - F(x)}{h} = \displaystyle\frac{dF}{dx} = f(x).\]</div>
+<p>This results is known as the <em>Fundamental theorem of Calculus</em>, and it say that the instantaneous rate of change of the area below a given curve <span class="math notranslate nohighlight">\(f\)</span> at a point <span class="math notranslate nohighlight">\(x\)</span> is given by the value of the function at that point. In other words, the area <span class="math notranslate nohighlight">\(F(x)\)</span> is a function that, if differentiated, gives the function <span class="math notranslate nohighlight">\(f(x)\)</span>.</p>
+<p>The function <span class="math notranslate nohighlight">\(F(x)\)</span> is then called an <strong>antiderivative</strong> of <span class="math notranslate nohighlight">\(f(x)\)</span>. In general, there is a <em>family</em> of antiderivates of <span class="math notranslate nohighlight">\(f(x)\)</span>, since <span class="math notranslate nohighlight">\(\displaystyle\frac{d(F(x) + C)}{dx}=\displaystyle\frac{dF}{dx}=f(x)\)</span>, independent of the value of the constant <span class="math notranslate nohighlight">\(C\)</span>. We therefore call <strong>indefinite integral</strong>:</p>
+<div class="math notranslate nohighlight">
+\[\int f(x)dx = F(x) + C\]</div>
+<p>It also directly follows that</p>
+<div class="math notranslate nohighlight">
+\[\int_b^c f(x)dx = F(c) - F(b)\]</div>
+<p>Thus the problem of calculating the area below the curve <span class="math notranslate nohighlight">\(f(x)\)</span> from <span class="math notranslate nohighlight">\(b\)</span> to <span class="math notranslate nohighlight">\(c\)</span>, which, by definition, is given by the infinite sum of terms <span class="math notranslate nohighlight">\(f(x_i)\Delta x_i\)</span> (with <span class="math notranslate nohighlight">\(\Delta x_i \rightarrow 0\)</span>), can be reduced to obtaining the antiderivative of <span class="math notranslate nohighlight">\(f\)</span>, <span class="math notranslate nohighlight">\(F(x)\)</span>, and subtracting the values of this function at points <span class="math notranslate nohighlight">\(c\)</span> and <span class="math notranslate nohighlight">\(b\)</span>. The main problem then becomes how to calculate the antiderivates of a given function <span class="math notranslate nohighlight">\(f(x)\)</span>, for which different methods can be used.</p>
+</section>
+<section id="calculating-integrals">
+<h2>Calculating integrals<a class="headerlink" href="#calculating-integrals" title="Link to this heading">#</a></h2>
+<p>The simplest integrals can be calculated by inverting the differentiation of fundamental functions, as</p>
+<div class="math notranslate nohighlight">
+\[\int x^n \,dx = \frac{x^{n+1}}{n+1} + C\]</div>
+<div class="math notranslate nohighlight">
+\[\int e^x \,dx = e^x + C\]</div>
+<div class="math notranslate nohighlight">
+\[\int \frac{1}{x}\,dx = \ln |x| + C\]</div>
+<div class="math notranslate nohighlight">
+\[\int \mbox{cos}(x)\, dx = \mbox{sin}(x) + C\]</div>
+<div class="math notranslate nohighlight">
+\[\int \mbox{sin}(x)\, dx = -\mbox{cos}(x) + C\]</div>
+<p>This are obtained simply because we know that the derivative of the functions on the righ-hand sides of the equations give exactly the function that is being integrated. When the integrals are not so simple, we must apply other methods to turn them into integrals we know.</p>
+<section id="integration-by-substitution">
+<h3>Integration by substitution<a class="headerlink" href="#integration-by-substitution" title="Link to this heading">#</a></h3>
+<p>This method consists in finding an appropriate substitution of the variable of integration so that we end up with a simpler integral to calculate.</p>
+<div class="admonition-examples admonition">
+<p class="admonition-title">Examples</p>
+<ul>
+<li><p><span class="math notranslate nohighlight">\(\int\frac{1}{\sqrt{x+4}}dx\)</span></p>
+<p>Note that if <span class="math notranslate nohighlight">\(u=x+4 \implies \displaystyle\frac{du}{dx}=1 \implies du=dx\)</span>. Thus, the integral can be given by:</p>
+<div class="math notranslate nohighlight">
+\[\int u^{-\frac{1}{2}}du = \frac{1}{\left(-\frac{1}{2}+1\right)} \cdot u^{\frac{1}{2}} + C = 2\sqrt{x+4} + C.\]</div>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>Note that three mathematicians die everytime we operate with <span class="math notranslate nohighlight">\(du\)</span> and <span class="math notranslate nohighlight">\(dx\)</span> as if they were numbers on a fraction. Formally, the correct substitution on the integral would be given by a function <span class="math notranslate nohighlight">\(u=u(x)\)</span>, so that <span class="math notranslate nohighlight">\( \int f(u(x)) u'(x)dx = \int f(u)du\)</span>. But we can procede with the heresy above as long as we have this in mind.</p>
+</div>
+  <br />
+</li>
+<li><p><span class="math notranslate nohighlight">\(\int xe^{(x^2-1)}dx\)</span></p>
+<p>See that <span class="math notranslate nohighlight">\(u=x^2-1 \implies \displaystyle\frac{du}{dx}=2x \implies  dx=\displaystyle\frac{du}{2x}\)</span>. Then:</p>
+<div class="math notranslate nohighlight">
+\[\int \frac{1}{2}e^{u}du = \frac{1}{2}e^u + C = \frac{e^{(x^2-1)}}{2} + C \]</div>
+</li>
+</ul>
+</div>
+</section>
+<section id="integration-by-parts">
+<h3>Integration by parts<a class="headerlink" href="#integration-by-parts" title="Link to this heading">#</a></h3>
+<p>When integrating by parts, we have to find an appropriate association <span class="math notranslate nohighlight">\((u=u(x), v=v(x))\)</span>, so that:</p>
+<div class="math notranslate nohighlight">
+\[\int uv'dx=uv-\int u'vdx\]</div>
+<div class="admonition-example admonition">
+<p class="admonition-title">Example</p>
+<ul>
+<li><p><span class="math notranslate nohighlight">\(\int x\ln x\, dx \)</span></p>
+<p>Note that</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{align}
+    u=\ln x \implies u' = \displaystyle\frac{1}{x} \\
+    v'=x \implies v=\displaystyle\frac{x^2}{2} \end{align}\end{split}\]</div>
+<p>Thus:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{align}\int x\ln x\, dx 
+&amp;= \frac{x^2}{2}\ln x - \int \frac{x^2}{2} \cdot \frac{1}{x}dx \\
+&amp;= \frac{x^2 \ln x}{2} - \frac{1}{2}\int xdx \\
+&amp;= \frac{x^2 \ln x}{2} - \frac{1}{4}x^2 + C \\
+&amp;= \frac{1}{4} x^2 (2\ln x - 1) + C \end{align}\end{split}\]</div>
+</li>
+<li><p><span class="math notranslate nohighlight">\(\int xe^x \,dx\)</span></p>
+<p>Note that</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{align}
+    u=x \implies u'=1 \\
+    v'=e^x \implies v=e^x \end{align}\end{split}\]</div>
+<p>Thus:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{align}\int xe^xdx  
+    &amp;= xe^x - \int e^x dx \hspace{0.1cm} \\
+    &amp;= xe^x-e^x + C \\
+    &amp;= e^x (x-1) + C \end{align}\end{split}\]</div>
+</li>
+</ul>
+</div>
+<div class="admonition tip">
+<p class="admonition-title">Tip</p>
+<p>For complicated integrals, computer algebra programs like Mathematica or Sympy can be helpful. An especially user-friendly option is the website WolframAlpha.</p>
+</div>
+</section>
+<section id="integration-by-partial-fractions">
+<h3>Integration by partial fractions<a class="headerlink" href="#integration-by-partial-fractions" title="Link to this heading">#</a></h3>
+<p>If the integral has the form</p>
+<div class="math notranslate nohighlight">
+\[\int \frac{P(x)}{Q(x)}dx\]</div>
+<p>where the degree of <span class="math notranslate nohighlight">\(P(x)\)</span> is smaller than the degree of <span class="math notranslate nohighlight">\(Q(x)\)</span>, and <span class="math notranslate nohighlight">\(Q(x)=a(x-x_1)(x-x_2)\cdots(x-x_n)\)</span>, we can write</p>
+<div class="math notranslate nohighlight">
+\[\frac{P(x)}{Q(x)} = \frac{1}{a}\left[\frac{A_1}{x-x_1}+\frac{A_2}{x-x_2}+\cdots+\frac{A_n}{x-x_n}\right]\]</div>
+<p>with <span class="math notranslate nohighlight">\(\{A_n\}\)</span> to be determined.</p>
+<div class="admonition-example admonition">
+<p class="admonition-title">Example</p>
+<ul>
+<li><p><span class="math notranslate nohighlight">\(\int \frac{1}{x(x-1)}\, dx\)</span></p>
+<p>First, note that <span class="math notranslate nohighlight">\(\displaystyle\frac{1}{x(x-1)} =  \displaystyle\frac{A}{x} + \displaystyle\frac{B}{(x-1)} \implies 1 = A(x-1) + Bx \implies 1 = x(A+B) - A\)</span>. By comparing the coeffients of the terms in <span class="math notranslate nohighlight">\(x\)</span> and the terms independent of <span class="math notranslate nohighlight">\(x\)</span>, we obtain:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{cases}
+    A + B &amp;= 0\\
+       -A &amp;= 1
+    \end{cases} \implies  
+    \begin{cases}
+    A &amp;= -1 \\
+    B &amp;= 1\end{cases}\end{split}\]</div>
+<p>Thus:</p>
+<div class="math notranslate nohighlight">
+\[\frac{1}{x(x-1)} = -\frac{1}{x} + \frac{1}{x-1}\]</div>
+<p>Then we will have for the integral:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{align} \int\frac{1}{x(x-1)}dx  
+    &amp;= \int\left(-\frac{1}{x}+\frac{1}{x-1}\right)dx \\
+    &amp;= -\int \frac{1}{x}dx + \int\frac{1}{x-1}dx \\
+    &amp;= - \ln|x| + \ln|x-1| + C \\
+    &amp;= \ln \left|\frac{x-1}{x}\right| + C\end{align}\end{split}\]</div>
+</li>
+</ul>
+</div>
+</section>
+</section>
+<section id="tutorial-01">
+<h2>Tutorial 01<a class="headerlink" href="#tutorial-01" title="Link to this heading">#</a></h2>
+<ol class="arabic">
+<li><p>An autocatalytic reaction uses its resulting product for the formation of a new product, as in the reaction</p>
+<div class="math notranslate nohighlight">
+\[A + X \rightarrow X\]</div>
+<p>If we assume that this reaction occurs in a closed vessel, then the reaction rate is given by</p>
+<div class="math notranslate nohighlight">
+\[
+       R(x) = kx(a - x)
+    \]</div>
+<p>for <span class="math notranslate nohighlight">\(0 \le x \le a\)</span>, where <span class="math notranslate nohighlight">\(a\)</span> is the initial concentration of <span class="math notranslate nohighlight">\(A\)</span> and <span class="math notranslate nohighlight">\(x\)</span> is the concentration of <span class="math notranslate nohighlight">\(X\)</span>.</p>
+<br/>
+<p>(a) Show that <span class="math notranslate nohighlight">\(R(x)\)</span> is a polynomial and determine its degree.
+<br/>
+(b) Without using any graphic visualiser, graph <span class="math notranslate nohighlight">\(R(x)\)</span> for <span class="math notranslate nohighlight">\(k = 2\)</span> and <span class="math notranslate nohighlight">\(a = 6\)</span>. Find the value of <span class="math notranslate nohighlight">\(x\)</span> at which the reaction rate is maximal.</p>
+<br/>
+</li>
+<li><p>Find the maximum/minimum value of the following polynomials</p>
+ <br/>
+<p>(a) <span class="math notranslate nohighlight">\(p(x) = x^2 - 2x - 3\)</span></p>
+<p>(b) <span class="math notranslate nohighlight">\(p(x) = -x^2 - 6x + 8\)</span></p>
+<p>(<em>Hint:</em> Calculate the roots and consider the symmetry of the parabola)</p>
+ <br/>
+</li>
+<li><p>There is a function that is frequently used to describe the per capita growth rate of organisms when the rate depends on the concentration of some nutrient and becomes saturated for large enough nutrient concentrations. If we denote the concentration of the nutrient by <span class="math notranslate nohighlight">\(N\)</span>, then the per capita growth rate <span class="math notranslate nohighlight">\(r(N)\)</span> is given by the <em>Monod growth function</em></p>
+<div class="math notranslate nohighlight">
+\[
+        r(N)=\frac{aN}{k+N},\ \ \ N \ge 0
+    \]</div>
+<p>where <span class="math notranslate nohighlight">\(a\)</span> and <span class="math notranslate nohighlight">\(k\)</span> are positive constants.</p>
+ <br/>
+<p>(a) Use Python or R to investigate the Monod growth function. Graph <span class="math notranslate nohighlight">\(r(N)\)</span> for (i) <span class="math notranslate nohighlight">\(a = 5\)</span> and <span class="math notranslate nohighlight">\(k = 1\)</span>, (ii) <span class="math notranslate nohighlight">\(a = 5\)</span> and <span class="math notranslate nohighlight">\(k = 3\)</span>, and (iii) <span class="math notranslate nohighlight">\(a = 8\)</span> and k = 1. Place all three graphs in the same coordinate system.</p>
+<p>(b) On the basis of the graphs in (a), describe in words what happens when you change <span class="math notranslate nohighlight">\(a\)</span>.</p>
+<p>(c) On the basis of the graphs in (a), describe in words what happens when you change <span class="math notranslate nohighlight">\(k\)</span>.</p>
+ <br/>
+</li>
+<li><p>The function</p>
+<div class="math notranslate nohighlight">
+\[
+        f(x) = \frac{x^n}{b^n + x^n},\ \ \ x \ge 0
+    \]</div>
+<p>where <span class="math notranslate nohighlight">\(n\)</span> is a positive integer and <span class="math notranslate nohighlight">\(b\)</span> is a positive real number, is used in biochemistry to model reaction rates as a function of the concentration of some reactants</p>
+ <br/>
+<p>(a) Use Python or R to graph <span class="math notranslate nohighlight">\(f(x)\)</span> for <span class="math notranslate nohighlight">\(n = 1, 2,\ \mbox{and}\ 3\)</span> in the same coordinate system when <span class="math notranslate nohighlight">\(b = 2\)</span>.</p>
+<p>(b) Where do the three graphs in (a) intersect?</p>
+<p>(c) What happens to <span class="math notranslate nohighlight">\(f(x)\)</span> as <span class="math notranslate nohighlight">\(x\)</span> gets larger?</p>
+<p>(d) For an arbitrary positive value of <span class="math notranslate nohighlight">\(b\)</span>, show that <span class="math notranslate nohighlight">\(f (b) = 1/2\)</span>. On the basis of this demonstration and your answer in C. ,  explain why <span class="math notranslate nohighlight">\(b\)</span> is called the half-saturation constant.</p>
+ <br/>
+</li>
+<li><p>The file <a class="reference external" href="https://imperialcollegelondon.box.com/s/940y521dq1owarx8pt4d4mu2d0f2cz85">words_moby_dick.csv</a> contains data on the cumulative distribution of the number of times specific words occur in the text of the novel <em>Moby Dick</em>, by Herman Melville. Assume that this distribution is a power law of the form</p>
+<div class="math notranslate nohighlight">
+\[
+        y = x^D
+    \]</div>
+<p>(a) Using the provided data, and R/Python, estimate the exponent of this power law.
+(<em>Hint:</em> If you plot this data in log-log scale, what should correspond to the exponent <span class="math notranslate nohighlight">\(D\)</span>?)</p>
+<p>(b) Can you think of a better way to estimate D?</p>
+ <br/>
+</li>
+<li><p>Assume that a population size at time <span class="math notranslate nohighlight">\(t\)</span> is <span class="math notranslate nohighlight">\(N(t)\)</span> and that</p>
+<div class="math notranslate nohighlight">
+\[
+        N(t) = 40 \cdot 2^t,\ \ \ t \ge 0
+    \]</div>
+<p>(a) Find the population size at time t = 0.</p>
+<p>(b) Show that</p>
+<div class="math notranslate nohighlight">
+\[
+        N(t) = 40\, \mbox{e}^{t\, ln 2},\ \ \ t \ge 0
+    \]</div>
+<p>(c) How long will it take until the population size reaches 1000?</p>
+<p>(<em>Hint</em>: Find <span class="math notranslate nohighlight">\(t\)</span> so that <span class="math notranslate nohighlight">\(N(t) = 1000\)</span>.)</p>
+<br/>
+</li>
+<li><p>(<em>Adapted from Moss, 1980</em>) Hall (1964) investigated the change in population size of the zooplankton species <em>Daphnia galeata mendota</em> in Base Line Lake, Michigan. The population size <span class="math notranslate nohighlight">\(N(t)\)</span> at time <span class="math notranslate nohighlight">\(t\)</span> was modeled by the equation</p>
+<div class="math notranslate nohighlight">
+\[ 
+        N(t) = N_0\mbox{e}^{rt}
+    \]</div>
+<p>where <span class="math notranslate nohighlight">\(N_0\)</span> denotes the population size at time <span class="math notranslate nohighlight">\(0\)</span>. The constant <span class="math notranslate nohighlight">\(r\)</span> is called the <strong>intrinsic rate of growth</strong>.</p>
+ <br/>
+<p>(a) Plot <span class="math notranslate nohighlight">\(N(t)\)</span> as a function of <span class="math notranslate nohighlight">\(t\)</span> if <span class="math notranslate nohighlight">\(N_0 = 100\)</span> and <span class="math notranslate nohighlight">\(r = 2\)</span>. Compare your graph against the graph of <span class="math notranslate nohighlight">\(N(t)\)</span> when <span class="math notranslate nohighlight">\(N_0 = 100\)</span> and <span class="math notranslate nohighlight">\(r = 3\)</span>. Which population grows faster?</p>
+<p>(b) The constant <span class="math notranslate nohighlight">\(r\)</span> is an important quantity because it describes how quickly the population changes. Suppose that you determine the size of the population at the beginning and at the end of a period of length <span class="math notranslate nohighlight">\(1\)</span>, and you find that at the beginning there were <span class="math notranslate nohighlight">\(200\)</span> individuals and after one unit of time there were <span class="math notranslate nohighlight">\(250\)</span> individuals. Determine <span class="math notranslate nohighlight">\(r\)</span>. (<em>Hint</em>: Consider the ratio <span class="math notranslate nohighlight">\(N(t+1)/N(t)\)</span>.)</p>
+ <br/>
+</li>
+<li><p>Fish are indeterminate growers; that is, they grow throughout their lifetime. The growth of fish can be described by the von Bertalanffy function</p>
+<div class="math notranslate nohighlight">
+\[
+        L(x) = L_{\infty}(1-\textrm{e}^{-kx} )
+    \]</div>
+<p>for <span class="math notranslate nohighlight">\(x \ge 0\)</span>, where <span class="math notranslate nohighlight">\(L(x)\)</span> is the length of the fish at age <span class="math notranslate nohighlight">\(x\)</span> and <span class="math notranslate nohighlight">\(k\)</span> and <span class="math notranslate nohighlight">\(L_{\infty}\)</span> are positive</p>
+ <br/>
+<p>(a) Using Python or R, graph <span class="math notranslate nohighlight">\(L(x)\)</span> for <span class="math notranslate nohighlight">\(L_{\infty} = 20\)</span>, for (i) <span class="math notranslate nohighlight">\(k = 1\)</span> and (ii) <span class="math notranslate nohighlight">\(k = 0.1\)</span>.</p>
+<p>(b) For <span class="math notranslate nohighlight">\(k = 1\)</span>, find <span class="math notranslate nohighlight">\(x\)</span> so that the length is 90% of <span class="math notranslate nohighlight">\(L_{\infty}\)</span>. Repeat for 99% of <span class="math notranslate nohighlight">\(L_{\infty}\)</span>. Can the fish ever attain length <span class="math notranslate nohighlight">\(L_{\infty}\)</span>? Interpret the meaning of <span class="math notranslate nohighlight">\(L_{\infty}\)</span>.</p>
+<p>(c) Compare the graphs obtained in A. Which growth curve reaches 90% of <span class="math notranslate nohighlight">\(L_{\infty}\)</span> faster? Can you explain what happens to the curve of <span class="math notranslate nohighlight">\(L(x)\)</span> when you vary <span class="math notranslate nohighlight">\(k\)</span> (for fixed <span class="math notranslate nohighlight">\(L_{\infty}\)</span>)?</p>
+ <br/>
+</li>
+<li><p>For the following pairs of functions, plot the ratio (quotient) between the two using R or Python. Based on the behaviour of the ratio when <span class="math notranslate nohighlight">\(x \rightarrow \infty\)</span> (very large values of <span class="math notranslate nohighlight">\(x\)</span>), how does each of these functions compare to the other in the velocity that they grow?</p>
+ <br/>  
+<p>(a) <span class="math notranslate nohighlight">\( f(x) = e^x \quad \text{ and } \quad g(x) = x^2 \)</span></p>
+<p>(b) <span class="math notranslate nohighlight">\( f(x) = e^x \quad \text{ and } \quad g(x) = x^{20} \)</span></p>
+<p>(c) <span class="math notranslate nohighlight">\( f(x) = x \quad \text{ and } \quad g(x) = \log(x) \)</span></p>
+ <br/>
+</li>
+<li><p>A community measure that takes both species abundance and species richness into account is the Shannon diversity index <span class="math notranslate nohighlight">\(H\)</span>. To calculate <span class="math notranslate nohighlight">\(H\)</span>, the proportion <span class="math notranslate nohighlight">\(p_i\)</span> of species <span class="math notranslate nohighlight">\(i\)</span> in the community is used. Assume that the community consists of <span class="math notranslate nohighlight">\(S\)</span> species. Then</p>
+<div class="math notranslate nohighlight">
+\[
+        H = -\sum_{i=1}^S p_i\, \textrm{ln}\, p_i = -( p_1 \textrm{ln}\, p_1 + p_2 \textrm{ln}\, p_2 + \cdots + p_S \textrm{ln}\, p_S )
+    \]</div>
+<br/>
+<p>(a) Assume that <span class="math notranslate nohighlight">\(S = 5\)</span> and that all species are equally abundant; that is, <span class="math notranslate nohighlight">\(p_1 = p_2 = \cdots = p_5\)</span>. Compute <span class="math notranslate nohighlight">\(H\)</span>.</p>
+<p>(b) Assume that <span class="math notranslate nohighlight">\(S = 10\)</span> and that all species are equally abundant; that is, <span class="math notranslate nohighlight">\(p_1 = p_2 = \cdots = p_{10}\)</span>. Compute <span class="math notranslate nohighlight">\(H\)</span>.</p>
+<p>(c) A measure of equitability (or evenness) of the species distribution can be measured by dividing the diversity index <span class="math notranslate nohighlight">\(H\)</span> by <span class="math notranslate nohighlight">\(\textrm{ln}\, S\)</span>. Compute <span class="math notranslate nohighlight">\(H/\textrm{ln}\, S\)</span> for <span class="math notranslate nohighlight">\(S = 5\)</span> and <span class="math notranslate nohighlight">\(S = 10\)</span>.</p>
+</li>
+</ol>
+</section>
+<section id="tutorial-05">
+<h2>Tutorial 05<a class="headerlink" href="#tutorial-05" title="Link to this heading">#</a></h2>
+<ol class="arabic">
+<li><p>Solve the following systems of linear equations
+<br/></p>
+<div class="math notranslate nohighlight">
+\[\begin{split}
+        \mbox{(a)}  \ 
+        \begin{cases}
+            5x - y + 2z = 6 \\
+            x + 2y - z = -1 \\
+            3x + 2y - 2z = 1\\
+        \end{cases}
+    \end{split}\]</div>
+ <br/>
+<div class="math notranslate nohighlight">
+\[\begin{split}
+        \mbox{(b)}  \ 
+        \begin{cases}
+            2x - y + 3z = 3 \\
+            2x + y + 4z = 4 \\
+            2x - 3y + 2z = 2\\
+        \end{cases}
+    \end{split}\]</div>
+ <br/>
+</li>
+<li><p>Three different species of insects are reared together in a laboratory cage. They are supplied with two different types of food each day. Each individual of species 1 consumes 3 units of food <span class="math notranslate nohighlight">\(A\)</span> and 5 units of food <span class="math notranslate nohighlight">\(B\)</span>, each individual of species 2 consumes 2 units of food <span class="math notranslate nohighlight">\(A\)</span> and 3 units of food <span class="math notranslate nohighlight">\(B\)</span>, and individual of species 3 consumes 1 unit of food <span class="math notranslate nohighlight">\(A\)</span> and 2 units of food <span class="math notranslate nohighlight">\(B\)</span>. Each day, 500 units of food <span class="math notranslate nohighlight">\(A\)</span> and 900 units of food <span class="math notranslate nohighlight">\(B\)</span> are supplied. How many individuals of each species can be reared together? Is there more than one solution? What happens if we add 550 units of a third type of food, called <span class="math notranslate nohighlight">\(C\)</span>, and each individual of species 1 consumes 2 units of food <span class="math notranslate nohighlight">\(C\)</span>, each individual of species 2 consumes 4 units of food <span class="math notranslate nohighlight">\(C\)</span>, and each individual of species 3 consumes 1 unit of food <span class="math notranslate nohighlight">\(C\)</span>?</p>
+ <br/>
+</li>
+<li><p>Networks are abstractions of real systems used to identify and study patterns of connectivity between the individual units that compose those systems. Networks (often referred to as <em>graphs</em>) are mathematically defined as a set of <em>nodes</em>, or vertices, (representing the individual units) and a set of <em>links</em>, or connections, between these nodes, which account for the interaction patterns within the studied systems. In Ecology, graphs have been widely used in studies investigating food-webs, plant-pollinator interactions, and disease transmission in populations.</p>
+ <br />
+<p>If the nodes of a network of size <span class="math notranslate nohighlight">\(N\)</span> are indexed with positive integers such as <span class="math notranslate nohighlight">\(i = 1, 2, \ldots, N\)</span>, an useful method for representing the structure of a network is to construct a matrix <span class="math notranslate nohighlight">\(A\)</span> for which the elements <span class="math notranslate nohighlight">\(a_{ij}\)</span> are given by:</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}a_{ij} = \begin{cases}1,\ \mbox{if nodes}\ i \ \mbox{and}\ j \ \mbox{are linked} \\ 0,\ \mbox{otherwise}  \end{cases}\end{split}\]</div>
+<p>Matrix <span class="math notranslate nohighlight">\(A\)</span> is called the <em>adjacency matrix</em> of the network. One example of a network with <span class="math notranslate nohighlight">\(6\)</span> nodes is shown below.</p>
+ <br/>  
+<img alt="../../_images/graph_example.png" src="../../_images/graph_example.png" />
+<p>(a) Based on the graphical representation of this network, write down its adjacency matrix.</p>
+ <br/>
+<p>(b) By the definition of the adjacency matrix, the sum of the matrix elements on row <span class="math notranslate nohighlight">\(i\)</span>, <span class="math notranslate nohighlight">\(k_i = \sum_{j=1}^N a_{ij}\)</span>, corresponds to the number of links of node <span class="math notranslate nohighlight">\(i\)</span>. The quantity <span class="math notranslate nohighlight">\(k_i\)</span> is called the <em>degree</em> of node <span class="math notranslate nohighlight">\(i\)</span>. Calculate the corresponding degrees for each node on the previous network.</p>
+ <br/>
+<p>(c) The Laplacian matrix is a mathematical object that can be used to find many useful properties of a network. By definition, <span class="math notranslate nohighlight">\(L\)</span> is constructed as</p>
+ <br/>  
+<div class="math notranslate nohighlight">
+\[ L = D - A \]</div>
+ <br/>  
+<p>Where <span class="math notranslate nohighlight">\(A\)</span> is the adjacency matrix, and <span class="math notranslate nohighlight">\(D\)</span> is the degree matrix. The degree matrix <span class="math notranslate nohighlight">\(D\)</span> is a diagonal matrix which contains information about the degree of each node. With your answer for itens (a) and (b), calculate <span class="math notranslate nohighlight">\(L\)</span>.</p>
+ <br/>
+<p>(d) Use R/Python to numerically calculate the eigenvalues of the Laplacian matrix.</p>
+ <br/>
+</li>
+<li><p>Write each system in matrix form
+<br/></p>
+<div class="math notranslate nohighlight">
+\[\begin{split}
+        \mbox{(a)}  \ 
+        \begin{cases}
+            2x_2-x_1 = x_3 \\
+            4x_1 + x_3 = 7x_2 \\
+            x_2 - x_1 = x_3\\
+        \end{cases}
+    \end{split}\]</div>
+ <br/>
+<div class="math notranslate nohighlight">
+\[\begin{split}
+        \mbox{(b)}  \ 
+        \begin{cases}
+            2x_1 - 3x_2 = 4 \\
+            -x_1 + x_2 = 3 \\
+            3x_1 = 4\\
+        \end{cases}
+    \end{split}\]</div>
+ <br/>
+</li>
+<li><p>Suppose that</p>
+ <br/>
+<div class="math notranslate nohighlight">
+\[\begin{split}A = \begin{pmatrix}
+            2 &amp; 4 \\
+            3 &amp; 6 
+            \end{pmatrix}
+        \ \ \ \text{,}\ \ \
+        X = \begin{pmatrix}
+            x \\
+            y 
+            \end{pmatrix} \ \ \ \text{and}\ \ \
+            B = \begin{pmatrix}
+            b_1 \\
+            b_2 \\
+            \end{pmatrix}\end{split}\]</div>
+<p>(a) Write <span class="math notranslate nohighlight">\(AX = B\)</span> as a system of linear equation.</p>
+ <br/>
+<p>(b) Show that if</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}
+        B = \begin{pmatrix}
+            3 \\
+            \frac92 \\
+            \end{pmatrix}
+    \end{split}\]</div>
+<p>then <span class="math notranslate nohighlight">\(AX = B\)</span> has infinitely many solutions. Graph the two straight lines associated with the corresponding system of linear equations, and explain why the system has infinitely many solutions.</p>
+ <br/>
+<p>(c) Find a column vector</p>
+ <br/>
+<div class="math notranslate nohighlight">
+\[\begin{split}B = \begin{pmatrix}
+            b_1 \\
+            b_2 \\
+            \end{pmatrix}\end{split}\]</div>
+<p>so that <span class="math notranslate nohighlight">\(AX = B\)</span> has no solutions.</p>
+ <br/>
+</li>
+<li><p>Assume that a population is divided into three age classes and that 80% of the individuals age 0 and 10% of the individuals age 1 survive until the end of the next breeding season. Assume further that individuals age 1 have an average of 1.6 offspring and individuals age 2 have an average of 3.9 offspring. If, at time 0, the population consists of 1000 individuals age 0, 100 individuals age 1, and 20 individuals age 2, find the projection matrix and the age distribution at time 3.</p>
+ <br/>
+</li>
+<li><p>Consider the following projection matrix representing a population with five age classes:</p>
+ <br/>  
+<div class="math notranslate nohighlight">
+\[\begin{split}
+        P = \begin{pmatrix}
+        0 &amp; 5 &amp; 3 &amp; 2 &amp; 1 \\
+        0.9 &amp; 0 &amp; 0 &amp; 0 &amp; 0 \\
+        0 &amp; 0.3 &amp; 0 &amp; 0 &amp; 0 \\
+        0 &amp; 0 &amp; 0.1 &amp; 0 &amp; 0 \\
+        0 &amp; 0 &amp; 0 &amp; 0.05 &amp; 0 \\
+    \end{pmatrix}
+    \end{split}\]</div>
+ <br/>
+<p>(a) Making use of Python or R code, begin with a population distributed as <span class="math notranslate nohighlight">\((0,0,0,0,10)\)</span> individuals at time <span class="math notranslate nohighlight">\(t=0\)</span>, project the population at time <span class="math notranslate nohighlight">\(t=20\)</span>, and plot the total number over time. Plot the logarithm of the total population over time.</p>
+ <br/>
+<p>(b) Repeat the above but begin with <span class="math notranslate nohighlight">\((80,16,5,1,1)\)</span> individuals. How do these results compare to the results in (a) ?</p>
+ <br/>
+</li>
+<li><p>In the problems below, find the eigenvalues <span class="math notranslate nohighlight">\(\lambda_1\)</span> and <span class="math notranslate nohighlight">\(\lambda_2\)</span> and corresponding eigenvectors <span class="math notranslate nohighlight">\(\mathbf{v_1}\)</span> and <span class="math notranslate nohighlight">\(\mathbf{v_2}\)</span> for each matrix <span class="math notranslate nohighlight">\(B\)</span>. Determine the equations of the lines through the origin in the direction of the eigenvectors <span class="math notranslate nohighlight">\(\mathbf{v_1}\)</span> and <span class="math notranslate nohighlight">\(\mathbf{v_2}\)</span>, and graph the lines together with the eigenvectors <span class="math notranslate nohighlight">\(\mathbf{v_1}\)</span> and <span class="math notranslate nohighlight">\(\mathbf{v_2}\)</span> and the vectors <span class="math notranslate nohighlight">\(A\mathbf{v_1}\)</span> and <span class="math notranslate nohighlight">\(A\mathbf{v_2}\)</span>.</p>
+ <br/>
+<div class="math notranslate nohighlight">
+\[\begin{split}
+    \text{(a)} \ \ B = \begin{pmatrix}
+    2 &amp; 3\\
+    0 &amp; -1
+    \end{pmatrix}
+    \end{split}\]</div>
+ <br/>
+<div class="math notranslate nohighlight">
+\[\begin{split}
+    \text{(b)} \ \ B = \begin{pmatrix}
+    3 &amp; 6\\
+    -1 &amp; -4
+    \end{pmatrix}  
+    \end{split}\]</div>
+ <br/>
+</li>
+<li><p>Suppse that</p>
+ <br/>
+<div class="math notranslate nohighlight">
+\[\begin{split}
+    P = \begin{pmatrix}
+    0.5 &amp; 2.3\\
+    a &amp; 0
+    \end{pmatrix}
+    \end{split}\]</div>
+ <br/>
+<p>is the projection matrix of a population with two age classes. For which values of <span class="math notranslate nohighlight">\(a\)</span> does this population grow?</p>
+ <br/>
+</li>
+<li><p>Suppose that</p>
+<br/>
+<div class="math notranslate nohighlight">
+\[\begin{split}
+    P = \begin{pmatrix}
+    0.5 &amp; 2.0\\
+    0.1 &amp; 0
+    \end{pmatrix}
+    \end{split}\]</div>
+<br/>
+<p>is the projection matrix of a population with two age classes.</p>
+<br/>
+<p>(a) If you were to manage this population, would you need to be concerned about its long-term survival?</p>
+<br/>
+<p>(b) Suppose that you can improve either the fecundity or the survival of the zero-year-olds, but due to physiological and environmental constraints, the fecundity of zero-year-olds will not exceed 1.5 and the survival of zero-year-olds will not exceed 0.4. Investigate how the growth rate of the population is affected by changing either the survival or the fecundity of zero-year-olds, or both. What would be the maximum achievable growth rate?</p>
+<br/>
+<p>(c) In real situations, what other factors might you need to consider when you decide on management strategies?</p>
+</li>
+</ol>
+</section>
+<section id="tutorial-06">
+<h2>Tutorial 06<a class="headerlink" href="#tutorial-06" title="Link to this heading">#</a></h2>
+<ol class="arabic">
+<li><p><strong>Island Model</strong> Assume that a species lives in a habitat that consists of many islands close to a mainland. The species occupies both the mainland and the islands, but, although it is present on the mainland at all times, it frequently goes extinct on the islands. Islands can be recolonized by migrants from the mainland. The following model keeps track of the fraction of islands occupied: Denote the fraction of islands occupied at time <span class="math notranslate nohighlight">\(t\)</span> by <span class="math notranslate nohighlight">\(p(t)\)</span>. Assume that each island experiences a constant risk of extinction and that vacant islands (the fraction <span class="math notranslate nohighlight">\(1-p\)</span>) are colonized from the mainland at a constant rate. Then</p>
+<div class="math notranslate nohighlight">
+\[
+        \frac{dp}{dt}=c(1-p)-ep
+    \]</div>
+<p>where <span class="math notranslate nohighlight">\(c\)</span> and <span class="math notranslate nohighlight">\(e\)</span> are positive constants.</p>
+ <br/>
+<p>(a) The gain from colonization is <span class="math notranslate nohighlight">\(f (p) = c(1 - p)\)</span> and the loss from extinction is <span class="math notranslate nohighlight">\(g(p) = ep\)</span>. Graph (without graph visualisers) <span class="math notranslate nohighlight">\(f(p)\)</span> and <span class="math notranslate nohighlight">\(g(p)\)</span> for <span class="math notranslate nohighlight">\(0 \le p \le 1\)</span> in the same coordinate system. Explain why the two graphs intersect whenever <span class="math notranslate nohighlight">\(e\)</span> and <span class="math notranslate nohighlight">\(c\)</span> are both positive. Compute the point of intersection and interpret its biological meaning.</p>
+  <br/>
+<p>(b) The parameter <span class="math notranslate nohighlight">\(c\)</span> measures how quickly a vacant island becomes colonized from the mainland. The closer the islands, the larger is the value of <span class="math notranslate nohighlight">\(c\)</span>. Use your graph in (a) to explain what happens to the point of intersection of the two lines as c increases. Interpret your result in biological terms.</p>
+ <br />  
+</li>
+<li><p><strong>Biotic Diversity</strong> (Adapted from Valentine, 1985.) Walker and Valentine (1984) suggested a model for species diversity which assumes that species extinction rates are independent of diversity but speciation rates are regulated by competition. Denoting the number of species at time <span class="math notranslate nohighlight">\(t\)</span> by <span class="math notranslate nohighlight">\(S(t)\)</span>, the speciation rate by <span class="math notranslate nohighlight">\(b\)</span>, and the extinction rate by <span class="math notranslate nohighlight">\(a\)</span>, they used the model</p>
+<div class="math notranslate nohighlight">
+\[
+        \frac{dS}{dt} = S\left[b\left(1-\frac{S}{K}\right)-a \right]
+    \]</div>
+<p>where <span class="math notranslate nohighlight">\(K\)</span> denotes the number of “niches”, or potential places for species in the ecosystem.</p>
+ <br/>
+<p>(a) Find possible equilibria (<span class="math notranslate nohighlight">\(dS/dt = 0\)</span>) under the condition <span class="math notranslate nohighlight">\(a &lt; b\)</span>.</p>
+ <br/>
+<p>(b) Use your result in (a) to explain the following statement by Valentine (1985):</p>
+<div class="admonition-quotation admonition">
+<p class="admonition-title">Quotation</p>
+<p>“In this situation, ecosystems are never ‘full’, with all potential niches occupied by species so long as the extinction rate is above zero.”</p>
+</div>
+ <br/>
+<p>(c) What happens when <span class="math notranslate nohighlight">\(a \ge b\)</span>?</p>
+ <br />  
+</li>
+<li><p>Find a point on the curve</p>
+<div class="math notranslate nohighlight">
+\[y = 1 - 3x^3\]</div>
+<p>whose tangent line is parallel to the line <span class="math notranslate nohighlight">\(y = -x\)</span>. Is there more than one such point? If so, find all other points with this property.</p>
+ <br />  
+</li>
+<li><p>Find a point on the curve</p>
+<div class="math notranslate nohighlight">
+\[y = 2x^3 - 4x + 1\]</div>
+<p>whose tangent line is parallel to the line <span class="math notranslate nohighlight">\(y-2x = 1\)</span>. Is there more than one such point? If so, find all other points with this property.</p>
+ <br />  
+</li>
+<li><p>In the following, use the product and quotient rules to find the derivative with respect to the independent variable.</p>
+ <br/>
+<p>(a) <span class="math notranslate nohighlight">\(f(x) = (2x^3 - 1)(3 + 2x^2)\)</span></p>
+ <br/>
+<p>(b) <span class="math notranslate nohighlight">\(h(s) = (4 - 3s^2 + 4s^3)^2\)</span></p>
+ <br/>
+<p>(c) <span class="math notranslate nohighlight">\(f(x) =\sqrt{3x}(x^2 - 1)\)</span></p>
+ <br/>
+<p>(d) <span class="math notranslate nohighlight">\(g(t) = 4(3t^2 - 1)(2t + 1)\)</span></p>
+ <br/>
+<p>(e) <span class="math notranslate nohighlight">\(f(x) = 3(x^2 + 2)(4x^2 - 5x^4) - 3\)</span></p>
+ <br/>
+<p>(f) <span class="math notranslate nohighlight">\(g(x) = \displaystyle\frac{1 - 4x^3}{1 - x}\)</span></p>
+ <br/>
+<p>(g) <span class="math notranslate nohighlight">\(h(x) = \displaystyle\frac{1 + 2x^2 - 4x^4}{3x^3 - 5x^5}\)</span></p>
+ <br/>
+<p>(h) <span class="math notranslate nohighlight">\(f(t) =\displaystyle\frac{3 - t^2}{(t + 1)^2}\)</span></p>
+ <br/>
+<p>(i) <span class="math notranslate nohighlight">\(h(s) = \displaystyle\frac{2s^3 - 4s^2 + 5s - 7}{(s^2 - 3)^2}\)</span></p>
+ <br/>
+<p>(j) <span class="math notranslate nohighlight">\(g(s) = \displaystyle\frac{s^{1/7} - s^{2/7}}{s^{3/7} + s^{4/7}}\)</span></p>
+ <br />  
+</li>
+<li><p><strong>Logistic Growth</strong></p>
+ <br/>
+<p>(a) Find the derivative of the logistic growth curve</p>
+<div class="math notranslate nohighlight">
+\[N(t)=\frac{K}{1+\left(\frac{K}{N(0)}-1\right)e^{-rt}}\]</div>
+<p>where <span class="math notranslate nohighlight">\(r\)</span> and <span class="math notranslate nohighlight">\(K\)</span> are positive constants and <span class="math notranslate nohighlight">\(N(0)\)</span> is the population size at time <span class="math notranslate nohighlight">\(0\)</span>.</p>
+ <br/>
+<p>(b) Show that <span class="math notranslate nohighlight">\(N(t)\)</span> satisfies the equation</p>
+<div class="math notranslate nohighlight">
+\[\frac{dN}{dt}=rN\left(1-\frac{N}{K}\right)\]</div>
+<p>[<em>Hint:</em> Use the function <span class="math notranslate nohighlight">\(N(t)\)</span> given in (a) for the right-hand side, and simplify until you obtain the derivative of N(t) that you computed in A.]</p>
+ <br/>
+<p>(c) What happens to the <em>per-capita</em> rate of growth <span class="math notranslate nohighlight">\(\frac{1}{N}\frac{dN}{dt}\)</span> as you increase N? What biological principle and what type of interaction is your model describing?</p>
+ <br />  
+</li>
+<li><p>In the following, find the derivative with respect to the independent variable (<em>Hint</em>: use the chain rule).</p>
+ <br/>
+<p>(a) <span class="math notranslate nohighlight">\(f(x) = \mbox{sin}(e^{2x} + x)\)</span></p>
+ <br/>
+<p>(b) <span class="math notranslate nohighlight">\(h(x) = \mbox{exp}[x^2 - 2 \mbox{cos}\, x]\)</span></p>
+ <br/>
+<p>(c) <span class="math notranslate nohighlight">\(f(x) =\displaystyle\frac{x-2^{−x}}
+{1+x2^{−x}}\)</span></p>
+ <br/>
+<p>(d) <span class="math notranslate nohighlight">\(g(t) = \mbox{log}\left(\displaystyle\frac{1 - t}{1 + 2t}\right)\)</span></p>
+ <br/>
+<p>(e) <span class="math notranslate nohighlight">\(f(t) = \mbox{sin}(\mbox{ln}(3t))\)</span></p>
+ <br/>
+<p>(f) <span class="math notranslate nohighlight">\(g(x) = \mbox{ln}(\mbox{cos}(1 - x))\)</span></p>
+ <br/>
+<p>(g) <span class="math notranslate nohighlight">\(h(x) = \displaystyle\frac{\sqrt{x^2 - 1}}{2+\sqrt{x^2+1}}\)</span></p>
+ <br/>
+<p>(h) <span class="math notranslate nohighlight">\(f(t) =\sqrt[3]{t^2+\sqrt{1-t}}\)</span></p>
+ <br/>
+</li>
+<li><p>The functional responses of some predators are sigmoidal; that is, the number of prey attacked per predator as a function of prey density has a sigmoidal shape. If we denote the prey density by <span class="math notranslate nohighlight">\(N\)</span>, the predator density by <span class="math notranslate nohighlight">\(P\)</span>, the time available for searching for prey by <span class="math notranslate nohighlight">\(T\)</span>, and the handling time of each prey item per predator by <span class="math notranslate nohighlight">\(T_h\)</span>, then the number of prey encounters per predator as a function of <span class="math notranslate nohighlight">\(N\)</span>, <span class="math notranslate nohighlight">\(T\)</span>, and <span class="math notranslate nohighlight">\(T_h\)</span> can be expressed as</p>
+<div class="math notranslate nohighlight">
+\[
+        f (N, T, T_h) = \frac{b^2N^2T}{1 + cN + bT_hN^2}
+    \]</div>
+<p>where <span class="math notranslate nohighlight">\(b\)</span> and <span class="math notranslate nohighlight">\(c\)</span> are positive constants.</p>
+ <br/>
+<p>(a) Calculate the first-order partial derivative of <span class="math notranslate nohighlight">\(f(N, T, T_h)\)</span> with respect to the prey density <span class="math notranslate nohighlight">\(N\)</span>. If <span class="math notranslate nohighlight">\(N\)</span> increases, what happens to <span class="math notranslate nohighlight">\(f(N, T, T_h)\)</span>?</p>
+ <br/>
+<p>(b) Calculate the first-order partial derivative of <span class="math notranslate nohighlight">\(f(N, T, T_h)\)</span> with respect to the the time <span class="math notranslate nohighlight">\(T\)</span> available for search. If <span class="math notranslate nohighlight">\(T\)</span> increases, what happens to <span class="math notranslate nohighlight">\(f(N, T, T_h)\)</span>?</p>
+ <br/>
+<p>(c) Calculate the first-order partial derivative of <span class="math notranslate nohighlight">\(f(N, T, T_h)\)</span> with respect to the handling time <span class="math notranslate nohighlight">\(T_h\)</span>. If <span class="math notranslate nohighlight">\(T_h\)</span> increases, what happens to <span class="math notranslate nohighlight">\(f(N, T, T_h)\)</span>?</p>
+<p>[<em>Hint:</em> The function <span class="math notranslate nohighlight">\(f(N, T, T_h)\)</span> will be an increasing (decreasing) function of the independent variable if the first-order partial derivative of <span class="math notranslate nohighlight">\(f\)</span> with respect to this variable is positive (negative).]</p>
+</li>
+</ol>
+</section>
+</section>
+
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+  <div class="page-toc tocsection onthispage">
+    <i class="fa-solid fa-list"></i> Contents
+  </div>
+  <nav class="bd-toc-nav page-toc">
+    <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#making-mathematical-statements">Making mathematical statements</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#sets-and-relations">Sets and relations</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#polynomial-functions">Polynomial functions</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#degree-0">Degree 0</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#degree-1">Degree 1</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#degree-2">Degree 2</a><ul class="nav section-nav flex-column">
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#concavity">Concavity</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#intercept-roots">Intercept &amp; Roots</a></li>
+<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#minimum-maximum">Minimum/ maximum</a></li>
+</ul>
+</li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#rational-functions">Rational functions</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#power-functions">Power functions</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#exponential-functions">Exponential functions</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#logarithmic-function">Logarithmic function</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#describing-shapes-and-patterns">Describing shapes and patterns</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#measuring-angles">Measuring angles</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#trigonometric-circle">Trigonometric circle</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#trigonometric-identities">Trigonometric identities</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#trigonometric-functions">Trigonometric functions</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#amplitude-and-period">Amplitude and Period</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#modelling-complex-periodic-phenomena">Modelling complex periodic phenomena</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#transformation-on-graphs">Transformation on graphs</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#translations">Translations</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#stretching-compression">Stretching/ Compression</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#reflection">Reflection</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#analysing-change-one-step-at-a-time">Analysing change, one step at a time</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#sequences">Sequences</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#limits">Limits</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#discrete-time-population-models">Discrete-time population models</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#density-dependent-population-growth">Density-dependent population growth</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#annual-plants">Annual plants</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#fibonacci-sequence">Fibonacci sequence</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#from-table-arrangements-to-powerful-tools">From table arrangements to powerful tools</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#systems-of-linear-equations">Systems of linear equations</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#matrix-operations">Matrix operations</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#inverse-matrix">Inverse matrix</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#determinant">Determinant</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#structured-models">Structured models</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#from-table-arrangements-to-powerful-tools-part-2">From table arrangements to powerful tools - part 2</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#linear-transformations">Linear transformations</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#eigenvalues-and-eigenvectors">Eigenvalues and eigenvectors</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#vector-decomposition">Vector decomposition</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#structured-models-revisited">Structured models revisited</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#analysing-change-continuously">Analysing change, continuously</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#average-rate-of-change">Average rate of change</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#instantaneous-rate-of-change">Instantaneous rate of change</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#calculating-derivatives">Calculating derivatives</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#chain-rule">Chain rule</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#higher-order-derivatives">Higher order derivatives</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#functions-of-several-variables">Functions of several variables</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#analysing-change-continuously-part-2">Analysing change, continuously (Part 2)</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#extreme-values-of-functions">Extreme values of functions</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#monotonicity-and-concavity">Monotonicity and concavity</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#finding-extrema">Finding extrema</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#inflection-points">Inflection points</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#series-expansions">Series expansions</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#calculating-accumulated-change">Calculating accumulated change</a></li>
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+    <h1>The Metabolic Basis of Everything</h1>
+    <!-- Table of contents -->
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+                <h2> Contents </h2>
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+                <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#introduction">Introduction</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#energy-and-metabolic-rate">Energy and Metabolic Rate</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#importance-of-size">Importance of Size</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#ecological-implications-of-size">Ecological Implications of Size</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#importance-of-temperature">Importance of Temperature</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#summary">Summary</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#discussion-questions">Discussion Questions</a></li>
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+                <article class="bd-article">
+                  
+  <section class="tex2jax_ignore mathjax_ignore" id="the-metabolic-basis-of-everything">
+<h1>The Metabolic Basis of Everything<a class="headerlink" href="#the-metabolic-basis-of-everything" title="Link to this heading">#</a></h1>
+<ul class="simple">
+<li><p>Scaling laws - space and time</p></li>
+<li><p>Biochemical kinetics and thermodynamics</p></li>
+<li><p>Implications for ecological and evolutionary dynamics (?)</p></li>
+<li><p>Applications/Examples</p></li>
+</ul>
+<section id="introduction">
+<h2>Introduction<a class="headerlink" href="#introduction" title="Link to this heading">#</a></h2>
+<p>Metabolism is central to the understanding of ecological and evolutionary processes. As Ludwig Boltzmann noted in 1886:</p>
+<blockquote>
+<div><p>“The struggle for existence of living beings is not for the fundamental constituents of food, but for the possession of the free energy obtained, chiefly by means of the green plant, from the transfer of radiant energy from the hot sun to the cold earth.”</p>
+</div></blockquote>
+<p>The metabolic theory of ecology (Brown et al., 2004) provides a framework for understanding how metabolic processes scale across levels of biological organization and influence ecological dynamics.</p>
+</section>
+<section id="energy-and-metabolic-rate">
+<h2>Energy and Metabolic Rate<a class="headerlink" href="#energy-and-metabolic-rate" title="Link to this heading">#</a></h2>
+<p>Energy is a measurable property of an object that determines its ability to perform work. Life’s purpose is to utilize energy for propagating itself through self-replication. Plants and other autotrophs harness photons via photosynthesis, converting light energy into chemical energy stored in glucose. Heterotrophs, such as animals, derive energy from breaking chemical bonds in organic molecules. ATP (adenosine triphosphate) is a critical energy carrier in cells, facilitating various biochemical processes.</p>
+<p>Metabolism, the set of chemical reactions within an organism, determines metabolic rate, which is the rate of energy use, often measured in J/s, kcal/day, or Watts. Organisms must balance energy consumption and expenditure:</p>
+<div class="math notranslate nohighlight">
+\[
+\text{Energy Consumed} = \text{Energy Used (Maintenance + Growth + Movement)}.
+\]</div>
+<p>Metabolic rates influence key life-history traits, including movement rates, development speed, lifespan, and reproductive output (Dell et al., 2011).</p>
+</section>
+<section id="importance-of-size">
+<h2>Importance of Size<a class="headerlink" href="#importance-of-size" title="Link to this heading">#</a></h2>
+<p>Metabolic rate ((B)) increases with body size ((M)) according to the allometric scaling law (Kleiber, 1947):</p>
+<div class="math notranslate nohighlight">
+\[
+\log_{10}(B) = \log_{10}(B_0) + b \log_{10}(M),
+\]</div>
+<p>or equivalently:</p>
+<div class="math notranslate nohighlight">
+\[
+B = B_0 M^b,
+\]</div>
+<p>where (b) is typically close to 3/4 for most organisms. This implies that larger organisms have higher absolute metabolic rates but lower mass-specific rates ((B/M)):</p>
+<div class="math notranslate nohighlight">
+\[
+\frac{B}{M} = B_0 M^{b-1}.
+\]</div>
+<p>For example, a mouse (0.1 kg) needs approximately 12.3 kcal/day, while its mass-specific rate is 123 kcal/(kg(\cdot)day). This reduction in per-mass energy use with size is hypothesized to reduce overheating risk and improve resource allocation efficiency (West et al., 2005).</p>
+<section id="ecological-implications-of-size">
+<h3>Ecological Implications of Size<a class="headerlink" href="#ecological-implications-of-size" title="Link to this heading">#</a></h3>
+<ul class="simple">
+<li><p>Larger organisms have lower mass-specific metabolic rates, enabling longer lifespans and greater reproductive investment (Savage et al., 2004).</p></li>
+<li><p>Population growth rate scales negatively with body size:</p></li>
+</ul>
+<div class="math notranslate nohighlight">
+\[
+    r_{\text{max}} = r_0 M^{-0.25}.
+\]</div>
+<ul class="simple">
+<li><p>Population density decreases with size (Damuth’s law).</p></li>
+<li><p>Larger carnivores are rarer than herbivores due to energy loss across trophic levels (Colinvaux, 1978).</p></li>
+</ul>
+</section>
+</section>
+<section id="importance-of-temperature">
+<h2>Importance of Temperature<a class="headerlink" href="#importance-of-temperature" title="Link to this heading">#</a></h2>
+<p>Temperature profoundly impacts metabolism, as reaction rates increase with temperature. The Boltzmann-Arrhenius equation describes the exponential increase in reaction rates (Dell et al., 2011):</p>
+<div class="math notranslate nohighlight">
+\[
+    k = A e^{-\frac{E_a}{k_B T}},
+\]</div>
+<p>where (k) is the reaction rate, (A) is a constant, (E_a) is the activation energy, (k_B) is the Boltzmann constant, and (T) is temperature in Kelvin. This relationship explains the characteristic thermal performance curve observed in biological processes, where rates increase with temperature to an optimum before declining due to enzyme denaturation or other stress factors (Johnson et al., 1974).</p>
+<ul class="simple">
+<li><p>Ectotherms, such as reptiles and insects, rely on external temperature to regulate their metabolic processes.</p></li>
+<li><p>Endotherms, such as mammals and birds, generate internal heat to maintain a stable body temperature, but this comes at a significant energetic cost (Bar-On et al., 2018).</p></li>
+</ul>
+</section>
+<section id="summary">
+<h2>Summary<a class="headerlink" href="#summary" title="Link to this heading">#</a></h2>
+<ul class="simple">
+<li><p>Metabolism drives growth rates and influences population dynamics through its scaling with size and temperature (Brown et al., 2004).</p></li>
+<li><p>Larger organisms exhibit slower per-mass metabolic rates but greater absolute energy use (Savage et al., 2004).</p></li>
+<li><p>Temperature affects metabolic rates exponentially, shaping life-history strategies and population dynamics (Dell et al., 2011).</p></li>
+</ul>
+</section>
+<section id="discussion-questions">
+<h2>Discussion Questions<a class="headerlink" href="#discussion-questions" title="Link to this heading">#</a></h2>
+<ol class="arabic simple">
+<li><p>Given the way resting metabolic rate scales with body size, what must consumption rate scale like?</p></li>
+<li><p>Why did we focus on resting (or “basal”) metabolic rate?</p></li>
+<li><p>How much energy would a resting rabbit (∼ 1 kg), a human (∼ 70 kg, and an Asian elephant (∼ 4000 kg) need for metabolic balance, at the individual level? What about per unit mass?</p></li>
+<li><p>How does the value you calculated in the previous question compare with the recommended calorie intake for humans? Is it lower or higher? Why?</p></li>
+<li><p>What are the optimal temperatures for the enzyme underlying bioluminescence in the two bacteria in the given example? Why might they have different thermal optima?</p></li>
+</ol>
+</section>
+</section>
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-    <h1>Model Fitting using Non-linear Least-squares</h1>
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-                <h2> Contents </h2>
-            </div>
-            <nav aria-label="Page">
-                <ul class="visible nav section-nav flex-column">
-<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#introduction">Introduction</a></li>
-<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#traits-data-as-an-example">Traits data as an example</a></li>
-<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#biochemical-kinetics">Biochemical Kinetics</a><ul class="nav section-nav flex-column">
-<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#generating-data">Generating data</a></li>
-</ul>
-</li>
-<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#fitting-the-model-using-nlls">Fitting the model using NLLS</a><ul class="nav section-nav flex-column">
-<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#visualizing-the-fit">Visualizing the fit</a></li>
-<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#summary-stats-of-the-fit">Summary stats of the fit</a></li>
-</ul>
-</li>
-<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#statistical-inference-using-nlls-fits">Statistical inference using NLLS fits</a><ul class="nav section-nav flex-column">
-<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#goodness-of-fit">Goodness of fit</a><ul class="nav section-nav flex-column">
-<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#r-squared-values">R-squared values</a></li>
-</ul>
-</li>
-<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#confidence-intervals">Confidence Intervals</a></li>
-</ul>
-</li>
-<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-starting-values-problem">The starting values problem</a><ul class="nav section-nav flex-column">
-<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#fitting-the-model-with-starting-values">Fitting the model with starting values</a></li>
-</ul>
-</li>
-<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#a-more-robust-nlls-algorithm">A more robust NLLS algorithm</a></li>
-<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#bounding-parameter-values">Bounding parameter values</a></li>
-<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#diagnostics-of-an-nlls-fit">Diagnostics of an NLLS fit</a></li>
-<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#allometric-scaling-of-traits">Allometric scaling of traits</a><ul class="nav section-nav flex-column">
-<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#nlls-fitting-using-a-model-object">NLLS fitting using a model object</a></li>
-<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#id1">Visualizing the fit</a></li>
-<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#summary-of-the-fit">Summary of the fit</a></li>
-<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#examine-the-residuals">Examine the residuals</a><ul class="nav section-nav flex-column">
-<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#exercises">Exercises <a id="Allom_Exercises"></a></a></li>
-</ul>
-</li>
-</ul>
-</li>
-<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#comparing-models">Comparing models</a><ul class="nav section-nav flex-column">
-<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#id2">Exercises <a id="ModelSelection_Exercises"></a></a></li>
-</ul>
-</li>
-<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#albatross-chick-growth">Albatross chick growth</a><ul class="nav section-nav flex-column">
-<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#fitting-the-three-models-using-nlls">Fitting the three models using NLLS</a></li>
-<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#id3">Exercises <a id="Albatross_Exercises"></a></a></li>
-</ul>
-</li>
-<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#aedes-aegypti-fecundity">Aedes aegypti fecundity</a><ul class="nav section-nav flex-column">
-<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#the-thermal-performance-curve-models">The Thermal Performance Curve models</a></li>
-<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#id4">Exercises <a id="Aedes_Exercises"></a></a></li>
-</ul>
-</li>
-<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#abundance-data-as-an-example">Abundance data as an example</a></li>
-<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#population-growth-rates">Population growth rates</a><ul class="nav section-nav flex-column">
-<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#basic-approach">Basic approach</a></li>
-<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#using-ols">Using OLS</a></li>
-<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#using-nlls">Using NLLS</a></li>
-<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#id5">Exercises</a></li>
-</ul>
-</li>
-<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#some-tips-and-tricks-for-nlls-fitting">Some tips and tricks for NLLS fitting</a><ul class="nav section-nav flex-column">
-<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#starting-values">Starting values</a><ul class="nav section-nav flex-column">
-<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#obtaining-them">Obtaining them</a></li>
-<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#sampling-them">Sampling them</a></li>
-</ul>
-</li>
-<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#bounding-parameters-revisited">Bounding parameters revisited</a></li>
-</ul>
-</li>
-<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#readings-and-resources">Readings and Resources</a></li>
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-  <section class="tex2jax_ignore mathjax_ignore" id="model-fitting-using-non-linear-least-squares">
-<h1>Model Fitting using Non-linear Least-squares<a class="headerlink" href="#model-fitting-using-non-linear-least-squares" title="Link to this heading">#</a></h1>
-<section id="introduction">
-<h2>Introduction<a class="headerlink" href="#introduction" title="Link to this heading">#</a></h2>
-<p>In this Chapter, you will learn to fit non-linear mathematical models to data using Non-Linear Least Squares (NLLS).</p>
-<p>Specifically, you will learn to</p>
-<ul class="simple">
-<li><p>Visualize the data and the mathematical model you want to fit to them</p></li>
-<li><p>Fit a non-linear model</p></li>
-<li><p>Assess the quality of the fit, and whether the model is appropriate for your data</p></li>
-<li><p>Compare and select between competing models</p></li>
-</ul>
-<p>We will work through various examples. These assume that you have at least a conceptual understanding of what Linear vs Non-linear models are, how they are fitted to data, and how the fits can be assessed statistically. You may want to see the <a class="reference external" href="https://github.com/mhasoba/TheMulQuaBio/tree/master/content/lectures/LinearModels">Linear Models lecture</a> (you can also watch the <a class="reference external" href="https://drive.google.com/drive/folders/12Sj56wHX6vcAnp9GE9qQ1gIXbn7QRHU2?usp=sharing">video</a>), and the <a class="reference external" href="https://github.com/mhasoba/TheMulQuaBio/blob/master/content/lectures/NLLS">NLLS Lecture</a> lecture first (you can also watch the <a class="reference external" href="https://web.microsoftstream.com/video/529b2560-4f45-4675-ae8b-28487d63ac41">video</a>).</p>
-<p>You may also (optionally) want to see the  <a class="reference external" href="https://github.com/mhasoba/TheMulQuaBio/tree/master/content/lectures/ModelFitting">lecture on model fitting in Ecology and Evolution in general</a>.</p>
-<p>We will use R. For starters, clear all variables and graphic devices and load necessary packages:</p>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-r notranslate"><div class="highlight"><pre><span></span><span class="nf">rm</span><span class="p">(</span><span class="n">list</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nf">ls</span><span class="p">())</span>
-<span class="nf">graphics.off</span><span class="p">()</span>
-</pre></div>
-</div>
-</div>
-</div>
-</section>
-<section id="traits-data-as-an-example">
-<h2>Traits data as an example<a class="headerlink" href="#traits-data-as-an-example" title="Link to this heading">#</a></h2>
-<p>Our first set of examples will focus on traits.</p>
-<p>A trait is any measurable feature of an individual organism. This includes physical traits (e.g., morphology, body mass, wing length), performance traits (e.g., biochemical kinetics, respiration rate, body velocity, fecundity), and behavioral traits (e.g., feeding preference, foraging strategy, mate choice). All natural populations show variation in traits across individuals. A trait is functional when it directly (e.g., mortality rate) or indirectly (e.g., somatic development or growth rate) determines individual fitness. Therefore, variation in (functional) traits can generate variation in the rate of increase and persistence of populations. When measured in the context of life cycles, without considering interactions with other organisms (e.g., predators or prey of the focal population), functional traits are typically called life history traits (such as mortality rate and fecundity). Other traits determine interactions both within the focal population (e.g., intra-specific interference or mating frequency) and between the focal population/species and others, including the species which may act as resources (prey, for example). Thus both life history and interaction traits determine population fitness and therefore abundance, which ultimately influences dynamics and functioning of the wider ecosystem, such as carbon fixation rate or disease transmission rate.</p>
-</section>
-<section id="biochemical-kinetics">
-<h2>Biochemical Kinetics<a class="headerlink" href="#biochemical-kinetics" title="Link to this heading">#</a></h2>
-<p>The properties of an organism’s metabolic pathways, and the underlying (enzyme-mediated) biochemical reactions (kinetics) are arguably its most fundamental “traits”, because these drive all “performance” traits, from photosynthesis and respiration, to movement and growth rate.</p>
-<p>The <a class="reference external" href="https://en.wikipedia.org/wiki/Michaelis%E2%80%93Menten_kinetics">Michaelis-Menten</a> model is widely used to quantify reaction kinetics data and estimate key biochemical parameters. This model relates biochemical reaction rate (<span class="math notranslate nohighlight">\(V\)</span>) (rate of formation of the product of the reaction), to concentration of the substrate (<span class="math notranslate nohighlight">\(S\)</span>):</p>
-<div class="math notranslate nohighlight" id="equation-eq-m-m">
-<span class="eqno">(7)<a class="headerlink" href="#equation-eq-m-m" title="Link to this equation">#</a></span>\[ 
-V = \frac{V_{\max} S}{K_M + S} 
-\]</div>
-<p>Here,</p>
-<ul class="simple">
-<li><p><span class="math notranslate nohighlight">\(V_{\max}\)</span> is the maximum rate that can be achieved in the reaction system, which happens at saturating substrate concentration (as <span class="math notranslate nohighlight">\(S\)</span> gets really large), and</p></li>
-<li><p><span class="math notranslate nohighlight">\(K_M\)</span> is the Michaelis or half-saturation constant, defined as the substrate concentration at which the reaction rate is half of <span class="math notranslate nohighlight">\(V_{\max }\)</span>. This parameter controls the overall shape of the curve, i.e., whether <span class="math notranslate nohighlight">\(V\)</span> approaches <span class="math notranslate nohighlight">\(V_{\max}\)</span> slowly or rapidly. In enzyme catalyzed reactions, it measures how loosely the substrate binds the enzyme: large <span class="math notranslate nohighlight">\(K_M\)</span> indicates loose binding of enzyme to substrate, small <span class="math notranslate nohighlight">\(K_M\)</span> indicates tight binding (it has units of the substrate concentration, <span class="math notranslate nohighlight">\(S\)</span>).</p></li>
-</ul>
-<p>Biochemical reactions involving a single substrate are often well fitted by the Michaelis-Menten kinetics.</p>
-<a class="reference internal image-reference" href="../_images/MM.png"><img alt="Michaelis-Menten model" src="../_images/MM.png" style="width: 400px;" /></a>
-<p><small><center>The Michaelis-Menten model.</center></small></p>
-<p>Let’s fit the Michaelis-Menten model to some data.</p>
-<section id="generating-data">
-<h3>Generating data<a class="headerlink" href="#generating-data" title="Link to this heading">#</a></h3>
-<p>Instead of using real experimental data, we will actually <em>generate</em> some “data” because that way we know exactly what the errors in the data are. You can also import and use your own dataset for the fitting steps further below.</p>
-<p>We can generate some data as follows.</p>
-<p>First, generate a sequence of substrate concentrations from 1 to 50 in jumps of 5, using <code class="docutils literal notranslate"><span class="pre">seq()</span></code> (look up the documentation for <code class="docutils literal notranslate"><span class="pre">seq()</span></code>).</p>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-r notranslate"><div class="highlight"><pre><span></span><span class="n">S_data</span><span class="w"> </span><span class="o">&lt;-</span><span class="w"> </span><span class="nf">seq</span><span class="p">(</span><span class="m">1</span><span class="p">,</span><span class="m">50</span><span class="p">,</span><span class="m">5</span><span class="p">)</span>
-<span class="n">S_data</span>
-</pre></div>
-</div>
-</div>
-<div class="cell_output docutils container">
-<div class="output text_html"><style>
-.list-inline {list-style: none; margin:0; padding: 0}
-.list-inline>li {display: inline-block}
-.list-inline>li:not(:last-child)::after {content: "\00b7"; padding: 0 .5ex}
-</style>
-<ol class=list-inline><li>1</li><li>6</li><li>11</li><li>16</li><li>21</li><li>26</li><li>31</li><li>36</li><li>41</li><li>46</li></ol>
-</div></div>
-</div>
-<p>Note that because we generated values only at intervals of, there will be 50/5 = 10 “substrate” values.</p>
-<p>Now generate a Michaelis-Menten reaction  velocity response with <code class="docutils literal notranslate"><span class="pre">V_max</span></code> = 12.5 and <code class="docutils literal notranslate"><span class="pre">K_M</span></code> = 7.1:</p>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-r notranslate"><div class="highlight"><pre><span></span><span class="n">V_data</span><span class="w"> </span><span class="o">&lt;-</span><span class="w"> </span><span class="p">((</span><span class="m">12.5</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">S_data</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="m">7.1</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">S_data</span><span class="p">))</span>
-<span class="nf">plot</span><span class="p">(</span><span class="n">S_data</span><span class="p">,</span><span class="w"> </span><span class="n">V_data</span><span class="p">)</span>
-</pre></div>
-</div>
-</div>
-<div class="cell_output docutils container">
-<a class="reference internal image-reference" href="../_images/bc7dddd31cca5899e87217186108ea51a733c2043f85c11dfa553b9946112263.png"><img alt="../_images/bc7dddd31cca5899e87217186108ea51a733c2043f85c11dfa553b9946112263.png" src="../_images/bc7dddd31cca5899e87217186108ea51a733c2043f85c11dfa553b9946112263.png" style="width: 240px; height: 240px;" /></a>
-</div>
-</div>
-<p>Note that our choice of <span class="math notranslate nohighlight">\(V_{\max} = 12.5\)</span> and <span class="math notranslate nohighlight">\(K_M = 7.1\)</span> is completely arbitrary. As long as we make sure that <span class="math notranslate nohighlight">\(V_{\max} &gt; 0\)</span>, <span class="math notranslate nohighlight">\(K_H &gt; 0\)</span>, and <span class="math notranslate nohighlight">\(K_M\)</span> lies well within the lower half of the the range of substrate concentrations (0-50), these “data” will be physically biologically sensible.</p>
-<p>Now let’s add some random (normally-distributed) fluctuations to the data to emulate experimental / measurement error:</p>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-r notranslate"><div class="highlight"><pre><span></span><span class="nf">set.seed</span><span class="p">(</span><span class="m">1456</span><span class="p">)</span><span class="w"> </span><span class="c1"># To get the same random fluctuations in the &quot;data&quot; every time</span>
-<span class="n">V_data</span><span class="w"> </span><span class="o">&lt;-</span><span class="w"> </span><span class="n">V_data</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nf">rnorm</span><span class="p">(</span><span class="m">10</span><span class="p">,</span><span class="m">0</span><span class="p">,</span><span class="m">1</span><span class="p">)</span><span class="w"> </span><span class="c1"># Add 10 random fluctuations  with standard deviation of 1 to emulate error</span>
-<span class="nf">plot</span><span class="p">(</span><span class="n">S_data</span><span class="p">,</span><span class="w"> </span><span class="n">V_data</span><span class="p">)</span>
-</pre></div>
-</div>
-</div>
-<div class="cell_output docutils container">
-<a class="reference internal image-reference" href="../_images/a15a3e554fe181291541b96d1906177d518594dc21520b339969b7cf2e7485b7.png"><img alt="../_images/a15a3e554fe181291541b96d1906177d518594dc21520b339969b7cf2e7485b7.png" src="../_images/a15a3e554fe181291541b96d1906177d518594dc21520b339969b7cf2e7485b7.png" style="width: 240px; height: 240px;" /></a>
-</div>
-</div>
-<p>That looks real!</p>
-</section>
-</section>
-<section id="fitting-the-model-using-nlls">
-<h2>Fitting the model using NLLS<a class="headerlink" href="#fitting-the-model-using-nlls" title="Link to this heading">#</a></h2>
-<p>Now, fit the model to the data:</p>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-r notranslate"><div class="highlight"><pre><span></span><span class="n">MM_model</span><span class="w"> </span><span class="o">&lt;-</span><span class="w"> </span><span class="nf">nls</span><span class="p">(</span><span class="n">V_data</span><span class="w"> </span><span class="o">~</span><span class="w"> </span><span class="n">V_max</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">S_data</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="p">(</span><span class="n">K_M</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">S_data</span><span class="p">))</span>
-</pre></div>
-</div>
-</div>
-<div class="cell_output docutils container">
-<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Warning message in nls(V_data ~ V_max * S_data/(K_M + S_data)):
-“No starting values specified for some parameters.
-Initializing ‘V_max’, ‘K_M’ to &#39;1.&#39;.
-Consider specifying &#39;start&#39; or using a selfStart model”
-</pre></div>
-</div>
-</div>
-</div>
-<p>This warning arises because <code class="docutils literal notranslate"><span class="pre">nls</span></code> requires “starting values” for the parameters (two in this case: <code class="docutils literal notranslate"><span class="pre">V_max</span></code> and <code class="docutils literal notranslate"><span class="pre">K_M</span></code>) to start searching for optimal combinations of parameter values (ones that minimize the RSS). Indeed, all NLLS fitting functions / algorithms require this. If you do not provide starting values, <code class="docutils literal notranslate"><span class="pre">nls</span></code> gives you a warning (as above) and uses a starting value of 1 for every parameter by default. For simple models, despite the warning, this works well enough.</p>
-<div class="admonition tip">
-<p class="admonition-title">Tip</p>
-<p>Before proceeding further, have a look at what <code class="docutils literal notranslate"><span class="pre">nls()</span></code>’s arguments are using <code class="docutils literal notranslate"><span class="pre">?nls</span></code>, or looking at the documentation <a class="reference external" href="https://www.rdocumentation.org/packages/stats/versions/3.6.2/topics/nls">online</a>.</p>
-</div>
-<p>We will address the issue of starting values soon enough, but first let’s look at how good the fit that we obtained looks.</p>
-<section id="visualizing-the-fit">
-<h3>Visualizing the fit<a class="headerlink" href="#visualizing-the-fit" title="Link to this heading">#</a></h3>
-<p>The first thing to do is to see how well the model fitted the data, for which plotting is the best  first option:</p>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-r notranslate"><div class="highlight"><pre><span></span><span class="nf">plot</span><span class="p">(</span><span class="n">S_data</span><span class="p">,</span><span class="n">V_data</span><span class="p">,</span><span class="w"> </span><span class="n">xlab</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s">&quot;Substrate Concentration&quot;</span><span class="p">,</span><span class="w"> </span><span class="n">ylab</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s">&quot;Reaction Rate&quot;</span><span class="p">)</span><span class="w">  </span><span class="c1"># first plot the data </span>
-<span class="nf">lines</span><span class="p">(</span><span class="n">S_data</span><span class="p">,</span><span class="nf">predict</span><span class="p">(</span><span class="n">MM_model</span><span class="p">),</span><span class="n">lty</span><span class="o">=</span><span class="m">1</span><span class="p">,</span><span class="n">col</span><span class="o">=</span><span class="s">&quot;blue&quot;</span><span class="p">,</span><span class="n">lwd</span><span class="o">=</span><span class="m">2</span><span class="p">)</span><span class="w"> </span><span class="c1"># now overlay the fitted model </span>
-</pre></div>
-</div>
-</div>
-<div class="cell_output docutils container">
-<a class="reference internal image-reference" href="../_images/acbea972718f3aae6efabc147714b63b6c1faec984a7d746caadf0b4330685b4.png"><img alt="../_images/acbea972718f3aae6efabc147714b63b6c1faec984a7d746caadf0b4330685b4.png" src="../_images/acbea972718f3aae6efabc147714b63b6c1faec984a7d746caadf0b4330685b4.png" style="width: 240px; height: 240px;" /></a>
-</div>
-</div>
-<p>This looks OK.</p>
-<div class="admonition note">
-<p class="admonition-title">Note</p>
-<p>We used the <code class="docutils literal notranslate"><span class="pre">predict()</span></code> function here just as we did in any of the linear models chapters (e.g., <a class="reference internal" href="MulExpl.html#mulexp-predicted-values"><span class="std std-ref">here</span></a>). In general, you can use most of the same commands/functions (e.g., <code class="docutils literal notranslate"><span class="pre">predict()</span></code> and <code class="docutils literal notranslate"><span class="pre">summary()</span></code>) on the output of a <code class="docutils literal notranslate"><span class="pre">nls()</span></code> model fitting object as you would on a <code class="docutils literal notranslate"><span class="pre">lm()</span></code> model fitting object. Please have a look at the <a class="reference external" href="https://stat.ethz.ch/R-manual/R-devel/library/stats/html/predict.nls.html">documentation</a> of the predict function for <code class="docutils literal notranslate"><span class="pre">nls</span></code> before proceeding.</p>
-</div>
-<p>However, the above approach for plotting is not the best way to do it, because <code class="docutils literal notranslate"><span class="pre">predict()</span></code>, without further arguments (see its <a class="reference external" href="https://stat.ethz.ch/R-manual/R-devel/library/stats/html/predict.nls.html">documentation</a>), by default only generates predicted values for the actual x-values (substrate) data used to fit the model. So if there are very few values in the original data, you will not get a smooth predicted curve (as you can see above). A better approach is to generate a sufficient number of x-axis values and then calculate the predicted line. Let’s do it:</p>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-r notranslate"><div class="highlight"><pre><span></span><span class="nf">coef</span><span class="p">(</span><span class="n">MM_model</span><span class="p">)</span><span class="w"> </span><span class="c1"># check the coefficients</span>
-</pre></div>
-</div>
-</div>
-<div class="cell_output docutils container">
-<div class="output text_html"><style>
-.dl-inline {width: auto; margin:0; padding: 0}
-.dl-inline>dt, .dl-inline>dd {float: none; width: auto; display: inline-block}
-.dl-inline>dt::after {content: ":\0020"; padding-right: .5ex}
-.dl-inline>dt:not(:first-of-type) {padding-left: .5ex}
-</style><dl class=dl-inline><dt>V_max</dt><dd>12.9636891133139</dd><dt>K_M</dt><dd>10.6072253230542</dd></dl>
-</div></div>
-</div>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-r notranslate"><div class="highlight"><pre><span></span><span class="n">Substrate2Plot</span><span class="w"> </span><span class="o">&lt;-</span><span class="w"> </span><span class="nf">seq</span><span class="p">(</span><span class="nf">min</span><span class="p">(</span><span class="n">S_data</span><span class="p">),</span><span class="w"> </span><span class="nf">max</span><span class="p">(</span><span class="n">S_data</span><span class="p">),</span><span class="n">len</span><span class="o">=</span><span class="m">200</span><span class="p">)</span><span class="w"> </span><span class="c1"># generate some new x-axis values just for plotting</span>
-</pre></div>
-</div>
-</div>
-</div>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-r notranslate"><div class="highlight"><pre><span></span><span class="n">Predict2Plot</span><span class="w"> </span><span class="o">&lt;-</span><span class="w"> </span><span class="nf">coef</span><span class="p">(</span><span class="n">MM_model</span><span class="p">)[</span><span class="s">&quot;V_max&quot;</span><span class="p">]</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">Substrate2Plot</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="p">(</span><span class="nf">coef</span><span class="p">(</span><span class="n">MM_model</span><span class="p">)[</span><span class="s">&quot;K_M&quot;</span><span class="p">]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">Substrate2Plot</span><span class="p">)</span><span class="w"> </span><span class="c1"># calculate the predicted values by plugging the fitted coefficients into the model equation </span>
-</pre></div>
-</div>
-</div>
-</div>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-r notranslate"><div class="highlight"><pre><span></span><span class="nf">plot</span><span class="p">(</span><span class="n">S_data</span><span class="p">,</span><span class="n">V_data</span><span class="p">,</span><span class="w"> </span><span class="n">xlab</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s">&quot;Substrate Concentration&quot;</span><span class="p">,</span><span class="w"> </span><span class="n">ylab</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s">&quot;Reaction Rate&quot;</span><span class="p">)</span><span class="w">  </span><span class="c1"># first plot the data </span>
-<span class="nf">lines</span><span class="p">(</span><span class="n">Substrate2Plot</span><span class="p">,</span><span class="w"> </span><span class="n">Predict2Plot</span><span class="p">,</span><span class="w"> </span><span class="n">lty</span><span class="o">=</span><span class="m">1</span><span class="p">,</span><span class="n">col</span><span class="o">=</span><span class="s">&quot;blue&quot;</span><span class="p">,</span><span class="n">lwd</span><span class="o">=</span><span class="m">2</span><span class="p">)</span><span class="w"> </span><span class="c1"># now overlay the fitted model</span>
-</pre></div>
-</div>
-</div>
-<div class="cell_output docutils container">
-<a class="reference internal image-reference" href="../_images/481d9f13bab6f26a1b6c86075db6cb5eb1f20c637f10ae096b305a02303da8fc.png"><img alt="../_images/481d9f13bab6f26a1b6c86075db6cb5eb1f20c637f10ae096b305a02303da8fc.png" src="../_images/481d9f13bab6f26a1b6c86075db6cb5eb1f20c637f10ae096b305a02303da8fc.png" style="width: 240px; height: 240px;" /></a>
-</div>
-</div>
-<p>That looks much better (smoother) than the plot above !</p>
-</section>
-<section id="summary-stats-of-the-fit">
-<h3>Summary stats of the fit<a class="headerlink" href="#summary-stats-of-the-fit" title="Link to this heading">#</a></h3>
-<p>Now lets get some stats of this NLLS fit. Having obtained the fit object (<code class="docutils literal notranslate"><span class="pre">MM_model</span></code>), we can use <code class="docutils literal notranslate"><span class="pre">summary()</span></code> just like we would for a <code class="docutils literal notranslate"><span class="pre">lm()</span></code> fit object:</p>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-r notranslate"><div class="highlight"><pre><span></span><span class="nf">summary</span><span class="p">(</span><span class="n">MM_model</span><span class="p">)</span>
-</pre></div>
-</div>
-</div>
-<div class="cell_output docutils container">
-<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Formula: V_data ~ V_max * S_data/(K_M + S_data)
-
-Parameters:
-      Estimate Std. Error t value Pr(&gt;|t|)    
-V_max   12.964      1.221  10.616 5.42e-06 ***
-K_M     10.607      3.266   3.248   0.0117 *  
----
-Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
-
-Residual standard error: 0.8818 on 8 degrees of freedom
-
-Number of iterations to convergence: 5 
-Achieved convergence tolerance: 9.503e-06
-</pre></div>
-</div>
-</div>
-</div>
-<p>This looks a lot like the output of a linear model, and to be specific, of a <a class="reference internal" href="#./14-regress.ipynb"><span class="xref myst">Linear Regression</span></a>. For starters, compare the above output with the output of <code class="docutils literal notranslate"><span class="pre">summary(genomeSizeModelDragon)</span></code> in <a class="reference internal" href="regress.html#regress-perform"><span class="std std-ref">this section</span></a> of the Linear Regression chapter.</p>
-<p>So here are the main things to note about the output of <code class="docutils literal notranslate"><span class="pre">summary()</span></code> of an <code class="docutils literal notranslate"><span class="pre">nls()</span></code> model object:</p>
-<ul class="simple">
-<li><p><code class="docutils literal notranslate"><span class="pre">Estimate</span></code>s are, as in the output of the <code class="docutils literal notranslate"><span class="pre">lm()</span></code> function for fitting linear models, the estimated values of the coefficients of the model that you fitted (<span class="math notranslate nohighlight">\(V_{\max}\)</span> and <span class="math notranslate nohighlight">\(K_M\)</span>). Note that although we generated our data using <span class="math notranslate nohighlight">\(V_{\max} = 12.5\)</span> and <span class="math notranslate nohighlight">\(K_M = 7.1\)</span>, the actual coefficients are quite different from what we are getting with the NLLS fitting (<span class="math notranslate nohighlight">\(\hat{V}_{\max} = 12.96\)</span> and <span class="math notranslate nohighlight">\(\hat{K}_M = 10.61\)</span>). This is because we added random (normally-distributed) errors when we generated the “data”. This tells you something about how experimental and/or measurement errors can distort your image of the underlying mechanism or process.</p></li>
-<li><p><code class="docutils literal notranslate"><span class="pre">Std.</span> <span class="pre">Error</span></code>, <code class="docutils literal notranslate"><span class="pre">t</span> <span class="pre">value</span></code>, and <code class="docutils literal notranslate"><span class="pre">Pr(&gt;|t|)</span></code> and <code class="docutils literal notranslate"><span class="pre">Residual</span> <span class="pre">standard</span> <span class="pre">error</span></code> have the same interpretation as in the output of <code class="docutils literal notranslate"><span class="pre">lm()</span></code> (please look back at the <a class="reference internal" href="#./14-regress.ipynb"><span class="xref myst">Linear Regression Chapter</span></a>)</p></li>
-<li><p><code class="docutils literal notranslate"><span class="pre">Number</span> <span class="pre">of</span> <span class="pre">iterations</span> <span class="pre">to</span> <span class="pre">convergence</span></code> tells you how many times the NLLS algorithm had to adjust the parameter values till it managed to find a solution that minimizes the Residual Sum of Squares (RSS)</p></li>
-<li><p><code class="docutils literal notranslate"><span class="pre">Achieved</span> <span class="pre">convergence</span> <span class="pre">tolerance</span></code> tells you on what basis the algorithm decided that it was close enough to the a solution; basically if the RSS does not improve more than a certain threshold despite parameter adjustments, the algorithm stops searching. This may or may not be close to an optimal solution (but in this case it is).</p></li>
-</ul>
-<p>The last two items are specific to the output of an <code class="docutils literal notranslate"><span class="pre">nls()</span></code> fitting <code class="docutils literal notranslate"><span class="pre">summary()</span></code>, because unlike Ordinary Least Squares (OLS), which is what we used for Linear regression, NLLS is not an <em>exact</em> procedure, and the fitting requires computer simulations; revisit the <a class="reference external" href="https://github.com/mhasoba/TheMulQuaBio/tree/master/content/lectures/NLLS">Lecture</a> for an explanation of this. This is all you need to know for now. As such, you do not need to report these last two items when presenting the results of an NLLS fit, but they are useful for problem solving in case the fitting does not work (more on this below).</p>
-<p>As noted above, you can use the same sort of commands on a <code class="docutils literal notranslate"><span class="pre">nls()</span></code> fitting result as you can on a <code class="docutils literal notranslate"><span class="pre">lm()</span></code> object.</p>
-<p>Thus, much of the output of NLLS fitting using <code class="docutils literal notranslate"><span class="pre">nls()</span></code> is similar to the output of an <code class="docutils literal notranslate"><span class="pre">lm()</span></code>, and can be further analyzed or processed using analogous functions such as <code class="docutils literal notranslate"><span class="pre">coef()</span></code>, <code class="docutils literal notranslate"><span class="pre">residuals()</span></code>, and <code class="docutils literal notranslate"><span class="pre">confint()</span></code>.</p>
-<p>For example, you can get just the values of the estimated coefficients using <code class="docutils literal notranslate"><span class="pre">coef()</span></code> as you did above for the plotting.</p>
-</section>
-</section>
-<section id="statistical-inference-using-nlls-fits">
-<h2>Statistical inference using NLLS fits<a class="headerlink" href="#statistical-inference-using-nlls-fits" title="Link to this heading">#</a></h2>
-<p>So what do we do with the results of an NLLS fit? What statistical inferences can be made? We will address this issue here at a basic level, and then revisit it using model selection further below (in the Allometric growth example).</p>
-<section id="goodness-of-fit">
-<h3>Goodness of fit<a class="headerlink" href="#goodness-of-fit" title="Link to this heading">#</a></h3>
-<p>Assessing whether an NLLS-fitted model is “significant” based on an F-value, by performing an Analysis of Variance (ANOVA) on the regression results as one can do for a Linear Model using the <a class="reference internal" href="regress.html#regress-perform"><span class="std std-ref"><code class="docutils literal notranslate"><span class="pre">anova()</span></code> function on a linear model fit</span></a> is not advisable. Try <code class="docutils literal notranslate"><span class="pre">anova(MM_model)</span></code>, and see what happens:</p>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-r notranslate"><div class="highlight"><pre><span></span><span class="nf">anova</span><span class="p">(</span><span class="n">MM_model</span><span class="p">)</span>
-</pre></div>
-</div>
-</div>
-<div class="cell_output docutils container">
-<div class="output traceback highlight-ipythontb notranslate"><div class="highlight"><pre><span></span><span class="n">Error</span> <span class="ow">in</span> <span class="n">anova</span><span class="o">.</span><span class="n">nls</span><span class="p">(</span><span class="n">MM_model</span><span class="p">):</span> <span class="n">anova</span> <span class="ow">is</span> <span class="n">only</span> <span class="n">defined</span> <span class="k">for</span> <span class="n">sequences</span> <span class="n">of</span> <span class="s2">&quot;nls&quot;</span> <span class="n">objects</span>
-<span class="ne">Traceback</span>:
-
-<span class="mi">1</span><span class="o">.</span> <span class="n">anova</span><span class="p">(</span><span class="n">MM_model</span><span class="p">)</span>
-<span class="mi">2</span><span class="o">.</span> <span class="n">anova</span><span class="o">.</span><span class="n">nls</span><span class="p">(</span><span class="n">MM_model</span><span class="p">)</span>
-<span class="mi">3</span><span class="o">.</span> <span class="n">stop</span><span class="p">(</span><span class="s2">&quot;anova is only defined for sequences of </span><span class="se">\&quot;</span><span class="s2">nls</span><span class="se">\&quot;</span><span class="s2"> objects&quot;</span><span class="p">)</span>
-</pre></div>
-</div>
-</div>
-</div>
-<p>So a ANOVA calculation is not implemented for a NLLS fit. The third item in the error message / traceback above says that you can however use <code class="docutils literal notranslate"><span class="pre">anova()</span></code> to compare two <em>nested</em> models. That is, you can do something like <code class="docutils literal notranslate"><span class="pre">anova(model1,model2)</span></code>, where model 1 is nested within model 1, as we did <a class="reference internal" href="#./14-regress.ipynb"><span class="xref myst">previously</span></a> to ask if a simpler model explains significantly more variation in the data than a more complicated one (with one or more addtional terms in the equation).</p>
-<p>To understand why you cannot use an ANOVA to assess the significance of a NLLS model fit, recall that the objective of an <a class="reference internal" href="regress.html#regress-perform"><span class="std std-ref"><code class="docutils literal notranslate"><span class="pre">anova()</span></code> function applied to  a linear model fit</span></a> is o compare the fitted model to a <em>null</em> model, asking whether you can reject that null model (hypothesis). The problem with doing so for a nonlinear model is that unfortunately it is not clear what the null hypothesis should be when the model is not a linear combination of terms (the definition of a linear model). For an F-statistic to be meaningful, the null hypothesis model must be nested <em>within</em> the alternative model (in which case you can indeed use the <code class="docutils literal notranslate"><span class="pre">anova()</span></code> function using <code class="docutils literal notranslate"><span class="pre">anova(Model1,Model2)</span></code>).</p>
-<p>To make this clear, let’s recall the equation for a simple linear regression:</p>
-<div class="math notranslate nohighlight">
-\[y = \beta_0 + \beta_1 x\]</div>
-<p>The null model for this is</p>
-<div class="math notranslate nohighlight">
-\[y = \beta_0\]</div>
-<p>This then allows the TSS, RSS and ESS to be calculated in a meaninglful way, as we <a class="reference internal" href="#./14-regress.ipynb"><span class="xref myst">did before</span></a>, leading to an ANOVA based assessment of significance of the regrssion moel fit to the data.</p>
-<p>Now consider a nonlinear model like the Michaelis-Menten eqn <a class="reference internal" href="#equation-eq-m-m">(7)</a>. <em>What would be an appropriate null model for this equation?</em></p>
-<p>Thus, the best ways to assess a NLLS model’s fit are:</p>
-<ul class="simple">
-<li><p>Compare it’s likelihood to those of other alternative models’ fits (which may include your best guess for a null model).</p></li>
-<li><p>Examine whether the fitted coefficients are reliable, i.e., are significant, based on their (low) standard errors, (high) t-values, and (low) p-values (see the example model’s summary above).</p></li>
-</ul>
-<section id="r-squared-values">
-<h4>R-squared values<a class="headerlink" href="#r-squared-values" title="Link to this heading">#</a></h4>
-<p>To put it simply, unlike an R<span class="math notranslate nohighlight">\(^2\)</span> value obtained by fitting a linear model, that obtained from NLLS fitting is not reliable, and should not be used. The reason for this is somewhat technical (e.g., see this <a class="reference external" href="https://dx.doi.org/10.1186%2F1471-2210-10-6">paper</a>) and we won’t go into it here. But basically, NLLS R<span class="math notranslate nohighlight">\(^2\)</span> values do not always accurately reflect the quality of fit, and definitely cannot be used to select between competing models (Model selection, as you learned <a class="reference internal" href="#./18-ModelSimp.ipynb"><span class="xref myst">previously</span></a>). Indeed R<span class="math notranslate nohighlight">\(^2\)</span> values obtained from NLLS fitting even be negative when the model fits very poorly! We will learn more about model selection with non-linear models later below.</p>
-</section>
-</section>
-<section id="confidence-intervals">
-<h3>Confidence Intervals<a class="headerlink" href="#confidence-intervals" title="Link to this heading">#</a></h3>
-<p>One particularly useful thing you can do after NLLS fitting is to calculate/construct the confidence intervals (CI’s) around the estimated parameters in our fitted model. This is analogous to how we would in the OLS fitting used for Linear Models:</p>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-r notranslate"><div class="highlight"><pre><span></span><span class="nf">confint</span><span class="p">(</span><span class="n">MM_model</span><span class="p">)</span>
-</pre></div>
-</div>
-</div>
-<div class="cell_output docutils container">
-<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Waiting for profiling to be done...
-</pre></div>
-</div>
-<div class="output text_html"><table class="dataframe">
-<caption>A matrix: 2 × 2 of type dbl</caption>
-<thead>
-	<tr><th></th><th scope=col>2.5%</th><th scope=col>97.5%</th></tr>
-</thead>
-<tbody>
-	<tr><th scope=row>V_max</th><td>10.640478</td><td>17.00502</td></tr>
-	<tr><th scope=row>K_M</th><td> 4.924547</td><td>22.39247</td></tr>
-</tbody>
-</table>
-</div></div>
-</div>
-<p>The <code class="docutils literal notranslate"><span class="pre">Waiting</span> <span class="pre">for</span> <span class="pre">profiling</span> <span class="pre">to</span> <span class="pre">be</span> <span class="pre">done...</span></code> message reflects the fact that calculating the standard errors from which the CI’s are calculated requires a particular computational procedure (which we will not go into here) when it comes to NLLS fits.</p>
-<p>Calculating confidence intervals can be useful because,</p>
-<ol class="arabic simple">
-<li><p>As you learned <a class="reference internal" href="t_F_tests.html#t-f-tests-ci"><span class="std std-ref">here</span></a>, <a class="reference internal" href="t_F_tests.html#t-f-tests-ci2"><span class="std std-ref">here</span></a>, and <a class="reference internal" href="anova.html#anova-ci"><span class="std std-ref">here</span></a> (among other places) you can use a coefficient/parameter estimate’s confidence intervals to test whether it is significantly different from some reference value. In our CI example above, the intervals for <code class="docutils literal notranslate"><span class="pre">K_M</span></code> do in fact include the original value of <span class="math notranslate nohighlight">\(K_M = 7.1\)</span> that we used to generate the data.</p></li>
-<li><p>Confidence intervals can also be used to do a quick (but not robust - see warning below) test of whether coefficient estimates of the same model coefficient obtained from different populations (samples) are significantly different from each other (If their CI’s don’t overlap, they are significantly different).</p></li>
-</ol>
-<div class="admonition warning">
-<p class="admonition-title">Warning</p>
-<p>You can compare estimates of the same coefficient (parameter) from samples of different populations: if their CI’s don’t overlap, this indicates a statistically significant difference between the two populations as far as that coefficient is concerned (e.g., for 95% CI’s this means at the 0.05 level of significance or p-value). However, the opposite is not necessarily true: when CI’s overlap, there may <em>still</em> be a statistically significant difference between the coefficients (and therefore, populations).</p>
-</div>
-</section>
-</section>
-<section id="the-starting-values-problem">
-<h2>The starting values problem<a class="headerlink" href="#the-starting-values-problem" title="Link to this heading">#</a></h2>
-<p>Now let’s revisit the issue of starting values in NLLS fitting. Previously, we fitted the Michaelis-Menten Model without any starting values, and R gave us a warning but managed to fit the model to our synthetic “data” using default starting values.</p>
-<section id="fitting-the-model-with-starting-values">
-<h3>Fitting the model with starting values<a class="headerlink" href="#fitting-the-model-with-starting-values" title="Link to this heading">#</a></h3>
-<p>Lets try the NLLS fitting again, but with some particular starting values:</p>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-r notranslate"><div class="highlight"><pre><span></span><span class="n">MM_model2</span><span class="w"> </span><span class="o">&lt;-</span><span class="w"> </span><span class="nf">nls</span><span class="p">(</span><span class="n">V_data</span><span class="w"> </span><span class="o">~</span><span class="w"> </span><span class="n">V_max</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">S_data</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="p">(</span><span class="n">K_M</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">S_data</span><span class="p">),</span><span class="w"> </span><span class="n">start</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nf">list</span><span class="p">(</span><span class="n">V_max</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">12</span><span class="p">,</span><span class="w"> </span><span class="n">K_M</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">7</span><span class="p">))</span>
-</pre></div>
-</div>
-</div>
-</div>
-<p>Note that unlike before, we got no warning message about starting values.</p>
-<p><em>As will become apparent below, using sensible starting values is important in NLLS fitting.</em></p>
-<div class="admonition note">
-<p class="admonition-title">Note</p>
-<p><em>How do you find “sensible” starting values for NLLS fitting?</em> This very much depends on your understanding of the mathematical model that is being fitted to the data, the mechanistic interpretation of its parameters, and the specific dataset. For example, in the Michaelis-Menten Model example, we can know that <span class="math notranslate nohighlight">\(V_{\max}\)</span> is maximum reaction velocity and <span class="math notranslate nohighlight">\(K_M\)</span> is the value of the substrate concentration at which half of <span class="math notranslate nohighlight">\(V_{\max}\)</span> is reached. So we can choose starting values by “eye-balling” a particular dataset and determining <em>approximately</em> what the <code class="docutils literal notranslate"><span class="pre">V_max</span></code>and <code class="docutils literal notranslate"><span class="pre">K_M</span></code> are be for that particular dataset. In this particular case, we chose <code class="docutils literal notranslate"><span class="pre">V_max</span></code> = 12 and <code class="docutils literal notranslate"><span class="pre">K_M</span></code> = 7 because looking at the data plot above, these values seem to be reasonable guesses for the two parameters.</p>
-</div>
-<p>Let’s compare the coefficient estimates from our two different model fits to the same dataset:</p>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-r notranslate"><div class="highlight"><pre><span></span><span class="nf">coef</span><span class="p">(</span><span class="n">MM_model</span><span class="p">)</span>
-<span class="nf">coef</span><span class="p">(</span><span class="n">MM_model2</span><span class="p">)</span>
-</pre></div>
-</div>
-</div>
-<div class="cell_output docutils container">
-<div class="output text_html"><style>
-.dl-inline {width: auto; margin:0; padding: 0}
-.dl-inline>dt, .dl-inline>dd {float: none; width: auto; display: inline-block}
-.dl-inline>dt::after {content: ":\0020"; padding-right: .5ex}
-.dl-inline>dt:not(:first-of-type) {padding-left: .5ex}
-</style><dl class=dl-inline><dt>V_max</dt><dd>12.9636891133139</dd><dt>K_M</dt><dd>10.6072253230542</dd></dl>
-</div><div class="output text_html"><style>
-.dl-inline {width: auto; margin:0; padding: 0}
-.dl-inline>dt, .dl-inline>dd {float: none; width: auto; display: inline-block}
-.dl-inline>dt::after {content: ":\0020"; padding-right: .5ex}
-.dl-inline>dt:not(:first-of-type) {padding-left: .5ex}
-</style><dl class=dl-inline><dt>V_max</dt><dd>12.9636297453629</dd><dt>K_M</dt><dd>10.6070555266004</dd></dl>
-</div></div>
-</div>
-<p>Not too different, but not exactly the same!</p>
-<p>In contrast, when you fit linear models you will get exactly the same coefficient estimates every single time, because OLS is an <em>exact</em> procedure.</p>
-<p>Now, let’s try even more different start values:</p>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-r notranslate"><div class="highlight"><pre><span></span><span class="n">MM_model3</span><span class="w"> </span><span class="o">&lt;-</span><span class="w"> </span><span class="nf">nls</span><span class="p">(</span><span class="n">V_data</span><span class="w"> </span><span class="o">~</span><span class="w"> </span><span class="n">V_max</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">S_data</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="p">(</span><span class="n">K_M</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">S_data</span><span class="p">),</span><span class="w"> </span><span class="n">start</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nf">list</span><span class="p">(</span><span class="n">V_max</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">.</span><span class="m">01</span><span class="p">,</span><span class="w"> </span><span class="n">K_M</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">20</span><span class="p">))</span>
-</pre></div>
-</div>
-</div>
-</div>
-<p>Compare the coefficients of this model fit to the two previous ones:</p>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-r notranslate"><div class="highlight"><pre><span></span><span class="nf">coef</span><span class="p">(</span><span class="n">MM_model</span><span class="p">)</span>
-<span class="nf">coef</span><span class="p">(</span><span class="n">MM_model2</span><span class="p">)</span>
-<span class="nf">coef</span><span class="p">(</span><span class="n">MM_model3</span><span class="p">)</span>
-</pre></div>
-</div>
-</div>
-<div class="cell_output docutils container">
-<div class="output text_html"><style>
-.dl-inline {width: auto; margin:0; padding: 0}
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-.dl-inline>dt::after {content: ":\0020"; padding-right: .5ex}
-.dl-inline>dt:not(:first-of-type) {padding-left: .5ex}
-</style><dl class=dl-inline><dt>V_max</dt><dd>12.9636891133139</dd><dt>K_M</dt><dd>10.6072253230542</dd></dl>
-</div><div class="output text_html"><style>
-.dl-inline {width: auto; margin:0; padding: 0}
-.dl-inline>dt, .dl-inline>dd {float: none; width: auto; display: inline-block}
-.dl-inline>dt::after {content: ":\0020"; padding-right: .5ex}
-.dl-inline>dt:not(:first-of-type) {padding-left: .5ex}
-</style><dl class=dl-inline><dt>V_max</dt><dd>12.9636297453629</dd><dt>K_M</dt><dd>10.6070555266004</dd></dl>
-</div><div class="output text_html"><style>
-.dl-inline {width: auto; margin:0; padding: 0}
-.dl-inline>dt, .dl-inline>dd {float: none; width: auto; display: inline-block}
-.dl-inline>dt::after {content: ":\0020"; padding-right: .5ex}
-.dl-inline>dt:not(:first-of-type) {padding-left: .5ex}
-</style><dl class=dl-inline><dt>V_max</dt><dd>0.816129930954726</dd><dt>K_M</dt><dd>-19.4993655809431</dd></dl>
-</div></div>
-</div>
-<p>The estimates in our latest model fit are completely different (in fact, <code class="docutils literal notranslate"><span class="pre">K_M</span></code> is negative)! Let’s plot this model’s and the first model’s fit together:</p>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-r notranslate"><div class="highlight"><pre><span></span><span class="nf">plot</span><span class="p">(</span><span class="n">S_data</span><span class="p">,</span><span class="n">V_data</span><span class="p">)</span><span class="w">  </span><span class="c1"># first plot the data </span>
-<span class="nf">lines</span><span class="p">(</span><span class="n">S_data</span><span class="p">,</span><span class="nf">predict</span><span class="p">(</span><span class="n">MM_model</span><span class="p">),</span><span class="n">lty</span><span class="o">=</span><span class="m">1</span><span class="p">,</span><span class="n">col</span><span class="o">=</span><span class="s">&quot;blue&quot;</span><span class="p">,</span><span class="n">lwd</span><span class="o">=</span><span class="m">2</span><span class="p">)</span><span class="w"> </span><span class="c1"># overlay the original model fit</span>
-<span class="nf">lines</span><span class="p">(</span><span class="n">S_data</span><span class="p">,</span><span class="nf">predict</span><span class="p">(</span><span class="n">MM_model3</span><span class="p">),</span><span class="n">lty</span><span class="o">=</span><span class="m">1</span><span class="p">,</span><span class="n">col</span><span class="o">=</span><span class="s">&quot;red&quot;</span><span class="p">,</span><span class="n">lwd</span><span class="o">=</span><span class="m">2</span><span class="p">)</span><span class="w"> </span><span class="c1"># overlay the latest model fit</span>
-</pre></div>
-</div>
-</div>
-<div class="cell_output docutils container">
-<a class="reference internal image-reference" href="../_images/726feccf2057c7208a1fd23f7b8f3fee3b06b1620b4f129b387698057852d7bd.png"><img alt="../_images/726feccf2057c7208a1fd23f7b8f3fee3b06b1620b4f129b387698057852d7bd.png" src="../_images/726feccf2057c7208a1fd23f7b8f3fee3b06b1620b4f129b387698057852d7bd.png" style="width: 240px; height: 240px;" /></a>
-</div>
-</div>
-<p>As you would have guessed from the really funky coefficient estimates that were obtained in <code class="docutils literal notranslate"><span class="pre">MM_model3</span></code>, this is a pretty poor model fit to the data, with the negative value of <code class="docutils literal notranslate"><span class="pre">K_M</span></code> causing the fitted version of the Michaelis-Menten model to behave strangely.</p>
-<p><em>Also, you could have used the nicer plotting approach that was introduced before.</em></p>
-<p>Let’s try with even more different starting values.</p>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-r notranslate"><div class="highlight"><pre><span></span><span class="nf">nls</span><span class="p">(</span><span class="n">V_data</span><span class="w"> </span><span class="o">~</span><span class="w"> </span><span class="n">V_max</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">S_data</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="p">(</span><span class="n">K_M</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">S_data</span><span class="p">),</span><span class="w"> </span><span class="n">start</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nf">list</span><span class="p">(</span><span class="n">V_max</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">,</span><span class="w"> </span><span class="n">K_M</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0.1</span><span class="p">))</span>
-</pre></div>
-</div>
-</div>
-<div class="cell_output docutils container">
-<div class="output traceback highlight-ipythontb notranslate"><div class="highlight"><pre><span></span><span class="n">Error</span> <span class="ow">in</span> <span class="n">nlsModel</span><span class="p">(</span><span class="n">formula</span><span class="p">,</span> <span class="n">mf</span><span class="p">,</span> <span class="n">start</span><span class="p">,</span> <span class="n">wts</span><span class="p">,</span> <span class="n">scaleOffset</span> <span class="o">=</span> <span class="n">scOff</span><span class="p">,</span> <span class="n">nDcentral</span> <span class="o">=</span> <span class="n">nDcntr</span><span class="p">):</span> <span class="n">singular</span> <span class="n">gradient</span> <span class="n">matrix</span> <span class="n">at</span> <span class="n">initial</span> <span class="n">parameter</span> <span class="n">estimates</span>
-<span class="ne">Traceback</span>:
-
-<span class="mi">1</span><span class="o">.</span> <span class="n">nls</span><span class="p">(</span><span class="n">V_data</span> <span class="o">~</span> <span class="n">V_max</span> <span class="o">*</span> <span class="n">S_data</span><span class="o">/</span><span class="p">(</span><span class="n">K_M</span> <span class="o">+</span> <span class="n">S_data</span><span class="p">),</span> <span class="n">start</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">V_max</span> <span class="o">=</span> <span class="mi">0</span><span class="p">,</span> 
- <span class="o">.</span>     <span class="n">K_M</span> <span class="o">=</span> <span class="mf">0.1</span><span class="p">))</span>
-<span class="mi">2</span><span class="o">.</span> <span class="n">nlsModel</span><span class="p">(</span><span class="n">formula</span><span class="p">,</span> <span class="n">mf</span><span class="p">,</span> <span class="n">start</span><span class="p">,</span> <span class="n">wts</span><span class="p">,</span> <span class="n">scaleOffset</span> <span class="o">=</span> <span class="n">scOff</span><span class="p">,</span> <span class="n">nDcentral</span> <span class="o">=</span> <span class="n">nDcntr</span><span class="p">)</span>
-<span class="mi">3</span><span class="o">.</span> <span class="n">stop</span><span class="p">(</span><span class="s2">&quot;singular gradient matrix at initial parameter estimates&quot;</span><span class="p">)</span>
-</pre></div>
-</div>
-</div>
-</div>
-<p>The <code class="docutils literal notranslate"><span class="pre">singular</span> <span class="pre">gradient</span> <span class="pre">matrix</span> <span class="pre">at</span> <span class="pre">initial</span> <span class="pre">parameter</span> <span class="pre">estimates</span></code> error arises from the fact that the starting values you provided were so far from the optimal solution, that the parameter searching in <code class="docutils literal notranslate"><span class="pre">nls()</span></code> failed at the very first step. The algorithm could not figure out where to go from those starting values. In fact, the starting value we gave it is biologically/ physically impossible, because <code class="docutils literal notranslate"><span class="pre">V_max</span></code> can’t equal 0.</p>
-<p>Let’s look at another pair of starting values that causes the model fitting to fail:</p>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-r notranslate"><div class="highlight"><pre><span></span><span class="nf">nls</span><span class="p">(</span><span class="n">V_data</span><span class="w"> </span><span class="o">~</span><span class="w"> </span><span class="n">V_max</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">S_data</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="p">(</span><span class="n">K_M</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">S_data</span><span class="p">),</span><span class="w"> </span><span class="n">start</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nf">list</span><span class="p">(</span><span class="n">V_max</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">-0.1</span><span class="p">,</span><span class="w"> </span><span class="n">K_M</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">100</span><span class="p">))</span>
-</pre></div>
-</div>
-</div>
-<div class="cell_output docutils container">
-<div class="output traceback highlight-ipythontb notranslate"><div class="highlight"><pre><span></span><span class="n">Error</span> <span class="ow">in</span> <span class="n">nls</span><span class="p">(</span><span class="n">V_data</span> <span class="o">~</span> <span class="n">V_max</span> <span class="o">*</span> <span class="n">S_data</span><span class="o">/</span><span class="p">(</span><span class="n">K_M</span> <span class="o">+</span> <span class="n">S_data</span><span class="p">),</span> <span class="n">start</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">V_max</span> <span class="o">=</span> <span class="o">-</span><span class="mf">0.1</span><span class="p">,</span> <span class="p">:</span> <span class="n">singular</span> <span class="n">gradient</span>
-<span class="ne">Traceback</span>:
-
-<span class="mi">1</span><span class="o">.</span> <span class="n">nls</span><span class="p">(</span><span class="n">V_data</span> <span class="o">~</span> <span class="n">V_max</span> <span class="o">*</span> <span class="n">S_data</span><span class="o">/</span><span class="p">(</span><span class="n">K_M</span> <span class="o">+</span> <span class="n">S_data</span><span class="p">),</span> <span class="n">start</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">V_max</span> <span class="o">=</span> <span class="o">-</span><span class="mf">0.1</span><span class="p">,</span> 
- <span class="o">.</span>     <span class="n">K_M</span> <span class="o">=</span> <span class="mi">100</span><span class="p">))</span>
-</pre></div>
-</div>
-</div>
-</div>
-<p>In this case, the model fitting did get started, but eventually failed, again because the starting values were too far from the (approximately) optimal values (<span class="math notranslate nohighlight">\(V_{\max} \approx 12.96, K_M \approx 10.61\)</span>).</p>
-<div class="admonition note">
-<p class="admonition-title">Note</p>
-<p>There are other types of errors (other than the “singular gadient matrix” one) that the nlls fitting can run into because of poor starting parameter values.</p>
-</div>
-<p>And what happens if we start really close to the optimal values? Let’s try:</p>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-r notranslate"><div class="highlight"><pre><span></span><span class="n">MM_model4</span><span class="w"> </span><span class="o">&lt;-</span><span class="w"> </span><span class="nf">nls</span><span class="p">(</span><span class="n">V_data</span><span class="w"> </span><span class="o">~</span><span class="w"> </span><span class="n">V_max</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">S_data</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="p">(</span><span class="n">K_M</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">S_data</span><span class="p">),</span><span class="w"> </span><span class="n">start</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nf">list</span><span class="p">(</span><span class="n">V_max</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">12.96</span><span class="p">,</span><span class="w"> </span><span class="n">K_M</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">10.61</span><span class="p">))</span>
-</pre></div>
-</div>
-</div>
-</div>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-r notranslate"><div class="highlight"><pre><span></span><span class="nf">coef</span><span class="p">(</span><span class="n">MM_model</span><span class="p">)</span>
-<span class="nf">coef</span><span class="p">(</span><span class="n">MM_model4</span><span class="p">)</span>
-</pre></div>
-</div>
-</div>
-<div class="cell_output docutils container">
-<div class="output text_html"><style>
-.dl-inline {width: auto; margin:0; padding: 0}
-.dl-inline>dt, .dl-inline>dd {float: none; width: auto; display: inline-block}
-.dl-inline>dt::after {content: ":\0020"; padding-right: .5ex}
-.dl-inline>dt:not(:first-of-type) {padding-left: .5ex}
-</style><dl class=dl-inline><dt>V_max</dt><dd>12.9636891133139</dd><dt>K_M</dt><dd>10.6072253230542</dd></dl>
-</div><div class="output text_html"><style>
-.dl-inline {width: auto; margin:0; padding: 0}
-.dl-inline>dt, .dl-inline>dd {float: none; width: auto; display: inline-block}
-.dl-inline>dt::after {content: ":\0020"; padding-right: .5ex}
-.dl-inline>dt:not(:first-of-type) {padding-left: .5ex}
-</style><dl class=dl-inline><dt>V_max</dt><dd>12.9636623970243</dd><dt>K_M</dt><dd>10.6071489188677</dd></dl>
-</div></div>
-</div>
-<p>The results of the first model fit and this last one are still not exactly the same! This drives home the point that NLLS is not an “exact” procedure. However, the differences between these two solutions are minuscule, so the main thing to take away is that if the starting values are reasonable, NLLS is <em>exact enough</em>.</p>
-<p>Note that and even if you started the NLLS fitting with the exact parameter values with which you generated the data before introducing errors (so use <code class="docutils literal notranslate"><span class="pre">start</span> <span class="pre">=</span> <span class="pre">list(V_max</span> <span class="pre">=</span> <span class="pre">12.5,</span> <span class="pre">K_M</span> <span class="pre">=</span> <span class="pre">7.1)</span></code> above instead), you would still get the same result for the coefficients (try it). This is because the NLLS fitting will converge back to the parameter estimates based on the actual data, errors and all.</p>
-</section>
-</section>
-<section id="a-more-robust-nlls-algorithm">
-<h2>A more robust NLLS algorithm<a class="headerlink" href="#a-more-robust-nlls-algorithm" title="Link to this heading">#</a></h2>
-<p>The standard NLLS function in R, <code class="docutils literal notranslate"><span class="pre">nls</span></code>, which we have been using so far, does the NLLS fitting by implementing an algorithm called the Gauss-Newton algorithm. While the Gauss-Newton algorithm works well for most simple non-linear models, it has a tendency to “get lost” or “stuck” while searching for optimal parameter estimates (that minimize the residual sum of squares, or RSS). Therefore, <code class="docutils literal notranslate"><span class="pre">nls</span></code> will often fail to fit your model to the data if you start off at starting values for the parameters that are too far from what the optimal values would be, as you saw above (e.g., when you got the <code class="docutils literal notranslate"><span class="pre">singular</span> <span class="pre">gradient</span> <span class="pre">matrix</span></code> error).</p>
-<p>Some nonlinear models are especially difficult for nls to fit to data because such model have a mathematical form that makes it hard to find parameter combinations that minimize the residual sum of squared (RSS). If this does not makes sense, don’t worry about it.</p>
-<p>One solution to this is to use a different algorithm than Gauss-Newton. <code class="docutils literal notranslate"><span class="pre">nls()</span></code> has one other algorithm that can be more robust in some situations, called the “port” algorithm. However, there is a better solution still: the Levenberg-Marqualdt algorithm, which is less likely to get stuck (is more robust than) than Gauss-Newton (or port). If you want to learn more about the technicalities of this, <a class="reference external" href="https://en.wikipedia.org/wiki/Gauss%E2%80%93Newton_algorithm">here</a> are <a class="reference external" href="https://en.wikipedia.org/wiki/Levenberg%E2%80%93Marquardt_algorithm">here</a> are good places to start (also see the Readings list at the end of this chapter).</p>
-<p>To be able to use nlsLM, we will need to switch to a different NLLS function called <code class="docutils literal notranslate"><span class="pre">nlsLM</span></code>. In order to be able to use <code class="docutils literal notranslate"><span class="pre">nlsLM</span></code>, you will need the <code class="docutils literal notranslate"><span class="pre">nls.lm</span></code> R package, which you can install using the method appropriate for your operating system  (e.g., linux users will launch R in <code class="docutils literal notranslate"><span class="pre">sudo</span></code> mode first) and then use:</p>
-<div class="highlight-r notranslate"><div class="highlight"><pre><span></span><span class="o">&gt;</span><span class="w"> </span><span class="nf">install.packages</span><span class="p">(</span><span class="s">&quot;minpack.lm&quot;</span><span class="p">)</span><span class="w"> </span>
-</pre></div>
-</div>
-<p>Now load the <code class="docutils literal notranslate"><span class="pre">minpack.lm</span></code> package:</p>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-r notranslate"><div class="highlight"><pre><span></span><span class="nf">require</span><span class="p">(</span><span class="s">&quot;minpack.lm&quot;</span><span class="p">)</span>
-</pre></div>
-</div>
-</div>
-</div>
-<p>Now let’s try it (using the same starting values as <code class="docutils literal notranslate"><span class="pre">MM_model2</span></code> above):</p>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-r notranslate"><div class="highlight"><pre><span></span><span class="n">MM_model5</span><span class="w"> </span><span class="o">&lt;-</span><span class="w"> </span><span class="nf">nlsLM</span><span class="p">(</span><span class="n">V_data</span><span class="w"> </span><span class="o">~</span><span class="w"> </span><span class="n">V_max</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">S_data</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="p">(</span><span class="n">K_M</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">S_data</span><span class="p">),</span><span class="w"> </span><span class="n">start</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nf">list</span><span class="p">(</span><span class="n">V_max</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">12</span><span class="p">,</span><span class="w"> </span><span class="n">K_M</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">7</span><span class="p">))</span>
-</pre></div>
-</div>
-</div>
-</div>
-<p>Now compare the <code class="docutils literal notranslate"><span class="pre">nls</span></code> and <code class="docutils literal notranslate"><span class="pre">nlsLM</span></code> fitted coefficients:</p>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-r notranslate"><div class="highlight"><pre><span></span><span class="nf">coef</span><span class="p">(</span><span class="n">MM_model2</span><span class="p">)</span>
-<span class="nf">coef</span><span class="p">(</span><span class="n">MM_model5</span><span class="p">)</span>
-</pre></div>
-</div>
-</div>
-<div class="cell_output docutils container">
-<div class="output text_html"><style>
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-.dl-inline>dt::after {content: ":\0020"; padding-right: .5ex}
-.dl-inline>dt:not(:first-of-type) {padding-left: .5ex}
-</style><dl class=dl-inline><dt>V_max</dt><dd>12.9636297453629</dd><dt>K_M</dt><dd>10.6070555266004</dd></dl>
-</div><div class="output text_html"><style>
-.dl-inline {width: auto; margin:0; padding: 0}
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-.dl-inline>dt:not(:first-of-type) {padding-left: .5ex}
-</style><dl class=dl-inline><dt>V_max</dt><dd>12.9636298178077</dd><dt>K_M</dt><dd>10.6070557350073</dd></dl>
-</div></div>
-</div>
-<p>Close enough.</p>
-<p>Now, let’s try fitting the model using all those starting parameter combinations that failed previously:</p>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-r notranslate"><div class="highlight"><pre><span></span><span class="n">MM_model6</span><span class="w"> </span><span class="o">&lt;-</span><span class="w"> </span><span class="nf">nlsLM</span><span class="p">(</span><span class="n">V_data</span><span class="w"> </span><span class="o">~</span><span class="w"> </span><span class="n">V_max</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">S_data</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="p">(</span><span class="n">K_M</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">S_data</span><span class="p">),</span><span class="w"> </span><span class="n">start</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nf">list</span><span class="p">(</span><span class="n">V_max</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">.</span><span class="m">01</span><span class="p">,</span><span class="w"> </span><span class="n">K_M</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">20</span><span class="p">))</span>
-</pre></div>
-</div>
-</div>
-</div>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-r notranslate"><div class="highlight"><pre><span></span><span class="n">MM_model7</span><span class="w"> </span><span class="o">&lt;-</span><span class="w"> </span><span class="nf">nlsLM</span><span class="p">(</span><span class="n">V_data</span><span class="w"> </span><span class="o">~</span><span class="w"> </span><span class="n">V_max</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">S_data</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="p">(</span><span class="n">K_M</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">S_data</span><span class="p">),</span><span class="w"> </span><span class="n">start</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nf">list</span><span class="p">(</span><span class="n">V_max</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0</span><span class="p">,</span><span class="w"> </span><span class="n">K_M</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0.1</span><span class="p">))</span>
-</pre></div>
-</div>
-</div>
-</div>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-r notranslate"><div class="highlight"><pre><span></span><span class="n">MM_model8</span><span class="w"> </span><span class="o">&lt;-</span><span class="w"> </span><span class="nf">nlsLM</span><span class="p">(</span><span class="n">V_data</span><span class="w"> </span><span class="o">~</span><span class="w"> </span><span class="n">V_max</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">S_data</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="p">(</span><span class="n">K_M</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">S_data</span><span class="p">),</span><span class="w"> </span><span class="n">start</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nf">list</span><span class="p">(</span><span class="n">V_max</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">-0.1</span><span class="p">,</span><span class="w"> </span><span class="n">K_M</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">100</span><span class="p">))</span>
-</pre></div>
-</div>
-</div>
-</div>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-r notranslate"><div class="highlight"><pre><span></span><span class="nf">coef</span><span class="p">(</span><span class="n">MM_model6</span><span class="p">)</span>
-<span class="nf">coef</span><span class="p">(</span><span class="n">MM_model7</span><span class="p">)</span>
-<span class="nf">coef</span><span class="p">(</span><span class="n">MM_model8</span><span class="p">)</span>
-</pre></div>
-</div>
-</div>
-<div class="cell_output docutils container">
-<div class="output text_html"><style>
-.dl-inline {width: auto; margin:0; padding: 0}
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-.dl-inline>dt::after {content: ":\0020"; padding-right: .5ex}
-.dl-inline>dt:not(:first-of-type) {padding-left: .5ex}
-</style><dl class=dl-inline><dt>V_max</dt><dd>12.9636487209143</dd><dt>K_M</dt><dd>10.6071097943627</dd></dl>
-</div><div class="output text_html"><style>
-.dl-inline {width: auto; margin:0; padding: 0}
-.dl-inline>dt, .dl-inline>dd {float: none; width: auto; display: inline-block}
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-.dl-inline>dt:not(:first-of-type) {padding-left: .5ex}
-</style><dl class=dl-inline><dt>V_max</dt><dd>12.9636360068123</dd><dt>K_M</dt><dd>10.6070734344093</dd></dl>
-</div><div class="output text_html"><style>
-.dl-inline {width: auto; margin:0; padding: 0}
-.dl-inline>dt, .dl-inline>dd {float: none; width: auto; display: inline-block}
-.dl-inline>dt::after {content: ":\0020"; padding-right: .5ex}
-.dl-inline>dt:not(:first-of-type) {padding-left: .5ex}
-</style><dl class=dl-inline><dt>V_max</dt><dd>12.9636401038953</dd><dt>K_M</dt><dd>10.6070851518134</dd></dl>
-</div></div>
-</div>
-<p>Nice, these all worked with <code class="docutils literal notranslate"><span class="pre">nlsLM</span></code> even though they had failed with <code class="docutils literal notranslate"><span class="pre">nls</span></code>!</p>
-<p>But <code class="docutils literal notranslate"><span class="pre">nlsLM</span></code> also has its limits. Let’s try more absurd starting values:</p>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-r notranslate"><div class="highlight"><pre><span></span><span class="nf">nlsLM</span><span class="p">(</span><span class="n">V_data</span><span class="w"> </span><span class="o">~</span><span class="w"> </span><span class="n">V_max</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">S_data</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="p">(</span><span class="n">K_M</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">S_data</span><span class="p">),</span><span class="w"> </span><span class="n">start</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nf">list</span><span class="p">(</span><span class="n">V_max</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">-10</span><span class="p">,</span><span class="w"> </span><span class="n">K_M</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">-100</span><span class="p">))</span>
-</pre></div>
-</div>
-</div>
-<div class="cell_output docutils container">
-<div class="output traceback highlight-ipythontb notranslate"><div class="highlight"><pre><span></span><span class="n">Error</span> <span class="ow">in</span> <span class="n">nlsModel</span><span class="p">(</span><span class="n">formula</span><span class="p">,</span> <span class="n">mf</span><span class="p">,</span> <span class="n">start</span><span class="p">,</span> <span class="n">wts</span><span class="p">):</span> <span class="n">singular</span> <span class="n">gradient</span> <span class="n">matrix</span> <span class="n">at</span> <span class="n">initial</span> <span class="n">parameter</span> <span class="n">estimates</span>
-<span class="ne">Traceback</span>:
-
-<span class="mi">1</span><span class="o">.</span> <span class="n">nlsLM</span><span class="p">(</span><span class="n">V_data</span> <span class="o">~</span> <span class="n">V_max</span> <span class="o">*</span> <span class="n">S_data</span><span class="o">/</span><span class="p">(</span><span class="n">K_M</span> <span class="o">+</span> <span class="n">S_data</span><span class="p">),</span> <span class="n">start</span> <span class="o">=</span> <span class="nb">list</span><span class="p">(</span><span class="n">V_max</span> <span class="o">=</span> <span class="o">-</span><span class="mi">10</span><span class="p">,</span> 
- <span class="o">.</span>     <span class="n">K_M</span> <span class="o">=</span> <span class="o">-</span><span class="mi">100</span><span class="p">))</span>
-<span class="mi">2</span><span class="o">.</span> <span class="n">nlsModel</span><span class="p">(</span><span class="n">formula</span><span class="p">,</span> <span class="n">mf</span><span class="p">,</span> <span class="n">start</span><span class="p">,</span> <span class="n">wts</span><span class="p">)</span>
-<span class="mi">3</span><span class="o">.</span> <span class="n">stop</span><span class="p">(</span><span class="s2">&quot;singular gradient matrix at initial parameter estimates&quot;</span><span class="p">)</span>
-</pre></div>
-</div>
-</div>
-</div>
-<p>Here again, NLLS fitting fails because these starting values are just too far from the best solution.</p>
-</section>
-<section id="bounding-parameter-values">
-<span id="modelfitting-nlls-bounding"></span><h2>Bounding parameter values<a class="headerlink" href="#bounding-parameter-values" title="Link to this heading">#</a></h2>
-<p>You can also bound the starting values, i.e., prevent them from exceeding some minimum and maximum value <em>during</em> the NLLS fitting process.</p>
-<p>For example let’s first re-run the fitting without bounding the parameters (and some relatively-far-from-optimal starting values):</p>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-r notranslate"><div class="highlight"><pre><span></span><span class="nf">nlsLM</span><span class="p">(</span><span class="n">V_data</span><span class="w"> </span><span class="o">~</span><span class="w"> </span><span class="n">V_max</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">S_data</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="p">(</span><span class="n">K_M</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">S_data</span><span class="p">),</span><span class="w"> </span><span class="n">start</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nf">list</span><span class="p">(</span><span class="n">V_max</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0.1</span><span class="p">,</span><span class="w"> </span><span class="n">K_M</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0.1</span><span class="p">))</span>
-</pre></div>
-</div>
-</div>
-<div class="cell_output docutils container">
-<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Nonlinear regression model
-  model: V_data ~ V_max * S_data/(K_M + S_data)
-   data: parent.frame()
-V_max   K_M 
-12.96 10.61 
- residual sum-of-squares: 6.22
-
-Number of iterations to convergence: 12 
-Achieved convergence tolerance: 1.49e-08
-</pre></div>
-</div>
-</div>
-</div>
-<p>Now, the same with parameter bounds:</p>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-r notranslate"><div class="highlight"><pre><span></span><span class="nf">nlsLM</span><span class="p">(</span><span class="n">V_data</span><span class="w"> </span><span class="o">~</span><span class="w"> </span><span class="n">V_max</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">S_data</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="p">(</span><span class="n">K_M</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">S_data</span><span class="p">),</span><span class="w"> </span><span class="n">start</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nf">list</span><span class="p">(</span><span class="n">V_max</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0.1</span><span class="p">,</span><span class="w"> </span><span class="n">K_M</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0.1</span><span class="p">),</span><span class="w"> </span><span class="n">lower</span><span class="o">=</span><span class="nf">c</span><span class="p">(</span><span class="m">0.4</span><span class="p">,</span><span class="m">0.4</span><span class="p">),</span><span class="w"> </span><span class="n">upper</span><span class="o">=</span><span class="nf">c</span><span class="p">(</span><span class="m">100</span><span class="p">,</span><span class="m">100</span><span class="p">))</span>
-</pre></div>
-</div>
-</div>
-<div class="cell_output docutils container">
-<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Nonlinear regression model
-  model: V_data ~ V_max * S_data/(K_M + S_data)
-   data: parent.frame()
-V_max   K_M 
-12.96 10.61 
- residual sum-of-squares: 6.22
-
-Number of iterations to convergence: 10 
-Achieved convergence tolerance: 1.49e-08
-</pre></div>
-</div>
-</div>
-</div>
-<p>So when you added bounds, the solution was found in two fewer iterations (not a spectacular improvement, but an improvement nevertheless).</p>
-<div class="admonition note">
-<p class="admonition-title">Note</p>
-<p>The <code class="docutils literal notranslate"><span class="pre">nls()</span></code> function too has an option to provide <code class="docutils literal notranslate"><span class="pre">lower</span></code> and <code class="docutils literal notranslate"><span class="pre">upper</span></code> parameter bounds, but that is only in effect available when using <code class="docutils literal notranslate"><span class="pre">algorithm</span> <span class="pre">=</span> <span class="pre">&quot;port&quot;</span></code> (only available for a particular algorithm).</p>
-</div>
-<p>However, if you bound the parameters too much (to excessively narrow ranges), the algorithm cannot search sufficient parameter space (combinations of parameters), and will fail to converge on a good solution. For example:</p>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-r notranslate"><div class="highlight"><pre><span></span><span class="nf">nlsLM</span><span class="p">(</span><span class="n">V_data</span><span class="w"> </span><span class="o">~</span><span class="w"> </span><span class="n">V_max</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">S_data</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="p">(</span><span class="n">K_M</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">S_data</span><span class="p">),</span><span class="w"> </span><span class="n">start</span><span class="w"> </span><span class="o">=</span><span class="w">  </span><span class="nf">list</span><span class="p">(</span><span class="n">V_max</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0.5</span><span class="p">,</span><span class="w"> </span><span class="n">K_M</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0.5</span><span class="p">),</span><span class="w"> </span><span class="n">lower</span><span class="o">=</span><span class="nf">c</span><span class="p">(</span><span class="m">0.4</span><span class="p">,</span><span class="m">0.4</span><span class="p">),</span><span class="w"> </span><span class="n">upper</span><span class="o">=</span><span class="nf">c</span><span class="p">(</span><span class="m">20</span><span class="p">,</span><span class="m">20</span><span class="p">))</span>
-</pre></div>
-</div>
-</div>
-<div class="cell_output docutils container">
-<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Nonlinear regression model
-  model: V_data ~ V_max * S_data/(K_M + S_data)
-   data: parent.frame()
-V_max   K_M 
-16.09 20.00 
- residual sum-of-squares: 9.227
-
-Number of iterations to convergence: 3 
-Achieved convergence tolerance: 1.49e-08
-</pre></div>
-</div>
-</div>
-</div>
-<p>Here the algorithm converged on a poor solution, and in fact took fewer iterations (3) than before to do so. This is because it could not explore sufficient parameter combinations of <code class="docutils literal notranslate"><span class="pre">V_max</span></code> and <code class="docutils literal notranslate"><span class="pre">K_M</span></code> as we have narrowed the range that both these parameters could be allowed to take during the optimization too much.</p>
-</section>
-<section id="diagnostics-of-an-nlls-fit">
-<h2>Diagnostics of an NLLS fit<a class="headerlink" href="#diagnostics-of-an-nlls-fit" title="Link to this heading">#</a></h2>
-<p>NLLS regression carries the same three key assumptions as Linear models:</p>
-<ul class="simple">
-<li><p>No (in practice, minimal) measurement error in explanatory/independent/predictor variable (<span class="math notranslate nohighlight">\(x\)</span>-axis variable)</p></li>
-<li><p>Data have constant normal variance — errors in the <span class="math notranslate nohighlight">\(y\)</span>-axis are homogeneously distributed over the <span class="math notranslate nohighlight">\(x\)</span>-axis range</p></li>
-<li><p>The measurement/observation errors are Normally distributed (Gaussian)</p></li>
-</ul>
-<p>At the very least, it is a good idea to plot the residuals of a fitted NLLS model. Let’s do that for our Michaelis-Menten Model fit:</p>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-r notranslate"><div class="highlight"><pre><span></span><span class="nf">hist</span><span class="p">(</span><span class="nf">residuals</span><span class="p">(</span><span class="n">MM_model6</span><span class="p">))</span>
-</pre></div>
-</div>
-</div>
-<div class="cell_output docutils container">
-<a class="reference internal image-reference" href="../_images/30bee34f202463767660b799988b63378eb84ff17a27a691206546e608b03fa2.png"><img alt="../_images/30bee34f202463767660b799988b63378eb84ff17a27a691206546e608b03fa2.png" src="../_images/30bee34f202463767660b799988b63378eb84ff17a27a691206546e608b03fa2.png" style="width: 240px; height: 240px;" /></a>
-</div>
-</div>
-<p>The residuals look OK. But this should not come as a surprise because we generated these “data” ourselves using normally-distributed errors!</p>
-<p>You may also want to look at further diagnostics, as we did <a class="reference internal" href="regress.html#regress-diagnostics"><span class="std std-ref">previously</span></a> in the case of Linear models. The most convenient way to do this is to use the <code class="docutils literal notranslate"><span class="pre">nlstools</span></code> package. We will not go into it here, but you can have a look at its <a class="reference external" href="https://rdrr.io/rforge/nlstools/">documentation</a>. Note that you will need to install this package as it is not one of the core (base) R packages. <code class="docutils literal notranslate"><span class="pre">nlstools</span></code> has some other handy utilities as well; for example, its <code class="docutils literal notranslate"><span class="pre">preview</span></code> command allows you to visualise how good the starting values are (by evaluating your non-linear function at those values) in advance of the actual fitting.</p>
-<div class="admonition note">
-<p class="admonition-title">Note</p>
-<p>For the remaining examples, we will switch to using <code class="docutils literal notranslate"><span class="pre">nlsLM</span></code> instead of <code class="docutils literal notranslate"><span class="pre">nls</span></code>.</p>
-</div>
-</section>
-<section id="allometric-scaling-of-traits">
-<h2>Allometric scaling of traits<a class="headerlink" href="#allometric-scaling-of-traits" title="Link to this heading">#</a></h2>
-<p>Now let’s move on to a very common class of traits in biology: physical traits like body weight, wing span, body length, limb length, eye size, ear width, etc.</p>
-<p>We will look at a very common phenomenon called <a class="reference external" href="https://en.wikipedia.org/wiki/Allometry">allometric scaling</a>. Allometric relationships between linear measurements such as body length, limb length, wing span, and thorax width are a good way to obtain estimates of body weights of individual organisms. We will look at allometric scaling of body weight vs. total body length in dragonflies and damselfiles.</p>
-<p>Allometric relationships take the form:</p>
-<div class="math notranslate nohighlight" id="equation-eq-allom">
-<span class="eqno">(8)<a class="headerlink" href="#equation-eq-allom" title="Link to this equation">#</a></span>\[
-y = a x^b
-\]</div>
-<p>where <span class="math notranslate nohighlight">\(x\)</span> and <span class="math notranslate nohighlight">\(y\)</span> are morphological measures (body length and body weight respectively, in our current example), the constant is the value of <span class="math notranslate nohighlight">\(y\)</span> at body length <span class="math notranslate nohighlight">\(x = 1\)</span> unit, and <span class="math notranslate nohighlight">\(b\)</span> is the scaling “exponent”. This is also called a power-law, because <span class="math notranslate nohighlight">\(y\)</span> relates to <span class="math notranslate nohighlight">\(x\)</span> through a simple power.</p>
-<p>Let’s fit a power low to a typical allometric relationship: The change in body weight vs change in body length. In general, this relationship is a allometry; that is, body weight does not increase proportionally with some measure of body length.</p>
-<p>First, let’s look at the data. You can get the data <a class="reference external" href="https://raw.githubusercontent.com/mhasoba/TheMulQuaBio/master/content/data/GenomeSize.csv">here</a> (first click on link and use “Save as” or <code class="docutils literal notranslate"><span class="pre">Ctrl+S</span></code> to download it as a csv).</p>
-<p><span class="math notranslate nohighlight">\(\star\)</span> Save the <code class="docutils literal notranslate"><span class="pre">GenomeSize.csv</span></code> data file to your <code class="docutils literal notranslate"><span class="pre">data</span></code> directory, and import it into your R workspace:</p>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-r notranslate"><div class="highlight"><pre><span></span><span class="n">MyData</span><span class="w"> </span><span class="o">&lt;-</span><span class="w"> </span><span class="nf">read.csv</span><span class="p">(</span><span class="s">&quot;../data/GenomeSize.csv&quot;</span><span class="p">)</span><span class="w"> </span><span class="c1"># using relative path assuming that your working directory is &quot;code&quot;</span>
-
-<span class="nf">head</span><span class="p">(</span><span class="n">MyData</span><span class="p">)</span>
-</pre></div>
-</div>
-</div>
-<div class="cell_output docutils container">
-<div class="output text_html"><table class="dataframe">
-<caption>A data.frame: 6 × 16</caption>
-<thead>
-	<tr><th></th><th scope=col>Suborder</th><th scope=col>Family</th><th scope=col>Species</th><th scope=col>GenomeSize</th><th scope=col>GenomeSE</th><th scope=col>GenomeN</th><th scope=col>BodyWeight</th><th scope=col>TotalLength</th><th scope=col>HeadLength</th><th scope=col>ThoraxLength</th><th scope=col>AdbdomenLength</th><th scope=col>ForewingLength</th><th scope=col>HindwingLength</th><th scope=col>ForewingArea</th><th scope=col>HindwingArea</th><th scope=col>MorphologyN</th></tr>
-	<tr><th></th><th scope=col>&lt;chr&gt;</th><th scope=col>&lt;chr&gt;</th><th scope=col>&lt;chr&gt;</th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;int&gt;</th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;int&gt;</th></tr>
-</thead>
-<tbody>
-	<tr><th scope=row>1</th><td>Anisoptera</td><td>Aeshnidae</td><td>Aeshna canadensis   </td><td>2.20</td><td>  NA</td><td>1</td><td>0.159</td><td>67.58</td><td>6.83</td><td>11.81</td><td>48.94</td><td>45.47</td><td>45.40</td><td>369.57</td><td>483.61</td><td>2</td></tr>
-	<tr><th scope=row>2</th><td>Anisoptera</td><td>Aeshnidae</td><td>Aeshna constricta   </td><td>1.76</td><td>0.06</td><td>4</td><td>0.228</td><td>71.97</td><td>6.84</td><td>10.72</td><td>54.41</td><td>46.00</td><td>45.48</td><td>411.15</td><td>517.38</td><td>3</td></tr>
-	<tr><th scope=row>3</th><td>Anisoptera</td><td>Aeshnidae</td><td>Aeshna eremita      </td><td>1.85</td><td>  NA</td><td>1</td><td>0.312</td><td>78.80</td><td>6.27</td><td>16.19</td><td>56.33</td><td>51.24</td><td>49.47</td><td>460.72</td><td>574.33</td><td>1</td></tr>
-	<tr><th scope=row>4</th><td>Anisoptera</td><td>Aeshnidae</td><td>Aeshna tuberculifera</td><td>1.78</td><td>0.10</td><td>2</td><td>0.218</td><td>72.44</td><td>6.62</td><td>12.53</td><td>53.29</td><td>49.84</td><td>48.82</td><td>468.74</td><td>591.42</td><td>2</td></tr>
-	<tr><th scope=row>5</th><td>Anisoptera</td><td>Aeshnidae</td><td>Aeshna umbrosa      </td><td>2.00</td><td>  NA</td><td>1</td><td>0.207</td><td>73.05</td><td>4.92</td><td>11.11</td><td>57.03</td><td>46.51</td><td>45.97</td><td>382.48</td><td>481.44</td><td>1</td></tr>
-	<tr><th scope=row>6</th><td>Anisoptera</td><td>Aeshnidae</td><td>Aeshna verticalis   </td><td>1.59</td><td>  NA</td><td>1</td><td>0.220</td><td>66.25</td><td>6.48</td><td>11.64</td><td>48.13</td><td>45.91</td><td>44.91</td><td>400.40</td><td>486.97</td><td>1</td></tr>
-</tbody>
-</table>
-</div></div>
-</div>
-<p><a class="reference external" href="https://en.wikipedia.org/wiki/Dragonfly">Anisoptera</a> are dragonflies, and <a class="reference external" href="https://en.wikipedia.org/wiki/Damselfly">Zygoptera</a> are Damselflies. The variables of interest are <code class="docutils literal notranslate"><span class="pre">BodyWeight</span></code> and <code class="docutils literal notranslate"><span class="pre">TotalLength</span></code>. Let’s use the dragonflies data subset.</p>
-<p>So subset the data accordingly and remove NAs:</p>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-r notranslate"><div class="highlight"><pre><span></span><span class="n">Data2Fit</span><span class="w"> </span><span class="o">&lt;-</span><span class="w"> </span><span class="nf">subset</span><span class="p">(</span><span class="n">MyData</span><span class="p">,</span><span class="n">Suborder</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="s">&quot;Anisoptera&quot;</span><span class="p">)</span>
-
-<span class="n">Data2Fit</span><span class="w"> </span><span class="o">&lt;-</span><span class="w"> </span><span class="n">Data2Fit</span><span class="p">[</span><span class="o">!</span><span class="nf">is.na</span><span class="p">(</span><span class="n">Data2Fit</span><span class="o">$</span><span class="n">TotalLength</span><span class="p">),]</span><span class="w"> </span><span class="c1"># remove NA&#39;s</span>
-</pre></div>
-</div>
-</div>
-</div>
-<p>Plot the data:</p>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-r notranslate"><div class="highlight"><pre><span></span><span class="nf">plot</span><span class="p">(</span><span class="n">Data2Fit</span><span class="o">$</span><span class="n">TotalLength</span><span class="p">,</span><span class="w"> </span><span class="n">Data2Fit</span><span class="o">$</span><span class="n">BodyWeight</span><span class="p">,</span><span class="w"> </span><span class="n">xlab</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s">&quot;Body Length&quot;</span><span class="p">,</span><span class="w"> </span><span class="n">ylab</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s">&quot;Body Weight&quot;</span><span class="p">)</span>
-</pre></div>
-</div>
-</div>
-<div class="cell_output docutils container">
-<a class="reference internal image-reference" href="../_images/62a33520032d38696c8900c1eed81e29fa2656b57073aa449f55c8cb56ca1cff.png"><img alt="../_images/62a33520032d38696c8900c1eed81e29fa2656b57073aa449f55c8cb56ca1cff.png" src="../_images/62a33520032d38696c8900c1eed81e29fa2656b57073aa449f55c8cb56ca1cff.png" style="width: 240px; height: 240px;" /></a>
-</div>
-</div>
-<p>Or, using <code class="docutils literal notranslate"><span class="pre">ggplot</span></code>:</p>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-r notranslate"><div class="highlight"><pre><span></span><span class="nf">library</span><span class="p">(</span><span class="s">&quot;ggplot2&quot;</span><span class="p">)</span>
-
-<span class="nf">ggplot</span><span class="p">(</span><span class="n">Data2Fit</span><span class="p">,</span><span class="w"> </span><span class="nf">aes</span><span class="p">(</span><span class="n">x</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">TotalLength</span><span class="p">,</span><span class="w"> </span><span class="n">y</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">BodyWeight</span><span class="p">))</span><span class="w"> </span><span class="o">+</span><span class="w"> </span>
-<span class="nf">geom_point</span><span class="p">(</span><span class="n">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="m">3</span><span class="p">),</span><span class="n">color</span><span class="o">=</span><span class="s">&quot;red&quot;</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nf">theme_bw</span><span class="p">()</span><span class="w"> </span><span class="o">+</span><span class="w"> </span>
-<span class="nf">labs</span><span class="p">(</span><span class="n">y</span><span class="o">=</span><span class="s">&quot;Body mass (mg)&quot;</span><span class="p">,</span><span class="w"> </span><span class="n">x</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s">&quot;Wing length (mm)&quot;</span><span class="p">)</span>
-</pre></div>
-</div>
-</div>
-<div class="cell_output docutils container">
-<a class="reference internal image-reference" href="../_images/ef82ab43f147b5c2c5e4fc67ba6e02bb1845afdc4ef72937a7427ddcab206ce1.png"><img alt="../_images/ef82ab43f147b5c2c5e4fc67ba6e02bb1845afdc4ef72937a7427ddcab206ce1.png" src="../_images/ef82ab43f147b5c2c5e4fc67ba6e02bb1845afdc4ef72937a7427ddcab206ce1.png" style="width: 240px; height: 240px;" /></a>
-</div>
-</div>
-<p>You can see these body weights of dragonflies does not increase proportionally with body length – they curve upwards w.r.t. wing length (so the allometric constant <span class="math notranslate nohighlight">\(b\)</span> in eqn <a class="reference internal" href="#equation-eq-allom">(8)</a> mustbe greater than 1), instead of increasing as a straight line (in which case <span class="math notranslate nohighlight">\(b = 1\)</span> (isometry, instead of allometry).</p>
-<p>Now fit the model to the data using NLLS:</p>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-r notranslate"><div class="highlight"><pre><span></span><span class="nf">nrow</span><span class="p">(</span><span class="n">Data2Fit</span><span class="p">)</span>
-</pre></div>
-</div>
-</div>
-<div class="cell_output docutils container">
-<div class="output text_html">60</div></div>
-</div>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-r notranslate"><div class="highlight"><pre><span></span><span class="n">PowFit</span><span class="w"> </span><span class="o">&lt;-</span><span class="w"> </span><span class="nf">nlsLM</span><span class="p">(</span><span class="n">BodyWeight</span><span class="w"> </span><span class="o">~</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">TotalLength</span><span class="o">^</span><span class="n">b</span><span class="p">,</span><span class="w"> </span><span class="n">data</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Data2Fit</span><span class="p">,</span><span class="w"> </span><span class="n">start</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nf">list</span><span class="p">(</span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">.</span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="n">b</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">.</span><span class="m">1</span><span class="p">))</span>
-</pre></div>
-</div>
-</div>
-</div>
-<section id="nlls-fitting-using-a-model-object">
-<h3>NLLS fitting using a model object<a class="headerlink" href="#nlls-fitting-using-a-model-object" title="Link to this heading">#</a></h3>
-<p>Another way to tell <code class="docutils literal notranslate"><span class="pre">nlsLM</span></code> which model to fit, is to first create a function object for the power law model:</p>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-r notranslate"><div class="highlight"><pre><span></span><span class="n">powMod</span><span class="w"> </span><span class="o">&lt;-</span><span class="w"> </span><span class="nf">function</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="w"> </span><span class="n">a</span><span class="p">,</span><span class="w"> </span><span class="n">b</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>
-<span class="w"> </span><span class="nf">return</span><span class="p">(</span><span class="n">a</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">x</span><span class="o">^</span><span class="n">b</span><span class="p">)</span>
-<span class="p">}</span>
-</pre></div>
-</div>
-</div>
-</div>
-<p>Now fit the model to the data using NLLS by calling the model:</p>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-r notranslate"><div class="highlight"><pre><span></span><span class="n">PowFit</span><span class="w"> </span><span class="o">&lt;-</span><span class="w"> </span><span class="nf">nlsLM</span><span class="p">(</span><span class="n">BodyWeight</span><span class="w"> </span><span class="o">~</span><span class="w"> </span><span class="nf">powMod</span><span class="p">(</span><span class="n">TotalLength</span><span class="p">,</span><span class="n">a</span><span class="p">,</span><span class="n">b</span><span class="p">),</span><span class="w"> </span><span class="n">data</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Data2Fit</span><span class="p">,</span><span class="w"> </span><span class="n">start</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nf">list</span><span class="p">(</span><span class="n">a</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">.</span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="n">b</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">.</span><span class="m">1</span><span class="p">))</span>
-</pre></div>
-</div>
-</div>
-</div>
-<p>Which gives the same result as before (you can check it).</p>
-</section>
-<section id="id1">
-<h3>Visualizing the fit<a class="headerlink" href="#id1" title="Link to this heading">#</a></h3>
-<p>The first thing to do is to see how well the model fitted the data, for which plotting is the best first option. So let’s visualize the fit. For this, first we need to generate a vector of body lengths (the x-axis variable) for plotting:</p>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-r notranslate"><div class="highlight"><pre><span></span><span class="n">Lengths</span><span class="w"> </span><span class="o">&lt;-</span><span class="w"> </span><span class="nf">seq</span><span class="p">(</span><span class="nf">min</span><span class="p">(</span><span class="n">Data2Fit</span><span class="o">$</span><span class="n">TotalLength</span><span class="p">),</span><span class="nf">max</span><span class="p">(</span><span class="n">Data2Fit</span><span class="o">$</span><span class="n">TotalLength</span><span class="p">),</span><span class="n">len</span><span class="o">=</span><span class="m">200</span><span class="p">)</span>
-</pre></div>
-</div>
-</div>
-</div>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-r notranslate"><div class="highlight"><pre><span></span><span class="nf">coef</span><span class="p">(</span><span class="n">PowFit</span><span class="p">)[</span><span class="s">&quot;a&quot;</span><span class="p">]</span>
-<span class="nf">coef</span><span class="p">(</span><span class="n">PowFit</span><span class="p">)[</span><span class="s">&quot;b&quot;</span><span class="p">]</span>
-</pre></div>
-</div>
-</div>
-<div class="cell_output docutils container">
-<div class="output text_html"><strong>a:</strong> 3.94068495559397e-06</div><div class="output text_html"><strong>b:</strong> 2.58504796499038</div></div>
-</div>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-r notranslate"><div class="highlight"><pre><span></span><span class="n">Predic2PlotPow</span><span class="w"> </span><span class="o">&lt;-</span><span class="w"> </span><span class="nf">powMod</span><span class="p">(</span><span class="n">Lengths</span><span class="p">,</span><span class="nf">coef</span><span class="p">(</span><span class="n">PowFit</span><span class="p">)[</span><span class="s">&quot;a&quot;</span><span class="p">],</span><span class="nf">coef</span><span class="p">(</span><span class="n">PowFit</span><span class="p">)[</span><span class="s">&quot;b&quot;</span><span class="p">])</span>
-</pre></div>
-</div>
-</div>
-</div>
-<p>Next, calculate the predicted line. For this, we will need to extract the coefficient from the model fit object using the <code class="docutils literal notranslate"><span class="pre">coef()</span></code>command.</p>
-<p>Now plot the data and the fitted model line:</p>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-r notranslate"><div class="highlight"><pre><span></span><span class="nf">plot</span><span class="p">(</span><span class="n">Data2Fit</span><span class="o">$</span><span class="n">TotalLength</span><span class="p">,</span><span class="w"> </span><span class="n">Data2Fit</span><span class="o">$</span><span class="n">BodyWeight</span><span class="p">)</span>
-<span class="nf">lines</span><span class="p">(</span><span class="n">Lengths</span><span class="p">,</span><span class="w"> </span><span class="n">Predic2PlotPow</span><span class="p">,</span><span class="w"> </span><span class="n">col</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s">&#39;blue&#39;</span><span class="p">,</span><span class="w"> </span><span class="n">lwd</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">2.5</span><span class="p">)</span>
-</pre></div>
-</div>
-</div>
-<div class="cell_output docutils container">
-<a class="reference internal image-reference" href="../_images/0be8dde533791c619dc1c867818fffc7f61221f70bdbc53767543dd1da5a96c3.png"><img alt="../_images/0be8dde533791c619dc1c867818fffc7f61221f70bdbc53767543dd1da5a96c3.png" src="../_images/0be8dde533791c619dc1c867818fffc7f61221f70bdbc53767543dd1da5a96c3.png" style="width: 240px; height: 240px;" /></a>
-</div>
-</div>
-</section>
-<section id="summary-of-the-fit">
-<h3>Summary of the fit<a class="headerlink" href="#summary-of-the-fit" title="Link to this heading">#</a></h3>
-<p>Now lets get some stats of this NLLS fit. Having obtained the fit object (<code class="docutils literal notranslate"><span class="pre">PowMod</span></code>), we can use <code class="docutils literal notranslate"><span class="pre">summary()</span></code> just like we would for a <code class="docutils literal notranslate"><span class="pre">lm()</span></code> fit object:</p>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-r notranslate"><div class="highlight"><pre><span></span><span class="nf">summary</span><span class="p">(</span><span class="n">PowFit</span><span class="p">)</span>
-</pre></div>
-</div>
-</div>
-<div class="cell_output docutils container">
-<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Formula: BodyWeight ~ powMod(TotalLength, a, b)
-
-Parameters:
-   Estimate Std. Error t value Pr(&gt;|t|)    
-a 3.941e-06  2.234e-06   1.764    0.083 .  
-b 2.585e+00  1.348e-01  19.174   &lt;2e-16 ***
----
-Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
-
-Residual standard error: 0.02807 on 58 degrees of freedom
-
-Number of iterations to convergence: 39 
-Achieved convergence tolerance: 1.49e-08
-</pre></div>
-</div>
-</div>
-</div>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-r notranslate"><div class="highlight"><pre><span></span><span class="nf">print</span><span class="p">(</span><span class="nf">confint</span><span class="p">(</span><span class="n">PowFit</span><span class="p">))</span>
-</pre></div>
-</div>
-</div>
-<div class="cell_output docutils container">
-<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Waiting for profiling to be done...
-</pre></div>
-</div>
-<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>          2.5%        97.5%
-a 1.171935e-06 1.205273e-05
-b 2.318292e+00 2.872287e+00
-</pre></div>
-</div>
-</div>
-</div>
-</section>
-<section id="examine-the-residuals">
-<h3>Examine the residuals<a class="headerlink" href="#examine-the-residuals" title="Link to this heading">#</a></h3>
-<p>As we did before, let’s plot the residuals around the fitted model:</p>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-r notranslate"><div class="highlight"><pre><span></span><span class="nf">hist</span><span class="p">(</span><span class="nf">residuals</span><span class="p">(</span><span class="n">PowFit</span><span class="p">))</span>
-</pre></div>
-</div>
-</div>
-<div class="cell_output docutils container">
-<a class="reference internal image-reference" href="../_images/89699fe8ad23aace87bf687cadb13c200d7ff2ac648c34480b70cfb64bc6cace.png"><img alt="../_images/89699fe8ad23aace87bf687cadb13c200d7ff2ac648c34480b70cfb64bc6cace.png" src="../_images/89699fe8ad23aace87bf687cadb13c200d7ff2ac648c34480b70cfb64bc6cace.png" style="width: 240px; height: 240px;" /></a>
-</div>
-</div>
-<p>The residuals look OK.</p>
-<div class="admonition tip">
-<p class="admonition-title">Tip</p>
-<p>Remember, when you write this analysis into a stand-alone R script, you should put all commands for loading packages (<code class="docutils literal notranslate"><span class="pre">library()</span></code>, <code class="docutils literal notranslate"><span class="pre">require()</span></code>) at the start of the script. *</p>
-</div>
-<section id="exercises">
-<h4>Exercises <a id='Allom_Exercises'></a><a class="headerlink" href="#exercises" title="Link to this heading">#</a></h4>
-<p>(a) Make the same plot as above, fitted line and all, in <code class="docutils literal notranslate"><span class="pre">ggplot</span></code>, and add (display) the equation you estimated to your new (ggplot) plot. The equation is: <span class="math notranslate nohighlight">\(\text{Weight} = 3.94 \times 10^{-06} \times \text{Length}^{2.59}\)</span></p>
-<p>(b) Try playing with the starting values, and see if you can “break” the model fitting – that is, change the starting values till the NLLS fitting does not converge on a solution.</p>
-<p>(c) Repeat the model fitting (including a-b above) using the Zygoptera data subset.</p>
-<p>(d) There is an alternative (and in fact, more commonly-used) approach for fitting the allometric model to data: using Ordinary Least Squares on bi-logarithamically transformed data. That is, if you take a log of both sides of the <a class="reference internal" href="#equation-eq-allom">(8)</a> we get,</p>
-<div class="math notranslate nohighlight">
-\[
-\log(y) = \log(a) + b \log(x)
-\]</div>
-<p>This is a straight line equation of the form <span class="math notranslate nohighlight">\(c = d + b z \)</span>, where <span class="math notranslate nohighlight">\(c = \log(c)\)</span>, <span class="math notranslate nohighlight">\(d = \log(a)\)</span>, <span class="math notranslate nohighlight">\(z = \log(x)\)</span>, and <span class="math notranslate nohighlight">\(b\)</span> is now the slope parameter. So you can use Ordinary Least Squares and the linear models framework (with <code class="docutils literal notranslate"><span class="pre">lm()</span></code>) in R to estimate the parameters of the allometric equation.</p>
-<p>In this exercise, try comparing the NLLS vs OLS methods to see how much difference you get in the parameter estimates between them. For example, see the methods used in this paper by <a class="reference external" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3465447/">Cohen et al 2012</a>.</p>
-<p>(e) The allometry between Body weight and Length is not the end of the story. You have a number of other linear morphological measurements (<code class="docutils literal notranslate"><span class="pre">HeadLength</span></code>, <code class="docutils literal notranslate"><span class="pre">ThoraxLength</span></code>, <code class="docutils literal notranslate"><span class="pre">AdbdomenLength</span></code>, <code class="docutils literal notranslate"><span class="pre">ForewingLength</span></code>, <code class="docutils literal notranslate"><span class="pre">HindwingLength</span></code>, <code class="docutils literal notranslate"><span class="pre">ForewingArea</span></code>, and <code class="docutils literal notranslate"><span class="pre">HindwingArea</span></code>) that can also be investigated. In this exercise, try two lines of investigation (again, repeated separately for Dragonflies and Damselfiles):</p>
-<p>(i) How do each of these measures allometrically scale with Body length (obtain estimates of scaling constant and exponent)? (Hint: you may want to use the <code class="docutils literal notranslate"><span class="pre">pairs()</span></code> command in R to get an overview of all the pairs of potential scaling relationships.</p>
-<p>(ii) Do any of the linear morphological measurements other than body length better predict Body weight? That is, does body weight scale more tightly with a linear morphological measurement other than total body length? You would use model selection here, which we will learn next. But for now, you can just look at and compare the <span class="math notranslate nohighlight">\(R^2\)</span> values of the models.</p>
-</section>
-</section>
-</section>
-<section id="comparing-models">
-<span id="model-fitting-r-comparing-models"></span><h2>Comparing models<a class="headerlink" href="#comparing-models" title="Link to this heading">#</a></h2>
-<p><em>How do we know that there isn’t a better or alternative model that adequately explains the pattern in your dataset?</em></p>
-<p>This is important consideration in all data analyses (and more generally, the scientific method!), so you must aim to compare your NLLS model with an one or more alternatives for a more extensive and reliable investigation of the problem.</p>
-<p>Let’s use model comparison to investigate whether the relationship between body weight and length we found above is indeed allometric. For this, we need an alternative model that can be fitted to the same data. Let’s try a quadratic curve, which is of the form:</p>
-<div class="math notranslate nohighlight">
-\[
-y = a + b x + c x^2
-\]</div>
-<p>This can also capture curvature in data, and is an alternative model to the <a class="reference internal" href="#equation-eq-allom">(8)</a>. Note that this mode is linear in its parameters (a linear model), which you can fit to the simply data using your familiar <code class="docutils literal notranslate"><span class="pre">lm()</span></code> function:</p>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-r notranslate"><div class="highlight"><pre><span></span><span class="n">QuaFit</span><span class="w"> </span><span class="o">&lt;-</span><span class="w"> </span><span class="nf">lm</span><span class="p">(</span><span class="n">BodyWeight</span><span class="w"> </span><span class="o">~</span><span class="w"> </span><span class="nf">poly</span><span class="p">(</span><span class="n">TotalLength</span><span class="p">,</span><span class="m">2</span><span class="p">),</span><span class="w"> </span><span class="n">data</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Data2Fit</span><span class="p">)</span>
-</pre></div>
-</div>
-</div>
-</div>
-<p>And like before, we obtain the predicted values (but this time using the <code class="docutils literal notranslate"><span class="pre">predict.lm</span></code> function):</p>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-r notranslate"><div class="highlight"><pre><span></span><span class="n">Predic2PlotQua</span><span class="w"> </span><span class="o">&lt;-</span><span class="w"> </span><span class="nf">predict.lm</span><span class="p">(</span><span class="n">QuaFit</span><span class="p">,</span><span class="w"> </span><span class="nf">data.frame</span><span class="p">(</span><span class="n">TotalLength</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Lengths</span><span class="p">))</span>
-</pre></div>
-</div>
-</div>
-</div>
-<p>Now let’s plot the two fitted models together:</p>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-r notranslate"><div class="highlight"><pre><span></span><span class="nf">plot</span><span class="p">(</span><span class="n">Data2Fit</span><span class="o">$</span><span class="n">TotalLength</span><span class="p">,</span><span class="w"> </span><span class="n">Data2Fit</span><span class="o">$</span><span class="n">BodyWeight</span><span class="p">)</span>
-<span class="nf">lines</span><span class="p">(</span><span class="n">Lengths</span><span class="p">,</span><span class="w"> </span><span class="n">Predic2PlotPow</span><span class="p">,</span><span class="w"> </span><span class="n">col</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s">&#39;blue&#39;</span><span class="p">,</span><span class="w"> </span><span class="n">lwd</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">2.5</span><span class="p">)</span>
-<span class="nf">lines</span><span class="p">(</span><span class="n">Lengths</span><span class="p">,</span><span class="w"> </span><span class="n">Predic2PlotQua</span><span class="p">,</span><span class="w"> </span><span class="n">col</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s">&#39;red&#39;</span><span class="p">,</span><span class="w"> </span><span class="n">lwd</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">2.5</span><span class="p">)</span>
-</pre></div>
-</div>
-</div>
-<div class="cell_output docutils container">
-<a class="reference internal image-reference" href="../_images/48ff509735afaf0c7bab4aea1745621a8e70318d0587256578347d6ac7012fa9.png"><img alt="../_images/48ff509735afaf0c7bab4aea1745621a8e70318d0587256578347d6ac7012fa9.png" src="../_images/48ff509735afaf0c7bab4aea1745621a8e70318d0587256578347d6ac7012fa9.png" style="width: 240px; height: 240px;" /></a>
-</div>
-</div>
-<p>Very similar fits, except that the quadratic model seems to deviate a bit from the data at the lower end of the data range. Let’s do a proper, formal model comparison now to check which model better-fits the data.</p>
-<p>First calculate the R<span class="math notranslate nohighlight">\(^2\)</span> values of the two fitted models:</p>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-r notranslate"><div class="highlight"><pre><span></span><span class="n">RSS_Pow</span><span class="w"> </span><span class="o">&lt;-</span><span class="w"> </span><span class="nf">sum</span><span class="p">(</span><span class="nf">residuals</span><span class="p">(</span><span class="n">PowFit</span><span class="p">)</span><span class="o">^</span><span class="m">2</span><span class="p">)</span><span class="w"> </span><span class="c1"># Residual sum of squares</span>
-<span class="n">TSS_Pow</span><span class="w"> </span><span class="o">&lt;-</span><span class="w"> </span><span class="nf">sum</span><span class="p">((</span><span class="n">Data2Fit</span><span class="o">$</span><span class="n">BodyWeight</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nf">mean</span><span class="p">(</span><span class="n">Data2Fit</span><span class="o">$</span><span class="n">BodyWeight</span><span class="p">))</span><span class="o">^</span><span class="m">2</span><span class="p">)</span><span class="w"> </span><span class="c1"># Total sum of squares</span>
-<span class="n">RSq_Pow</span><span class="w"> </span><span class="o">&lt;-</span><span class="w"> </span><span class="m">1</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="p">(</span><span class="n">RSS_Pow</span><span class="o">/</span><span class="n">TSS_Pow</span><span class="p">)</span><span class="w"> </span><span class="c1"># R-squared value</span>
-
-<span class="n">RSS_Qua</span><span class="w"> </span><span class="o">&lt;-</span><span class="w"> </span><span class="nf">sum</span><span class="p">(</span><span class="nf">residuals</span><span class="p">(</span><span class="n">QuaFit</span><span class="p">)</span><span class="o">^</span><span class="m">2</span><span class="p">)</span><span class="w"> </span><span class="c1"># Residual sum of squares</span>
-<span class="n">TSS_Qua</span><span class="w"> </span><span class="o">&lt;-</span><span class="w"> </span><span class="nf">sum</span><span class="p">((</span><span class="n">Data2Fit</span><span class="o">$</span><span class="n">BodyWeight</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nf">mean</span><span class="p">(</span><span class="n">Data2Fit</span><span class="o">$</span><span class="n">BodyWeight</span><span class="p">))</span><span class="o">^</span><span class="m">2</span><span class="p">)</span><span class="w"> </span><span class="c1"># Total sum of squares</span>
-<span class="n">RSq_Qua</span><span class="w"> </span><span class="o">&lt;-</span><span class="w"> </span><span class="m">1</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="p">(</span><span class="n">RSS_Qua</span><span class="o">/</span><span class="n">TSS_Qua</span><span class="p">)</span><span class="w"> </span><span class="c1"># R-squared value</span>
-
-<span class="n">RSq_Pow</span><span class="w"> </span>
-<span class="n">RSq_Qua</span>
-</pre></div>
-</div>
-</div>
-<div class="cell_output docutils container">
-<div class="output text_html">0.90054752976309</div><div class="output text_html">0.900302864503218</div></div>
-</div>
-<p>Not very useful. In general, R<span class="math notranslate nohighlight">\(^2\)</span> is a good measure of model fit, but cannot be used for model selection – especially not here, given the tiny difference in R<span class="math notranslate nohighlight">\(^2\)</span>’s.</p>
-<p>Instead, as explained in the <a class="reference external" href="https://github.com/mhasoba/TheMulQuaBio/blob/master/lectures/ModelFitting">lecture</a>, we can use the Akaike Information Criterion (AIC):</p>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-r notranslate"><div class="highlight"><pre><span></span><span class="n">n</span><span class="w"> </span><span class="o">&lt;-</span><span class="w"> </span><span class="nf">nrow</span><span class="p">(</span><span class="n">Data2Fit</span><span class="p">)</span><span class="w"> </span><span class="c1">#set sample size</span>
-<span class="n">pPow</span><span class="w"> </span><span class="o">&lt;-</span><span class="w"> </span><span class="nf">length</span><span class="p">(</span><span class="nf">coef</span><span class="p">(</span><span class="n">PowFit</span><span class="p">))</span><span class="w"> </span><span class="c1"># get number of parameters in power law model</span>
-<span class="n">pQua</span><span class="w"> </span><span class="o">&lt;-</span><span class="w"> </span><span class="nf">length</span><span class="p">(</span><span class="nf">coef</span><span class="p">(</span><span class="n">QuaFit</span><span class="p">))</span><span class="w"> </span><span class="c1"># get number of parameters in quadratic model</span>
-
-<span class="n">AIC_Pow</span><span class="w"> </span><span class="o">&lt;-</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">2</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="nf">log</span><span class="p">((</span><span class="m">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="kc">pi</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="nf">log</span><span class="p">(</span><span class="n">RSS_Pow</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">pPow</span>
-<span class="n">AIC_Qua</span><span class="w"> </span><span class="o">&lt;-</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">2</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="nf">log</span><span class="p">((</span><span class="m">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="kc">pi</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="nf">log</span><span class="p">(</span><span class="n">RSS_Qua</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">pQua</span>
-<span class="n">AIC_Pow</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">AIC_Qua</span>
-</pre></div>
-</div>
-</div>
-<div class="cell_output docutils container">
-<div class="output text_html">-2.14742608125084</div></div>
-</div>
-<p>Of course, as you might have suspected, we can do this using an in-built function in R!</p>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-r notranslate"><div class="highlight"><pre><span></span><span class="nf">AIC</span><span class="p">(</span><span class="n">PowFit</span><span class="p">)</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nf">AIC</span><span class="p">(</span><span class="n">QuaFit</span><span class="p">)</span>
-</pre></div>
-</div>
-</div>
-<div class="cell_output docutils container">
-<div class="output text_html">-2.1474260812509</div></div>
-</div>
-<ul class="simple">
-<li><p>So which model wins? * As we had dicussed in the NLLS lecture, a rule of thumb is that a AIC value difference (typically denoted as <span class="math notranslate nohighlight">\(\Delta\)</span>AIC) &gt;-2 (more negative is better) is a acceptable cutoff for calling a winner. So the power law (allometric model) is a better fit here. Read the <a class="reference external" href="https://github.com/mhasoba/TheMulQuaBio/blob/master/readings/Modelling/JohnsonOmland2004.pdf">Johnson &amp; Omland paper</a> for more on model selection in Ecology and Evolution.</p></li>
-</ul>
-<section id="id2">
-<h3>Exercises <a id='ModelSelection_Exercises'></a><a class="headerlink" href="#id2" title="Link to this heading">#</a></h3>
-<p>(a) Calculate the Bayesian Information Criterion (BIC), also know as the Schwarz Criterion (see your Lecture notes and the <a class="reference external" href="https://github.com/mhasoba/TheMulQuaBio/blob/master/readings/Modelling/JohnsonOmland2004.pdf">Johnson &amp; Omland paper</a>, and use <span class="math notranslate nohighlight">\(\Delta\)</span>BIC to select the better fitting model.</p>
-<p>(b) Fit a straight line to the same data and compare with the allometric and quadratic models.</p>
-<p>(c) Repeat the model comparison (incuding 1-2 above) using the Damselflies (Zygoptera) data subset – does the allometric model still win?</p>
-<p>(d) Repeat exercise (e)(i) and (ii) from the <a class="reference internal" href="#Allom_Exercises"><span class="xref myst">above set</span></a>, but with model comparison (e.g., again using a quadratic as an alternative model) to establish that the relationships are indeed allometric.</p>
-<p>(e) Repeat exercise (e)(ii) from the <a class="reference internal" href="#Allom_Exercises"><span class="xref myst">above set</span></a>, but with model comparison to establish which linear measurement is the best predictor of Body weight.</p>
-</section>
-</section>
-<section id="albatross-chick-growth">
-<h2>Albatross chick growth<a class="headerlink" href="#albatross-chick-growth" title="Link to this heading">#</a></h2>
-<p>Now let’s look at a different trait example: the growth of an individual albatross chick (you can find similar data for vector and non-vector arthropods in <a class="reference external" href="https://vectorbyte.org/">VecTraits</a>). First load and plot the data:</p>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-r notranslate"><div class="highlight"><pre><span></span><span class="n">alb</span><span class="w"> </span><span class="o">&lt;-</span><span class="w"> </span><span class="nf">read.csv</span><span class="p">(</span><span class="n">file</span><span class="o">=</span><span class="s">&quot;../data/albatross_grow.csv&quot;</span><span class="p">)</span>
-<span class="n">alb</span><span class="w"> </span><span class="o">&lt;-</span><span class="w"> </span><span class="nf">subset</span><span class="p">(</span><span class="n">x</span><span class="o">=</span><span class="n">alb</span><span class="p">,</span><span class="w"> </span><span class="o">!</span><span class="nf">is.na</span><span class="p">(</span><span class="n">alb</span><span class="o">$</span><span class="n">wt</span><span class="p">))</span>
-<span class="nf">plot</span><span class="p">(</span><span class="n">alb</span><span class="o">$</span><span class="n">age</span><span class="p">,</span><span class="w"> </span><span class="n">alb</span><span class="o">$</span><span class="n">wt</span><span class="p">,</span><span class="w"> </span><span class="n">xlab</span><span class="o">=</span><span class="s">&quot;age (days)&quot;</span><span class="p">,</span><span class="w"> </span><span class="n">ylab</span><span class="o">=</span><span class="s">&quot;weight (g)&quot;</span><span class="p">,</span><span class="w"> </span><span class="n">xlim</span><span class="o">=</span><span class="nf">c</span><span class="p">(</span><span class="m">0</span><span class="p">,</span><span class="w"> </span><span class="m">100</span><span class="p">))</span>
-</pre></div>
-</div>
-</div>
-<div class="cell_output docutils container">
-<a class="reference internal image-reference" href="../_images/e16d450a550af34914299d36b906315861aec8037e6a2b9f81c94cca60a797d4.png"><img alt="../_images/e16d450a550af34914299d36b906315861aec8037e6a2b9f81c94cca60a797d4.png" src="../_images/e16d450a550af34914299d36b906315861aec8037e6a2b9f81c94cca60a797d4.png" style="width: 240px; height: 240px;" /></a>
-</div>
-</div>
-<section id="fitting-the-three-models-using-nlls">
-<h3>Fitting the three models using NLLS<a class="headerlink" href="#fitting-the-three-models-using-nlls" title="Link to this heading">#</a></h3>
-<p>Let’s fit multiple models to this dataset.</p>
-<p>The Von Bertalanffy model is commonly used for modelling the growth of an individual. It’s formulation is:</p>
-<div class="math notranslate nohighlight">
-\[
-W(t) = \rho (L_{\infty}(1-e^{-Kt})+L_0 e^{-Kt})^3
-\]</div>
-<p>If we pull out <span class="math notranslate nohighlight">\(L_{\infty}\)</span> and define <span class="math notranslate nohighlight">\(c=L_0/L_{\infty}\)</span> and <span class="math notranslate nohighlight">\(W_{\infty}=\rho L_{\infty}^3\)</span> this equation becomes:</p>
-<div class="math notranslate nohighlight">
-\[
-W(t) = W_{\infty}(1-e^{-Kt}+ c e^{-Kt})^3.
-\]</div>
-<p><span class="math notranslate nohighlight">\(W_{\infty}\)</span> is interpreted as the mean asymptotic weight, and <span class="math notranslate nohighlight">\(c\)</span> the ratio between the initial and final lengths. This second equation is the one we will fit.</p>
-<p>We will compare this model against the classical Logistic growth equation and a straight line.</p>
-<p>The logistic equation is:</p>
-<div class="math notranslate nohighlight">
-\[
-N_t = \frac{N_0 K e^{r t}}{K + N_0 (e^{r t} - 1)}
-\]</div>
-<p>Here <span class="math notranslate nohighlight">\(N_t\)</span> is population size at time <span class="math notranslate nohighlight">\(t\)</span>, <span class="math notranslate nohighlight">\(N_0\)</span> is initial population size, <span class="math notranslate nohighlight">\(r\)</span> is maximum growth rate (AKA <span class="math notranslate nohighlight">\(r_\text{max}\)</span>), and <span class="math notranslate nohighlight">\(K\)</span> is carrying capacity.</p>
-<p>First, as we did before, let’s define the R functions for the two models:</p>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-r notranslate"><div class="highlight"><pre><span></span><span class="n">logistic1</span><span class="w"> </span><span class="o">&lt;-</span><span class="w"> </span><span class="nf">function</span><span class="p">(</span><span class="n">t</span><span class="p">,</span><span class="w"> </span><span class="n">r</span><span class="p">,</span><span class="w"> </span><span class="n">K</span><span class="p">,</span><span class="w"> </span><span class="n">N0</span><span class="p">){</span>
-<span class="w"> </span><span class="n">N0</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">K</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="nf">exp</span><span class="p">(</span><span class="n">r</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">t</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">K</span><span class="o">+</span><span class="n">N0</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="p">(</span><span class="nf">exp</span><span class="p">(</span><span class="n">r</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">t</span><span class="p">)</span><span class="m">-1</span><span class="p">))</span>
-<span class="p">}</span>
-
-<span class="n">vonbert.w</span><span class="w"> </span><span class="o">&lt;-</span><span class="w"> </span><span class="nf">function</span><span class="p">(</span><span class="n">t</span><span class="p">,</span><span class="w"> </span><span class="n">Winf</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="p">,</span><span class="w"> </span><span class="n">K</span><span class="p">){</span>
-<span class="w"> </span><span class="n">Winf</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="p">(</span><span class="m">1</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nf">exp</span><span class="p">(</span><span class="o">-</span><span class="n">K</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">t</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="nf">exp</span><span class="p">(</span><span class="o">-</span><span class="n">K</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">t</span><span class="p">))</span><span class="o">^</span><span class="m">3</span>
-<span class="p">}</span>
-</pre></div>
-</div>
-</div>
-</div>
-<p>For the straight line, we use simply use R’s <code class="docutils literal notranslate"><span class="pre">lm()</span></code> function, as that is a linear least squares problem. Using NLLS will give (approximately) the same answer, of course. Now fit all 3 models using least squares.</p>
-<p>We will scale the data before fitting to improve the stability of the estimates:</p>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-r notranslate"><div class="highlight"><pre><span></span><span class="n">scale</span><span class="w"> </span><span class="o">&lt;-</span><span class="w"> </span><span class="m">4000</span>
-
-<span class="n">alb.lin</span><span class="w"> </span><span class="o">&lt;-</span><span class="w"> </span><span class="nf">lm</span><span class="p">(</span><span class="n">wt</span><span class="o">/</span><span class="n">scale</span><span class="w"> </span><span class="o">~</span><span class="w"> </span><span class="n">age</span><span class="p">,</span><span class="w"> </span><span class="n">data</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">alb</span><span class="p">)</span>
-</pre></div>
-</div>
-</div>
-</div>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-r notranslate"><div class="highlight"><pre><span></span><span class="n">alb.log</span><span class="w"> </span><span class="o">&lt;-</span><span class="w"> </span><span class="nf">nlsLM</span><span class="p">(</span><span class="n">wt</span><span class="o">/</span><span class="n">scale</span><span class="o">~</span><span class="nf">logistic1</span><span class="p">(</span><span class="n">age</span><span class="p">,</span><span class="w"> </span><span class="n">r</span><span class="p">,</span><span class="w"> </span><span class="n">K</span><span class="p">,</span><span class="w"> </span><span class="n">N0</span><span class="p">),</span><span class="w"> </span><span class="n">start</span><span class="o">=</span><span class="nf">list</span><span class="p">(</span><span class="n">K</span><span class="o">=</span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="n">r</span><span class="o">=</span><span class="m">0.1</span><span class="p">,</span><span class="w"> </span><span class="n">N0</span><span class="o">=</span><span class="m">0.1</span><span class="p">),</span><span class="w"> </span><span class="n">data</span><span class="o">=</span><span class="n">alb</span><span class="p">)</span>
-
-<span class="n">alb.vb</span><span class="w"> </span><span class="o">&lt;-</span><span class="w"> </span><span class="nf">nlsLM</span><span class="p">(</span><span class="n">wt</span><span class="o">/</span><span class="n">scale</span><span class="o">~</span><span class="nf">vonbert.w</span><span class="p">(</span><span class="n">age</span><span class="p">,</span><span class="w"> </span><span class="n">Winf</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="p">,</span><span class="w"> </span><span class="n">K</span><span class="p">),</span><span class="w"> </span><span class="n">start</span><span class="o">=</span><span class="nf">list</span><span class="p">(</span><span class="n">Winf</span><span class="o">=</span><span class="m">0.75</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="o">=</span><span class="m">0.01</span><span class="p">,</span><span class="w"> </span><span class="n">K</span><span class="o">=</span><span class="m">0.01</span><span class="p">),</span><span class="w"> </span><span class="n">data</span><span class="o">=</span><span class="n">alb</span><span class="p">)</span>
-</pre></div>
-</div>
-</div>
-</div>
-<p>Next let’s calculate predictions for each of the models across a range of ages.</p>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-r notranslate"><div class="highlight"><pre><span></span><span class="n">ages</span><span class="w"> </span><span class="o">&lt;-</span><span class="w"> </span><span class="nf">seq</span><span class="p">(</span><span class="m">0</span><span class="p">,</span><span class="w"> </span><span class="m">100</span><span class="p">,</span><span class="w"> </span><span class="n">length</span><span class="o">=</span><span class="m">1000</span><span class="p">)</span>
-
-<span class="n">pred.lin</span><span class="w"> </span><span class="o">&lt;-</span><span class="w"> </span><span class="nf">predict</span><span class="p">(</span><span class="n">alb.lin</span><span class="p">,</span><span class="w"> </span><span class="n">newdata</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nf">list</span><span class="p">(</span><span class="n">age</span><span class="o">=</span><span class="n">ages</span><span class="p">))</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">scale</span>
-
-<span class="n">pred.log</span><span class="w"> </span><span class="o">&lt;-</span><span class="w"> </span><span class="nf">predict</span><span class="p">(</span><span class="n">alb.log</span><span class="p">,</span><span class="w"> </span><span class="n">newdata</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nf">list</span><span class="p">(</span><span class="n">age</span><span class="o">=</span><span class="n">ages</span><span class="p">))</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">scale</span>
-
-<span class="n">pred.vb</span><span class="w"> </span><span class="o">&lt;-</span><span class="w"> </span><span class="nf">predict</span><span class="p">(</span><span class="n">alb.vb</span><span class="p">,</span><span class="w"> </span><span class="n">newdata</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nf">list</span><span class="p">(</span><span class="n">age</span><span class="o">=</span><span class="n">ages</span><span class="p">))</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">scale</span>
-</pre></div>
-</div>
-</div>
-</div>
-<p>And finally plot the data with the fits:</p>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-r notranslate"><div class="highlight"><pre><span></span><span class="nf">plot</span><span class="p">(</span><span class="n">alb</span><span class="o">$</span><span class="n">age</span><span class="p">,</span><span class="w"> </span><span class="n">alb</span><span class="o">$</span><span class="n">wt</span><span class="p">,</span><span class="w"> </span><span class="n">xlab</span><span class="o">=</span><span class="s">&quot;age (days)&quot;</span><span class="p">,</span><span class="w"> </span><span class="n">ylab</span><span class="o">=</span><span class="s">&quot;weight (g)&quot;</span><span class="p">,</span><span class="w"> </span><span class="n">xlim</span><span class="o">=</span><span class="nf">c</span><span class="p">(</span><span class="m">0</span><span class="p">,</span><span class="m">100</span><span class="p">))</span>
-<span class="nf">lines</span><span class="p">(</span><span class="n">ages</span><span class="p">,</span><span class="w"> </span><span class="n">pred.lin</span><span class="p">,</span><span class="w"> </span><span class="n">col</span><span class="o">=</span><span class="m">2</span><span class="p">,</span><span class="w"> </span><span class="n">lwd</span><span class="o">=</span><span class="m">2</span><span class="p">)</span>
-<span class="nf">lines</span><span class="p">(</span><span class="n">ages</span><span class="p">,</span><span class="w"> </span><span class="n">pred.log</span><span class="p">,</span><span class="w"> </span><span class="n">col</span><span class="o">=</span><span class="m">3</span><span class="p">,</span><span class="w"> </span><span class="n">lwd</span><span class="o">=</span><span class="m">2</span><span class="p">)</span>
-<span class="nf">lines</span><span class="p">(</span><span class="n">ages</span><span class="p">,</span><span class="w"> </span><span class="n">pred.vb</span><span class="p">,</span><span class="w"> </span><span class="n">col</span><span class="o">=</span><span class="m">4</span><span class="p">,</span><span class="w"> </span><span class="n">lwd</span><span class="o">=</span><span class="m">2</span><span class="p">)</span>
-
-<span class="nf">legend</span><span class="p">(</span><span class="s">&quot;topleft&quot;</span><span class="p">,</span><span class="w"> </span><span class="n">legend</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nf">c</span><span class="p">(</span><span class="s">&quot;linear&quot;</span><span class="p">,</span><span class="w"> </span><span class="s">&quot;logistic&quot;</span><span class="p">,</span><span class="w"> </span><span class="s">&quot;Von Bert&quot;</span><span class="p">),</span><span class="w"> </span><span class="n">lwd</span><span class="o">=</span><span class="m">2</span><span class="p">,</span><span class="w"> </span><span class="n">lty</span><span class="o">=</span><span class="m">1</span><span class="p">,</span><span class="w"> </span><span class="n">col</span><span class="o">=</span><span class="m">2</span><span class="o">:</span><span class="m">4</span><span class="p">)</span>
-</pre></div>
-</div>
-</div>
-<div class="cell_output docutils container">
-<a class="reference internal image-reference" href="../_images/e07fedb57f96ba1b189dc08a6174b7d077c9712ccf807124dc76121881e3b188.png"><img alt="../_images/e07fedb57f96ba1b189dc08a6174b7d077c9712ccf807124dc76121881e3b188.png" src="../_images/e07fedb57f96ba1b189dc08a6174b7d077c9712ccf807124dc76121881e3b188.png" style="width: 240px; height: 240px;" /></a>
-</div>
-</div>
-<p>Next examine the residuals between the 3 models:</p>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-r notranslate"><div class="highlight"><pre><span></span><span class="nf">par</span><span class="p">(</span><span class="n">mfrow</span><span class="o">=</span><span class="nf">c</span><span class="p">(</span><span class="m">3</span><span class="p">,</span><span class="m">1</span><span class="p">),</span><span class="w"> </span><span class="n">bty</span><span class="o">=</span><span class="s">&quot;n&quot;</span><span class="p">)</span>
-<span class="nf">plot</span><span class="p">(</span><span class="n">alb</span><span class="o">$</span><span class="n">age</span><span class="p">,</span><span class="w"> </span><span class="nf">resid</span><span class="p">(</span><span class="n">alb.lin</span><span class="p">),</span><span class="w"> </span><span class="n">main</span><span class="o">=</span><span class="s">&quot;LM resids&quot;</span><span class="p">,</span><span class="w"> </span><span class="n">xlim</span><span class="o">=</span><span class="nf">c</span><span class="p">(</span><span class="m">0</span><span class="p">,</span><span class="m">100</span><span class="p">))</span>
-<span class="nf">plot</span><span class="p">(</span><span class="n">alb</span><span class="o">$</span><span class="n">age</span><span class="p">,</span><span class="w"> </span><span class="nf">resid</span><span class="p">(</span><span class="n">alb.log</span><span class="p">),</span><span class="w"> </span><span class="n">main</span><span class="o">=</span><span class="s">&quot;Logisitic resids&quot;</span><span class="p">,</span><span class="w"> </span><span class="n">xlim</span><span class="o">=</span><span class="nf">c</span><span class="p">(</span><span class="m">0</span><span class="p">,</span><span class="m">100</span><span class="p">))</span>
-<span class="nf">plot</span><span class="p">(</span><span class="n">alb</span><span class="o">$</span><span class="n">age</span><span class="p">,</span><span class="w"> </span><span class="nf">resid</span><span class="p">(</span><span class="n">alb.vb</span><span class="p">),</span><span class="w"> </span><span class="n">main</span><span class="o">=</span><span class="s">&quot;VB resids&quot;</span><span class="p">,</span><span class="w"> </span><span class="n">xlim</span><span class="o">=</span><span class="nf">c</span><span class="p">(</span><span class="m">0</span><span class="p">,</span><span class="m">100</span><span class="p">))</span>
-</pre></div>
-</div>
-</div>
-<div class="cell_output docutils container">
-<a class="reference internal image-reference" href="../_images/74a99a960d28e3b168a641f1b3a4429c3f6a7f96f52456fdeca8bd8f624d9954.png"><img alt="../_images/74a99a960d28e3b168a641f1b3a4429c3f6a7f96f52456fdeca8bd8f624d9954.png" src="../_images/74a99a960d28e3b168a641f1b3a4429c3f6a7f96f52456fdeca8bd8f624d9954.png" style="width: 240px; height: 240px;" /></a>
-</div>
-</div>
-<p>The residuals for all 3 models still exhibit some patterns. In particular, the data seems to go down near the end of the observation period, but none of these models can capture that behavior.</p>
-<p>Finally, let’s compare the 3 models using a simpler approach than the AIC/BIC one that we used <a class="reference internal" href="#Allom_Exercises"><span class="xref myst">above</span></a> by calculating adjusted Sums of Squared Errors (SSE’s):</p>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-r notranslate"><div class="highlight"><pre><span></span><span class="n">n</span><span class="w"> </span><span class="o">&lt;-</span><span class="w"> </span><span class="nf">length</span><span class="p">(</span><span class="n">alb</span><span class="o">$</span><span class="n">wt</span><span class="p">)</span>
-<span class="nf">list</span><span class="p">(</span><span class="n">lin</span><span class="o">=</span><span class="nf">signif</span><span class="p">(</span><span class="nf">sum</span><span class="p">(</span><span class="nf">resid</span><span class="p">(</span><span class="n">alb.lin</span><span class="p">)</span><span class="o">^</span><span class="m">2</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">n</span><span class="m">-2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="m">2</span><span class="p">),</span><span class="w"> </span><span class="m">3</span><span class="p">),</span><span class="w"> </span>
-<span class="w"> </span><span class="n">log</span><span class="o">=</span><span class="w"> </span><span class="nf">signif</span><span class="p">(</span><span class="nf">sum</span><span class="p">(</span><span class="nf">resid</span><span class="p">(</span><span class="n">alb.log</span><span class="p">)</span><span class="o">^</span><span class="m">2</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">n</span><span class="m">-2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="m">3</span><span class="p">),</span><span class="w"> </span><span class="m">3</span><span class="p">),</span><span class="w"> </span>
-<span class="w"> </span><span class="n">vb</span><span class="o">=</span><span class="w"> </span><span class="nf">signif</span><span class="p">(</span><span class="nf">sum</span><span class="p">(</span><span class="nf">resid</span><span class="p">(</span><span class="n">alb.vb</span><span class="p">)</span><span class="o">^</span><span class="m">2</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">n</span><span class="m">-2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="m">3</span><span class="p">),</span><span class="w"> </span><span class="m">3</span><span class="p">))</span><span class="w">   </span>
-</pre></div>
-</div>
-</div>
-<div class="cell_output docutils container">
-<div class="output text_html"><dl>
-	<dt>$lin</dt>
-		<dd>0.00958</dd>
-	<dt>$log</dt>
-		<dd>0.0056</dd>
-	<dt>$vb</dt>
-		<dd>0.00628</dd>
-</dl>
-</div></div>
-</div>
-<p>The adjusted SSE accounts for sample size and number of parameters by dividing the RSS by the residual degrees of freedom. Adjusted SSE can also be used for model selection like AIC/BIC (but is less robust than AIC/BIC). The residual degrees of freedom is calculated as the number of response values (sample size, <span class="math notranslate nohighlight">\(n\)</span>) minus 2, times the number of fitted coefficients <span class="math notranslate nohighlight">\(m\)</span> (= 2 or 3 in this case) estimated.</p>
-<p>The logistic model has the lowest adjusted SSE, so it’s the best by this measure. It is also, visually, a better fit.</p>
-</section>
-<section id="id3">
-<h3>Exercises <a id='Albatross_Exercises'></a><a class="headerlink" href="#id3" title="Link to this heading">#</a></h3>
-<p>(a) Use AIC/BIC to perform model selection on the Albatross data as we did for the trait allometry example.</p>
-<p>(b) Write this example as a self-sufficient R script, with ggplot istead of base plotting</p>
-</section>
-</section>
-<section id="aedes-aegypti-fecundity">
-<h2>Aedes aegypti fecundity<a class="headerlink" href="#aedes-aegypti-fecundity" title="Link to this heading">#</a></h2>
-<p>Now let’s look at a disease vector example. These data measure the reponse of * Aedes aegypti * fecundity to temperature.</p>
-<p>First load and visualize the data:</p>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-r notranslate"><div class="highlight"><pre><span></span><span class="n">aedes</span><span class="w"> </span><span class="o">&lt;-</span><span class="w"> </span><span class="nf">read.csv</span><span class="p">(</span><span class="n">file</span><span class="o">=</span><span class="s">&quot;../data/aedes_fecund.csv&quot;</span><span class="p">)</span>
-
-<span class="nf">plot</span><span class="p">(</span><span class="n">aedes</span><span class="o">$</span><span class="bp">T</span><span class="p">,</span><span class="w"> </span><span class="n">aedes</span><span class="o">$</span><span class="n">EFD</span><span class="p">,</span><span class="w"> </span><span class="n">xlab</span><span class="o">=</span><span class="s">&quot;temperature (C)&quot;</span><span class="p">,</span><span class="w"> </span><span class="n">ylab</span><span class="o">=</span><span class="s">&quot;Eggs/day&quot;</span><span class="p">)</span>
-</pre></div>
-</div>
-</div>
-<div class="cell_output docutils container">
-<a class="reference internal image-reference" href="../_images/607fd57884a48ff887e971fbd1fa9514522df46a71bb50f7d17949bccf7ae2ae.png"><img alt="../_images/607fd57884a48ff887e971fbd1fa9514522df46a71bb50f7d17949bccf7ae2ae.png" src="../_images/607fd57884a48ff887e971fbd1fa9514522df46a71bb50f7d17949bccf7ae2ae.png" style="width: 240px; height: 240px;" /></a>
-</div>
-</div>
-<section id="the-thermal-performance-curve-models">
-<span id="model-fitting-nlls-tpcs"></span><h3>The Thermal Performance Curve models<a class="headerlink" href="#the-thermal-performance-curve-models" title="Link to this heading">#</a></h3>
-<p>Let’s define some models for Thermal Performance Curves:</p>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-r notranslate"><div class="highlight"><pre><span></span><span class="n">quad1</span><span class="w"> </span><span class="o">&lt;-</span><span class="w"> </span><span class="nf">function</span><span class="p">(</span><span class="bp">T</span><span class="p">,</span><span class="w"> </span><span class="n">T0</span><span class="p">,</span><span class="w"> </span><span class="n">Tm</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="p">){</span>
-<span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="p">(</span><span class="bp">T</span><span class="o">-</span><span class="n">T0</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="p">(</span><span class="bp">T</span><span class="o">-</span><span class="n">Tm</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="nf">as.numeric</span><span class="p">(</span><span class="bp">T</span><span class="o">&lt;</span><span class="n">Tm</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="nf">as.numeric</span><span class="p">(</span><span class="bp">T</span><span class="o">&gt;</span><span class="n">T0</span><span class="p">)</span>
-<span class="p">}</span>
-</pre></div>
-</div>
-</div>
-</div>
-<p>Instead of using the inbuilt quadratic function in R, we we defined our own to make it easier to choose starting values, and so that we can force the function to be equal to zero above and below the minimum and maximum temperature thresholds (more on this below).</p>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-r notranslate"><div class="highlight"><pre><span></span><span class="n">briere</span><span class="w"> </span><span class="o">&lt;-</span><span class="w"> </span><span class="nf">function</span><span class="p">(</span><span class="bp">T</span><span class="p">,</span><span class="w"> </span><span class="n">T0</span><span class="p">,</span><span class="w"> </span><span class="n">Tm</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="p">){</span>
-<span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="bp">T</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="p">(</span><span class="bp">T</span><span class="o">-</span><span class="n">T0</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="p">(</span><span class="nf">abs</span><span class="p">(</span><span class="n">Tm</span><span class="o">-</span><span class="bp">T</span><span class="p">)</span><span class="o">^</span><span class="p">(</span><span class="m">1</span><span class="o">/</span><span class="m">2</span><span class="p">))</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="nf">as.numeric</span><span class="p">(</span><span class="bp">T</span><span class="o">&lt;</span><span class="n">Tm</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="nf">as.numeric</span><span class="p">(</span><span class="bp">T</span><span class="o">&gt;</span><span class="n">T0</span><span class="p">)</span>
-<span class="p">}</span>
-</pre></div>
-</div>
-</div>
-</div>
-<p>The Briere function is a commonly used model for temperature dependence of insect traits. See here <a class="reference internal" href="Appendix-MiniProj.html#miniproj-tpcs-models"><span class="std std-ref">section</span></a> for more info. Unlike the original <a class="reference internal" href="Appendix-MiniProj.html#miniproj-tpcs-models"><span class="std std-ref">model definition</span></a>, we have used <code class="docutils literal notranslate"><span class="pre">abs()</span></code> to allow the NLLS algorithm to explore the full parameter space of <span class="math notranslate nohighlight">\(T_m\)</span>; if we did not do this, the NLLS could fail as soon as a value of <span class="math notranslate nohighlight">\(T_m &lt; T\)</span> was reached during the optimization, because the <a class="reference external" href="https://en.wikipedia.org/wiki/Square_root">square root of a negative number is complex</a>. Another way to deal with this issue is to set parameter bounds on <span class="math notranslate nohighlight">\(T_m\)</span> so that it is can never be less than T. However, this is a more technical approach that we will not go into here.</p>
-<p>As in the case of the albatross growth data, we will also compare the above two models with a * straight line * (again, its a linear model, so we can just use <code class="docutils literal notranslate"><span class="pre">lm()</span></code> without needing to define a function for it).</p>
-<p>Now fit all 3 models using least squares. Although it’s not as necessary here (as the data don’t have as large values as the albatross example), lets again scale the data first:</p>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-r notranslate"><div class="highlight"><pre><span></span><span class="n">scale</span><span class="w"> </span><span class="o">&lt;-</span><span class="w"> </span><span class="m">20</span>
-
-<span class="n">aed.lin</span><span class="w"> </span><span class="o">&lt;-</span><span class="w"> </span><span class="nf">lm</span><span class="p">(</span><span class="n">EFD</span><span class="o">/</span><span class="n">scale</span><span class="w"> </span><span class="o">~</span><span class="w"> </span><span class="bp">T</span><span class="p">,</span><span class="w"> </span><span class="n">data</span><span class="o">=</span><span class="n">aedes</span><span class="p">)</span>
-
-<span class="n">aed.quad</span><span class="w"> </span><span class="o">&lt;-</span><span class="w"> </span><span class="nf">nlsLM</span><span class="p">(</span><span class="n">EFD</span><span class="o">/</span><span class="n">scale</span><span class="o">~</span><span class="nf">quad1</span><span class="p">(</span><span class="bp">T</span><span class="p">,</span><span class="w"> </span><span class="n">T0</span><span class="p">,</span><span class="w"> </span><span class="n">Tm</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="p">),</span><span class="w"> </span><span class="n">start</span><span class="o">=</span><span class="nf">list</span><span class="p">(</span><span class="n">T0</span><span class="o">=</span><span class="m">10</span><span class="p">,</span><span class="w"> </span><span class="n">Tm</span><span class="o">=</span><span class="m">40</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="o">=</span><span class="m">0.01</span><span class="p">),</span><span class="w"> </span><span class="n">data</span><span class="o">=</span><span class="n">aedes</span><span class="p">)</span>
-
-<span class="n">aed.br</span><span class="w"> </span><span class="o">&lt;-</span><span class="w"> </span><span class="nf">nlsLM</span><span class="p">(</span><span class="n">EFD</span><span class="o">/</span><span class="n">scale</span><span class="o">~</span><span class="nf">briere</span><span class="p">(</span><span class="bp">T</span><span class="p">,</span><span class="w"> </span><span class="n">T0</span><span class="p">,</span><span class="w"> </span><span class="n">Tm</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="p">),</span><span class="w"> </span><span class="n">start</span><span class="o">=</span><span class="nf">list</span><span class="p">(</span><span class="n">T0</span><span class="o">=</span><span class="m">10</span><span class="p">,</span><span class="w"> </span><span class="n">Tm</span><span class="o">=</span><span class="m">40</span><span class="p">,</span><span class="w"> </span><span class="n">c</span><span class="o">=</span><span class="m">0.1</span><span class="p">),</span><span class="w"> </span><span class="n">data</span><span class="o">=</span><span class="n">aedes</span><span class="p">)</span>
-</pre></div>
-</div>
-</div>
-</div>
-</section>
-<section id="id4">
-<h3>Exercises <a id='Aedes_Exercises'></a><a class="headerlink" href="#id4" title="Link to this heading">#</a></h3>
-<p>(a) Complete the * Aedes * data analysis by fitting the models, calculating predictions and then comparing models. Write a single, self-standing script for it. Which model fits best? By what measure?</p>
-<p>(b) In this script, use ggplot instead of base plotting.</p>
-</section>
-</section>
-<section id="abundance-data-as-an-example">
-<h2>Abundance data as an example<a class="headerlink" href="#abundance-data-as-an-example" title="Link to this heading">#</a></h2>
-<p>Fluctuations in the abundance (density) of single populations may play a crucial role in ecosystem dynamics and emergent functional characteristics, such as rates of carbon fixation or disease transmission. For example, if vector population densities or their traits change at the same or shorter timescales than the rate of disease transmission, then (vector) abundance fluctuations can cause significant fluctuations in disease transmission rates. Indeed, most disease vectors are small ectotherms with short generation times and greater sensitivity to environmental conditions than their (invariably larger, longer-lived, and often, endothermic) hosts. So understanding how populations vary over time, space, and with respect to environmental variables such as temperature and precipitation is key. We will look at fitting models to the growth of a single population here.</p>
-</section>
-<section id="population-growth-rates">
-<span id="model-fitting-r-population-growth"></span><h2>Population growth rates<a class="headerlink" href="#population-growth-rates" title="Link to this heading">#</a></h2>
-<p>A population grows exponentially while its abundance is low and resources are not limiting (the Malthusian principle). This growth then slows and eventually stops as resources become limiting. There may also be a time lag before the population growth really takes off at the start. We will focus on microbial (specifically, bacterial) growth rates. Bacterial growth in batch culture follows a distinct set of phases; lag phase, exponential phase and stationary phase. During the lag phase a suite of transcriptional machinery is activated, including genes involved in nutrient uptake and metabolic changes, as bacteria prepare for growth. During the exponential growth phase, bacteria divide at a constant rate, the population doubling with each generation. When the carrying capacity of the media is reached, growth slows and the number of cells in the culture stabilizes, beginning the stationary phase.</p>
-<p>Traditionally, microbial growth rates were measured by plotting cell numbers or culture density against time on a semi-log graph and fitting a straight line through the exponential growth phase – the slope of the line gives the maximum growth rate (<span class="math notranslate nohighlight">\(r_{max}\)</span>). Models have since been developed which we can use to describe the whole sigmoidal bacterial growth curve (e.g., using NLLS). Here we will take a look at these different approaches, from applying linear models to the exponential phase, through to fitting non-linear models to the full growth curve.</p>
-<p>Let’s first generate some “data” on the number of bacterial cells as a function of time that we can play with:</p>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-r notranslate"><div class="highlight"><pre><span></span><span class="n">t</span><span class="w"> </span><span class="o">&lt;-</span><span class="w"> </span><span class="nf">seq</span><span class="p">(</span><span class="m">0</span><span class="p">,</span><span class="w"> </span><span class="m">22</span><span class="p">,</span><span class="w"> </span><span class="m">2</span><span class="p">)</span>
-<span class="n">N</span><span class="w"> </span><span class="o">&lt;-</span><span class="w"> </span><span class="nf">c</span><span class="p">(</span><span class="m">32500</span><span class="p">,</span><span class="w"> </span><span class="m">33000</span><span class="p">,</span><span class="w"> </span><span class="m">38000</span><span class="p">,</span><span class="w"> </span><span class="m">105000</span><span class="p">,</span><span class="w"> </span><span class="m">445000</span><span class="p">,</span><span class="w"> </span><span class="m">1430000</span><span class="p">,</span><span class="w"> </span><span class="m">3020000</span><span class="p">,</span><span class="w"> </span><span class="m">4720000</span><span class="p">,</span><span class="w"> </span><span class="m">5670000</span><span class="p">,</span><span class="w"> </span><span class="m">5870000</span><span class="p">,</span><span class="w"> </span><span class="m">5930000</span><span class="p">,</span><span class="w"> </span><span class="m">5940000</span><span class="p">)</span>
-
-<span class="nf">set.seed</span><span class="p">(</span><span class="m">1234</span><span class="p">)</span><span class="w"> </span><span class="c1"># To ensure we always get the same random sequence in this example &quot;dataset&quot; </span>
-
-<span class="n">data</span><span class="w"> </span><span class="o">&lt;-</span><span class="w"> </span><span class="nf">data.frame</span><span class="p">(</span><span class="n">t</span><span class="p">,</span><span class="w"> </span><span class="n">N</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="p">(</span><span class="m">1</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nf">rnorm</span><span class="p">(</span><span class="nf">length</span><span class="p">(</span><span class="n">t</span><span class="p">),</span><span class="w"> </span><span class="n">sd</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">0.1</span><span class="p">)))</span><span class="w"> </span><span class="c1"># add some random error</span>
-
-<span class="nf">names</span><span class="p">(</span><span class="n">data</span><span class="p">)</span><span class="w"> </span><span class="o">&lt;-</span><span class="w"> </span><span class="nf">c</span><span class="p">(</span><span class="s">&quot;Time&quot;</span><span class="p">,</span><span class="w"> </span><span class="s">&quot;N&quot;</span><span class="p">)</span>
-
-<span class="nf">head</span><span class="p">(</span><span class="n">data</span><span class="p">)</span>
-</pre></div>
-</div>
-</div>
-<div class="cell_output docutils container">
-<div class="output text_html"><table class="dataframe">
-<caption>A data.frame: 6 × 2</caption>
-<thead>
-	<tr><th></th><th scope=col>Time</th><th scope=col>N</th></tr>
-	<tr><th></th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th></tr>
-</thead>
-<tbody>
-	<tr><th scope=row>1</th><td> 0</td><td>  28577.04</td></tr>
-	<tr><th scope=row>2</th><td> 2</td><td>  33915.52</td></tr>
-	<tr><th scope=row>3</th><td> 4</td><td>  42120.88</td></tr>
-	<tr><th scope=row>4</th><td> 6</td><td>  80370.17</td></tr>
-	<tr><th scope=row>5</th><td> 8</td><td> 464096.05</td></tr>
-	<tr><th scope=row>6</th><td>10</td><td>1502365.99</td></tr>
-</tbody>
-</table>
-</div></div>
-</div>
-<p>Note how we added some random “sampling” error using <code class="docutils literal notranslate"><span class="pre">N</span> <span class="pre">*</span> <span class="pre">(1</span> <span class="pre">+</span> <span class="pre">rnorm(length(t),</span> <span class="pre">sd</span> <span class="pre">=</span> <span class="pre">.1))</span></code>.</p>
-<p>This means that we are adding an error at each time point <span class="math notranslate nohighlight">\(t\)</span> (let’s call this fluctuation <span class="math notranslate nohighlight">\(\epsilon_t\)</span>) as a <em>percentage</em> of the population (<span class="math notranslate nohighlight">\(N_t\)</span>) at that time point in a vectorized way. That is, we are performing the operation <span class="math notranslate nohighlight">\(N_t \times (1 + \epsilon_t)\)</span> at all time points at one go. This is important to note because this is often the way that errors appear – proportional to the value being measured.</p>
-<p>Now let’s plot these data:</p>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-r notranslate"><div class="highlight"><pre><span></span><span class="nf">ggplot</span><span class="p">(</span><span class="n">data</span><span class="p">,</span><span class="w"> </span><span class="nf">aes</span><span class="p">(</span><span class="n">x</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Time</span><span class="p">,</span><span class="w"> </span><span class="n">y</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">N</span><span class="p">))</span><span class="w"> </span><span class="o">+</span><span class="w"> </span>
-<span class="w"> </span><span class="nf">geom_point</span><span class="p">(</span><span class="n">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">3</span><span class="p">)</span><span class="w"> </span><span class="o">+</span>
-<span class="w"> </span><span class="nf">labs</span><span class="p">(</span><span class="n">x</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s">&quot;Time (Hours)&quot;</span><span class="p">,</span><span class="w"> </span><span class="n">y</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s">&quot;Population size (cells)&quot;</span><span class="p">)</span>
-</pre></div>
-</div>
-</div>
-<div class="cell_output docutils container">
-<a class="reference internal image-reference" href="../_images/4ae8e201b47c8f68aca1aa6a0fb4b1cca033788530e60a72200bc6b61a3974b7.png"><img alt="../_images/4ae8e201b47c8f68aca1aa6a0fb4b1cca033788530e60a72200bc6b61a3974b7.png" src="../_images/4ae8e201b47c8f68aca1aa6a0fb4b1cca033788530e60a72200bc6b61a3974b7.png" style="width: 240px; height: 240px;" /></a>
-</div>
-</div>
-<section id="basic-approach">
-<h3>Basic approach<a class="headerlink" href="#basic-approach" title="Link to this heading">#</a></h3>
-<p>The size of an exponentially growing population (<span class="math notranslate nohighlight">\(N\)</span>) at any given time (<span class="math notranslate nohighlight">\(t\)</span>) is given by:</p>
-<p><span class="math notranslate nohighlight">\(
-N(t) = N_0 e^{rt} ,
-\)</span></p>
-<p>where <span class="math notranslate nohighlight">\(N_0\)</span> is the initial population size and <span class="math notranslate nohighlight">\(r\)</span> is the growth rate. We can re-arrange this to give:</p>
-<p><span class="math notranslate nohighlight">\(
-r = \frac{\log(N(t)) - \log(N_0)}{t} ,
-\)</span></p>
-<p>That is, in exponential growth at a constant rate, the growth rate can be simply calculated as the difference in the log of two population sizes, over time. We will log-transform the data and estimate by eye where growth looks exponential.</p>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-r notranslate"><div class="highlight"><pre><span></span><span class="n">data</span><span class="o">$</span><span class="n">LogN</span><span class="w"> </span><span class="o">&lt;-</span><span class="w"> </span><span class="nf">log</span><span class="p">(</span><span class="n">data</span><span class="o">$</span><span class="n">N</span><span class="p">)</span>
-
-<span class="c1"># visualise</span>
-<span class="nf">ggplot</span><span class="p">(</span><span class="n">data</span><span class="p">,</span><span class="w"> </span><span class="nf">aes</span><span class="p">(</span><span class="n">x</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">t</span><span class="p">,</span><span class="w"> </span><span class="n">y</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">LogN</span><span class="p">))</span><span class="w"> </span><span class="o">+</span><span class="w"> </span>
-<span class="w"> </span><span class="nf">geom_point</span><span class="p">(</span><span class="n">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">3</span><span class="p">)</span><span class="w"> </span><span class="o">+</span>
-<span class="w"> </span><span class="nf">labs</span><span class="p">(</span><span class="n">x</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s">&quot;Time (Hours)&quot;</span><span class="p">,</span><span class="w"> </span><span class="n">y</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s">&quot;log(cell number)&quot;</span><span class="p">)</span>
-</pre></div>
-</div>
-</div>
-<div class="cell_output docutils container">
-<a class="reference internal image-reference" href="../_images/540dbe7279dd824ca2e1d12e0fde94cc96e35c386c83d1eec23cb841cc09bfc0.png"><img alt="../_images/540dbe7279dd824ca2e1d12e0fde94cc96e35c386c83d1eec23cb841cc09bfc0.png" src="../_images/540dbe7279dd824ca2e1d12e0fde94cc96e35c386c83d1eec23cb841cc09bfc0.png" style="width: 240px; height: 240px;" /></a>
-</div>
-</div>
-<p>By eye the logged data looks fairly linear (beyond the initial “lag phase” of growth; see below) between hours 5 and 10, so we’ll use that time-period to calculate the growth rate.</p>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-r notranslate"><div class="highlight"><pre><span></span><span class="p">(</span><span class="n">data</span><span class="p">[</span><span class="n">data</span><span class="o">$</span><span class="n">Time</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">10</span><span class="p">,]</span><span class="o">$</span><span class="n">LogN</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">data</span><span class="p">[</span><span class="n">data</span><span class="o">$</span><span class="n">Time</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="m">6</span><span class="p">,]</span><span class="o">$</span><span class="n">LogN</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="m">10-6</span><span class="p">)</span>
-</pre></div>
-</div>
-</div>
-<div class="cell_output docutils container">
-<div class="output text_html">0.732038333517017</div></div>
-</div>
-<p>This is our first, most basic estimate of <span class="math notranslate nohighlight">\(r\)</span>.</p>
-<p>Or, we can decide not to eyeball the data, but just pick the maximum observed gradient of the curve. For this, we can use the the <code class="docutils literal notranslate"><span class="pre">diff()</span></code> function:</p>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-r notranslate"><div class="highlight"><pre><span></span><span class="nf">diff</span><span class="p">(</span><span class="n">data</span><span class="o">$</span><span class="n">LogN</span><span class="p">)</span>
-</pre></div>
-</div>
-</div>
-<div class="cell_output docutils container">
-<div class="output text_html"><style>
-.list-inline {list-style: none; margin:0; padding: 0}
-.list-inline>li {display: inline-block}
-.list-inline>li:not(:last-child)::after {content: "\00b7"; padding: 0 .5ex}
-</style>
-<ol class=list-inline><li>0.171269154259665</li><li>0.216670872460636</li><li>0.646099642770272</li><li>1.75344839347772</li><li>1.17470494059035</li><li>0.639023867964838</li><li>0.44952974020198</li><li>0.181493481601755</li><li>-0.000450183952025895</li><li>0.0544907101941003</li><li>-0.0546009242768832</li></ol>
-</div></div>
-</div>
-<p>This gives all the (log) population size differences between successive timepoint pairs. The max of this is what we want, divided by the time-step.</p>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-r notranslate"><div class="highlight"><pre><span></span><span class="nf">max</span><span class="p">(</span><span class="nf">diff</span><span class="p">(</span><span class="n">data</span><span class="o">$</span><span class="n">LogN</span><span class="p">))</span><span class="o">/</span><span class="m">2</span><span class="w"> </span><span class="c1"># 2 is the difference in any successive pair of timepoints</span>
-</pre></div>
-</div>
-</div>
-<div class="cell_output docutils container">
-<div class="output text_html">0.87672419673886</div></div>
-</div>
-</section>
-<section id="using-ols">
-<h3>Using OLS<a class="headerlink" href="#using-ols" title="Link to this heading">#</a></h3>
-<p>But we can do better than this. To account for some error in measurement, we shouldn’t really take the data points directly, but fit a linear model through them instead, where the slope gives our growth rate. This is pretty much the “traditional” way of calculating microbial growth rates – draw a straight line through the linear part of the log-transformed data.</p>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-r notranslate"><div class="highlight"><pre><span></span><span class="n">lm_growth</span><span class="w"> </span><span class="o">&lt;-</span><span class="w"> </span><span class="nf">lm</span><span class="p">(</span><span class="n">LogN</span><span class="w"> </span><span class="o">~</span><span class="w"> </span><span class="n">Time</span><span class="p">,</span><span class="w"> </span><span class="n">data</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">data</span><span class="p">[</span><span class="n">data</span><span class="o">$</span><span class="n">Time</span><span class="w"> </span><span class="o">&gt;</span><span class="w"> </span><span class="m">2</span><span class="w"> </span><span class="o">&amp;</span><span class="w"> </span><span class="n">data</span><span class="o">$</span><span class="n">Time</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="m">12</span><span class="p">,])</span>
-<span class="nf">summary</span><span class="p">(</span><span class="n">lm_growth</span><span class="p">)</span>
-</pre></div>
-</div>
-</div>
-<div class="cell_output docutils container">
-<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Call:
-lm(formula = LogN ~ Time, data = data[data$Time &gt; 2 &amp; data$Time &lt; 
-    12, ])
-
-Residuals:
-       3        4        5        6 
- 0.21646 -0.38507  0.12076  0.04785 
-
-Coefficients:
-            Estimate Std. Error t value Pr(&gt;|t|)   
-(Intercept)   7.9366     0.5350  14.835  0.00451 **
-Time          0.6238     0.0728   8.569  0.01335 * 
----
-Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
-
-Residual standard error: 0.3256 on 2 degrees of freedom
-Multiple R-squared:  0.9735,	Adjusted R-squared:  0.9602 
-F-statistic: 73.42 on 1 and 2 DF,  p-value: 0.01335
-</pre></div>
-</div>
-</div>
-</div>
-<p>Npw we get <span class="math notranslate nohighlight">\(r \approx 0.62\)</span>, which is probably closer to the “truth”.</p>
-<p>But this is still not ideal because we only guessed the exponential phase by eye. We could do it better by iterating through different windows of points, comparing the slopes and finding which the highest is to give the maximum growth rate, <span class="math notranslate nohighlight">\(r_{max}\)</span>. This is called a “rolling regression”.</p>
-<p>Or better still, we can fit a more appropriate mathematical model using NLLS!</p>
-</section>
-<section id="using-nlls">
-<h3>Using NLLS<a class="headerlink" href="#using-nlls" title="Link to this heading">#</a></h3>
-<p>For starters, a classical, (somewhat) mechanistic model is the logistic equation:</p>
-<div class="math notranslate nohighlight" id="equation-eq-logist-growth-sol">
-<span class="eqno">(9)<a class="headerlink" href="#equation-eq-logist-growth-sol" title="Link to this equation">#</a></span>\[
-N_t = \frac{N_0 K e^{r t}}{K + N_0 (e^{r t} - 1)}
-\]</div>
-<p>Here <span class="math notranslate nohighlight">\(N_t\)</span> is population size at time <span class="math notranslate nohighlight">\(t\)</span>, <span class="math notranslate nohighlight">\(N_0\)</span> is initial population size, <span class="math notranslate nohighlight">\(r\)</span> is maximum growth rate (AKA <span class="math notranslate nohighlight">\(r_\text{max}\)</span>), and <span class="math notranslate nohighlight">\(K\)</span> is carrying capacity (maximum possible abundance of the population). Note that this model is actually the solution to the differential equation that defines the classic logistic population growth model (eqn. <a class="reference internal" href="Appendix-Maths.html#equation-eq-logist-growth">(13)</a>).</p>
-<div class="admonition note">
-<p class="admonition-title">Note</p>
-<p>The derivation of eqn. <a class="reference internal" href="#equation-eq-logist-growth-sol">(9)</a> is covered <a class="reference internal" href="Appendix-Maths.html#logistic-population-growth"><span class="std std-ref">here</span></a>. But you don’t need to know the derivation to fit eqn. <a class="reference internal" href="#equation-eq-logist-growth-sol">(9)</a> to data.</p>
-</div>
-<p>Let’s fit it to the data. First, we need to define it as a function object:</p>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-r notranslate"><div class="highlight"><pre><span></span><span class="n">logistic_model</span><span class="w"> </span><span class="o">&lt;-</span><span class="w"> </span><span class="nf">function</span><span class="p">(</span><span class="n">t</span><span class="p">,</span><span class="w"> </span><span class="n">r_max</span><span class="p">,</span><span class="w"> </span><span class="n">K</span><span class="p">,</span><span class="w"> </span><span class="n">N_0</span><span class="p">){</span><span class="w"> </span><span class="c1"># The classic logistic equation</span>
-<span class="w"> </span><span class="nf">return</span><span class="p">(</span><span class="n">N_0</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">K</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="nf">exp</span><span class="p">(</span><span class="n">r_max</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">t</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">K</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="n">N_0</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="p">(</span><span class="nf">exp</span><span class="p">(</span><span class="n">r_max</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">t</span><span class="p">)</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="m">1</span><span class="p">)))</span>
-<span class="p">}</span>
-</pre></div>
-</div>
-</div>
-</div>
-<p>Now fit it:</p>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-r notranslate"><div class="highlight"><pre><span></span><span class="c1"># first we need some starting parameters for the model</span>
-<span class="n">N_0_start</span><span class="w"> </span><span class="o">&lt;-</span><span class="w"> </span><span class="nf">min</span><span class="p">(</span><span class="n">data</span><span class="o">$</span><span class="n">N</span><span class="p">)</span><span class="w"> </span><span class="c1"># lowest population size</span>
-<span class="n">K_start</span><span class="w"> </span><span class="o">&lt;-</span><span class="w"> </span><span class="nf">max</span><span class="p">(</span><span class="n">data</span><span class="o">$</span><span class="n">N</span><span class="p">)</span><span class="w"> </span><span class="c1"># highest population size</span>
-<span class="n">r_max_start</span><span class="w"> </span><span class="o">&lt;-</span><span class="w"> </span><span class="m">0.62</span><span class="w"> </span><span class="c1"># use our estimate from the OLS fitting from above</span>
-
-<span class="n">fit_logistic</span><span class="w"> </span><span class="o">&lt;-</span><span class="w"> </span><span class="nf">nlsLM</span><span class="p">(</span><span class="n">N</span><span class="w"> </span><span class="o">~</span><span class="w"> </span><span class="nf">logistic_model</span><span class="p">(</span><span class="n">t</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Time</span><span class="p">,</span><span class="w"> </span><span class="n">r_max</span><span class="p">,</span><span class="w"> </span><span class="n">K</span><span class="p">,</span><span class="w"> </span><span class="n">N_0</span><span class="p">),</span><span class="w"> </span><span class="n">data</span><span class="p">,</span>
-<span class="w">      </span><span class="nf">list</span><span class="p">(</span><span class="n">r_max</span><span class="o">=</span><span class="n">r_max_start</span><span class="p">,</span><span class="w"> </span><span class="n">N_0</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">N_0_start</span><span class="p">,</span><span class="w"> </span><span class="n">K</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">K_start</span><span class="p">))</span>
-
-<span class="nf">summary</span><span class="p">(</span><span class="n">fit_logistic</span><span class="p">)</span>
-</pre></div>
-</div>
-</div>
-<div class="cell_output docutils container">
-<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Formula: N ~ logistic_model(t = Time, r_max, K, N_0)
-
-Parameters:
-       Estimate Std. Error t value Pr(&gt;|t|)    
-r_max 6.309e-01  3.791e-02  16.641 4.56e-08 ***
-N_0   3.317e+03  1.451e+03   2.286   0.0481 *  
-K     5.538e+06  7.192e+04  76.995 5.32e-14 ***
----
-Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
-
-Residual standard error: 119200 on 9 degrees of freedom
-
-Number of iterations to convergence: 12 
-Achieved convergence tolerance: 1.49e-08
-</pre></div>
-</div>
-</div>
-</div>
-<p>We did not pay much attention to what starting values we used in the simpler example of fitting the allometric model because the power-law model is easy to fit using NLLS, and starting far from the optimal parameters does not matter too much. Here, we used the actual data to generate more realistic start values for each of the three parameters (<code class="docutils literal notranslate"><span class="pre">r_max</span></code>, <code class="docutils literal notranslate"><span class="pre">N_0</span></code>, <code class="docutils literal notranslate"><span class="pre">K</span></code>) of the Logistic equation.</p>
-<p>Now, plot the fit:</p>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-r notranslate"><div class="highlight"><pre><span></span><span class="n">timepoints</span><span class="w"> </span><span class="o">&lt;-</span><span class="w"> </span><span class="nf">seq</span><span class="p">(</span><span class="m">0</span><span class="p">,</span><span class="w"> </span><span class="m">22</span><span class="p">,</span><span class="w"> </span><span class="m">0.1</span><span class="p">)</span>
-
-<span class="n">logistic_points</span><span class="w"> </span><span class="o">&lt;-</span><span class="w"> </span><span class="nf">logistic_model</span><span class="p">(</span><span class="n">t</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">timepoints</span><span class="p">,</span><span class="w"> </span>
-<span class="w">         </span><span class="n">r_max</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nf">coef</span><span class="p">(</span><span class="n">fit_logistic</span><span class="p">)[</span><span class="s">&quot;r_max&quot;</span><span class="p">],</span><span class="w"> </span>
-<span class="w">         </span><span class="n">K</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nf">coef</span><span class="p">(</span><span class="n">fit_logistic</span><span class="p">)[</span><span class="s">&quot;K&quot;</span><span class="p">],</span><span class="w"> </span>
-<span class="w">         </span><span class="n">N_0</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nf">coef</span><span class="p">(</span><span class="n">fit_logistic</span><span class="p">)[</span><span class="s">&quot;N_0&quot;</span><span class="p">])</span>
-<span class="n">df1</span><span class="w"> </span><span class="o">&lt;-</span><span class="w"> </span><span class="nf">data.frame</span><span class="p">(</span><span class="n">timepoints</span><span class="p">,</span><span class="w"> </span><span class="n">logistic_points</span><span class="p">)</span>
-<span class="n">df1</span><span class="o">$</span><span class="n">model</span><span class="w"> </span><span class="o">&lt;-</span><span class="w"> </span><span class="s">&quot;Logistic equation&quot;</span>
-<span class="nf">names</span><span class="p">(</span><span class="n">df1</span><span class="p">)</span><span class="w"> </span><span class="o">&lt;-</span><span class="w"> </span><span class="nf">c</span><span class="p">(</span><span class="s">&quot;Time&quot;</span><span class="p">,</span><span class="w"> </span><span class="s">&quot;N&quot;</span><span class="p">,</span><span class="w"> </span><span class="s">&quot;model&quot;</span><span class="p">)</span>
-
-<span class="nf">ggplot</span><span class="p">(</span><span class="n">data</span><span class="p">,</span><span class="w"> </span><span class="nf">aes</span><span class="p">(</span><span class="n">x</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Time</span><span class="p">,</span><span class="w"> </span><span class="n">y</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">N</span><span class="p">))</span><span class="w"> </span><span class="o">+</span>
-<span class="w"> </span><span class="nf">geom_point</span><span class="p">(</span><span class="n">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">3</span><span class="p">)</span><span class="w"> </span><span class="o">+</span>
-<span class="w"> </span><span class="nf">geom_line</span><span class="p">(</span><span class="n">data</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">df1</span><span class="p">,</span><span class="w"> </span><span class="nf">aes</span><span class="p">(</span><span class="n">x</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Time</span><span class="p">,</span><span class="w"> </span><span class="n">y</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">N</span><span class="p">,</span><span class="w"> </span><span class="n">col</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">model</span><span class="p">),</span><span class="w"> </span><span class="n">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span><span class="p">)</span><span class="w"> </span><span class="o">+</span>
-<span class="w"> </span><span class="nf">theme</span><span class="p">(</span><span class="n">aspect.ratio</span><span class="o">=</span><span class="m">1</span><span class="p">)</span><span class="o">+</span><span class="w"> </span><span class="c1"># make the plot square </span>
-<span class="w"> </span><span class="nf">labs</span><span class="p">(</span><span class="n">x</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s">&quot;Time&quot;</span><span class="p">,</span><span class="w"> </span><span class="n">y</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s">&quot;Cell number&quot;</span><span class="p">)</span>
-</pre></div>
-</div>
-</div>
-<div class="cell_output docutils container">
-<a class="reference internal image-reference" href="../_images/73a141ad43288a93443fc5f111d64fa95abaa4e394d7244dec032feb2224a16f.png"><img alt="../_images/73a141ad43288a93443fc5f111d64fa95abaa4e394d7244dec032feb2224a16f.png" src="../_images/73a141ad43288a93443fc5f111d64fa95abaa4e394d7244dec032feb2224a16f.png" style="width: 240px; height: 240px;" /></a>
-</div>
-</div>
-<p>That looks nice, and the <span class="math notranslate nohighlight">\(r_{max}\)</span> estimate we get (0.64) is fairly close to what we got above with OLS fitting.</p>
-<p>Note that we’ve done this fitting to the original non transformed data, whilst the linear regressions earlier were on log transformed data. What would this function look like on a log-transformed axis?</p>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-r notranslate"><div class="highlight"><pre><span></span><span class="nf">ggplot</span><span class="p">(</span><span class="n">data</span><span class="p">,</span><span class="w"> </span><span class="nf">aes</span><span class="p">(</span><span class="n">x</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Time</span><span class="p">,</span><span class="w"> </span><span class="n">y</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">LogN</span><span class="p">))</span><span class="w"> </span><span class="o">+</span>
-<span class="w"> </span><span class="nf">geom_point</span><span class="p">(</span><span class="n">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">3</span><span class="p">)</span><span class="w"> </span><span class="o">+</span>
-<span class="w"> </span><span class="nf">geom_line</span><span class="p">(</span><span class="n">data</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">df1</span><span class="p">,</span><span class="w"> </span><span class="nf">aes</span><span class="p">(</span><span class="n">x</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Time</span><span class="p">,</span><span class="w"> </span><span class="n">y</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nf">log</span><span class="p">(</span><span class="n">N</span><span class="p">),</span><span class="w"> </span><span class="n">col</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">model</span><span class="p">),</span><span class="w"> </span><span class="n">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span><span class="p">)</span><span class="w"> </span><span class="o">+</span>
-<span class="w"> </span><span class="nf">theme</span><span class="p">(</span><span class="n">aspect.ratio</span><span class="o">=</span><span class="m">1</span><span class="p">)</span><span class="o">+</span><span class="w"> </span>
-<span class="w"> </span><span class="nf">labs</span><span class="p">(</span><span class="n">x</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s">&quot;Time&quot;</span><span class="p">,</span><span class="w"> </span><span class="n">y</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s">&quot;log(Cell number)&quot;</span><span class="p">)</span>
-</pre></div>
-</div>
-</div>
-<div class="cell_output docutils container">
-<a class="reference internal image-reference" href="../_images/89c23c6a8be4677efa95ae5dbc04c19756d27258bec66c534ff42afd910753ec.png"><img alt="../_images/89c23c6a8be4677efa95ae5dbc04c19756d27258bec66c534ff42afd910753ec.png" src="../_images/89c23c6a8be4677efa95ae5dbc04c19756d27258bec66c534ff42afd910753ec.png" style="width: 240px; height: 240px;" /></a>
-</div>
-</div>
-<p>The model actually diverges from the data at the lower end! This was not visible in the previous plot where you examined the model in linear scale (without taking a log) because the deviation of the model is small, and only becomes clear in the log scale. This is because of the way logarithms work. Let’s have a look at this in our Cell counts “data”:</p>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-r notranslate"><div class="highlight"><pre><span></span><span class="nf">ggplot</span><span class="p">(</span><span class="n">data</span><span class="p">,</span><span class="w"> </span><span class="nf">aes</span><span class="p">(</span><span class="n">x</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">N</span><span class="p">,</span><span class="w"> </span><span class="n">y</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">LogN</span><span class="p">))</span><span class="w"> </span><span class="o">+</span>
-<span class="w"> </span><span class="nf">geom_point</span><span class="p">(</span><span class="n">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">3</span><span class="p">)</span><span class="w"> </span><span class="o">+</span>
-<span class="w"> </span><span class="nf">theme</span><span class="p">(</span><span class="n">aspect.ratio</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span><span class="p">)</span><span class="o">+</span><span class="w"> </span>
-<span class="w"> </span><span class="nf">labs</span><span class="p">(</span><span class="n">x</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s">&quot;N&quot;</span><span class="p">,</span><span class="w"> </span><span class="n">y</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s">&quot;log(N)&quot;</span><span class="p">)</span>
-</pre></div>
-</div>
-</div>
-<div class="cell_output docutils container">
-<a class="reference internal image-reference" href="../_images/cd2722c00c37cf632ed71a362df9dc0d431acd13dcc2a813c6007b2ffde9fd82.png"><img alt="../_images/cd2722c00c37cf632ed71a362df9dc0d431acd13dcc2a813c6007b2ffde9fd82.png" src="../_images/cd2722c00c37cf632ed71a362df9dc0d431acd13dcc2a813c6007b2ffde9fd82.png" style="width: 240px; height: 240px;" /></a>
-</div>
-</div>
-<p>As you can see the logarithm is a strongly nonlinear transformation of any sequence of real numbers, with small numbers close to zero yielding disproportionately large deviations.</p>
-<div class="admonition note">
-<p class="admonition-title">Note</p>
-<p>You may play with increasing the error (by increasing the value of <code class="docutils literal notranslate"><span class="pre">sd</span></code> in synthetic data generation step above) and re-evaluating all the subsequent model fitting steps above. However, note that above some values of <code class="docutils literal notranslate"><span class="pre">sd</span></code>, you will start to get negative values of populations, especially at early time points, which will raise issues with taking a logarithm.</p>
-</div>
-<p>The above seen deviation of the Logistic model from the data is because this model assumes that the population is growing right from the start (Time = 0), while in “reality” (in our synthetic “data”), this is not what’s happening; the population takes a while to grow truly exponentially (i.e., there is a time lag in the population growth). This time lag is seen frequently in the lab, and is also expected in nature, because when bacteria encounter fresh growth media (in the lab) or a new resource/environment (in the field), they take some time to acclimate, activating genes involved in nutrient uptake and metabolic processes, before beginning exponential growth. This is called the lag phase and can be seen in our example data where exponential growth doesn’t properly begin until around the 4th hour.</p>
-<p>To capture the lag phase, more complicated bacterial growth models have been designed.</p>
-<p>One of these is the modified Gompertz model (Zwietering et. al., 1990), which has been used frequently in the literature to model bacterial growth:</p>
-<div class="math notranslate nohighlight" id="equation-eq-gompertz">
-<span class="eqno">(10)<a class="headerlink" href="#equation-eq-gompertz" title="Link to this equation">#</a></span>\[ 
-  \log(N_t) = N_0 + (N_{max} - N_0) e^{-e^{r_{max} \exp(1) \frac{t_{lag} - t}{(N_{max} - N_0) \log(10)} + 1}}
-\]</div>
-<p>Here maximum growth rate (<span class="math notranslate nohighlight">\(r_{max}\)</span>) is the tangent to the inflection point, <span class="math notranslate nohighlight">\(t_{lag}\)</span> is the x-axis intercept to this tangent (duration of the delay before the population starts growing exponentially) and <span class="math notranslate nohighlight">\(\log\left(\frac{N_{max}}{N_0}\right)\)</span> is the asymptote of the log-transformed population growth trajectory, i.e., the log ratio of maximum population density <span class="math notranslate nohighlight">\(N_{max}\)</span> (aka “carrying capacity”) and initial cell (Population) <span class="math notranslate nohighlight">\(N_0\)</span> density.</p>
-<div class="admonition note">
-<p class="admonition-title">Note</p>
-<p>Note that unlike the Logistic growth model above, the Gompertz model is in the log scale. This is because the model is not derived from a differential equation, but was designed * specifically * to be fitted to log-transformed data.</p>
-</div>
-<p>Now let’s fit and compare the two alternative nonlinear growth models: Logistic and Gompertz.</p>
-<p>First, specify the function object for the Gompertz model (we already defined the function for the Logistic model above):</p>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-r notranslate"><div class="highlight"><pre><span></span><span class="n">gompertz_model</span><span class="w"> </span><span class="o">&lt;-</span><span class="w"> </span><span class="nf">function</span><span class="p">(</span><span class="n">t</span><span class="p">,</span><span class="w"> </span><span class="n">r_max</span><span class="p">,</span><span class="w"> </span><span class="n">K</span><span class="p">,</span><span class="w"> </span><span class="n">N_0</span><span class="p">,</span><span class="w"> </span><span class="n">t_lag</span><span class="p">){</span><span class="w"> </span><span class="c1"># Modified gompertz growth model (Zwietering 1990)</span>
-<span class="w"> </span><span class="nf">return</span><span class="p">(</span><span class="n">N_0</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="n">K</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">N_0</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="nf">exp</span><span class="p">(</span><span class="o">-</span><span class="nf">exp</span><span class="p">(</span><span class="n">r_max</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="nf">exp</span><span class="p">(</span><span class="m">1</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="p">(</span><span class="n">t_lag</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">t</span><span class="p">)</span><span class="o">/</span><span class="p">((</span><span class="n">K</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">N_0</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="nf">log</span><span class="p">(</span><span class="m">10</span><span class="p">))</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="m">1</span><span class="p">)))</span>
-<span class="p">}</span><span class="w">   </span>
-</pre></div>
-</div>
-</div>
-</div>
-<p>Again, note that unlike the Logistic growth function above, this function has been written in the log scale.</p>
-<p>Now let’s generate some starting values for the NLLS fitting of the Gompertz model.</p>
-<p>As we did above for the logistic equation, let’s derive the starting values by using the actual data:</p>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-r notranslate"><div class="highlight"><pre><span></span><span class="n">N_0_start</span><span class="w"> </span><span class="o">&lt;-</span><span class="w"> </span><span class="nf">min</span><span class="p">(</span><span class="n">data</span><span class="o">$</span><span class="n">LogN</span><span class="p">)</span><span class="w"> </span><span class="c1"># lowest population size, note log scale</span>
-<span class="n">K_start</span><span class="w"> </span><span class="o">&lt;-</span><span class="w"> </span><span class="nf">max</span><span class="p">(</span><span class="n">data</span><span class="o">$</span><span class="n">LogN</span><span class="p">)</span><span class="w"> </span><span class="c1"># highest population size, note log scale</span>
-<span class="n">r_max_start</span><span class="w"> </span><span class="o">&lt;-</span><span class="w"> </span><span class="m">0.62</span><span class="w"> </span><span class="c1"># use our previous estimate from the OLS fitting from above</span>
-<span class="n">t_lag_start</span><span class="w"> </span><span class="o">&lt;-</span><span class="w"> </span><span class="n">data</span><span class="o">$</span><span class="n">Time</span><span class="p">[</span><span class="nf">which.max</span><span class="p">(</span><span class="nf">diff</span><span class="p">(</span><span class="nf">diff</span><span class="p">(</span><span class="n">data</span><span class="o">$</span><span class="n">LogN</span><span class="p">)))]</span><span class="w"> </span><span class="c1"># find last timepoint of lag phase</span>
-</pre></div>
-</div>
-</div>
-</div>
-<ul class="simple">
-<li><p>So how did we find a reasonable time lag from the data? *</p></li>
-</ul>
-<p>Let’s break the last command down:</p>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-r notranslate"><div class="highlight"><pre><span></span><span class="nf">diff</span><span class="p">(</span><span class="n">data</span><span class="o">$</span><span class="n">LogN</span><span class="p">)</span><span class="w"> </span><span class="c1"># same as what we did above - get differentials</span>
-</pre></div>
-</div>
-</div>
-<div class="cell_output docutils container">
-<div class="output text_html"><style>
-.list-inline {list-style: none; margin:0; padding: 0}
-.list-inline>li {display: inline-block}
-.list-inline>li:not(:last-child)::after {content: "\00b7"; padding: 0 .5ex}
-</style>
-<ol class=list-inline><li>0.171269154259665</li><li>0.216670872460636</li><li>0.646099642770272</li><li>1.75344839347772</li><li>1.17470494059035</li><li>0.639023867964838</li><li>0.44952974020198</li><li>0.181493481601755</li><li>-0.000450183952025895</li><li>0.0544907101941003</li><li>-0.0546009242768832</li></ol>
-</div></div>
-</div>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-r notranslate"><div class="highlight"><pre><span></span><span class="nf">diff</span><span class="p">(</span><span class="nf">diff</span><span class="p">(</span><span class="n">data</span><span class="o">$</span><span class="n">LogN</span><span class="p">))</span><span class="w"> </span><span class="c1"># get the differentials of the differentials (approx 2nd order derivatives)</span>
-</pre></div>
-</div>
-</div>
-<div class="cell_output docutils container">
-<div class="output text_html"><style>
-.list-inline {list-style: none; margin:0; padding: 0}
-.list-inline>li {display: inline-block}
-.list-inline>li:not(:last-child)::after {content: "\00b7"; padding: 0 .5ex}
-</style>
-<ol class=list-inline><li>0.0454017182009707</li><li>0.429428770309636</li><li>1.10734875070745</li><li>-0.578743452887371</li><li>-0.535681072625511</li><li>-0.189494127762858</li><li>-0.268036258600224</li><li>-0.181943665553781</li><li>0.0549408941461262</li><li>-0.109091634470984</li></ol>
-</div></div>
-</div>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-r notranslate"><div class="highlight"><pre><span></span><span class="nf">which.max</span><span class="p">(</span><span class="nf">diff</span><span class="p">(</span><span class="nf">diff</span><span class="p">(</span><span class="n">data</span><span class="o">$</span><span class="n">LogN</span><span class="p">)))</span><span class="w"> </span><span class="c1"># find the timepoint where this 2nd order derivative really takes off </span>
-</pre></div>
-</div>
-</div>
-<div class="cell_output docutils container">
-<div class="output text_html">3</div></div>
-</div>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-r notranslate"><div class="highlight"><pre><span></span><span class="n">data</span><span class="o">$</span><span class="n">Time</span><span class="p">[</span><span class="nf">which.max</span><span class="p">(</span><span class="nf">diff</span><span class="p">(</span><span class="nf">diff</span><span class="p">(</span><span class="n">data</span><span class="o">$</span><span class="n">LogN</span><span class="p">)))]</span><span class="w"> </span><span class="c1"># This then is a good guess for the last timepoint of the lag phase</span>
-</pre></div>
-</div>
-</div>
-<div class="cell_output docutils container">
-<div class="output text_html">4</div></div>
-</div>
-<p>Now fit the model using these start values:</p>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-r notranslate"><div class="highlight"><pre><span></span><span class="n">fit_gompertz</span><span class="w"> </span><span class="o">&lt;-</span><span class="w"> </span><span class="nf">nlsLM</span><span class="p">(</span><span class="n">LogN</span><span class="w"> </span><span class="o">~</span><span class="w"> </span><span class="nf">gompertz_model</span><span class="p">(</span><span class="n">t</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Time</span><span class="p">,</span><span class="w"> </span><span class="n">r_max</span><span class="p">,</span><span class="w"> </span><span class="n">K</span><span class="p">,</span><span class="w"> </span><span class="n">N_0</span><span class="p">,</span><span class="w"> </span><span class="n">t_lag</span><span class="p">),</span><span class="w"> </span><span class="n">data</span><span class="p">,</span>
-<span class="w">      </span><span class="nf">list</span><span class="p">(</span><span class="n">t_lag</span><span class="o">=</span><span class="n">t_lag_start</span><span class="p">,</span><span class="w"> </span><span class="n">r_max</span><span class="o">=</span><span class="n">r_max_start</span><span class="p">,</span><span class="w"> </span><span class="n">N_0</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">N_0_start</span><span class="p">,</span><span class="w"> </span><span class="n">K</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">K_start</span><span class="p">))</span>
-</pre></div>
-</div>
-</div>
-</div>
-<p>You might one or more warning(s) that the model fitting iterations generated NaNs during the fitting procedure for these data (because at some point the NLLS fitting algorithm “wandered” to a combination of K and N_0 values that yields a NaN for log(K/N_0)).</p>
-<p>You can ignore these warning in this case. But not always – sometimes these NaNs mean that the equation is wrongly written, or that it generates NaNs across the whole range of the x-values, in which case the model is inappropriate for these data.</p>
-<p>Get the model summary:</p>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-r notranslate"><div class="highlight"><pre><span></span><span class="nf">summary</span><span class="p">(</span><span class="n">fit_gompertz</span><span class="p">)</span>
-</pre></div>
-</div>
-</div>
-<div class="cell_output docutils container">
-<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Formula: LogN ~ gompertz_model(t = Time, r_max, K, N_0, t_lag)
-
-Parameters:
-      Estimate Std. Error t value Pr(&gt;|t|)    
-t_lag  4.80680    0.18433   26.08 5.02e-09 ***
-r_max  1.86616    0.08749   21.33 2.45e-08 ***
-N_0   10.39142    0.05998  173.24 1.38e-15 ***
-K     15.54956    0.05056  307.57  &lt; 2e-16 ***
----
-Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
-
-Residual standard error: 0.09418 on 8 degrees of freedom
-
-Number of iterations to convergence: 10 
-Achieved convergence tolerance: 1.49e-08
-</pre></div>
-</div>
-</div>
-</div>
-<p>And see how the fits of the two nonlinear models compare:</p>
-<div class="cell docutils container">
-<div class="cell_input docutils container">
-<div class="highlight-r notranslate"><div class="highlight"><pre><span></span><span class="n">timepoints</span><span class="w"> </span><span class="o">&lt;-</span><span class="w"> </span><span class="nf">seq</span><span class="p">(</span><span class="m">0</span><span class="p">,</span><span class="w"> </span><span class="m">24</span><span class="p">,</span><span class="w"> </span><span class="m">0.1</span><span class="p">)</span>
-
-<span class="n">logistic_points</span><span class="w"> </span><span class="o">&lt;-</span><span class="w"> </span><span class="nf">log</span><span class="p">(</span><span class="nf">logistic_model</span><span class="p">(</span><span class="n">t</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">timepoints</span><span class="p">,</span><span class="w"> </span>
-<span class="w">          </span><span class="n">r_max</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nf">coef</span><span class="p">(</span><span class="n">fit_logistic</span><span class="p">)[</span><span class="s">&quot;r_max&quot;</span><span class="p">],</span><span class="w"> </span>
-<span class="w">          </span><span class="n">K</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nf">coef</span><span class="p">(</span><span class="n">fit_logistic</span><span class="p">)[</span><span class="s">&quot;K&quot;</span><span class="p">],</span><span class="w"> </span>
-<span class="w">          </span><span class="n">N_0</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nf">coef</span><span class="p">(</span><span class="n">fit_logistic</span><span class="p">)[</span><span class="s">&quot;N_0&quot;</span><span class="p">]))</span>
-
-<span class="n">gompertz_points</span><span class="w"> </span><span class="o">&lt;-</span><span class="w"> </span><span class="nf">gompertz_model</span><span class="p">(</span><span class="n">t</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">timepoints</span><span class="p">,</span><span class="w"> </span>
-<span class="w">         </span><span class="n">r_max</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nf">coef</span><span class="p">(</span><span class="n">fit_gompertz</span><span class="p">)[</span><span class="s">&quot;r_max&quot;</span><span class="p">],</span><span class="w"> </span>
-<span class="w">         </span><span class="n">K</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nf">coef</span><span class="p">(</span><span class="n">fit_gompertz</span><span class="p">)[</span><span class="s">&quot;K&quot;</span><span class="p">],</span><span class="w"> </span>
-<span class="w">         </span><span class="n">N_0</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nf">coef</span><span class="p">(</span><span class="n">fit_gompertz</span><span class="p">)[</span><span class="s">&quot;N_0&quot;</span><span class="p">],</span><span class="w"> </span>
-<span class="w">         </span><span class="n">t_lag</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nf">coef</span><span class="p">(</span><span class="n">fit_gompertz</span><span class="p">)[</span><span class="s">&quot;t_lag&quot;</span><span class="p">])</span>
-
-<span class="n">df1</span><span class="w"> </span><span class="o">&lt;-</span><span class="w"> </span><span class="nf">data.frame</span><span class="p">(</span><span class="n">timepoints</span><span class="p">,</span><span class="w"> </span><span class="n">logistic_points</span><span class="p">)</span>
-<span class="n">df1</span><span class="o">$</span><span class="n">model</span><span class="w"> </span><span class="o">&lt;-</span><span class="w"> </span><span class="s">&quot;Logistic model&quot;</span>
-<span class="nf">names</span><span class="p">(</span><span class="n">df1</span><span class="p">)</span><span class="w"> </span><span class="o">&lt;-</span><span class="w"> </span><span class="nf">c</span><span class="p">(</span><span class="s">&quot;Time&quot;</span><span class="p">,</span><span class="w"> </span><span class="s">&quot;LogN&quot;</span><span class="p">,</span><span class="w"> </span><span class="s">&quot;model&quot;</span><span class="p">)</span>
-
-<span class="n">df2</span><span class="w"> </span><span class="o">&lt;-</span><span class="w"> </span><span class="nf">data.frame</span><span class="p">(</span><span class="n">timepoints</span><span class="p">,</span><span class="w"> </span><span class="n">gompertz_points</span><span class="p">)</span>
-<span class="n">df2</span><span class="o">$</span><span class="n">model</span><span class="w"> </span><span class="o">&lt;-</span><span class="w"> </span><span class="s">&quot;Gompertz model&quot;</span>
-<span class="nf">names</span><span class="p">(</span><span class="n">df2</span><span class="p">)</span><span class="w"> </span><span class="o">&lt;-</span><span class="w"> </span><span class="nf">c</span><span class="p">(</span><span class="s">&quot;Time&quot;</span><span class="p">,</span><span class="w"> </span><span class="s">&quot;LogN&quot;</span><span class="p">,</span><span class="w"> </span><span class="s">&quot;model&quot;</span><span class="p">)</span>
-
-<span class="n">model_frame</span><span class="w"> </span><span class="o">&lt;-</span><span class="w"> </span><span class="nf">rbind</span><span class="p">(</span><span class="n">df1</span><span class="p">,</span><span class="w"> </span><span class="n">df2</span><span class="p">)</span>
-
-<span class="nf">ggplot</span><span class="p">(</span><span class="n">data</span><span class="p">,</span><span class="w"> </span><span class="nf">aes</span><span class="p">(</span><span class="n">x</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Time</span><span class="p">,</span><span class="w"> </span><span class="n">y</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">LogN</span><span class="p">))</span><span class="w"> </span><span class="o">+</span>
-<span class="w"> </span><span class="nf">geom_point</span><span class="p">(</span><span class="n">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">3</span><span class="p">)</span><span class="w"> </span><span class="o">+</span>
-<span class="w"> </span><span class="nf">geom_line</span><span class="p">(</span><span class="n">data</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">model_frame</span><span class="p">,</span><span class="w"> </span><span class="nf">aes</span><span class="p">(</span><span class="n">x</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">Time</span><span class="p">,</span><span class="w"> </span><span class="n">y</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">LogN</span><span class="p">,</span><span class="w"> </span><span class="n">col</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">model</span><span class="p">),</span><span class="w"> </span><span class="n">size</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="m">1</span><span class="p">)</span><span class="w"> </span><span class="o">+</span>
-<span class="w"> </span><span class="nf">theme_bw</span><span class="p">()</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="c1"># make the background white</span>
-<span class="w"> </span><span class="nf">theme</span><span class="p">(</span><span class="n">aspect.ratio</span><span class="o">=</span><span class="m">1</span><span class="p">)</span><span class="o">+</span><span class="w"> </span><span class="c1"># make the plot square </span>
-<span class="w"> </span><span class="nf">labs</span><span class="p">(</span><span class="n">x</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s">&quot;Time&quot;</span><span class="p">,</span><span class="w"> </span><span class="n">y</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="s">&quot;log(Abundance)&quot;</span><span class="p">)</span>
-</pre></div>
-</div>
-</div>
-<div class="cell_output docutils container">
-<a class="reference internal image-reference" href="../_images/371f77d6a4ca196b815bbac92b38d545d96ddde547615cd0b7ff22683f30173b.png"><img alt="../_images/371f77d6a4ca196b815bbac92b38d545d96ddde547615cd0b7ff22683f30173b.png" src="../_images/371f77d6a4ca196b815bbac92b38d545d96ddde547615cd0b7ff22683f30173b.png" style="width: 240px; height: 240px;" /></a>
-</div>
-</div>
-<p>Clearly, the Gompertz model fits way better than the logistic growth equation in this case! Note also that there is a big difference in the fitted value of <span class="math notranslate nohighlight">\(r_{max}\)</span> from the two models; the value is much lower from the Logistic model because it ignores the lag phase, including it into the exponential growth phase.</p>
-<p>You can now perform model selection like you did above in the allometric scaling example.</p>
-</section>
-<section id="id5">
-<h3>Exercises<a class="headerlink" href="#id5" title="Link to this heading">#</a></h3>
-<p>(a) Calculate the confidence intervals on the parameters of each of the two fitted models, and use model selection (using AIC and/or BIC) as you did before to see if you can determine the best-fitting model among the three.</p>
-<p>(b) Alternatively, for a different random sequence of fluctuations, one or more of the models may fail to fit (a <code class="docutils literal notranslate"><span class="pre">singular</span> <span class="pre">gradiant</span> <span class="pre">matrix</span></code> error). Try repeating the above fitting with a different random seed (change the integers given to the <code class="docutils literal notranslate"><span class="pre">random.seed(</span> <span class="pre">)</span></code> function), or increase the sampling error by increasing the standard deviation and see if it happens. If/when the NLLS optimization does fail to converge (the RSS minimum was not found), you can try to fix it by changing the starting values.</p>
-<p>(c) Repeat the model comparison exercise 1000 times (You will have to write a loop), and determine if/whether one model generally wins more often than the others. Note that each run will generate a slightly different dataset, because we are adding a vector of random errors every time the “data” are generated. This may result in failure of the NLLS fitting to converge, in which case you will need to use the <a class="reference external" href="https://nbviewer.jupyter.org/github/mhasoba/TheMulQuaBio/blob/master/notebooks/07-R.ipynb"><code class="docutils literal notranslate"><span class="pre">try()</span></code> or <code class="docutils literal notranslate"><span class="pre">tryCatch</span></code> functions</a>.</p>
-<p>(d) Repeat (b), but increase the error by increasing the standard deviation of the normal error distribution, and see if there are differences in the robustness of the models to sampling/experimental errors. You may also want to try changing the distribution of the errors to some non-normal distribution and see what happens.</p>
-</section>
-</section>
-<section id="some-tips-and-tricks-for-nlls-fitting">
-<h2>Some tips and tricks for NLLS fitting<a class="headerlink" href="#some-tips-and-tricks-for-nlls-fitting" title="Link to this heading">#</a></h2>
-<section id="starting-values">
-<span id="model-fitting-nlls-starting-values"></span><h3>Starting values<a class="headerlink" href="#starting-values" title="Link to this heading">#</a></h3>
-<p>The main challenge for NLLS fitting is finding starting (initial) values for the parameters, which the algorithm needs to proceed with the fitting/optimization. Inappropriate starting values can result in the algorithm finding parameter combinations represent convergence to a local optimum rather than the (globally) optimal solution. Starting parameter estimates can also result in or complete “divergence”, i.e., the search results in a combination of parameters that cause mathematical “singularity” (e.g., log(0) or division by zero).</p>
-<section id="obtaining-them">
-<h4>Obtaining them<a class="headerlink" href="#obtaining-them" title="Link to this heading">#</a></h4>
-<p>Finding the starting values is a <a class="reference external" href="https://en.wikipedia.org/wiki/Non-linear_least_squares#Initial_parameter_estimates">bit of an art</a>. There is no method for finding starting values that works universally (across different types of models).</p>
-<p>The one universal rule though, is that finding starting values requires you to understand the meaning of each of the parameters in your model. So, for example, in the population <a class="reference internal" href="#model-fitting-r-population-growth"><span class="std std-ref">growth rate example</span></a>, the parameters in both the nonlinear models that <a class="reference internal" href="#model-fitting-r-population-growth"><span class="std std-ref">we covered</span></a> (Logistic growth, eqn. <a class="reference internal" href="#equation-eq-logist-growth-sol">(9)</a> , Gompertz model; eqn. <a class="reference internal" href="#equation-eq-gompertz">(10)</a>) have a clear meaning.</p>
-<p>Furthermore, you will typically need to determine starting values <em>specific</em> to each model <em>and</em> each dataset that that you are wanting to fit that model to (e.g., every distinct functional response dataset to be <a class="reference internal" href="Appendix-MiniProj.html#miniproject-fr-models"><span class="std std-ref">fitted to the Holling Type II model</span></a>). To do so, understanding how each parameter in the model corresponds to features of the actual data is really necessary.</p>
-<p>For example, in the Gompertz population growth rate model (eqn. <a class="reference internal" href="#equation-eq-gompertz">(10)</a>), your starting values generator function would, for each dataset,</p>
-<ul class="simple">
-<li><p>Calculate a starting value for <span class="math notranslate nohighlight">\(r_{max}\)</span> by searching for the steepest slope of the growth curve (e.g., with a rolling OLS regression)</p></li>
-<li><p>Calculate a starting value of <span class="math notranslate nohighlight">\(t_{lag}\)</span> by intersecting the fitted line with the x (time)-axis</p></li>
-<li><p>Calculate a starting value for the asymptote <span class="math notranslate nohighlight">\(K\)</span> as the highest data (abundance) value in the dataset.</p></li>
-</ul>
-<div class="admonition tip">
-<p class="admonition-title">Tip</p>
-<p>Ideally, you should write a separate a function that calculates starting values for the model parameters.</p>
-</div>
-</section>
-<section id="sampling-them">
-<h4>Sampling them<a class="headerlink" href="#sampling-them" title="Link to this heading">#</a></h4>
-<p>Once you have worked out how to generate starting values for each non-linear model and dataset, a good next step for optimizing the fitting across multiple datasets (and thus maximize how many datasets are successfully fitted to the model) is to <em>rerun fitting attempts multiple times, sampling each of the starting values (simultaneously) randomly</em> (that is, randomly vary the set of starting values a bit each time). This sampling of starting values will increase the likelihood of the NLLS optimization algorithm finding a solution (optimal combination of parameters), and not getting stuck in a combination of parameters somewhere far away from that optimal solution.</p>
-<p>In particular,</p>
-<ul class="simple">
-<li><p>You can choose a Gaussian/Normal distribution if you have high confidence in mean value of parameter, or</p></li>
-<li><p>You can uniform distribution if you have low confidence in the mean, but higher confidence in the range of values that the parameter can take.
-In both cases, the <em>mean</em> of the sampling distribution will be the starting value you inferred from the model and the data (previous section).</p></li>
-</ul>
-<p>Furthermore,</p>
-<ul class="simple">
-<li><p>Whichever distribution you choose (gaussian vs uniform), you will need to determine what range of values to restrict each parameter’s samples to. In the case of the normal distribution, this is determined by what standard deviation parameter (you choose), and in the case of the uniform distribution, this is determined by what lower and upper bound (you choose). Generally, a good approach is to set the bound to be some <em>percent</em> (say 5-10%) of the parameter’s (mean) starting value. In both cases the chosen range to restrict the sampling to would typically be some subset of the model’s <em>parameter bounds</em> (next section).</p></li>
-<li><p><em>How many times to re-run</em> the fitting for a single dataset and model?* – this depends on how “difficult” the model is, and how much computational power you have.</p></li>
-</ul>
-<div class="admonition tip">
-<p class="admonition-title">Tip</p>
-<p>For the sampling of starting values, recall that you learned to generate random numbers from probability distributions in both the <a class="reference internal" href="R.html#r-random-numbers"><span class="std std-ref">R</span></a> and <a class="reference internal" href="Python.html#python-scipy-stats"><span class="std std-ref">Python</span></a> chapters).</p>
-</div>
-<p>You may also try and use a more sophisticated approach such as <a class="reference external" href="https://en.wikipedia.org/wiki/Hyperparameter_optimization#Approaches">grid searching</a> for varying your starting values randomly. An example is in the <a class="reference internal" href="#ModelFitting-MLE-LikelihoodSurface"><span class="xref myst">MLE chapter</span></a>.</p>
-</section>
-</section>
-<section id="bounding-parameters-revisited">
-<h3>Bounding parameters revisited<a class="headerlink" href="#bounding-parameters-revisited" title="Link to this heading">#</a></h3>
-<p>At the start, we looked at an <a class="reference internal" href="#modelfitting-nlls-bounding"><span class="std std-ref">example</span></a> of NLLS fitting where we bounded the parameters. It can be a good idea to restrict the range of values that the each of the model’s parameters can take <em>during any one fitting/optimization run</em>. To “bound” a parameter in this way means to give it upper and lower limits. By doing so, during one optimization/fitting (e.g., one call to <code class="docutils literal notranslate"><span class="pre">nlsLM</span></code>, to fit one model, to one dataset), the fitting algorithm does not allow a parameter to go outside some limits. This reduces the chances of the optimization getting stuck too far from the solution, or failing completely due to some mathematical singularity (e.g., log(0)).</p>
-<p>The bounds are typically fixed for each parameter of a model <em>at the level of the model</em> (e.g., they do not change based on each dataset). For example, in the Gompertz model for growth rates (eqn. <a class="reference internal" href="#equation-eq-gompertz">(10)</a>), you can limit the growth rate parameter to never be negative (the bounds would be <span class="math notranslate nohighlight">\([0,\infty]\)</span>), or restrict it further to be some value between zero and an upper limit (say, 10) that you know organismal growth rates cannot exceed (the bounds would in this case would be <span class="math notranslate nohighlight">\([0,10]\)</span>).</p>
-<p>However, as we saw in the Michaelis-Menten model fitting <a class="reference internal" href="#modelfitting-nlls-bounding"><span class="std std-ref">example</span></a>, bounding the parameters too much (excessively narrow ranges) can result in poor solutions because the algorithm cannot explore sufficient parameter space.</p>
-<div class="admonition tip">
-<p class="admonition-title">Tip</p>
-<p>The values of the parameter bounds you choose, of course, may depend on the <em>units of measurement</em> of the data. For example, in <a class="reference external" href="https://en.wikipedia.org/wiki/International_System_of_Units">SI</a>, growth rates in the Logistic or Gompertz models would be in units of s<span class="math notranslate nohighlight">\(^{-1}\)</span>).</p>
-</div>
-<p>Irrespective of which computer language the NLLS fitting algorithm is implemented in (<code class="docutils literal notranslate"><span class="pre">nlsLM</span></code>  in R or <code class="docutils literal notranslate"><span class="pre">lmfit</span></code> in Python), the fitting command/method will have options for setting the parameter bounds. In particular,</p>
-<ul class="simple">
-<li><p>For <code class="docutils literal notranslate"><span class="pre">nlsLM</span></code> in R, look up <a class="reference external" href="https://www.rdocumentation.org/packages/minpack.lm/versions/1.2-1/topics/nlsLM">https://www.rdocumentation.org/packages/minpack.lm/versions/1.2-1/topics/nlsLM</a> (the <code class="docutils literal notranslate"><span class="pre">lower</span></code> and <code class="docutils literal notranslate"><span class="pre">upper</span></code> arguments to the function).</p></li>
-<li><p>For <code class="docutils literal notranslate"><span class="pre">lmfit</span></code> in Python, look up <a class="reference external" href="https://lmfit.github.io/lmfit-py/parameters.html">https://lmfit.github.io/lmfit-py/parameters.html</a> (and in particular, <a class="reference external" href="https://lmfit.github.io/lmfit-py/parameters.html#lmfit.parameter.Parameter">https://lmfit.github.io/lmfit-py/parameters.html#lmfit.parameter.Parameter</a>) (the <code class="docutils literal notranslate"><span class="pre">min</span></code> and <code class="docutils literal notranslate"><span class="pre">max</span></code> arguments).</p></li>
-</ul>
-<p><em>Bounding the parameter values has nothing to do, per se, with sampling the starting values of each parameter, though if you choose to sample starting values (explained in previous section), you need to make sure that the samples don’t exceed the pre-set bounds (explained in this section).</em></p>
-<div class="admonition note">
-<p class="admonition-title">Note</p>
-<p>Python’s <code class="docutils literal notranslate"><span class="pre">lmfit</span></code> has an option to also internally vary the parameter. So by using a sampling approach as described in the previous section, <em>and</em> allowing the parameter to vary (note that <code class="docutils literal notranslate"><span class="pre">vary=True</span></code> is the default) within <code class="docutils literal notranslate"><span class="pre">lmfit</span></code>, you will be in essence be imposing sampling twice. This may or may not improve fitting performance – try it out both ways.</p>
-</div>
-</section>
-</section>
-<section id="readings-and-resources">
-<h2>Readings and Resources<a class="headerlink" href="#readings-and-resources" title="Link to this heading">#</a></h2>
-<ul class="simple">
-<li><p>Motulsky, Harvey, and Arthur Christopoulos. Fitting models to biological data using linear and nonlinear regression: a practical guide to curve fitting. OUP USA, 2004: <a class="reference external" href="https://www.facm.ucl.ac.be/cooperation/Vietnam/WBI-Vietnam-October-2011/Modelling/RegressionBook.pdf">https://www.facm.ucl.ac.be/cooperation/Vietnam/WBI-Vietnam-October-2011/Modelling/RegressionBook.pdf</a></p></li>
-<li><p>These are a pretty good series of notes on NLLS (even if you are using R instead of Python): <a class="reference external" href="https://lmfit.github.io/lmfit-py/intro.html">https://lmfit.github.io/lmfit-py/intro.html</a></p></li>
-<li><p>Another technical description of NLLS  algorithms: <a class="reference external" href="https://www.gnu.org/software/gsl/doc/html/nls.html">https://www.gnu.org/software/gsl/doc/html/nls.html</a></p></li>
-<li><p>Johnson, J. B. &amp; Omland, K. S. 2004 Model selection in ecology and evolution. Trends Ecol. Evol. 19, 101–108.</p></li>
-<li><p>The <em>nlstools</em> package for NLLS fit diagnostics: <a class="reference external" href="https://rdrr.io/rforge/nlstools">https://rdrr.io/rforge/nlstools</a></p>
-<ul>
-<li><p>The original paper: <a class="reference external" href="http://dx.doi.org/10.18637/jss.v066.i05">http://dx.doi.org/10.18637/jss.v066.i05</a></p></li>
-</ul>
-</li>
-</ul>
-</section>
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-<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#introduction">Introduction</a></li>
-<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#traits-data-as-an-example">Traits data as an example</a></li>
-<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#biochemical-kinetics">Biochemical Kinetics</a><ul class="nav section-nav flex-column">
-<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#generating-data">Generating data</a></li>
-</ul>
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-<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#fitting-the-model-using-nlls">Fitting the model using NLLS</a><ul class="nav section-nav flex-column">
-<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#visualizing-the-fit">Visualizing the fit</a></li>
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-<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#statistical-inference-using-nlls-fits">Statistical inference using NLLS fits</a><ul class="nav section-nav flex-column">
-<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#goodness-of-fit">Goodness of fit</a><ul class="nav section-nav flex-column">
-<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#r-squared-values">R-squared values</a></li>
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-<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#confidence-intervals">Confidence Intervals</a></li>
-</ul>
-</li>
-<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-starting-values-problem">The starting values problem</a><ul class="nav section-nav flex-column">
-<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#fitting-the-model-with-starting-values">Fitting the model with starting values</a></li>
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-</li>
-<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#a-more-robust-nlls-algorithm">A more robust NLLS algorithm</a></li>
-<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#bounding-parameter-values">Bounding parameter values</a></li>
-<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#diagnostics-of-an-nlls-fit">Diagnostics of an NLLS fit</a></li>
-<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#allometric-scaling-of-traits">Allometric scaling of traits</a><ul class="nav section-nav flex-column">
-<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#nlls-fitting-using-a-model-object">NLLS fitting using a model object</a></li>
-<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#id1">Visualizing the fit</a></li>
-<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#summary-of-the-fit">Summary of the fit</a></li>
-<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#examine-the-residuals">Examine the residuals</a><ul class="nav section-nav flex-column">
-<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#exercises">Exercises <a id="Allom_Exercises"></a></a></li>
-</ul>
-</li>
-</ul>
-</li>
-<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#comparing-models">Comparing models</a><ul class="nav section-nav flex-column">
-<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#id2">Exercises <a id="ModelSelection_Exercises"></a></a></li>
-</ul>
-</li>
-<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#albatross-chick-growth">Albatross chick growth</a><ul class="nav section-nav flex-column">
-<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#fitting-the-three-models-using-nlls">Fitting the three models using NLLS</a></li>
-<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#id3">Exercises <a id="Albatross_Exercises"></a></a></li>
-</ul>
-</li>
-<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#aedes-aegypti-fecundity">Aedes aegypti fecundity</a><ul class="nav section-nav flex-column">
-<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#the-thermal-performance-curve-models">The Thermal Performance Curve models</a></li>
-<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#id4">Exercises <a id="Aedes_Exercises"></a></a></li>
-</ul>
-</li>
-<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#abundance-data-as-an-example">Abundance data as an example</a></li>
-<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#population-growth-rates">Population growth rates</a><ul class="nav section-nav flex-column">
-<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#basic-approach">Basic approach</a></li>
-<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#using-ols">Using OLS</a></li>
-<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#using-nlls">Using NLLS</a></li>
-<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#id5">Exercises</a></li>
-</ul>
-</li>
-<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#some-tips-and-tricks-for-nlls-fitting">Some tips and tricks for NLLS fitting</a><ul class="nav section-nav flex-column">
-<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#starting-values">Starting values</a><ul class="nav section-nav flex-column">
-<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#obtaining-them">Obtaining them</a></li>
-<li class="toc-h4 nav-item toc-entry"><a class="reference internal nav-link" href="#sampling-them">Sampling them</a></li>
-</ul>
-</li>
-<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#bounding-parameters-revisited">Bounding parameters revisited</a></li>
-</ul>
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diff --git a/notebooks/ModelSimp.html b/notebooks/ModelSimp.html
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-    <title>Model simplification &#8212; TheMulQuaBio</title>
+    <title>Model simplification &#8212; MulQuaBio</title>
   
   
   
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@@ -188,23 +188,9 @@
 <li class="toctree-l1"><a class="reference internal" href="Python.html">Biological Computing in Python</a></li>
 <li class="toctree-l1"><a class="reference internal" href="R.html">Biological Computing in R</a></li>
 </ul>
-<p aria-level="2" class="caption" role="heading"><span class="caption-text">Maths for Biologists</span></p>
-<ul class="nav bd-sidenav">
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture01/01_Making_mathematical_statements.html">01 - Making mathematical statements</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture02/02_Describing_shapes_and_patterns.html">02   - Describing shapes and patterns</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture03/03_Analysing_change%2C_one_step_at_a_time.html">03   - Analysing change, one step at a time</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture04/04_From_table_arrangements_to_powerful_tools.html">04 - From table arrangements to powerful tools</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture05/05_From_table_arrangements_to_powerful_tools_part2.html">05 - From table arrangements to powerful tools - part 2</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture06/06_Analysing_change_continuously.html">06 - Analysing change, continuously</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture07/07_Analysing_change_continuously_part2.html">07 - Analysing change, continuously (Part 2)</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture08/08_Calculating_accumulated_change.html">08 - Calculating accumulated change</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/tutorials/lecture01/Tutorial_01.html">Tutorial 01</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/tutorials/lecture05/Tutorial_05.html">Tutorial 05</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/tutorials/lecture06/Tutorial_06.html">Tutorial 06</a></li>
-</ul>
 <p aria-level="2" class="caption" role="heading"><span class="caption-text">Basic Data Analyses and Statistics</span></p>
 <ul class="current nav bd-sidenav">
-<li class="toctree-l1"><a class="reference internal" href="../Stats-Intro.html">Basic Data Science and Statistics: Introduction</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../intro-Stats.html">Introduction</a></li>
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@@ -217,13 +203,14 @@
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 <ul class="nav bd-sidenav">
 <li class="toctree-l1"><a class="reference internal" href="glm.html">Generalised Linear Models</a></li>
-<li class="toctree-l1"><a class="reference internal" href="ModelFitting-NLLS.html">Model Fitting using Non-linear Least-squares</a></li>
+<li class="toctree-l1"><a class="reference internal" href="NLLS.html">Model Fitting using Non-linear Least-squares</a></li>
 </ul>
 <p aria-level="2" class="caption" role="heading"><span class="caption-text">Appendices</span></p>
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 <li class="toctree-l1"><a class="reference internal" href="Appendix-JupyIntro.html">Introduction to Jupyter</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-Data-Python.html">Data analyses  with Python &amp; Jupyter</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-Maths.html">Mathematical models in Jupyter</a></li>
+<li class="toctree-l1"><a class="reference internal" href="Appendix-MathsForBiologists/MathsForBiologists.html">Maths for Biologists</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-Databases.html">Databases <span class="tocSkip"></span></a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-NLLS-Python.html">Model fitting in Python</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-MiniProj.html">The Computing Miniproject</a></li>
@@ -283,7 +270,7 @@
       
       
       
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+      <li><a href="https://github.com/MulQuaBio/MQB" target="_blank"
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    title="Source repository"
    data-bs-placement="left" data-bs-toggle="tooltip"
@@ -300,7 +287,7 @@
       
       
       
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+      <li><a href="https://github.com/MulQuaBio/MQB/issues/new?title=Issue%20on%20page%20%2Fnotebooks/ModelSimp.html&body=Your%20issue%20content%20here." target="_blank"
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+By MulQuaBio collective!
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-    <title>Linear Models: Multiple explanatory variables &#8212; TheMulQuaBio</title>
+    <title>Linear Models: Multiple explanatory variables &#8212; MulQuaBio</title>
   
   
   
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+    <img src="../_static/logo.png" class="logo__image only-light" alt="MulQuaBio - Home"/>
+    <script>document.write(`<img src="../_static/logo.png" class="logo__image only-dark" alt="MulQuaBio - Home"/>`);</script>
   
   
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 <ul class="current nav bd-sidenav">
-<li class="toctree-l1"><a class="reference internal" href="../Stats-Intro.html">Basic Data Science and Statistics: Introduction</a></li>
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 <li class="toctree-l1"><a class="reference internal" href="Data_R.html">Data Management and Visualization</a></li>
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 <li class="toctree-l1"><a class="reference internal" href="Appendix-Maths.html">Mathematical models in Jupyter</a></li>
+<li class="toctree-l1"><a class="reference internal" href="Appendix-MathsForBiologists/MathsForBiologists.html">Maths for Biologists</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-Databases.html">Databases <span class="tocSkip"></span></a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-NLLS-Python.html">Model fitting in Python</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-MiniProj.html">The Computing Miniproject</a></li>
@@ -283,7 +270,7 @@
       
       
       
-      <li><a href="https://github.com/mhasoba/TheMulQuaBio" target="_blank"
+      <li><a href="https://github.com/MulQuaBio/MQB" target="_blank"
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+      <li><a href="https://github.com/MulQuaBio/MQB/issues/new?title=Issue%20on%20page%20%2Fnotebooks/MulExpl.html&body=Your%20issue%20content%20here." target="_blank"
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    title="Open an issue"
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   <div class="footer-item">
     
 <p class="component-author">
-By TheMulQuaBio collective!
+By MulQuaBio collective!
 </p>
 
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-    <title>Linear Models: Multiple variables with interactions &#8212; TheMulQuaBio</title>
+    <title>Linear Models: Multiple variables with interactions &#8212; MulQuaBio</title>
   
   
   
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 <p aria-level="2" class="caption" role="heading"><span class="caption-text">Basic Data Analyses and Statistics</span></p>
 <ul class="current nav bd-sidenav">
-<li class="toctree-l1"><a class="reference internal" href="../Stats-Intro.html">Basic Data Science and Statistics: Introduction</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../intro-Stats.html">Introduction</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Data_R.html">Data Management and Visualization</a></li>
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 <li class="toctree-l1"><a class="reference internal" href="Appendix-Maths.html">Mathematical models in Jupyter</a></li>
+<li class="toctree-l1"><a class="reference internal" href="Appendix-MathsForBiologists/MathsForBiologists.html">Maths for Biologists</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-Databases.html">Databases <span class="tocSkip"></span></a></li>
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@@ -283,7 +270,7 @@
       
       
       
-      <li><a href="https://github.com/mhasoba/TheMulQuaBio" target="_blank"
+      <li><a href="https://github.com/MulQuaBio/MQB" target="_blank"
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    title="Source repository"
    data-bs-placement="left" data-bs-toggle="tooltip"
@@ -300,7 +287,7 @@
       
       
       
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+      <li><a href="https://github.com/MulQuaBio/MQB/issues/new?title=Issue%20on%20page%20%2Fnotebooks/MulExplInter.html&body=Your%20issue%20content%20here." target="_blank"
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@@ -983,7 +970,7 @@ <h2>Model 2 (ANCOVA): Body Weight in Odonata<a class="headerlink" href="#model-2
   <div class="footer-item">
     
 <p class="component-author">
-By TheMulQuaBio collective!
+By MulQuaBio collective!
 </p>
 
   </div>
diff --git a/notebooks/NLLS.html b/notebooks/NLLS.html
index cdf8da86..6f6ee293 100644
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-    <title>Model Fitting using Non-linear Least-squares &#8212; TheMulQuaBio</title>
+    <title>Model Fitting using Non-linear Least-squares &#8212; MulQuaBio</title>
   
   
   
@@ -153,8 +153,8 @@
       
     
     
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-    <script>document.write(`<img src="../_static/logo.png" class="logo__image only-dark" alt="TheMulQuaBio - Home"/>`);</script>
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+    <script>document.write(`<img src="../_static/logo.png" class="logo__image only-dark" alt="MulQuaBio - Home"/>`);</script>
   
   
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 </ul>
-<p aria-level="2" class="caption" role="heading"><span class="caption-text">Maths for Biologists</span></p>
-<ul class="nav bd-sidenav">
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture01/01_Making_mathematical_statements.html">01 - Making mathematical statements</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture02/02_Describing_shapes_and_patterns.html">02   - Describing shapes and patterns</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture03/03_Analysing_change%2C_one_step_at_a_time.html">03   - Analysing change, one step at a time</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture04/04_From_table_arrangements_to_powerful_tools.html">04 - From table arrangements to powerful tools</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture05/05_From_table_arrangements_to_powerful_tools_part2.html">05 - From table arrangements to powerful tools - part 2</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture06/06_Analysing_change_continuously.html">06 - Analysing change, continuously</a></li>
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-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture08/08_Calculating_accumulated_change.html">08 - Calculating accumulated change</a></li>
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-</ul>
 <p aria-level="2" class="caption" role="heading"><span class="caption-text">Basic Data Analyses and Statistics</span></p>
 <ul class="nav bd-sidenav">
-<li class="toctree-l1"><a class="reference internal" href="../Stats-Intro.html">Basic Data Science and Statistics: Introduction</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../intro-Stats.html">Introduction</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Data_R.html">Data Management and Visualization</a></li>
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 <li class="toctree-l1"><a class="reference internal" href="Appendix-Maths.html">Mathematical models in Jupyter</a></li>
+<li class="toctree-l1"><a class="reference internal" href="Appendix-MathsForBiologists/MathsForBiologists.html">Maths for Biologists</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-Databases.html">Databases <span class="tocSkip"></span></a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-NLLS-Python.html">Model fitting in Python</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-MiniProj.html">The Computing Miniproject</a></li>
@@ -283,7 +270,7 @@
       
       
       
-      <li><a href="https://github.com/mhasoba/TheMulQuaBio" target="_blank"
+      <li><a href="https://github.com/MulQuaBio/MQB" target="_blank"
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    title="Source repository"
    data-bs-placement="left" data-bs-toggle="tooltip"
@@ -300,7 +287,7 @@
       
       
       
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+      <li><a href="https://github.com/MulQuaBio/MQB/issues/new?title=Issue%20on%20page%20%2Fnotebooks/NLLS.html&body=Your%20issue%20content%20here." target="_blank"
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    title="Open an issue"
    data-bs-placement="left" data-bs-toggle="tooltip"
@@ -2418,7 +2405,7 @@ <h2>Readings and Resources<a class="headerlink" href="#readings-and-resources" t
   <div class="footer-item">
     
 <p class="component-author">
-By TheMulQuaBio collective!
+By MulQuaBio collective!
 </p>
 
   </div>
diff --git a/notebooks/Populations.html b/notebooks/Populations.html
new file mode 100644
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+    <meta name="viewport" content="width=device-width, initial-scale=1.0" /><meta name="viewport" content="width=device-width, initial-scale=1" />
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+    <h1>Single Populations</h1>
+    <!-- Table of contents -->
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+        <div id="jb-print-toc">
+            
+        </div>
+    </div>
+</div>
+
+              
+                
+<div id="searchbox"></div>
+                <article class="bd-article">
+                  
+  <section class="tex2jax_ignore mathjax_ignore" id="single-populations">
+<h1>Single Populations<a class="headerlink" href="#single-populations" title="Link to this heading">#</a></h1>
+<ul class="simple">
+<li><p>Single populations</p>
+<ul>
+<li><p>Are there <em>any</em> single populations?</p></li>
+<li><p>Exponential growth</p>
+<ul>
+<li><p>Metabolic basis</p></li>
+</ul>
+</li>
+<li><p>From populations to alleles</p>
+<ul>
+<li><p>Basic pop gen</p></li>
+</ul>
+</li>
+<li><p>Logistic growth</p>
+<ul>
+<li><p>Density dependence</p></li>
+<li><p>Logistic growth as an “effective” model</p>
+<ul>
+<li><p>SIS model (VJ)</p></li>
+</ul>
+</li>
+<li><p>Bifurcations (+Intro to theory - VJ)</p></li>
+</ul>
+</li>
+<li><p>Evolutionary dynamics</p></li>
+<li><p>Stochasticity</p></li>
+<li><p>Applications/Examples</p></li>
+<li><p>Stage and age structured populations</p>
+<ul>
+<li><p>Stage- vs Age-structured models</p></li>
+<li><p>IPMs</p></li>
+</ul>
+</li>
+<li><p>Applications/Examples</p></li>
+</ul>
+</li>
+</ul>
+</section>
+
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diff --git a/notebooks/Python.html b/notebooks/Python.html
index c171d69f..0b464164 100644
--- a/notebooks/Python.html
+++ b/notebooks/Python.html
@@ -8,7 +8,7 @@
     <meta charset="utf-8" />
     <meta name="viewport" content="width=device-width, initial-scale=1.0" /><meta name="viewport" content="width=device-width, initial-scale=1" />
 
-    <title>Biological Computing in Python &#8212; TheMulQuaBio</title>
+    <title>Biological Computing in Python &#8212; MulQuaBio</title>
   
   
   
@@ -153,8 +153,8 @@
       
     
     
-    <img src="../_static/logo.png" class="logo__image only-light" alt="TheMulQuaBio - Home"/>
-    <script>document.write(`<img src="../_static/logo.png" class="logo__image only-dark" alt="TheMulQuaBio - Home"/>`);</script>
+    <img src="../_static/logo.png" class="logo__image only-light" alt="MulQuaBio - Home"/>
+    <script>document.write(`<img src="../_static/logo.png" class="logo__image only-dark" alt="MulQuaBio - Home"/>`);</script>
   
   
 </a></div>
@@ -188,23 +188,9 @@
 <li class="toctree-l1 current active"><a class="current reference internal" href="#">Biological Computing in Python</a></li>
 <li class="toctree-l1"><a class="reference internal" href="R.html">Biological Computing in R</a></li>
 </ul>
-<p aria-level="2" class="caption" role="heading"><span class="caption-text">Maths for Biologists</span></p>
-<ul class="nav bd-sidenav">
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture01/01_Making_mathematical_statements.html">01 - Making mathematical statements</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture02/02_Describing_shapes_and_patterns.html">02   - Describing shapes and patterns</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture03/03_Analysing_change%2C_one_step_at_a_time.html">03   - Analysing change, one step at a time</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture04/04_From_table_arrangements_to_powerful_tools.html">04 - From table arrangements to powerful tools</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture05/05_From_table_arrangements_to_powerful_tools_part2.html">05 - From table arrangements to powerful tools - part 2</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture06/06_Analysing_change_continuously.html">06 - Analysing change, continuously</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture07/07_Analysing_change_continuously_part2.html">07 - Analysing change, continuously (Part 2)</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture08/08_Calculating_accumulated_change.html">08 - Calculating accumulated change</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/tutorials/lecture01/Tutorial_01.html">Tutorial 01</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/tutorials/lecture05/Tutorial_05.html">Tutorial 05</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/tutorials/lecture06/Tutorial_06.html">Tutorial 06</a></li>
-</ul>
 <p aria-level="2" class="caption" role="heading"><span class="caption-text">Basic Data Analyses and Statistics</span></p>
 <ul class="nav bd-sidenav">
-<li class="toctree-l1"><a class="reference internal" href="../Stats-Intro.html">Basic Data Science and Statistics: Introduction</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../intro-Stats.html">Introduction</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Data_R.html">Data Management and Visualization</a></li>
 <li class="toctree-l1"><a class="reference internal" href="ExpDesign.html">Experimental design and Data exploration</a></li>
 <li class="toctree-l1"><a class="reference internal" href="t_F_tests.html">Basic hypothesis testing: <span class="math notranslate nohighlight">\(t\)</span> and <span class="math notranslate nohighlight">\(F\)</span> tests</a></li>
@@ -224,6 +210,7 @@
 <li class="toctree-l1"><a class="reference internal" href="Appendix-JupyIntro.html">Introduction to Jupyter</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-Data-Python.html">Data analyses  with Python &amp; Jupyter</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-Maths.html">Mathematical models in Jupyter</a></li>
+<li class="toctree-l1"><a class="reference internal" href="Appendix-MathsForBiologists/MathsForBiologists.html">Maths for Biologists</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-Databases.html">Databases <span class="tocSkip"></span></a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-NLLS-Python.html">Model fitting in Python</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-MiniProj.html">The Computing Miniproject</a></li>
@@ -283,7 +270,7 @@
       
       
       
-      <li><a href="https://github.com/mhasoba/TheMulQuaBio" target="_blank"
+      <li><a href="https://github.com/MulQuaBio/MQB" target="_blank"
    class="btn btn-sm btn-source-repository-button dropdown-item"
    title="Source repository"
    data-bs-placement="left" data-bs-toggle="tooltip"
@@ -300,7 +287,7 @@
       
       
       
-      <li><a href="https://github.com/mhasoba/TheMulQuaBio/issues/new?title=Issue%20on%20page%20%2Fnotebooks/Python.html&body=Your%20issue%20content%20here." target="_blank"
+      <li><a href="https://github.com/MulQuaBio/MQB/issues/new?title=Issue%20on%20page%20%2Fnotebooks/Python.html&body=Your%20issue%20content%20here." target="_blank"
    class="btn btn-sm btn-source-issues-button dropdown-item"
    title="Open an issue"
    data-bs-placement="left" data-bs-toggle="tooltip"
@@ -5914,8 +5901,8 @@ <h5>Area under a curve<a class="headerlink" href="#area-under-a-curve" title="Li
 <section id="the-lotka-volterra-model">
 <h5>The Lotka-Volterra model<a class="headerlink" href="#the-lotka-volterra-model" title="Link to this heading">#</a></h5>
 <p>Now let’s try numerical integration in Python for solving a classical model in biology — the Lotka-Volterra (LV) model for a predator-prey system in two-dimensional space (e.g., on land). The LV model is:</p>
-<div class="amsmath math notranslate nohighlight" id="equation-bb744b22-a1e1-42c5-85ee-49f2c4291131">
-<span class="eqno">(2)<a class="headerlink" href="#equation-bb744b22-a1e1-42c5-85ee-49f2c4291131" title="Permalink to this equation">#</a></span>\[\begin{align}
+<div class="amsmath math notranslate nohighlight" id="equation-104ba973-4989-4e8d-a86f-d10fe6dd762b">
+<span class="eqno">(2)<a class="headerlink" href="#equation-104ba973-4989-4e8d-a86f-d10fe6dd762b" title="Permalink to this equation">#</a></span>\[\begin{align}
     \frac{dR}{dt} &amp;= r R - a C R  \nonumber \\
     \frac{dC}{dt} &amp;= - z C + e a C R
 \end{align}\]</div>
@@ -6473,8 +6460,8 @@ <h2>Practicals<a class="headerlink" href="#id3" title="Link to this heading">#</
 <h3>The Lotka-Volterra model revisited<a class="headerlink" href="#the-lotka-volterra-model-revisited" title="Link to this heading">#</a></h3>
 <p><em>What happens when the Consumer population <span class="math notranslate nohighlight">\(C\)</span> goes extinct in the Lotka-Volterra that you encountered above</em>?</p>
 <p>If the consumer population goes extinct, the resources grow to infinity! This is not biologically realistic, of course. To address this, we can introduce density dependence to the resource population <span class="math notranslate nohighlight">\(\left(1 - \frac{R} {K}\right)\)</span>, which changes the ODE system to</p>
-<div class="amsmath math notranslate nohighlight" id="equation-5f8b751a-d8ac-4348-b587-69078f47ffcc">
-<span class="eqno">(3)<a class="headerlink" href="#equation-5f8b751a-d8ac-4348-b587-69078f47ffcc" title="Permalink to this equation">#</a></span>\[\begin{align}
+<div class="amsmath math notranslate nohighlight" id="equation-ca156554-f81c-4c18-9527-765c91f1b59c">
+<span class="eqno">(3)<a class="headerlink" href="#equation-ca156554-f81c-4c18-9527-765c91f1b59c" title="Permalink to this equation">#</a></span>\[\begin{align}
     \frac{dR}{dt} &amp;= r R \left(1 - \frac{R} {K}\right) - a C R\\
     \frac{dC}{dt} &amp;= - z C + e a C R
 \end{align}\]</div>
@@ -6504,8 +6491,8 @@ <h3>Groupwork practical: Discrete time LV Model<a class="headerlink" href="#grou
 <ul class="simple">
 <li><p>Write a discrete-time version of the LV model called <code class="docutils literal notranslate"><span class="pre">LV3.py</span></code>. The discrete-time model is:</p></li>
 </ul>
-<div class="amsmath math notranslate nohighlight" id="equation-83b75028-7db5-45b8-a6df-048c58a39d99">
-<span class="eqno">(4)<a class="headerlink" href="#equation-83b75028-7db5-45b8-a6df-048c58a39d99" title="Permalink to this equation">#</a></span>\[\begin{align} 
+<div class="amsmath math notranslate nohighlight" id="equation-acf926f0-cdca-4f9c-bc13-ca0e84ee8ef3">
+<span class="eqno">(4)<a class="headerlink" href="#equation-acf926f0-cdca-4f9c-bc13-ca0e84ee8ef3" title="Permalink to this equation">#</a></span>\[\begin{align} 
     R_{t+1} &amp;= R_t (1 + r \left(1 - \frac{R_t}{K}\right) - a C_t)\\ 
     C_{t+1} &amp;= C_t (1 - z + e a R_t) 
 \end{align}\]</div>
@@ -6516,14 +6503,14 @@ <h3>Groupwork practical: Discrete time LV model with stochasticity<a class="head
 <ul class="simple">
 <li><p>Write a version of the discrete-time model (which you implemented in <code class="docutils literal notranslate"><span class="pre">LV3.py</span></code>) simulation with a random gaussian fluctuation in resource’s growth rate at each time-step:</p></li>
 </ul>
-<div class="amsmath math notranslate nohighlight" id="equation-daa2e4a7-710a-4d8f-872b-ca05d1ee0366">
-<span class="eqno">(5)<a class="headerlink" href="#equation-daa2e4a7-710a-4d8f-872b-ca05d1ee0366" title="Permalink to this equation">#</a></span>\[\begin{align}
+<div class="amsmath math notranslate nohighlight" id="equation-e081c3b1-2820-4b57-a126-cf5632841895">
+<span class="eqno">(5)<a class="headerlink" href="#equation-e081c3b1-2820-4b57-a126-cf5632841895" title="Permalink to this equation">#</a></span>\[\begin{align}
         R_{t+1} &amp;= R_t (1 + (r + \epsilon) \left(1 - \frac{R_t}{K}\right)- a C_t)\\
         C_{t+1} &amp;= C_t (1 - z + e a R_t)
 \end{align}\]</div>
 <p>where <span class="math notranslate nohighlight">\(\epsilon\)</span> is a random fluctuation drawn from a gaussian distribution (use <code class="docutils literal notranslate"><span class="pre">sc.stats</span></code> or <code class="docutils literal notranslate"><span class="pre">np.random</span></code>). Include this script in <code class="docutils literal notranslate"> <span class="pre">run_LV.py</span></code>, and profile it as well. You can also add fluctuations to both populations simultaneously this way:</p>
-<div class="amsmath math notranslate nohighlight" id="equation-c0dc250c-9ccd-4488-aee8-f7e228035562">
-<span class="eqno">(6)<a class="headerlink" href="#equation-c0dc250c-9ccd-4488-aee8-f7e228035562" title="Permalink to this equation">#</a></span>\[\begin{align}
+<div class="amsmath math notranslate nohighlight" id="equation-1a79e2fd-68ff-484b-8a9b-0d0d4046d001">
+<span class="eqno">(6)<a class="headerlink" href="#equation-1a79e2fd-68ff-484b-8a9b-0d0d4046d001" title="Permalink to this equation">#</a></span>\[\begin{align}
     R_{t+1} &amp;= R_t (1 + (r + \epsilon)  \left(1 - \frac{R_t}{K}\right) - a C_t)\\
     C_{t+1} &amp;= C_t (1 - (z + \epsilon) + e a R_t)
 \end{align}\]</div>
@@ -7956,7 +7943,7 @@ <h3>Other stuff<a class="headerlink" href="#other-stuff" title="Link to this hea
   <div class="footer-item">
     
 <p class="component-author">
-By TheMulQuaBio collective!
+By MulQuaBio collective!
 </p>
 
   </div>
diff --git a/notebooks/R.html b/notebooks/R.html
index 1e276b76..deb5c806 100644
--- a/notebooks/R.html
+++ b/notebooks/R.html
@@ -8,7 +8,7 @@
     <meta charset="utf-8" />
     <meta name="viewport" content="width=device-width, initial-scale=1.0" /><meta name="viewport" content="width=device-width, initial-scale=1" />
 
-    <title>Biological Computing in R &#8212; TheMulQuaBio</title>
+    <title>Biological Computing in R &#8212; MulQuaBio</title>
   
   
   
@@ -60,7 +60,7 @@
     <script>DOCUMENTATION_OPTIONS.pagename = 'notebooks/R';</script>
     <link rel="index" title="Index" href="../genindex.html" />
     <link rel="search" title="Search" href="../search.html" />
-    <link rel="next" title="01 - Making mathematical statements" href="../mathscourse/lecture01/01_Making_mathematical_statements.html" />
+    <link rel="next" title="Introduction" href="../intro-Stats.html" />
     <link rel="prev" title="Biological Computing in Python" href="Python.html" />
   <meta name="viewport" content="width=device-width, initial-scale=1"/>
   <meta name="docsearch:language" content="en"/>
@@ -153,8 +153,8 @@
       
     
     
-    <img src="../_static/logo.png" class="logo__image only-light" alt="TheMulQuaBio - Home"/>
-    <script>document.write(`<img src="../_static/logo.png" class="logo__image only-dark" alt="TheMulQuaBio - Home"/>`);</script>
+    <img src="../_static/logo.png" class="logo__image only-light" alt="MulQuaBio - Home"/>
+    <script>document.write(`<img src="../_static/logo.png" class="logo__image only-dark" alt="MulQuaBio - Home"/>`);</script>
   
   
 </a></div>
@@ -188,23 +188,9 @@
 <li class="toctree-l1"><a class="reference internal" href="Python.html">Biological Computing in Python</a></li>
 <li class="toctree-l1 current active"><a class="current reference internal" href="#">Biological Computing in R</a></li>
 </ul>
-<p aria-level="2" class="caption" role="heading"><span class="caption-text">Maths for Biologists</span></p>
-<ul class="nav bd-sidenav">
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture01/01_Making_mathematical_statements.html">01 - Making mathematical statements</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture02/02_Describing_shapes_and_patterns.html">02   - Describing shapes and patterns</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture03/03_Analysing_change%2C_one_step_at_a_time.html">03   - Analysing change, one step at a time</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture04/04_From_table_arrangements_to_powerful_tools.html">04 - From table arrangements to powerful tools</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture05/05_From_table_arrangements_to_powerful_tools_part2.html">05 - From table arrangements to powerful tools - part 2</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture06/06_Analysing_change_continuously.html">06 - Analysing change, continuously</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture07/07_Analysing_change_continuously_part2.html">07 - Analysing change, continuously (Part 2)</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture08/08_Calculating_accumulated_change.html">08 - Calculating accumulated change</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/tutorials/lecture01/Tutorial_01.html">Tutorial 01</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/tutorials/lecture05/Tutorial_05.html">Tutorial 05</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/tutorials/lecture06/Tutorial_06.html">Tutorial 06</a></li>
-</ul>
 <p aria-level="2" class="caption" role="heading"><span class="caption-text">Basic Data Analyses and Statistics</span></p>
 <ul class="nav bd-sidenav">
-<li class="toctree-l1"><a class="reference internal" href="../Stats-Intro.html">Basic Data Science and Statistics: Introduction</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../intro-Stats.html">Introduction</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Data_R.html">Data Management and Visualization</a></li>
 <li class="toctree-l1"><a class="reference internal" href="ExpDesign.html">Experimental design and Data exploration</a></li>
 <li class="toctree-l1"><a class="reference internal" href="t_F_tests.html">Basic hypothesis testing: <span class="math notranslate nohighlight">\(t\)</span> and <span class="math notranslate nohighlight">\(F\)</span> tests</a></li>
@@ -224,6 +210,7 @@
 <li class="toctree-l1"><a class="reference internal" href="Appendix-JupyIntro.html">Introduction to Jupyter</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-Data-Python.html">Data analyses  with Python &amp; Jupyter</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-Maths.html">Mathematical models in Jupyter</a></li>
+<li class="toctree-l1"><a class="reference internal" href="Appendix-MathsForBiologists/MathsForBiologists.html">Maths for Biologists</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-Databases.html">Databases <span class="tocSkip"></span></a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-NLLS-Python.html">Model fitting in Python</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-MiniProj.html">The Computing Miniproject</a></li>
@@ -283,7 +270,7 @@
       
       
       
-      <li><a href="https://github.com/mhasoba/TheMulQuaBio" target="_blank"
+      <li><a href="https://github.com/MulQuaBio/MQB" target="_blank"
    class="btn btn-sm btn-source-repository-button dropdown-item"
    title="Source repository"
    data-bs-placement="left" data-bs-toggle="tooltip"
@@ -300,7 +287,7 @@
       
       
       
-      <li><a href="https://github.com/mhasoba/TheMulQuaBio/issues/new?title=Issue%20on%20page%20%2Fnotebooks/R.html&body=Your%20issue%20content%20here." target="_blank"
+      <li><a href="https://github.com/MulQuaBio/MQB/issues/new?title=Issue%20on%20page%20%2Fnotebooks/R.html&body=Your%20issue%20content%20here." target="_blank"
    class="btn btn-sm btn-source-issues-button dropdown-item"
    title="Open an issue"
    data-bs-placement="left" data-bs-toggle="tooltip"
@@ -5117,11 +5104,11 @@ <h3>Sweave and knitr, and beyond<a class="headerlink" href="#sweave-and-knitr-an
       </div>
     </a>
     <a class="right-next"
-       href="../mathscourse/lecture01/01_Making_mathematical_statements.html"
+       href="../intro-Stats.html"
        title="next page">
       <div class="prev-next-info">
         <p class="prev-next-subtitle">next</p>
-        <p class="prev-next-title">01 - Making mathematical statements</p>
+        <p class="prev-next-title">Introduction</p>
       </div>
       <i class="fa-solid fa-angle-right"></i>
     </a>
@@ -5283,7 +5270,7 @@ <h3>Sweave and knitr, and beyond<a class="headerlink" href="#sweave-and-knitr-an
   <div class="footer-item">
     
 <p class="component-author">
-By TheMulQuaBio collective!
+By MulQuaBio collective!
 </p>
 
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diff --git a/notebooks/ShellScripting.html b/notebooks/ShellScripting.html
index d7775167..7b1957fc 100644
--- a/notebooks/ShellScripting.html
+++ b/notebooks/ShellScripting.html
@@ -8,7 +8,7 @@
     <meta charset="utf-8" />
     <meta name="viewport" content="width=device-width, initial-scale=1.0" /><meta name="viewport" content="width=device-width, initial-scale=1" />
 
-    <title>Shell scripting &#8212; TheMulQuaBio</title>
+    <title>Shell scripting &#8212; MulQuaBio</title>
   
   
   
@@ -153,8 +153,8 @@
       
     
     
-    <img src="../_static/logo.png" class="logo__image only-light" alt="TheMulQuaBio - Home"/>
-    <script>document.write(`<img src="../_static/logo.png" class="logo__image only-dark" alt="TheMulQuaBio - Home"/>`);</script>
+    <img src="../_static/logo.png" class="logo__image only-light" alt="MulQuaBio - Home"/>
+    <script>document.write(`<img src="../_static/logo.png" class="logo__image only-dark" alt="MulQuaBio - Home"/>`);</script>
   
   
 </a></div>
@@ -188,23 +188,9 @@
 <li class="toctree-l1"><a class="reference internal" href="Python.html">Biological Computing in Python</a></li>
 <li class="toctree-l1"><a class="reference internal" href="R.html">Biological Computing in R</a></li>
 </ul>
-<p aria-level="2" class="caption" role="heading"><span class="caption-text">Maths for Biologists</span></p>
-<ul class="nav bd-sidenav">
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture01/01_Making_mathematical_statements.html">01 - Making mathematical statements</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture02/02_Describing_shapes_and_patterns.html">02   - Describing shapes and patterns</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture03/03_Analysing_change%2C_one_step_at_a_time.html">03   - Analysing change, one step at a time</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture04/04_From_table_arrangements_to_powerful_tools.html">04 - From table arrangements to powerful tools</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture05/05_From_table_arrangements_to_powerful_tools_part2.html">05 - From table arrangements to powerful tools - part 2</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture06/06_Analysing_change_continuously.html">06 - Analysing change, continuously</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture07/07_Analysing_change_continuously_part2.html">07 - Analysing change, continuously (Part 2)</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture08/08_Calculating_accumulated_change.html">08 - Calculating accumulated change</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/tutorials/lecture01/Tutorial_01.html">Tutorial 01</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/tutorials/lecture05/Tutorial_05.html">Tutorial 05</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/tutorials/lecture06/Tutorial_06.html">Tutorial 06</a></li>
-</ul>
 <p aria-level="2" class="caption" role="heading"><span class="caption-text">Basic Data Analyses and Statistics</span></p>
 <ul class="nav bd-sidenav">
-<li class="toctree-l1"><a class="reference internal" href="../Stats-Intro.html">Basic Data Science and Statistics: Introduction</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../intro-Stats.html">Introduction</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Data_R.html">Data Management and Visualization</a></li>
 <li class="toctree-l1"><a class="reference internal" href="ExpDesign.html">Experimental design and Data exploration</a></li>
 <li class="toctree-l1"><a class="reference internal" href="t_F_tests.html">Basic hypothesis testing: <span class="math notranslate nohighlight">\(t\)</span> and <span class="math notranslate nohighlight">\(F\)</span> tests</a></li>
@@ -224,6 +210,7 @@
 <li class="toctree-l1"><a class="reference internal" href="Appendix-JupyIntro.html">Introduction to Jupyter</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-Data-Python.html">Data analyses  with Python &amp; Jupyter</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-Maths.html">Mathematical models in Jupyter</a></li>
+<li class="toctree-l1"><a class="reference internal" href="Appendix-MathsForBiologists/MathsForBiologists.html">Maths for Biologists</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-Databases.html">Databases <span class="tocSkip"></span></a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-NLLS-Python.html">Model fitting in Python</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-MiniProj.html">The Computing Miniproject</a></li>
@@ -283,7 +270,7 @@
       
       
       
-      <li><a href="https://github.com/mhasoba/TheMulQuaBio" target="_blank"
+      <li><a href="https://github.com/MulQuaBio/MQB" target="_blank"
    class="btn btn-sm btn-source-repository-button dropdown-item"
    title="Source repository"
    data-bs-placement="left" data-bs-toggle="tooltip"
@@ -300,7 +287,7 @@
       
       
       
-      <li><a href="https://github.com/mhasoba/TheMulQuaBio/issues/new?title=Issue%20on%20page%20%2Fnotebooks/ShellScripting.html&body=Your%20issue%20content%20here." target="_blank"
+      <li><a href="https://github.com/MulQuaBio/MQB/issues/new?title=Issue%20on%20page%20%2Fnotebooks/ShellScripting.html&body=Your%20issue%20content%20here." target="_blank"
    class="btn btn-sm btn-source-issues-button dropdown-item"
    title="Open an issue"
    data-bs-placement="left" data-bs-toggle="tooltip"
@@ -1023,7 +1010,7 @@ <h2>Readings &amp; Resources<a class="headerlink" href="#readings-resources" tit
   <div class="footer-item">
     
 <p class="component-author">
-By TheMulQuaBio collective!
+By MulQuaBio collective!
 </p>
 
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diff --git a/notebooks/Space.html b/notebooks/Space.html
new file mode 100644
index 00000000..800300ff
--- /dev/null
+++ b/notebooks/Space.html
@@ -0,0 +1,518 @@
+
+<!DOCTYPE html>
+
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+<html lang="en" data-content_root="../" >
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+    <meta name="viewport" content="width=device-width, initial-scale=1.0" /><meta name="viewport" content="width=device-width, initial-scale=1" />
+
+    <title>Spatial Ecology and Evolution &#8212; MulQuaBio</title>
+  
+  
+  
+  <script data-cfasync="false">
+    document.documentElement.dataset.mode = localStorage.getItem("mode") || "";
+    document.documentElement.dataset.theme = localStorage.getItem("theme") || "";
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+    <h1>The Turbulent Marriage of Ecology and Evolutionary Biology</h1>
+    <!-- Table of contents -->
+    <div id="print-main-content">
+        <div id="jb-print-toc">
+            
+            <div>
+                <h2> Contents </h2>
+            </div>
+            <nav aria-label="Page">
+                <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#introduction">Introduction</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-roots-of-disharmony">The Roots of Disharmony</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#contemporary-ecology-vis-a-vis-evolutionary-biology">Contemporary ecology vis-à-vis evolutionary biology</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#an-early-association-and-subsequent-divergence-1850-1950">An early association and subsequent divergence: 1850-1950</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#marriage-and-the-quest-for-harmony-1950-1985">Marriage and the quest for harmony: 1950-1985</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-marriage-of-ecology-and-evolutionary-biology-the-ups-the-downs-and-a-taxonomy">The marriage of ecology and evolutionary biology: the ups, the downs, and a taxonomy</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#the-issue-of-ecological-vs-evolutionary-time-scales">The issue of ecological vs. evolutionary time scales</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#demographic-considerations-and-measures-of-fitness">Demographic considerations and measures of fitness</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#the-role-of-environmental-heterogeneity">The role of environmental heterogeneity</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#taxonomy-of-evolutionary-ecology">Taxonomy of evolutionary ecology**</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#discussion">Discussion</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#has-the-marriage-come-of-age">Has The Marriage Come Of Age?</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#on-the-dullness-of-ecology">On the Dullness of Ecology</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#conclusion">Conclusion</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#references">References</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#footnotes">Footnotes</a></li>
+</ul>
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+                  
+  <section class="tex2jax_ignore mathjax_ignore" id="the-turbulent-marriage-of-ecology-and-evolutionary-biology">
+<h1>The Turbulent Marriage of Ecology and Evolutionary Biology<a class="headerlink" href="#the-turbulent-marriage-of-ecology-and-evolutionary-biology" title="Link to this heading">#</a></h1>
+<blockquote class="epigraph">
+<div><p>‘If there’s no meaning in it,’ said the King, ‘that saves a world of trouble, you know, as we needn’t try to find any…’</p>
+<p class="attribution">—Alice in Wonderland
+</p>
+</div></blockquote>
+<section id="introduction">
+<h2>Introduction<a class="headerlink" href="#introduction" title="Link to this heading">#</a></h2>
+<p>Few would hesitate in identifying ecology and evolutionary biology as distinct field of research with somwhat disparate goals. In this  chapter, we will examine the alliance between these two, which have been distinct programs of research for more than a century now. In particular, we will focus on the most theoretically and empirically tangible part of this alliance: the unification of population ecology and population genetics.</p>
+<p>We are generally taught Ecological and Evoluonary theory and practice separately; one need only flip though an introductory text in biology to see how well-established this separation is. Indeed, a majority of the practitioners of either discipline continue to conduct research with limited perspectives from the other, perhaps, albeit subliminally, taking the stance of the King of Hearts in Alice quoted above! In this introductory chapter we will examine whether this lack of collaboration hinders each discipline’s progress, what the roots for discord are, and the current status of the collaboration (the “Marriage”).</p>
+<p>This is an appropriate time to make this evaluation, as the last two decades decade or so in particular have seen a resurgence of theoretical and empirical work that combines ecological and evolutionary concepts, theory and empirical data (and apporaches, driven by advances in molecular bilogy and muti-omics). Another, more philosophical motivation for this chapter is that ecology and evolution have been noticeably underrepresented in accounts of the philosophy of biology <span id="id1">[<a class="reference internal" href="#id134" title="Marjorie Grene and David Depew. The philosophy of biology: an episodic history. Cambridge University Press, 2004.">Grene and Depew, 2004</a>, <a class="reference internal" href="#id192" title="David L Hull. Philosophy of biological science. Prentice-Hall, Englewood Cliffs, N.J, 1974.">Hull, 1974</a>, <a class="reference internal" href="#id136" title="Elliott Sober. The nature of selection: evolutionary theory in philosophical focus. University of Chicago Press, Chicago, 1993.">Sober, 1993</a>]</span>. The reasons for this ostensible disinterest are worth exploring. Both cultural and conceptual differences between these disciplines have hindered their synthesis, and that this discord, and the nature of the current synthesis themselves are fertile grounds for philosophical analyses. Moreover, by highlighting the interdependence of ecological and evolutionary theory, and the importance of a synthesis to the overall progress of biology, we will also reexamine the view that ecology appears dull to philosophers because in its classical mold it is theoretically and causally redundant within biology <span id="id2">[<a class="reference internal" href="#id195" title="Yrjö Haila and P Taylor. The philosophical dullness of classical ecology, and a Levinsian alternative. Biol. Philos., 16(1):93–102, 2001.">Haila and Taylor, 2001</a>]</span>.</p>
+<p>Ecology today comprises a heterogeneous array of disciplines, transcending levels of complexity from individuals and populations at one end to ecosystems at the other. An attempt to examine the role of evolutionary biology at each of these and vice versa is beyond the scope of this Chapter. We will therefore focus mainly on two fields that form the traditional foundation of ecology and evolutionary biology, and which have a long history of exchange and attempted union: population biology and population genetics. Nevertheless, instances where such a union has obvious implications for higher levels of complexity will be dwelt upon.</p>
+<p>In the following sections we will address:</p>
+<ul class="simple">
+<li><p>Goals of contemporary ecology and evolutionary biology.</p></li>
+<li><p>A brief review of historical relationship with focus on perceptible roots of discord as well as attempts for collaboration between them<a class="footnote-reference brackets" href="#id198" id="id3" role="doc-noteref"><span class="fn-bracket">[</span>2<span class="fn-bracket">]</span></a>.</p></li>
+<li><p>Current sources of discord between the two disciplines.</p></li>
+<li><p>Based upon the approach taken towards resolving these differences, a classification of current research themes in evolutionary ecology.</p></li>
+<li><p>The current status of this synthesis in light of recent theoretical and empirical advances.</p></li>
+<li><p>And finally, a discussion in light of the preceding results.</p></li>
+</ul>
+</section>
+<section id="the-roots-of-disharmony">
+<h2>The Roots of Disharmony<a class="headerlink" href="#the-roots-of-disharmony" title="Link to this heading">#</a></h2>
+<section id="contemporary-ecology-vis-a-vis-evolutionary-biology">
+<h3>Contemporary ecology vis-à-vis evolutionary biology<a class="headerlink" href="#contemporary-ecology-vis-a-vis-evolutionary-biology" title="Link to this heading">#</a></h3>
+<p>Ecology today covers a wide range of topics ranging from the level of the individual, to ecosystems. This naturally implies a varied set of objectives, but a common underlying theme can still be identified as being along the lines of what the inventor of the term Haeckel (see below) originally defined it: the relationship between organisms and their environment. These two elements also comprise distinct ways in which things “ecological” are taken to mean: to some, ecology is synonymous with the characteristics of individuals, populations, communities or ecosystems of organisms, while to others, ecology is the environment that affects the organisms. This environment of course, can be biotic as well as abiotic. Disciplines in contemporary ecology are largely devoted to analyses of patterns observed in nature, their origins, and their changes with respect to the environment. Compared to evolutionary biology, ecology tends to dwell more in the present, its practitioners being most concerned with the relationship between organisms and their biotic as well as abiotic environment in “ecological time scales” (temporal scales in the order of one of few generations of the studied organisms). The common approach used to study these aspects is a mechanistic, physical one, with a tacit assumption of purely non-evolutionary response, and without consideration of the genetic variation (Mayr 1997)(pp. 210-211). Relative to evolutionary biology, the hope for general principles that govern these relationships have spurred the development of very detailed analyses of organism-environment interactions.</p>
+<p>Evolutionary biology on the other hand, consists of both prospective and retrospective theoretical frameworks that are concerned with the origins of, and changes in living entities over varying time scales. At one end lies the mathematical theory of genetics that forms a basis for microevolutionary (evolution on relatively short temporal scales) research. At the same time, the fecund collaboration of molecular biology and classical evolutionary theory (Huxley 1942) now also contributes to macroevolutionary (evolution over longer, generally geological time scales) research through an increasing collaboration between phylogenetics, systematics, and paleontology. I now present a brief historical background with the objective of exploring the roots of discord as well as collaboration between these two disciplines.</p>
+</section>
+<section id="an-early-association-and-subsequent-divergence-1850-1950">
+<h3>An early association and subsequent divergence: 1850-1950<a class="headerlink" href="#an-early-association-and-subsequent-divergence-1850-1950" title="Link to this heading">#</a></h3>
+<p><span id="id4">Darwin [<a class="reference internal" href="#id129" title="Charles Darwin. On the origin of species by means of natural selection, or the preservation of favoured races in the struggle for life. P. F. Collier &amp; Son co., New York, 6 edition, 1859.">1859</a>]</span>, in the process of laying the foundations of evolutionary biology did so with explicit and deep consideration of the relationship between the organism and the environment. Arguably, this prompted Haeckel to coin the term <em>ökologie</em> in 1866 <span id="id5">[<a class="reference internal" href="#id131" title="Ernst Heinrich Philipp August Haeckel. Generelle morphologie der organismen: allgemeine grundzüge der organischen formen-wissenschaft, mechanisch begründet durch die von Charles Darwin reformirte descendenz-theorie. G. Reimer, Berlin, 1866.">Haeckel, 1866</a>]</span><a class="footnote-reference brackets" href="#id199" id="id6" role="doc-noteref"><span class="fn-bracket">[</span>3<span class="fn-bracket">]</span></a>, in whose words ecology was</p>
+<blockquote class="epigraph">
+<div><p>“<em>…the investigation of the total relations of the animal both to it’s inorganic and to its organic environment ….in other words, ecology is the study of all those complex interrelationships referred to by Darwin as the conditions for the struggle for existence…</em>”</p>
+</div></blockquote>
+<p>(quoted in <span id="id7">Stauffer and Stauffer [<a class="reference internal" href="#id132" title="R C Robert C Stauffer and B Y Robert C Stauffer. Haeckel, Darwin, and ecology. Q. Rev. Biol., 32(2):138–144, 1957.">1957</a>]</span>).</p>
+<p>Naturally, the first step towards achieving this goal was to study the effect of the environment on individuals and populations.</p>
+<p>The period from Haeckel’s definition to the early 19th century was a crucial period for the discipline, although the term “ecology” was not used much. This was when the tradition of detailed botanical and zoological study of morphology, anatomy and physiology of the individual in relation to its adaptive environment was established. This was also the period when Herbert Spencer published his ideas about the emergence of order in nature <span id="id8">[<a class="reference internal" href="#id133" title="Herbert Spencer. The principles of biology. William and Norgate, London etc., 1864.">Spencer, 1864</a>]</span>. Spencer’s writings strongly influenced the approach of future generations by embedding the notion of stability and the “balance of nature” in the ecological psyche. By the beginning of the 20th century, when the units of study in ecology had already been defined at multiple levels beyond the individual (population, communities, ecosystems, etc), and population as well as community ecology we coming to the fore as distinct disciplines, a strong belief in the equilibrium state of nature and its units had already taken hold. This was followed by the adoption of a physical view by many ecologists, wherein living units were transformers of energy enmeshed in larger systems with considerable resilience to perturbation, and where details of the evolutionary process could be glossed in interest of a more systemic approach <span id="id9">[<a class="reference internal" href="#id130" title="Sharon E Kingsland. Modeling nature episodes in the history of population ecology. University of Chicago Press, Chicago, 1985.">Kingsland, 1985</a>]</span>(“The beginnings of “systems ecology””; pp. 34-49)</p>
+<p>Apart from the systems approach to ecology, this was also the period when faith in the pre-adaptation of species and the slowness of evolution took firm hold. These beliefs freed the ecologist of evolutionary concerns, and allowed a single-minded commitment to the study of the nature of the adaptations in relation to the environment. The roots of the viewpoint that ecology and evolution operate at vastly different time scales are probably to be found in fact that including Darwin’s work, early evolutionary ideas were based on the fossil record, and tended to be “gradualist” <span id="id10">[<a class="reference internal" href="#id128" title="Steven M Stanley, J Roughgarden, RM May, and SA Levin. Fossils, macroevolution, and theoretical ecology. Perspectives in ecological theory, pages 125–134, 1989.">Stanley <em>et al.</em>, 1989</a>]</span>. In Darwin’s <span id="id11">[<a class="reference internal" href="#id129" title="Charles Darwin. On the origin of species by means of natural selection, or the preservation of favoured races in the struggle for life. P. F. Collier &amp; Son co., New York, 6 edition, 1859.">Darwin, 1859</a>]</span>(pp. 126-127) words,</p>
+<blockquote class="epigraph">
+<div><p>I do believe that natural selection will generally act very slowly, only at long intervals of time, and only on a few of the inhabitants of the same region. I further believe that these slow, intermittent results accord well with what geology tells us of the rate and manner at which the inhabitants of the world have changed.</p>
+</div></blockquote>
+<p>This view was reinforced by the fact that examples of natural life tangible to early observers mostly happened to be relatively long-lived vertebrates. In contrast, modern biologists we can see evolution in action due to advances in molecular and phenotyping technologies, especially in microbial populations (something we will revisit later in this chapter). Thus, in its development through the first part of the 20th century, the rapidly proliferating branches of ecology had largely diverged from evolutionary biology with only sporadic exchange between the two disciplines. Ecologists during time focused increasingly on the effect of the abiotic environment and biotic interactions on populations to the exclusion of any considerations of genetics or evolution, trying to deal with the onerous task of defining the characteristics and emergent properties of the various levels from individuals to ecosystems.</p>
+<p>The “The golden age of theoretical ecology” from 1920 to 1940 <span id="id12">[<a class="reference internal" href="#id172" title="Francesco M Scudo and James R Ziegler. The Golden age of theoretical ecology, 1923-1940: a collection of works by V. Volterra, V. A. Kostitzin, A. J. Lotka, and A. N. Kolmogoroff. Springer-Verlag, Berlin, 1978.">Scudo and Ziegler, 1978</a>]</span>, was marked by the rediscovery of the Verhulst logistic equation by Raymond Pearl in the 1920’s and its extension to interacting populations, as well as experimental evaluation <span id="id13">[<a class="reference internal" href="#id145" title="G F Gause. Experimental analysis of vito volterra's mathematical theory of the struggle for existence. Science, 79(2036):16–17, January 1934.">Gause, 1934</a>]</span> (e.g., <span id="id14">[<a class="reference internal" href="#id130" title="Sharon E Kingsland. Modeling nature episodes in the history of population ecology. University of Chicago Press, Chicago, 1985.">Kingsland, 1985</a>]</span>). The approach of theoretical ecology then was to define state variables representing properties of the population as a whole with a tacit assumption of genetic homogeneity. For example, the coupled differential equations of the classic Lotka-Volterra equations of competition between two species with population sizes <span class="math notranslate nohighlight">\(N_1\)</span>, <span class="math notranslate nohighlight">\(N_2\)</span>,</p>
+<div class="math notranslate nohighlight" id="equation-eq-turbulent-lv-competition">
+<span class="eqno">()<a class="headerlink" href="#equation-eq-turbulent-lv-competition" title="Link to this equation">#</a></span>\[\begin{split}\begin{align}
+    \frac{dN_1}{dt} &amp;= r_1 N_1 \left( 1 - \frac{N_1 + \alpha_{12} N_2}{K_1} \right) \\
+    \frac{dN_2}{dt} &amp;= r_2 N_2 \left( 1 - \frac{N_2 + \alpha_{21} N_1}{K_2} \right),
+\end{align}\end{split}\]</div>
+<p>include the following immutable parameters: <span class="math notranslate nohighlight">\(r_1\)</span>, <span class="math notranslate nohighlight">\(r_2\)</span> (intrinsic growth rates of species’ 1 &amp; 2), <span class="math notranslate nohighlight">\(K_1\)</span>, <span class="math notranslate nohighlight">\(K_2\)</span> (carrying capacities), and <span class="math notranslate nohighlight">\(α_{12}\)</span>, <span class="math notranslate nohighlight">\(α_{21}\)</span> (coefficients of competition).</p>
+<p>It is clear that many of the pioneers of theoretical population biology during this period were, to various degrees, motivated by Darwin’s theory of natural selection and the then prevalent schools of evolutionary thought<a class="footnote-reference brackets" href="#id200" id="id15" role="doc-noteref"><span class="fn-bracket">[</span>4<span class="fn-bracket">]</span></a>. This period also coincided with the mathematical fusion of Mendelian genetics and Darwin’s theory by Fisher, Haldane and Wright <span id="id16">[<a class="reference internal" href="#id168" title="Julian Huxley. Evolution, the modern synthesis. G. Allen &amp; Unwin ltd, London, 1942.">Huxley, 1942</a>]</span>(the “Modern Synthesis” of evolutionary biology).</p>
+<div class="cell docutils container">
+<div class="cell_input docutils container">
+<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
+<span class="kn">from</span> <span class="nn">scipy.integrate</span> <span class="kn">import</span> <span class="n">odeint</span>
+
+<span class="c1"># Define Lotka-Volterra competition model</span>
+<span class="k">def</span> <span class="nf">lotka_volterra</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="n">r1</span><span class="p">,</span> <span class="n">r2</span><span class="p">,</span> <span class="n">K1</span><span class="p">,</span> <span class="n">K2</span><span class="p">,</span> <span class="n">alpha12</span><span class="p">,</span> <span class="n">alpha21</span><span class="p">):</span>
+    <span class="n">N1</span><span class="p">,</span> <span class="n">N2</span> <span class="o">=</span> <span class="n">y</span>
+    <span class="n">dN1_dt</span> <span class="o">=</span> <span class="n">r1</span> <span class="o">*</span> <span class="n">N1</span> <span class="o">*</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="p">(</span><span class="n">N1</span> <span class="o">+</span> <span class="n">alpha12</span> <span class="o">*</span> <span class="n">N2</span><span class="p">)</span> <span class="o">/</span> <span class="n">K1</span><span class="p">)</span>
+    <span class="n">dN2_dt</span> <span class="o">=</span> <span class="n">r2</span> <span class="o">*</span> <span class="n">N2</span> <span class="o">*</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="p">(</span><span class="n">N2</span> <span class="o">+</span> <span class="n">alpha21</span> <span class="o">*</span> <span class="n">N1</span><span class="p">)</span> <span class="o">/</span> <span class="n">K2</span><span class="p">)</span>
+    <span class="k">return</span> <span class="p">[</span><span class="n">dN1_dt</span><span class="p">,</span> <span class="n">dN2_dt</span><span class="p">]</span>
+
+<span class="c1"># Parameters</span>
+<span class="n">r1</span><span class="p">,</span> <span class="n">r2</span> <span class="o">=</span> <span class="mf">0.5</span><span class="p">,</span> <span class="mf">0.3</span>  <span class="c1"># Growth rates</span>
+<span class="n">K1</span><span class="p">,</span> <span class="n">K2</span> <span class="o">=</span> <span class="mi">50</span><span class="p">,</span> <span class="mi">30</span>    <span class="c1"># Carrying capacities</span>
+<span class="n">alpha12</span><span class="p">,</span> <span class="n">alpha21</span> <span class="o">=</span> <span class="mf">0.6</span><span class="p">,</span> <span class="mf">0.8</span>  <span class="c1"># Competition coefficients</span>
+<span class="n">y0</span> <span class="o">=</span> <span class="p">[</span><span class="mi">10</span><span class="p">,</span> <span class="mi">5</span><span class="p">]</span>  <span class="c1"># Initial populations</span>
+<span class="n">t</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">50</span><span class="p">,</span> <span class="mi">500</span><span class="p">)</span>  <span class="c1"># Time range</span>
+
+<span class="c1"># Solve ODEs</span>
+<span class="n">result</span> <span class="o">=</span> <span class="n">odeint</span><span class="p">(</span><span class="n">lotka_volterra</span><span class="p">,</span> <span class="n">y0</span><span class="p">,</span> <span class="n">t</span><span class="p">,</span> <span class="n">args</span><span class="o">=</span><span class="p">(</span><span class="n">r1</span><span class="p">,</span> <span class="n">r2</span><span class="p">,</span> <span class="n">K1</span><span class="p">,</span> <span class="n">K2</span><span class="p">,</span> <span class="n">alpha12</span><span class="p">,</span> <span class="n">alpha21</span><span class="p">))</span>
+
+<span class="c1"># Plot results</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mi">6</span><span class="p">))</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">result</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Species 1 (N1)&quot;</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">result</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;Species 2 (N2)&quot;</span><span class="p">,</span> <span class="n">lw</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">&quot;Lotka-Volterra Competition Dynamics&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;Time&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;Population Size&quot;</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">grid</span><span class="p">()</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
+</pre></div>
+</div>
+</div>
+<div class="cell_output docutils container">
+<img alt="../_images/0e3205167ee6fe026989b4ed40257f1e3dd61651f3a9d9a9350f32e0b5ebc84a.png" src="../_images/0e3205167ee6fe026989b4ed40257f1e3dd61651f3a9d9a9350f32e0b5ebc84a.png" />
+</div>
+</div>
+<p>Both population ecology and population genetics had, and still do, the same mathematical foundations at their core. For example, consider the case of an asexual haploid population with discrete, non-overlapping generations. Let time be measured in generations, such that population size at time <span class="math notranslate nohighlight">\(t\)</span>, <span class="math notranslate nohighlight">\(N(t)\)</span>, is the result of the previous generation’s population size, <span class="math notranslate nohighlight">\(N(t-1)\)</span> multiplied by a growth factor which combines survival and reproduction, say <span class="math notranslate nohighlight">\(w\)</span>:</p>
+<div class="math notranslate nohighlight" id="equation-eq-turbulent-fitness">
+<span class="eqno">()<a class="headerlink" href="#equation-eq-turbulent-fitness" title="Link to this equation">#</a></span>\[\begin{split}
+\begin{align}
+    N(t) &amp;= w N(t-1)\\
+        &amp;= w^2 N(t-2)\\
+        &amp;= ~...\\
+        &amp;= w^t N(0),
+\end{align}
+\end{split}\]</div>
+<p>yielding the classical model of population growth in discrete time. It is straightforward to extend equation <code class="xref eq docutils literal notranslate"><span class="pre">(eq:turbulent:fitness)</span></code> in population genetics terms. The growth factor <span class="math notranslate nohighlight">\(w\)</span> is the absolute fitness of the population, and can be written as <span class="math notranslate nohighlight">\(w = 1+s\)</span>, such that the population increases if <span class="math notranslate nohighlight">\(s&gt;0\)</span> and decreases if <span class="math notranslate nohighlight">\(s&lt;0\)</span>. The change in population size in one generation then is,</p>
+<div class="math notranslate nohighlight" id="equation-eq-turbulent-fitness-2">
+<span class="eqno">()<a class="headerlink" href="#equation-eq-turbulent-fitness-2" title="Link to this equation">#</a></span>\[\begin{split}
+\begin{align}
+\Delta N(t) 
+  &amp;= N(t+1) - N(t)\\ 
+  &amp;= (1 + s)N(t) - N(t)\\ 
+  &amp;= sN(t).
+\end{align}
+\end{split}\]</div>
+<div class="admonition note">
+<p class="admonition-title">Note</p>
+<p>You will learn about this fitness equation and its usage in ecology as well as evolutionary biology in the next Chapter.</p>
+</div>
+<p>Now, suppose the population contains <span class="math notranslate nohighlight">\(k\)</span> genotypes (alleles), each with its own growth rate (fitness), <span class="math notranslate nohighlight">\(w_i, \mid i = 1,2,...,k\)</span>. Let <span class="math notranslate nohighlight">\(n_i(t)\)</span> denote the population size of genotype <span class="math notranslate nohighlight">\(i\)</span> at time (=generation) <span class="math notranslate nohighlight">\(t\)</span>. Then, total population size at time <span class="math notranslate nohighlight">\(t\)</span> is <span class="math notranslate nohighlight">\(N(t) = \sum_in_i(t)\)</span>, and the mean fitness of the population is</p>
+<div class="math notranslate nohighlight" id="equation-eq-turbulent-fitness-mean">
+<span class="eqno">()<a class="headerlink" href="#equation-eq-turbulent-fitness-mean" title="Link to this equation">#</a></span>\[
+    \overline{W}(t) = \sum_{i=1}^{k} W_i \frac{n_i(t)}{N(t)}.
+\]</div>
+<p><span class="math notranslate nohighlight">\(\overline{W}\)</span> will will change over generations with changes in <span class="math notranslate nohighlight">\(n_i\)</span>. A continuous time extension of the population growth model shows the relationship between the population biologist’s Malthusian parameter <span class="math notranslate nohighlight">\(r\)</span> (also known as the intrinsic growth rate) and fitness <span class="math notranslate nohighlight">\(W\)</span>. Consider the population growth in continuous time <span class="math notranslate nohighlight">\(\lim_{\mathrm{d}t \to 0} \frac{\mathrm{d}N}{\mathrm{d}t} = rN.
+\)</span>, which has the solution (next Chapter),</p>
+<div class="math notranslate nohighlight" id="equation-eq-turbulent-exponential-growth">
+<span class="eqno">()<a class="headerlink" href="#equation-eq-turbulent-exponential-growth" title="Link to this equation">#</a></span>\[
+    N(t) = N(0)e^{rt}.
+\]</div>
+<p>This is the familiar equation of exponential population growth. Then by comparing eqns. <a class="reference internal" href="#equation-eq-turbulent-fitness">()</a> and <a class="reference internal" href="#equation-eq-turbulent-exponential-growth">()</a>, <span class="math notranslate nohighlight">\(w = e^{r t}\)</span>, or <span class="math notranslate nohighlight">\(r = \ln(w)\)</span>. If <span class="math notranslate nohighlight">\(w\)</span> is close to 1 (population size is not changing), i.e., <span class="math notranslate nohighlight">\(s&lt;&lt;1\)</span>, <span class="math notranslate nohighlight">\(\ln{w} = \ln(1+s) \approx s\)</span> i.e., <span class="math notranslate nohighlight">\(r \approx s\)</span>, thus allowing the Malthusian parameter to be used as a measure of fitness. This illustrates the interchangeability of the core concepts in population genetics and ecology, under some assumptions.</p>
+<p>During this coincident period of the development of population genetics and population biology however, little effort was made to combine the theories, although there were voices such as <span id="id17">[<a class="reference internal" href="#id180" title="Charles S Elton. Animal ecology. Macmillan, New York, 1927.">Elton, 1927</a>]</span> and <span id="id18">[<a class="reference internal" href="#id145" title="G F Gause. Experimental analysis of vito volterra's mathematical theory of the struggle for existence. Science, 79(2036):16–17, January 1934.">Gause, 1934</a>]</span> that espoused the importance of exchange between the two disciplines. In part, this was because the extensions of models such as eqns. <a class="reference internal" href="#equation-eq-turbulent-lv-competition">()</a> by themselves posed mathematical challenges without introducing the added complexity of genetics. Another reason is that these early voices of evolutionary ecology were drowned in the clamor created by the rapid proliferation of ecological subdisciplines <span id="id19">[<a class="reference internal" href="#id130" title="Sharon E Kingsland. Modeling nature episodes in the history of population ecology. University of Chicago Press, Chicago, 1985.">Kingsland, 1985</a>]</span>. For example, Gause, who was clearly motivated by Darwin’s theory (see introductory remarks in <span id="id20">Gause [<a class="reference internal" href="#id145" title="G F Gause. Experimental analysis of vito volterra's mathematical theory of the struggle for existence. Science, 79(2036):16–17, January 1934.">1934</a>]</span>) expressed the desire to eventually combine the population ecology of Volterra with the mathematical genetics of Haldane and others, but apparently never attempted this <span id="id21">[<a class="reference internal" href="#id130" title="Sharon E Kingsland. Modeling nature episodes in the history of population ecology. University of Chicago Press, Chicago, 1985.">Kingsland, 1985</a>]</span>(p. 160). A rare example of such work appears to be that of V. A. Kostinzin, whose efforts however passed largely unnoticed, and were cut short by the Second World War <span id="id22">[<a class="reference internal" href="#id130" title="Sharon E Kingsland. Modeling nature episodes in the history of population ecology. University of Chicago Press, Chicago, 1985.">Kingsland, 1985</a>]</span>(p. 144)<a class="footnote-reference brackets" href="#id201" id="id23" role="doc-noteref"><span class="fn-bracket">[</span>5<span class="fn-bracket">]</span></a>.</p>
+<p>On the other hand, the direction taken by evolutionary theorists during the early 20th century also resulted in disassociation from many aspects of ecology. The foundations of Mendelian population genetics were built by a bare-bones approach of achieving mathematical tractability by judicious approximation and selective omission of ecological reality. For instance, as shown above (and you will learn more about later), approximation of the Malthusian parameter <span class="math notranslate nohighlight">\(r\)</span> by the selection coefficient <span class="math notranslate nohighlight">\(s\)</span> involved (and still does in classical population genetics theory) the assumption of a fixed population size. The Hardy-Weinberg theorem (next Chapter) served as a useful model by allowing an estimation of expected allele frequencies in diploid populations under assumption of panmictic, random mating, infinitely sized populations that experience no selection, mutation, emigration or immigration. A relaxation of different assumptions then allowed a modeling of the various agents of evolution, an approach that is still used in classical population genetics.</p>
+<p>For example, a traditional approach to modeling the role of natural selection in a diploid sexual population is to track the changes in allele frequencies for a single locus and two alleles. Consider two alleles <span class="math notranslate nohighlight">\(1\)</span> &amp; <span class="math notranslate nohighlight">\(2\)</span>, with frequencies <span class="math notranslate nohighlight">\(p\)</span> &amp; <span class="math notranslate nohighlight">\(q\)</span>, respectively. Assume an infinite population size, which assures that there is no random loss of alleles (no genetic drift), and that the condition of fixed population size derived in eqns. <a class="reference internal" href="#equation-eq-turbulent-fitness">()</a> - <a class="reference internal" href="#equation-eq-turbulent-exponential-growth">()</a> is satisfied. Furthermore, assume completely random mating followed by selective mortality, such that zygotes in every generation are in Hardy-Weinberg (HW) proportions of <span class="math notranslate nohighlight">\(p^2\)</span>, <span class="math notranslate nohighlight">\(2pq\)</span> and <span class="math notranslate nohighlight">\(q^2\)</span> before selection. Denoting population sizes of the three genotypes as <span class="math notranslate nohighlight">\(N_{11}\)</span>, <span class="math notranslate nohighlight">\(N_{12}\)</span>, <span class="math notranslate nohighlight">\(N_{22}\)</span>, and assigning fitnesses <span class="math notranslate nohighlight">\(w_{11}\)</span>, <span class="math notranslate nohighlight">\(w_{12}\)</span> and <span class="math notranslate nohighlight">\(w_{22}\)</span> (loss of zygotes from mortality followed by reproductive gain) respectively, the number of individuals of each genotype in the next generation are, <span class="math notranslate nohighlight">\(N_{11}' = w_{11} N_{11}\)</span>, <span class="math notranslate nohighlight">\(N_{12}' = w_{12} N_{12}\)</span>, and <span class="math notranslate nohighlight">\(N_{22}' = w_{22} N_{22}\)</span>. Then the frequency of allele <em>1</em> in the next generation is,</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}\begin{align}
+    p' &amp;= \frac{N_{11}' + \tfrac12 N_{12}'}{N_{11}' + N_{12}' + N_{22}'}\\
+       &amp;= \frac{w_{11}N_{11} + w_{12}\tfrac12N_{12}}
+       {w_{11}N_{11} + w_{12}N_{12} + w_{22}N_{22}}\\
+       &amp;= \frac{w_{11}p^{2} + w_{12}pq}
+         {w_{11}p^{2} + w_{12}\bigl(2pq\bigr) + w_{22}q^{2}}.
+\end{align}\end{split}\]</div>
+<p>Now, dividing through by <span class="math notranslate nohighlight">\(w_{11}\)</span> adds the convenience of reducing the fitnesses to dimensionless quantities by redefining them <em>relative</em> to fitness of the <span class="math notranslate nohighlight">\(w11\)</span> genotype. The percentage advantage or disadvantage can then be denoted by the coefficient of selection, <span class="math notranslate nohighlight">\(s\)</span> which appeared in an <em>absolute</em> context in eqn. <a class="reference internal" href="#equation-eq-turbulent-fitness-2">()</a> above. For example, assuming a multiplicative fitness scheme,<br />
+$<span class="math notranslate nohighlight">\(\begin{array}
+    W_{11} &amp;= \frac{w_{11}}{w_{11}} = 1,\\
+    W_{12} &amp;= \frac{w_{12}}{w_{11}} = 1 - s,\\
+    W_{22} &amp;= \frac{w_{22}}{w_{11}} = (1 - s)^{2}.
+\end{array} \)</span>$</p>
+<p>Plugging these fitness values into eqn. <a class="reference internal" href="#equation-eq-turbulent-fitness">()</a> and remembering that <span class="math notranslate nohighlight">\(q = 1 - p\)</span>, yields after some algebraic manipulation,</p>
+<div class="math notranslate nohighlight">
+\[
+\Delta p = p' - p 
+  = p(1 - p) \frac{s}{ 1 - s + s p }.
+\]</div>
+<p>Thus by introducing relative fitnesses, and retaining all other, albeit ecologically unrealistic assumptions for HW equilibrium, a simple two parameter model of natural selection can be analyzed. The issue of population size and the manner in which fitness is measured, are important issues in the marriage of population biology and population genetics, and will be considered in the next section. Furthermore, population geneticists during this period increasingly channeled their efforts towards extending one locus two allele cases towards more genetically realistic multi-locus models involving epistasis and genetic recombination <span id="id24">[<a class="reference internal" href="#id197" title="William B Provine. The origins of theoretical population genetics. University of Chicago Press, Chicago, 2001.">Provine, 2001</a>]</span>. For mathematical tractability, this had to be at the cost of ecological reality such as demographic fluctuations and environmental heterogeneity, topics very much at the heart of ecological research.</p>
+<p>Evolutionary theory was not entirely oblivious to ecological reality though, and two distinct topics in particular impinged on evolutionary thought. One was demography, the evolutionary importance of which was championed by <span id="id25">[<a class="reference internal" href="#id127" title="Charles S Elton. Animal ecology and evolution. The Clarendon press, Oxford, 1930.">Elton, 1930</a>]</span>, and the other was ecological heterogeneity and niche partitioning, on which Gause <span id="id26">[<a class="reference internal" href="#id145" title="G F Gause. Experimental analysis of vito volterra's mathematical theory of the struggle for existence. Science, 79(2036):16–17, January 1934.">Gause, 1934</a>]</span> did seminal ecological work, and which really came to the fore in evolutionary ecology with Lack’s work on the Galapagos finches in the 1940’s <span id="id27">[<a class="reference internal" href="#id126" title="David Lambert Lack. Darwin's finches. University Press, Cambridge Eng., 1947.">Lack, 1947</a>]</span>. In the “Causes of Evolution” Haldane paid some attention to conjectures by <span id="id28">[<a class="reference internal" href="#id127" title="Charles S Elton. Animal ecology and evolution. The Clarendon press, Oxford, 1930.">Elton, 1930</a>]</span> and other evolutionary-minded ecologists about the role of finite population size and random extinction in evolution, but concluded that these factors had a minor role to play <span id="id29">[<a class="reference internal" href="#id191" title="J B S Haldane. The causes of evolution. Longmans, Green and co, London, 1932.">Haldane, 1932</a>]</span>(pp. 201-204). By the 1930’s, groundwork had been laid for modeling the effects of finite population sizes and spatial heterogeneity <span id="id30">[<a class="reference internal" href="#id160" title="Ronald A Fisher. The genetical theory of natural selection. Clarendon Press, Oxford, 1930.">Fisher, 1930</a>, <a class="reference internal" href="#id191" title="J B S Haldane. The causes of evolution. Longmans, Green and co, London, 1932.">Haldane, 1932</a>]</span>, which was developed further by the succeeding generation of population geneticists using stochastic theory. Wright’s “Shifting Balance Theory” for evolution, which invokes both finite population sizes and spatial heterogeneity is perhaps the most ecologically realistic population genetics theory coming from an evolutionary theorist of the period <span id="id31">[<a class="reference internal" href="#id170" title="Sewall Wright. Evolution in Mendelian populations. Bull. Math. Biol., 52(1-2):241–295, March 1931.">Wright, 1931</a>]</span>.<br />
+It is worth noting that by the early 20th century, the two aspects of ecology mentioned above, i.e., the study of populations, and the study of the environment, were clearly being given different weightage by different scientists in ecology and evolutionary biology. This is discussed further below, and it will be seen in the next section that this fact has had a strong impact on how evolutionary ecology or the synthesis of population biology and genetics is pursued today.</p>
+</section>
+</section>
+<section id="marriage-and-the-quest-for-harmony-1950-1985">
+<h2>Marriage and the quest for harmony: 1950-1985<a class="headerlink" href="#marriage-and-the-quest-for-harmony-1950-1985" title="Link to this heading">#</a></h2>
+<p>{cite:t}(Kingsland1985)(pp. 143-144) observes that there was a widespread disinterest apparent in the activities and views of ecologists and evolutionary biologists towards the other field from the 1920’s into the 1950’s. From closely entwined origins, ecology and evolutionary biology had diverged in many respects, resulting in deep-rooted cultural differences<a class="footnote-reference brackets" href="#id202" id="id32" role="doc-noteref"><span class="fn-bracket">[</span>6<span class="fn-bracket">]</span></a>. The historical background above indicates various factors that contributed to the divergence of the two fields. Most of these persist today, and will be considered in more detail in the next section.</p>
+<p>Around 1950, a renewed awareness of the chasm that had developed between evolutionary biology and ecology arose, with work such as David Lack’s research on the Galapagos finches reiterating the importance of ecological heterogeneity, biotic interactions, and niche partitioning <span id="id33">[<a class="reference internal" href="#id126" title="David Lambert Lack. Darwin's finches. University Press, Cambridge Eng., 1947.">Lack, 1947</a>]</span>. A shift away from the essentially equilibrium “balance of nature” view also played an important part in this. While a large body of ecologists continued to pursue the program of physical ecology involving detailed studies of the interface between the environment and ecological units ranging from individual to whole ecosystems <span id="id34">[<a class="reference internal" href="#id125" title="Alfred J Lotka. Elements of mathematical biology. Dover Publications, 1956.">Lotka, 1956</a>, <a class="reference internal" href="#id156" title="Eugene Pleasants Odum. Fundamentals of ecology. Saunders, Philadelphia, 1953.">Odum, 1953</a>, <a class="reference internal" href="#id164" title="F Weiling. Lotka, a. j.: elements of mathematical biology. dover publications inc., New‐York 1956; XXX + 465 S., 72 abb., 36 tabellen und 4 übersichtstabellen im anhang, preis \$ 2,45. Biom. Z., 7(3):208–208, January 1965.">Weiling, 1965</a>]</span>, the view had become distinctly more dynamic for many others. In part this was due to the publication of the “The Theory of Island Biogeography” by MacArthur and Wilson <span id="id35">[<a class="reference internal" href="#id149" title="Robert H MacArthur and Edward O Wilson. The theory of island biogeography. Princeton University Press, Princeton, N.J, 1967.">MacArthur and Wilson, 1967</a>]</span> which simultaneously introduced two ideas atypical to the ecological thought of that time; a neutral theory of maintenance of communities immigration-extinction dynamics, and a strong synthesis of ecological and evolutionary thought. MacArthur’s other work was also very influential, and within a short period of less than two decades, he initiated considerable shifts in the ecological community’s viewpoints about theory as well as empirical work <span id="id36">[<a class="reference internal" href="#id130" title="Sharon E Kingsland. Modeling nature episodes in the history of population ecology. University of Chicago Press, Chicago, 1985.">Kingsland, 1985</a>]</span>. He made many contributions to the theoretical fusion of population genetics and population biology, and had a big impact in drawing the attention of the ecologists towards evolutionary thinking. As Rosenzweig <span id="id37">[<a class="reference internal" href="#id123" title="M L Rosenzweig. No Title. Evol. Ecol., 1:1–3, 1987.">Rosenzweig, 1987</a>]</span> says,</p>
+<blockquote>
+<div><p><em>MacArthur’s greatest contribution may have been that he seems to have got through to ecologists that they ignore evolution at the peril of missing the most predictive aspects and beguiling issues in ecology.</em></p>
+</div></blockquote>
+<p>Additionally, the interface between genetics and ecology was also beginning to be seen as an exciting and important area of theoretical and empirical work as the discovery of the genetic code, and advances in bio-molecular techniques allowed hitherto impossible quantification of genetic variation in nature <span id="id38">[<a class="reference internal" href="#id194" title="E B Ford. Ecological Genetics. Chapman &amp; Hall, London, 1964.">Ford, 1964</a>, <a class="reference internal" href="#id122" title="Richard C Lewontin. The genetic basis of evolutionary change. Columbia University Press, New York, 1974.">Lewontin, 1974</a>]</span>. This spurred on the efforts to combine models of population genetics and ecology. The period from 1960 onwards saw frequent appearances of papers, seminars, and books that displayed the intention to bring the theories and practices of ecology and evolution together. These included the works of both, ecological minded evolutionary biologists and vice versa <span id="id39">[<a class="reference internal" href="#id117">Bradshaw and Shorrocks, 1984</a>, <a class="reference internal" href="#id115" title="Peter F Brussard and R W Allard. Ecological genetics: the interface. Springer-Verlag, New York, 1978.">Brussard and Allard, 1978</a>, <a class="reference internal" href="#id190" title="Martin L Cody and Jared M Diamond. Ecology and evolution of communities. Belknap Press of Harvard University Press, Cambridge, Mass, 1975.">Cody and Diamond, 1975</a>, <a class="reference internal" href="#id113" title="Joseph H Connell, David B Mertz, and William W Murdoch. Readings in ecology and ecological genetics. Harper &amp; Row, New York, 1970.">Connell <em>et al.</em>, 1970</a>, <a class="reference internal" href="#id193" title="Peter S Dawson and Charles Everett King. Readings in population biology. Prentice-Hall, Englewood Cliffs, N.J, 1971.">Dawson and King, 1971</a>, <a class="reference internal" href="#id114" title="Ellen T Drake. Evolution and environment: a symposium presented on the occasion of the one hundredth anniversary of the foundation of Peabody Museum of Natural History at Yale University. Yale University Press, New Haven, 1968.">Drake, 1968</a>, <a class="reference internal" href="#id177" title="John Merritt Emlen. Ecology: an evolutionary approach. Addison-Wesley Pub. Co, Reading, Mass, 1973.">Emlen, 1973</a>, <a class="reference internal" href="#id194" title="E B Ford. Ecological Genetics. Chapman &amp; Hall, London, 1964.">Ford, 1964</a>, <a class="reference internal" href="#id116" title="Samuel Karlin and Eviatar Nevo. Population genetics and ecology: proceedings of the conference held in Israel, March 1975. Academic Press, New York, 1976.">Karlin and Nevo, 1976</a>, <a class="reference internal" href="#id124" title="Richard Levins. Evolution in changing environments. Monographs in Population Biology. Princeton University Press, Princeton, NJ, August 1968.">Levins, 1968</a>, <a class="reference internal" href="#id171" title="Richard C Lewontin. Population biology and evolution: proceedings of the international symposium, June 7-9, 1967, Syracuse, New York. Syracuse University Press, Syracuse, N.Y., 1968.">Lewontin, 1968</a>, <a class="reference internal" href="#id122" title="Richard C Lewontin. The genetic basis of evolutionary change. Columbia University Press, New York, 1974.">Lewontin, 1974</a>, <a class="reference internal" href="#id120" title="Eric R Pianka and Herbert Parkes Riley. Evolutionary ecology. Dickenson Pub. Co, San Francisco, 1970.">Pianka and Riley, 1970</a>, <a class="reference internal" href="#id173" title="J Roughgarden. Theory of Population Genetics and Evolutionary Ecology: an Introduction. Macmillan Publishing Co., Inc., New York, 1979.">Roughgarden, 1979</a>, <a class="reference internal" href="#id119" title="K P Sammeta and R Levins. Genetics and ecology. Annu. Rev. Genet., 4(4):469–488, January 1970.">Sammeta and Levins, 1970</a>, <a class="reference internal" href="#id121" title="Otto Thomas Solbrig. Demography and evolution in plant populations. University of California Press, Berkeley, 1980.">Solbrig, 1980</a>, <a class="reference internal" href="#id176" title="Otto Thomas Solbrig and Dorothy J Solbrig. Introduction to population biology &amp; evolution. Addison-Wesley Pub. Co, Reading, Mass, 1979.">Solbrig and Solbrig, 1979</a>, <a class="reference internal" href="#id146" title="Bernard Stonehouse and Christopher M Perrins. Evolutionary ecology. University Park Press, Baltimore, Md, 1977.">Stonehouse and Perrins, 1977</a>, <a class="reference internal" href="#id112" title="Donald R Strong. Ecological communities: conceptual issues and the evidence. Princeton University Press, Princeton, N.J, 1984.">Strong, 1984</a>, <a class="reference internal" href="#id118" title="K Wöhrmann and V Loeschcke. Population biology and evolution. Springer-Verlag, Berlin, 1984.">Wöhrmann and Loeschcke, 1984</a>]</span>. The first textbooks explicitly attempting to present an evolutionary ecology perspective also appeared during this period <span id="id40">[<a class="reference internal" href="#id187" title="Philip W Hedrick. Population biology: the evolution and ecology of populations. Jones and Bartlett Publishers, Boston, 1984.">Hedrick, 1984</a>, <a class="reference internal" href="#id184" title="Robert H MacArthur and Joseph H Connell. The biology of populations. Wiley, New York, 1966.">MacArthur and Connell, 1966</a>, <a class="reference internal" href="#id120" title="Eric R Pianka and Herbert Parkes Riley. Evolutionary ecology. Dickenson Pub. Co, San Francisco, 1970.">Pianka and Riley, 1970</a>]</span>).</p>
+<p>Naturally, these forays into evolutionary ecology reflected the difference in approach of scientists. Introducing the aims and scope of the journal Evolutionary Ecology <span id="id41">[<a class="reference internal" href="#id123" title="M L Rosenzweig. No Title. Evol. Ecol., 1:1–3, 1987.">Rosenzweig, 1987</a>]</span> says,</p>
+<blockquote>
+<div><p><em>…some of our articles will examine the evolution of natural ecosystems. Others will stress how the external environment, biotic or abiotic, influences the course of evolution. Finally, there will be some that study how the environment actually molds the evolutionary mechanisms themselves.</em></p>
+</div></blockquote>
+<p>Among these, the most common theme was the role of the abiotic and biotic milieu in guiding evolution, well expressed in Hutchinson’s <span id="id42">[<a class="reference internal" href="#id111" title="G Evelyn Hutchinson. The ecological theater and the evolutionary play. Yale University Press, New Haven, 1965.">Hutchinson, 1965</a>]</span>metaphor of “<em>The ecological theater and the evolutionary play”.</em> This focus on ecology as a background for evolutionary processes continues to be a dominant theme in evolutionary ecology (see below)<a class="footnote-reference brackets" href="#id203" id="id43" role="doc-noteref"><span class="fn-bracket">[</span>7<span class="fn-bracket">]</span></a>.</p>
+<p>As always, there was a difference in approach between ecologists and evolutionary biologists towards the synthesis. Whereas ecologists focused on evolution with respect to population dynamics, interspecific interactions and environmental aspects external to focal populations (e.g., heterogeneity of the biotic and abiotic habitat), ecologically oriented evolutionary geneticists such as Lewontin and Ford made detailed investigations of genetic variation in nature with a more population genetics approach. The term “Ecological Genetics” was coined by Ford in 1964, a field which in his words, <em>“…deals with the adjustments and adaptations of wild populations to their environment”</em> <span id="id44">[<a class="reference internal" href="#id194" title="E B Ford. Ecological Genetics. Chapman &amp; Hall, London, 1964.">Ford, 1964</a>]</span>. This program was bolstered by further development of population genetics theory on evolution in spatially heterogeneous environments, that had been initially formalized by Wright, Fisher and Haldane <span id="id45">[<a class="reference internal" href="#id197" title="William B Provine. The origins of theoretical population genetics. University of Chicago Press, Chicago, 2001.">Provine, 2001</a>]</span><a class="footnote-reference brackets" href="#id204" id="id46" role="doc-noteref"><span class="fn-bracket">[</span>8<span class="fn-bracket">]</span></a>.</p>
+<p>A significant development during this period, which had a huge impact on the evolutionary ecology synthesis, was the development of quantitative genetics theory <span id="id47">[<a class="reference internal" href="#id110" title="D S Falconer. Introduction to quantitative genetics. Ronald Press Co, Burnt Mill, Harlow, Essex, England, 1960.">Falconer, 1960</a>]</span>. Based on Fisher’s <span id="id48">[<a class="reference internal" href="#id160" title="Ronald A Fisher. The genetical theory of natural selection. Clarendon Press, Oxford, 1930.">Fisher, 1930</a>]</span> initial work and the ideas of the biometrician school of genetics <span id="id49">[<a class="reference internal" href="#id197" title="William B Provine. The origins of theoretical population genetics. University of Chicago Press, Chicago, 2001.">Provine, 2001</a>]</span>, this involved modeling evolution of quantitative, polygenic traits without explicit description of the underlying genetics, by using statistical descriptors (e.g., mean &amp; variance) of trait values. This approach was extremely useful from the viewpoint of empirical work, as it generated predictions about expected changes in continuously distributed trait values and indirect measurements of the strength of selection and fitness components from field data. Quantitative genetics therefore quickly became an integral component of ecological genetics. In a way, quantitative genetics was the ideal middle ground between population genetics and population ecology, as it involved a simplification of certain genetic as well as ecological complexities, yet addressing realistic scenarios. For example, consider a trait value <span class="math notranslate nohighlight">\(\theta\)</span>, which is continuously distributed in a population as a Gaussian function <code class="docutils literal notranslate"><span class="pre">p(θ)</span></code>(i.e, the trait values are assumed to be normally distributed). Assume that there is no gene flow or mutation affecting the character, nor linkage disequilibrium <span id="id50">[<a class="reference internal" href="#id109" title="Montogemery Slatkin. Ecological Character Displacement. Ecology, 61(1):163, February 1980.">Slatkin, 1980</a>]</span>. As before, Let <span class="math notranslate nohighlight">\(w(\theta)\)</span> be the absolute fitness of an individual with trait <span class="math notranslate nohighlight">\(\theta\)</span>. Various selection models can be used to generate values of <span class="math notranslate nohighlight">\(w(θ)\)</span> (see below). Assuming that the <span class="math notranslate nohighlight">\(w(\theta)\)</span> functions are given, the change in mean and variance of the trait values are respectively,</p>
+<div class="math notranslate nohighlight">
+\[
+\overline{z}(t+1) 
+  = \frac{\sigma^{2}}{\sigma^{2} + \sigma_{e}^{2}}
+    \int \theta  p(\theta) w(\theta) \frac{d\theta}{ \overline{W} }
+\]</div>
+<p>and</p>
+<div class="math notranslate nohighlight">
+\[
+\sigma^{2}(t+1)
+  = h^{4} \int \bigl[\theta - \overline{\theta}(t)\bigr]^{2}  
+    p(\theta) w(\theta) \frac{d\theta}{\overline{W}}
+    \;+\;
+    \Bigl(1 \;-\; \frac{\sigma^{2}}{\sigma^{2} + \sigma_{e}^{2}}\Bigr) 
+    \sigma^{2}(t).
+\]</div>
+<p>where <span class="math notranslate nohighlight">\(\sigma_e\)</span> is the environmental component of the variance, and <span class="math notranslate nohighlight">\(\overline{W} = \int p(\theta) w(\theta) d \theta\)</span>. The term <span class="math notranslate nohighlight">\(\frac{\sigma^{2}}{\sigma^{2} + \sigma_{e}^{2}}\)</span> is traditionally called the heritability (<span class="math notranslate nohighlight">\(h^2\)</span>). Such models can be used to examine the effect of a wide range of selection mechanisms, depending upon what functions are used to define the <span class="math notranslate nohighlight">\(w(\theta)\)</span>’s. Quantitative genetics circumvented some of the problems faced by previous attempts to model selection under frequency and density dependent population growth <span id="id51">[<a class="reference internal" href="#id122" title="Richard C Lewontin. The genetic basis of evolutionary change. Columbia University Press, New York, 1974.">Lewontin, 1974</a>, <a class="reference internal" href="#id163" title="R H Robert H Macarthur. Some generalized theorems of natural selection. Proc. Natl. Acad. Sci. …, 18(11):1893–1897, November 1962.">Macarthur, 1962</a>]</span><a class="footnote-reference brackets" href="#id205" id="id52" role="doc-noteref"><span class="fn-bracket">[</span>9<span class="fn-bracket">]</span></a> by allowing tractable modeling of continuously distributed traits with definable fitness measures (see below). Its application to modeling niche width, species interactions, character displacement, and niche partitioning had a huge impact on the progress of the evolutionary ecology synthesis in the 1970’s <span id="id53">[<a class="reference internal" href="#id143" title="Jonathan Roughgarden. Evolution of niche width. Am. Nat., 106(952):683–718, November 1972.">Roughgarden, 1972</a>, <a class="reference internal" href="#id106" title="M Slatkin and J Maynard Smith. Models of coevolution. Q. Rev. Biol., 54(3):233–263, 1979.">Slatkin and Maynard Smith, 1979</a>, <a class="reference internal" href="#id108" title="Montgomery Slatkin. Selection and polygenic characters. Proc. Natl. Acad. …, 66(1):87–93, 1970.">Slatkin, 1970</a>, <a class="reference internal" href="#id107" title="Montgomery Slatkin. The evolutionary response to frequency- and density-dependent interactions. Am. Nat., 114(3):384–398, 1979.">Slatkin, 1979</a>, <a class="reference internal" href="#id109" title="Montogemery Slatkin. Ecological Character Displacement. Ecology, 61(1):163, February 1980.">Slatkin, 1980</a>]</span> by creating the first realistic evolutionary versions of hitherto purely demographic models <span id="id54">[<a class="reference internal" href="#id149" title="Robert H MacArthur and Edward O Wilson. The theory of island biogeography. Princeton University Press, Princeton, N.J, 1967.">MacArthur and Wilson, 1967</a>]</span>.</p>
+<p>Another important step towards a demographically realistic evolutionary theory was the extension of models of selection to continuously breeding, age-structured populations <span id="id55">[<a class="reference internal" href="#id105" title="Brian Charlesworth. Evolution in age-structured populations. Cambridge University Press, Cambridge Eng., 1980.">Charlesworth, 1980</a>]</span>. The temporal aspect of environmental heterogeneity and its effect on evolution also received attention during this period from both ecologists and evolutionary biologists. The approach again, was rather different, with population biologists tending to use demographically explicit models <span id="id56">[<a class="reference internal" href="#id124" title="Richard Levins. Evolution in changing environments. Monographs in Population Biology. Princeton University Press, Princeton, NJ, August 1968.">Levins, 1968</a>, <a class="reference internal" href="#id104" title="M Slatkin and R Lande. Niche width in a fluctuating environment: density independent model. Am. Nat., 110(971):31–55, 1976.">Slatkin and Lande, 1976</a>]</span>, whereas population geneticists, following Kimura’s neutral theory of molecular evolution <span id="id57">[<a class="reference internal" href="#id103" title="M Kimura. The Neutral Theory of Molecular Evolution. Cambridge University Press, Cambridge, 1983.">Kimura, 1983</a>]</span>, focused on genetic polymorphism, using stochastic theory to address questions of evolution under fluctuating selection by building on preliminary work by Haldane and Jayakar <span id="id58">[<a class="reference internal" href="#id101" title="John H Gillespie. The causes of molecular evolution. Oxford University Press, Oxford, U.K., 1991.">Gillespie, 1991</a>, <a class="reference internal" href="#id102" title="J B S Haldane and S D Jayakar. Polymorphism due to selection of varying direction. J. Genet., 58:237–242, 1963.">Haldane and Jayakar, 1963</a>]</span>. The idea of temporally fluctuating environments was also central to the development of theory on the evolution of sex <span id="id59">[<a class="reference internal" href="#id142" title="John Maynard Smith. The evolution of sex. Cambridge University Press, Cambridge Eng., 1978.">Maynard Smith, 1978</a>]</span>.</p>
+<p>However, along with this upsurge of exchange between ecology and evolutionary biology from the 1950’s onwards, other issues within the two disciplines increasingly claimed attention, contributing further to the overall divergence of the two. Evolutionary biologists on one hand, enthused by advances in molecular biology were increasingly caught up in the gene-centered view of life, which was essentially reductionist <code class="docutils literal notranslate"><span class="pre">Sarkar1998</span></code>. Apart from the different ways, as mentioned above, in which ecological realism was brought in by population geneticists into their models, a large body of biologists delved deeper into genetic models of selection and evolution, ignoring many aspects of genotype environment relationships. This reductionism has been lamented as being one of the causes of disunity between population biology and population genetics <span id="id60">[<a class="reference internal" href="#id100" title="Francisco Jose Ayala and Theodosius Grigorievich Dobzhansky. Studies in the philosophy of biology: reduction and related problems. University of California Press, Berkeley, 1974.">Ayala and Dobzhansky, 1974</a>, <a class="reference internal" href="#id124" title="Richard Levins. Evolution in changing environments. Monographs in Population Biology. Princeton University Press, Princeton, NJ, August 1968.">Levins, 1968</a>, <a class="reference internal" href="#id99" title="Richard Levins. Toward a population biology, still. In Rama S Singh and Marcy K Uyenoyama, editors, The Evolution of Population Biology, pages 21–48. Cambridge University Press, Cambridge, January 2004.">Levins, 2004</a>]</span>.</p>
+<p>In ecology on the other hand, debates raged over the nature of community assembly, measures of stability in biological systems, the relative role of extrinsic vs. intrinsic factors driving ecological dynamics, the relationship between species diversity and community stability, the role of history in ecological patterns, and importance of processes at different spatial scales in determining diversity. Along the way, ecological sub-disciplines multiplied, with the addition of fields such as metapopulation ecology, metacommunity ecology, spatial ecology and landscape ecology. Unlike in evolutionary biology, a large rift also developed between theory and empirical work, with field biologists finding it tough going to glean testable predictions from complex, mechanistic models often inspired by physics, raising deep philosophical concerns about the role of theory in ecology and the place of ecology in biology <span id="id61">[<a class="reference internal" href="#id151">Kareiva <em>et al.</em>, 1989</a>, <a class="reference internal" href="#id130" title="Sharon E Kingsland. Modeling nature episodes in the history of population ecology. University of Chicago Press, Chicago, 1985.">Kingsland, 1985</a>, <a class="reference internal" href="#id98" title="Kim Sterelny and Paul E Griffiths. Sex and death: an introduction to philosophy of biology. University of Chicago Press, Chicago, Ill, 1999.">Sterelny and Griffiths, 1999</a>]</span>. Field ecology became increasingly preoccupied with conducting detailed characterizations of ecological patterns at various levels of complexity ranging from individuals to ecosystems. The hope was that cataloging patterns and processes would somehow reveal general principles. This was the period when global patterns such as “Rapoport’s rule” <span id="id62">[<a class="reference internal" href="#id154" title="K J Gaston, T M Blackburn, and J I Spicer. Rapoport's rule: time for an epitaph? Trends Ecol. Evol., 13(2):70–74, 1998.">Gaston <em>et al.</em>, 1998</a>]</span> and “Bergmann’s rule” <span id="id63">[<a class="reference internal" href="#id97" title="R P Freckleton, P H Harvey, and M Pagel. Bergmann's rule and body size in mammals. Am. Nat., 161(5):821–825, 2003.">Freckleton <em>et al.</em>, 2003</a>]</span> were described. A formidable array of statistical techniques were also developed during this period to allow a mechanistic characterization of organism environment interrelationships. Evolutionary ecology took somewhat of a back-seat in these pursuits; especially because apart from a few exceptions, most of the focal biological systems for empirical work were vertebrates, where the role of evolution was considered insignificant at ecological time scales. Instead, complexity of the systems being studied commanded almost complete attention.</p>
+<p>Nevertheless, in the meantime the unification of population genetics and population biology continued by a small group of biologists. From the 1980’s onwards in particular, the effort came closer to achieving a more seamless union due to the use of increasingly sophisticated mathematical treatment and careful empirical testing with new model systems by a new breed of biologists who dwelt comfortably in both ecological and evolutionary hemispheres. In particular, the convergence of multilocus population genetic and quantitative genetic models (Burger 2000)(<a class="reference external" href="https://paperpile.com/c/lwDPbc/fycI">https://paperpile.com/c/lwDPbc/fycI</a>) allowed greater ecological flexibility as well as genetic reality to evolutionary ecology modeling (Michael Doebeli 1996b, 1996a)(<a class="reference external" href="https://paperpile.com/c/lwDPbc/Abha+dbVj">https://paperpile.com/c/lwDPbc/Abha+dbVj</a>). In the process, these efforts diversified into various distinct approaches, which will be considered in the following section.</p>
+</section>
+<section id="the-marriage-of-ecology-and-evolutionary-biology-the-ups-the-downs-and-a-taxonomy">
+<h2>The marriage of ecology and evolutionary biology: the ups, the downs, and a taxonomy<a class="headerlink" href="#the-marriage-of-ecology-and-evolutionary-biology-the-ups-the-downs-and-a-taxonomy" title="Link to this heading">#</a></h2>
+<p>We can now attempt a classification of contemporary approaches in evolutionary ecology, based on three themes that appear to have traditionally been at the core of dissent between ecology and evolutionary biology: the issue of time-scale, the role of demography, and the effect of environmental heterogeneity. We will discuss these before presenting the classification.</p>
+<p>Of the three issues, “time-scale” is mainly a cultural issue, referring to the importance given to history in population biology. Although there is no definable boundary between micro- vs. macro- evolutionary, or ecological vs. evolutionary time scales, the question of relative importance of processes at different time scales have generated distinct disciplines and this issue has apparently been a key factor affecting the unification of ecology and evolutionary biology.</p>
+<p>The other two criteria are conceptual. The issue of demography is closely tied to the concept of fitness, and the two can be discussed together. Richard Levins gave a succinct account of these conceptual issues more than forty years ago, during an early phase of the evolutionary ecology synthesis <span id="id64">[<a class="reference internal" href="#id124" title="Richard Levins. Evolution in changing environments. Monographs in Population Biology. Princeton University Press, Princeton, NJ, August 1968.">Levins, 1968</a>]</span>(p. 5):</p>
+<blockquote>
+<div><p><em>Darwinian fitness (Wright’s <span class="math notranslate nohighlight">\(\overbar{W}\)</span>) has to be interpreted in terms of its ecological components, such as the intrinsic capacity for increase (Andrewartha and Birch’s <span class="math notranslate nohighlight">\(r_0\)</span>) and the carrying capacity of the environment for a given genotype (<span class="math notranslate nohighlight">\(K\)</span>). …
+While the population geneticist’s models generally assume stable age distributions, fixed populations sizes, and constant environments in order to study the pattern of genetic heterogeneity in single species, the population ecologist considers genetically uniform populations in multispecies systems in a heterogeneous environment</em>.</p>
+</div></blockquote>
+<section id="the-issue-of-ecological-vs-evolutionary-time-scales">
+<h3>The issue of ecological vs. evolutionary time scales<a class="headerlink" href="#the-issue-of-ecological-vs-evolutionary-time-scales" title="Link to this heading">#</a></h3>
+<p>As mentioned above, the early school of evolutionary thought was essentially gradualist, with a firm belief in the slow and steady progress of evolution over the geological time scale, a viewpoint that was prevalent till relatively recently <span id="id65">[]</span>. A competing view of punctuated equilibrium too exists, but this does not alter the argument that is being made here. The relevance of micro-evolutionary processes to observed macroevolutionary patterns, is still a contentious issue in evolutionary biology because of the observed opposition of trends at different time scales <span id="id66">[]</span>. As such, this issue does not affect the evolutionary ecology synthesis much because evolutionary biologists naturally emphasize lineage-dependent processes and incorporate them into microevolutionary studies in an ecological context. However, a related view in ecology, that ecological and evolutionary time scales are independent of each other, has had considerable influence on the importance given to evolutionary processes in ecology. Tied to this is the strong faith among many ecologists of the existence of adaptation and co-adaptation in natural populations and their persistence over long, geologically significant periods of time <span id="id67">[]</span>. For example Pimm <span id="id68">[]</span> discusses the stability and persistence of populations with respect to time scales ranging from centuries to millennia or more, invoking evolution only as a purely historical process. Similarly, Tokeshi <span id="id69">[<a class="reference internal" href="#id166" title="M Tokeshi. Species coexistence: ecological and evolutionary perspectives. Blackwell Science, Oxford, 1999.">Tokeshi, 1999</a>]</span> discusses evolution in community structure on purely geological time scales. These perspectives naturally cause omission of evolutionary thought from ecological research.</p>
+<p>We have already mentioned above that a view of disjunct temporal processes is rooted in the tradition in biology to study a small set of vertebrate model systems <span id="id70">[<a class="reference internal" href="#id92" title="Xavier Bonnet, Richard Shine, and Olivier Lourdais. Taxonomic chauvinism. Trends Ecol. Evol., 17(1):1–3, January 2002.">Bonnet <em>et al.</em>, 2002</a>, <a class="reference internal" href="#id93" title="Samraat Pawar. Taxonomic chauvinism and the methodologically challenged. Bioscience, 53(9):861–864, 2003.">Pawar, 2003</a>]</span>. There is no denying that evolutionary history is an important factor to consider in ecology, and has been given due consideration. However, the role of microevolution in ecological patterns and processes has not been well evaluated till the two decade or so. In the last two decades or so, ecological geneticists and molecular ecologists have shown ample evidence for the existence of functional polymorphism and gene frequency shifts in nature. Not surprisingly, there is now an increasing awareness of the need to integrate macro- and microevolutionary processes in ecology at different scales of organization <span id="id71">[<a class="reference internal" href="#id155" title="J Cavender-Bares and A Wilczek. Integrating micro- and macroevolutionary processes in community ecology. Ecology, 84(3):592–597, 2003.">Cavender-Bares and Wilczek, 2003</a>]</span>. In addition, the use of microbial model systems in ecology has also allowed many ecological questions to be answered in an evolutionary context, often adding important insights relevant to other systems where such methods are impossible <span id="id72">[<a class="reference internal" href="#id161" title="Christine M Jessup, Rees Kassen, Samantha E Forde, Ben Kerr, Angus Buckling, Paul B Rainey, and Brendan J M Bohannan. Big questions, small worlds: microbial model systems in ecology. Trends Ecol. Evol., 19(4):189–197, April 2004.">Jessup <em>et al.</em>, 2004</a>]</span>. Perhaps ironically, the empirical insights from contemporary microbial work have come more than 60 years after Gause’s <span id="id73">[<a class="reference internal" href="#id145" title="G F Gause. Experimental analysis of vito volterra's mathematical theory of the struggle for existence. Science, 79(2036):16–17, January 1934.">Gause, 1934</a>]</span> ecological experiments with microorganisms, which too had a strong evolutionary context. No doubt advances in molecular biology techniques have a large part to play in this, but this also reflects the reticence of ecologists to deviate from established model biological systems.</p>
+<p>The argument here of course, is not that research on elephants and tigers be completely abandoned in favor of laboratory cultures; it is a plea for a pluralism of empirical approach that will provide insights for finding avenues of escape from the theoretical cul-de-sac that ecologists often find themselves due to the inability to test crucial aspects of their models <span id="id74">[<a class="reference internal" href="#id151">Kareiva <em>et al.</em>, 1989</a>]</span>. Indeed, in the subsequent discussion on the empirical support for evolutionary ecology theory, there will be a striking preponderance of work on rapidly evolving model organisms such as bacteria and <em>Drosophila</em> that have provided new insights into questions that have traditionally challenged ecologists.</p>
+</section>
+<section id="demographic-considerations-and-measures-of-fitness">
+<h3>Demographic considerations and measures of fitness<a class="headerlink" href="#demographic-considerations-and-measures-of-fitness" title="Link to this heading">#</a></h3>
+<p>From a population biology perspective, many questions are raised by the treatment of demography and the closely tied concept of fitness in evolutionary biology. The issue of fitness in particular, is a controversial one even within evolutionary biology <span id="id75">[<a class="reference internal" href="#id196" title="A Ariew and R C C Lewontin. The confusions of fitness. Br. J. Philos. …, 55(2):347–363, 2004.">Ariew and Lewontin, 2004</a>, <a class="reference internal" href="#id95">Lewontin <em>et al.</em>, 2004</a>]</span>. For our purposes however, we will focus on three aspects of population genetics that have particularly troubled population ecologists. The first two of these are visible in the illustration of the traditional population genetics approach in the equations above. The problems are,</p>
+<p>(i) population sizes are assumed to be infinite, which circumvents the complications of random birth-death processes in finite populations (called “ecological drift” or “demographic stochasticity”) <span id="id76">[<a class="reference internal" href="#id94" title="P A Abrams, Stephen P Hubbell, and C de Mazancourt. The unified neutral theory of biodiversity and biogeography. Nature, 412(5536):1772, 2001.">Abrams <em>et al.</em>, 2001</a>]</span>.
+(ii) The Malthusian parameter <span class="math notranslate nohighlight">\(r\)</span> is converted into a fitness measure <span class="math notranslate nohighlight">\(s\)</span> by assuming fixed population size, which allows the convenience of reducing the parameters to just genotype proportions and relative fitnesses, and
+(iii) When considering inter-specific interactions, the mean fitness (eqn. <code class="xref eq docutils literal notranslate"> </code>) of species’ population becomes an absolute measure, creating the need to develop a general algorithm for translating absolute fitnesses between populations. As will be emphasized below, the use of quantitative genetic models helped resolve the problem theoretically, but the practicability of measuring fitness components in the field remains a difficult issue.</p>
+<p>Population geneticists dealt with the issue of infinite population size by introducing population size as a statistical parameter that affected the rate of random genetic drift– effective population size, <span class="math notranslate nohighlight">\(N_e\)</span> <span id="id77">[<a class="reference internal" href="#id103" title="M Kimura. The Neutral Theory of Molecular Evolution. Cambridge University Press, Cambridge, 1983.">Kimura, 1983</a>]</span>. Although it was not an entirely satisfactory solution from a population biology perspective as it is only a phenomenological model of demographic stochasticity, it was acceptable in ecology because it could still be used to consider the evolutionary consequences of finite population size in terms of extinction probabilities <span id="id78">[<a class="reference internal" href="#id179" title="Graeme Caughley. Directions in Conservation Biology. J. Anim. Ecol., 63(2):215, April 1994.">Caughley, 1994</a>]</span>. The issue of fixed population sizes on the other hand, raised some crucial questions for population biologists. What happens when populations change in size as they inevitably do? Or when generations are not discrete, but overlapping? Or when population change is density dependent? Extensions of population genetics theory to address questions of population change and overlapping generations had been accomplished by the 1970’s, albeit with some issues still unresolved <span id="id79">[<a class="reference internal" href="#id105" title="Brian Charlesworth. Evolution in age-structured populations. Cambridge University Press, Cambridge Eng., 1980.">Charlesworth, 1980</a>, <a class="reference internal" href="#id95">Lewontin <em>et al.</em>, 2004</a>]</span>.</p>
+<p>Evolution under density dependent population growth was first modeled by Kostitzin between 1935-40 <span id="id80">[<a class="reference internal" href="#id96">Christiansen <em>et al.</em>, 2004</a>]</span>, and Fisher <span id="id81">[<a class="reference internal" href="#id160" title="Ronald A Fisher. The genetical theory of natural selection. Clarendon Press, Oxford, 1930.">Fisher, 1930</a>]</span> by assuming weak selection, and fitnesses dependent upon density dependent death rates. Subsequently, MacArthur <span id="id82">[<a class="reference internal" href="#id163" title="R H Robert H Macarthur. Some generalized theorems of natural selection. Proc. Natl. Acad. Sci. …, 18(11):1893–1897, November 1962.">Macarthur, 1962</a>]</span> used a similar approach, but used the then familiar <span class="math notranslate nohighlight">\(r\)</span> &amp; <span class="math notranslate nohighlight">\(K\)</span> parameterization of the Pearl-Verhulst logistic growth equation of density dependent growth,</p>
+<div class="math notranslate nohighlight">
+\[
+    \dfrac{dN}{dt} = rN\left( 1-\frac{N}{K}\right)
+\]</div>
+<p>This general approach, which was developed further by others <span id="id83">[<a class="reference internal" href="#id96">Christiansen <em>et al.</em>, 2004</a>]</span> will be briefly illustrated here. Consider a haploid, asexual population with only two alleles 1 &amp; 2 at a single locus. Assuming that the population is under density dependent growth, the fitnesses can then be defined as (cf eqns 8-10),</p>
+<div class="math notranslate nohighlight">
+\[
+    w_{1} = 1 + r_{1}\Bigl(1 - \frac{N}{K}\Bigr)
+    \quad \text{and} \quad
+    w_{2} = 1 + r_{2}\Bigl(1 - \frac{N}{K}\Bigr)
+\]</div>
+<p>Frequency dependent selection and genotype-specific carrying capacities can now be included by extending these eqns to,</p>
+<div class="math notranslate nohighlight">
+\[
+    w_{1} = 1 + r_{1} \Bigl(1 - \frac{N_{1} + N_{2}}{K_{1}}\Bigr)
+    \quad \text{and} \quad
+    w_{2} = 1 + r_{2} \Bigl(1 - \frac{N_{1} + N_{2}}{K_{2}}\Bigr)
+\]</div>
+<p>which amounts to rephrasing the Lotka-Volterra equations (eqns. <code class="xref eq docutils literal notranslate"> </code>) for competition of two species to competition between two genotypes of one species, assuming the intra-specific competition parameters to be <span class="math notranslate nohighlight">\(\alpha_{12} = α_{21} = 1\)</span>. Extensions of such models to overlapping generations under different single-population scenarios, and to coevolution of two or more species have provided many interesting insights, and contributed to the unification of population genetics and ecology <span id="id84">[<a class="reference internal" href="#id96">Christiansen <em>et al.</em>, 2004</a>]</span>. The first major result of this research, of course, was the development in the 1970’s of the ideas <span class="math notranslate nohighlight">\(r\)</span>-<span class="math notranslate nohighlight">\(K\)</span> selection and the role of the environment in shaping life history strategies <span id="id85">[<a class="reference internal" href="#id96">Christiansen <em>et al.</em>, 2004</a>, <a class="reference internal" href="#id149" title="Robert H MacArthur and Edward O Wilson. The theory of island biogeography. Princeton University Press, Princeton, N.J, 1967.">MacArthur and Wilson, 1967</a>, <a class="reference internal" href="#id173" title="J Roughgarden. Theory of Population Genetics and Evolutionary Ecology: an Introduction. Macmillan Publishing Co., Inc., New York, 1979.">Roughgarden, 1979</a>]</span>.</p>
+<p>However, modeling fitnesses under frequency as well as density dependent growth in multiple populations does create the third problem mentioned above: that of absolute fitnesses. The interpretation of density dependent fitnesses becomes difficult in multi-species systems because mapping the effect of each populations absolute mean fitness on another population becomes context dependent because fitnesses are a function of the environment under this scheme (because they are tied to <span class="math notranslate nohighlight">\(K\)</span>), instead of being an inherent property of individuals’ genotype <span id="id86">[<a class="reference internal" href="#id196" title="A Ariew and R C C Lewontin. The confusions of fitness. Br. J. Philos. …, 55(2):347–363, 2004.">Ariew and Lewontin, 2004</a>, <a class="reference internal" href="#id95">Lewontin <em>et al.</em>, 2004</a>]</span>. This makes it difficult to develop a framework for making general predictions about the evolutionary outcomes of species interactions without taking the situation-specific environment-phenotype-genotype map into consideration. For example, if we want to extend eqns. <code class="xref eq docutils literal notranslate"> </code> to two haploid asexual species <span class="math notranslate nohighlight">\(A\)</span> &amp; <span class="math notranslate nohighlight">\(B\)</span>, each experiencing intra- and inter-specific density as well as frequency dependent selection, we could begin with, assuming one locus and two alleles, and using the Lotka-Volterra scheme,</p>
+<div class="math notranslate nohighlight">
+\[\begin{split}
+\begin{aligned}
+w_{1A}
+  &amp;= 1 + r_{1A}
+     \;-\;
+     \frac{r_{1A}}{K_{1A}}\;N_{A}
+     \;-\;
+     \frac{r_{1A}}{K_{1A}}
+     \Bigl(\alpha_{1A1B}\,N_{1B}
+            \;+\;
+            \alpha_{1A2B}\,N_{2B}
+     \Bigr)
+  \\[6pt]
+  &amp;= 1 + r_{1A}
+     \;-\;
+     \frac{r_{1A}}{K_{1A}}
+     \Bigl[
+        N_{A}
+        \;-\;
+        \bigl(\alpha_{1A1B}\,p_{B}
+               \;+\;
+               \alpha_{1A2B}\,\bigl(1 - p_{B}\bigr)
+        \bigr)\,N_{B}
+     \Bigr]
+\end{aligned}
+\end{split}\]</div>
+<p>and similarly,</p>
+<div class="math notranslate nohighlight">
+\[
+w_{2A}
+  = 1 + r_{2A}
+    \;-\;
+    \frac{r_{2A}}{K_{2A}}
+    \Bigl[
+      N_{A}
+      \;-\;
+      \bigl(\alpha_{2A1B}\,p_{B}
+             \;+\;
+             \alpha_{2A2B}\,\bigl(1 - p_{B}\bigr)
+      \bigr)\,N_{B}
+    \Bigr]
+\]</div>
+<p>where <span class="math notranslate nohighlight">\(w_{1A}\)</span> and <span class="math notranslate nohighlight">\(w_{2A}\)</span> are as before, the fitnesses of the two genotypes, while <span class="math notranslate nohighlight">\(r_{1A}\)</span>, <span class="math notranslate nohighlight">\(r_{2A}\)</span> &amp; <span class="math notranslate nohighlight">\(K_{1A}\)</span>, <span class="math notranslate nohighlight">\(K_{2A}\)</span> are the genotype-specific intrinsic rate of increase and carrying capacity respectively. This model assumes that a genotype in a species <span class="math notranslate nohighlight">\(A\)</span> interacts with all available genotypes of species <span class="math notranslate nohighlight">\(B\)</span> in the community in proportion to their frequency. Thus, <span class="math notranslate nohighlight">\(\alpha_{1A 1B}\)</span> and <span class="math notranslate nohighlight">\(\alpha_{1A 2B}\)</span> represent the competition coefficients of genotype 1 in species <span class="math notranslate nohighlight">\(A\)</span> with genotype 1 and 2 in species <span class="math notranslate nohighlight">\(B\)</span>, respectively. These are multiplied by the frequency of the respective genotype (<span class="math notranslate nohighlight">\(p\)</span>, <span class="math notranslate nohighlight">\(1-p\)</span>). The competition coefficient introduces the effect of interspecific frequency dependence relative to intraspecific density dependence <span id="id87">[<a class="reference internal" href="#id90" title="Mark S Laska and J Timothy Wootton. Theoretical concepts and empirical approaches to measuring interaction strength. Ecology, 79(2):461–476, March 1998.">Laska and Wootton, 1998</a>, <a class="reference internal" href="#id91">MacArthur and Deevey, 1972</a>]</span>. If <span class="math notranslate nohighlight">\(\lapha\)</span> were zero, the interacting populations would experience purely intra-specific density dependent growth and selection. Rewriting eqn. (4) eq in terms of eqns. (12 &amp; 13) gives the mean fitness of the population:</p>
+<div class="math notranslate nohighlight">
+\[
+\overline{W}_{A}
+  = 1 + \overline{r}_{A}
+    \;-\;
+    \frac{\overline{r}_{A}}{K_{A}}
+    \Bigl(
+      N_{A}
+      \;-\;
+      \overline{\alpha}_{AB}\,N_{B}
+    \Bigr),
+\]</div>
+<p>where</p>
+<p>![][image40],								      (17)</p>
+<p>![][image41],							      (18)</p>
+<p>and</p>
+<p>![][image42]				      (19)</p>
+<p>In each generation, the population growth is a function of the mean fitness![][image43]of the population in the previous generation, made up of the relative contributions of the two genotypes. For example, if allele A is fixed in the species <em>A</em>, its population will grow at a density dependent growth determined by the <em>r</em> and <em>K</em> values of the genotype 1<em>A</em> and its interaction with genotypes of species <em>B</em>. Eqns.(14-19) illustrate the difficulty with measuring fitness in such situations. In this model, one would have to keep track of no less than 10 parameters to evaluate the evolutionary outcome of interaction between just two species! And this is assuming that <em>α</em> is a reasonable representation of both direct and indirect interactions in nature. Computer simulations and mathematical analyses (with some reparameterization) of these models do yield insights, but measuring all the components of fitness in such a situation to be an almost impossible task in reality, let alone defining a more general scheme for translating absolute fitnesses between species.</p>
+<p>The problem is considerably simplified by using a quantitative genetics framework (see eqns. (9-10) above). For example, for the two competing species in the above example, the fitness functions from eqns (14 &amp; 16) could be easily modified to feed into eqns. (9 &amp; 10), thus creating a quantitative map between fitnesses of the two populations generated by frequency- and density-dependent interactions within and between species. However, the empirical problem of measuring all components of fitness remains a difficult problem. On the whole then, the manner in which fitnesses are dealt with will be an important criterion for the classification presented below. It will become apparent that certain model systems that have been used in recent empirical work are relatively amenable to estimation of fitness components. This has gone a long way in testing some theoretical predictions.</p>
+<p>It should also be noted that in addition to examining the role of demography in evolution, it is possible to flip the question over, and focus on the effects of evolution on population dynamics. Such work has also emerged in recent years and is an area of active theoretical development and empirical research, adding yet another facet to evolutionary ecology.</p>
+</section>
+<section id="the-role-of-environmental-heterogeneity">
+<h3>The role of environmental heterogeneity<a class="headerlink" href="#the-role-of-environmental-heterogeneity" title="Link to this heading">#</a></h3>
+<p>The ‘problem’ of complex population behavior and the role of “external” vs. “internal” factors play an important role in ecology (Hastings et al. 1993)(<a class="reference external" href="https://paperpile.com/c/lwDPbc/GV43">https://paperpile.com/c/lwDPbc/GV43</a>). In search for answers to such questions, ecologists have expended considerable effort over the last century for a detailed characterization of heterogeneity in nature. Areas such as landscape and spatial ecology have steadily built a realistic characterization of heterogeneity in nature, and to many biologists, ecology is synonymous with environmental characteristics; the ecology of a species is at least as likely to connote the organism’s environment as its inherent properties, such as life history characteristics. As was pointed out above, the field of ecological genetics has developed around this view of ecology.<br />
+The incorporation of environmental heterogeneity into a unified evolutionary ecology framework is far from complete though. This is one area where the flow of information must largely be from the direction of ecology (in its environmental avatar) towards evolution. As Levins (Levins 2004)(<a class="reference external" href="https://paperpile.com/c/lwDPbc/Uyrg">https://paperpile.com/c/lwDPbc/Uyrg</a>) points out, “ ‘<em>There has always been a much finer sophistication of our understanding of genetic variation than of the environment,</em>’ ‘<em>…a successful study of evolution requires the recognition of the complexity, not just of the genotype, but also of the environment and of the whole organism in its development and its physiological flux’</em> “. This is of course, a reiteration of the argument made above about the effect of genetic reductionism on the conceptual unification of population genetics and population biology. Practitioners of ecological genetics do indeed often pay just lip service to environmental heterogeneity, using a set approach of employing statistical tools to fit observed genetic variation to predictions of traditional population genetics models, most of which have tenuous connections with ecological reality. Statistically reasonable fits may be observed, but the results do not reveal whether the model being tested describes the real mechanisms for the maintenance of genetic variation. Although this is a general concern in all scientific disciplines, it is also a function of how much of the universe of possibilities has been explored while formulating theories.<br />
+In the quest for a formal description of all the nuances of environment complexity, ecologists have been no laggards; the mathematical structure of many aspects of the environment in which evolution can potentially operate has been well characterized in both time (Vasseur and Yodzis 2004)(<a class="reference external" href="https://paperpile.com/c/lwDPbc/dKQx">https://paperpile.com/c/lwDPbc/dKQx</a>) and space (Keitt et al. 2002)(<a class="reference external" href="https://paperpile.com/c/lwDPbc/3b7Q">https://paperpile.com/c/lwDPbc/3b7Q</a>). For example, consider Figure 1, which shows an empirical observation of temporal environmental pattern: seasonal variation in temperature across latitudes. It is well known that the amplitude of seasonal cycles as well as random environmental noise increases away from the latitudes (Vasseur and Yodzis 2004)(<a class="reference external" href="https://paperpile.com/c/lwDPbc/dKQx">https://paperpile.com/c/lwDPbc/dKQx</a>).</p>
+<p>Fig. 1. Variation in temperature across months over 11 years for four sites spanning 400 of latitude, from New Zealand (46.70S), to Papua New Guinea (6.730S). All sites are from a similar altitude (0–20 m above sea level (S. S. Pawar 2005)(<a class="reference external" href="https://paperpile.com/c/lwDPbc/BCIv">https://paperpile.com/c/lwDPbc/BCIv</a>). Note that the fluctuations increase away from the equator.</p>
+<p>![][image44]</p>
+<p>This observation is easily translated into a model of fluctuating selection (Bürger and Gimelfarb 2002)(<a class="reference external" href="https://paperpile.com/c/lwDPbc/3fia">https://paperpile.com/c/lwDPbc/3fia</a>). Consider a selectively optimum trait value <em>θ</em>, such as the one introduced in eqn.(9). Assume that the position of the optimum fluctuates cyclically about the midpoint of the range of trait values (<em>θo</em>), with a scaled random perturbation around it. This can be modeled by assuming that at time <em>t</em> the optimum <em>θt</em> is drawn from a normal distribution with mean</p>
+<p>![][image45]										      (20)</p>
+<p>where <em>A</em> is the amplitude and <em>L</em> the length of the selection cycle relative to the organism’s generation time. The perturbations have a standard deviation![][image46], where <em>d</em> is a measure of the magnitude of stochasticity. If <em>d</em>=0, there is purely cyclical selection; if, in addition, <em>A</em>=0, there is pure Gaussian stabilizing selection. Figure 2 shows the temporal pattern of cyclical selection generated from this model with arbitrary parameter values. This matches the empirical pattern in Figure 1 quite well.</p>
+<p>Fig. 2. Cyclical selection with random fluctuations over 100 generations, based on the model in eqn. (20).</p>
+<p>![untitled][image47]</p>
+<p>This idea can then be further developed to consider the effect of such environmental grain on evolution as well as persistence of populations. After the initial beginning in the 1960’s of work on the effect of fluctuating environments on adaptation and persistence of populations, the role of periodic instead of completely random fluctuations, partly motivated by empirical patterns like the one in Figure 1, increasingly drew attention over the last decade or so, yielding new insights (Kirzhner, Korol, and Nevo 1998; Wichmann et al. 2003; Bürger and Gimelfarb 2002; Lande and Shannon 1996)(<a class="reference external" href="https://paperpile.com/c/lwDPbc/032V+9RCS+3fia+fzdV">https://paperpile.com/c/lwDPbc/032V+9RCS+3fia+fzdV</a>). Detailed examination of such empirical patterns of environmental heterogeneity can also help resolve questions about the likelihood of seeing certain evolutionary processes in nature. For example, Gillespie (J. H. Gillespie, Singh, and Uyenoyama 2004)(<a class="reference external" href="https://paperpile.com/c/lwDPbc/Nb5C">https://paperpile.com/c/lwDPbc/Nb5C</a>) long argued that widely accepted population genetic models of adaptive evolution in DNA sequences need to be reevaluated in light of the fact that certain patterns of fluctuating selection can generate patterns of polymorphism akin to those from models of neutral evolution, a claim that remains largely untested. The evolutionary ecology synthesis will definitely benefit from the use of environmental information; the extent to which this has been achieved is considered in the classification that follows.</p>
+</section>
+<section id="taxonomy-of-evolutionary-ecology">
+<h3>Taxonomy of evolutionary ecology**<a class="headerlink" href="#taxonomy-of-evolutionary-ecology" title="Link to this heading">#</a></h3>
+<p>classification of contemporary approaches that combine ecological and evolutionary perspectives from population genetics and population biology are presented in Table 1. Keeping the tradition initiated by stalwarts of the evolutionary ecology synthesis, I drop the term “evolutionary ecology” in favor of “population biology” for this classification (Levins 2004; Lewontin et al. 2004; Travis et al. 1989)(<a class="reference external" href="https://paperpile.com/c/lwDPbc/Uyrg+fPH1+bZbA">https://paperpile.com/c/lwDPbc/Uyrg+fPH1+bZbA</a>).</p>
+<p><strong>Table 1</strong>. A taxonomy of research that unifies ecological and evolutionary perspectives in biology. The issues considered for classification are: approach towards time-scale, demographic considerations, fitness measures, and treatment of ecological heterogeneity.</p>
+<div class="pst-scrollable-table-container"><table class="table">
+<thead>
+<tr class="row-odd"><th class="head text-left"><p>Category</p></th>
+<th class="head text-left"><p>Scientific approach</p></th>
+</tr>
+</thead>
+<tbody>
+<tr class="row-even"><td class="text-left"><p><strong>I. Paleo-population biology</strong></p></td>
+<td class="text-left"><p><em>Retrospective studies considering historical effects on properties of current populations.</em></p></td>
+</tr>
+<tr class="row-odd"><td class="text-left"><p><strong>(a) <em>Non-demographic schema</em></strong></p></td>
+<td class="text-left"><p>Reconstruction of evolutionary relationships without paleo-demographic inferences, with focus on extant populations of two or more species. Fitness concepts used are absolute and often qualitative. Structural or spatial aspects of habitat interpretable as niches play an important part.</p></td>
+</tr>
+<tr class="row-even"><td class="text-left"><p><strong>(b) <em>Demographic schema</em></strong></p></td>
+<td class="text-left"><p>Reconstruction of phylogenies and gene genealogies with paleo-demographic and paleo-ecological inferences with focus on one or more extant species’ populations. Fitness measures and ecological heterogeneity can be considered only in extant populations.</p></td>
+</tr>
+<tr class="row-odd"><td class="text-left"><p><strong>II. Population biology</strong></p></td>
+<td class="text-left"><p><em>Essentially prospective studies restricted to “microevolutionary” or “ecological” time scales.</em></p></td>
+</tr>
+<tr class="row-even"><td class="text-left"><p><strong>(a) <em>Non-demographic schema</em></strong></p></td>
+<td class="text-left"><p>Study of populations without explicit consideration of demography, using traditional population genetic tools. Relative fitnesses are used, assuming fixed population sizes. Ecological heterogeneity, if considered, can be spatial and/or temporal.</p></td>
+</tr>
+<tr class="row-odd"><td class="text-left"><p><strong>(b) <em>Indirect-demographic schema</em></strong></p></td>
+<td class="text-left"><p>Study of populations with inference of demographic characteristics (in general, population size), indirectly, by using population genetics theory to estimate N<em>e</em>. Fitness measures are absolute, and generally qualitative. Ecological heterogeneity is generally spatial.</p></td>
+</tr>
+<tr class="row-even"><td class="text-left"><p><strong>(c) <em>Demographic schema</em></strong></p></td>
+<td class="text-left"><p>Entail explicit consideration of the interplay between evolutionary changes and demography of populations. Fitness measures vary, often involving density as well as frequency dependence, or are absolute. Ecological heterogeneity, if considered, can be spatial and/or temporal.</p></td>
+</tr>
+</tbody>
+</table>
+</div>
+<p>Each of these research programs is now discussed with reference to the nature of scientific approach, the conceptual issues it addresses, its theory and supporting empirical data. Together, these disciplines comprise a number of research topics and a huge body of literature. This discussion below does not claim to be an exhaustive survey of all topics. Within each research program however, poorly explored avenues of interesting research are also mentioned.</p>
+<p><strong>I. Paleo-population biology</strong><br />
+One solution towards fusing evolution and ecology is to incorporate historical processes by reconstructing the evolutionary history and paleo-ecology of populations and drawing inferences about contemporary properties of organisms. Two distinct approaches are currently being used:</p>
+<p><strong>(a) <em>Non-demographic schema</em></strong></p>
+<p><strong>Approach</strong>.This program of research is an outcome of the above-mentioned ‘historical’ viewpoint mentioned that has been so prevalent in ecology. This program was invigorated in the 1990’s by the introduction of phylogenetic techniques, thus allowing a more reliable reconstruction of evolutionary history (Eggleton and Vane-Wright 1994; Losos 1996)(<a class="reference external" href="https://paperpile.com/c/lwDPbc/6vwA+rB1C">https://paperpile.com/c/lwDPbc/6vwA+rB1C</a>). This method is non-demographic in the sense that population size considerations rarely enter the analyses, either in the historical reconstructions, or in the study of contemporary adaptations. The concepts of fitness used here are absolute and qualitative, with the outcome of interactions between populations described in terms of the apparent level of fitness in populations.</p>
+<p><strong>Research themes:</strong> <em>theory and empirical results</em>. This approach is strongly retrospective, generally involving verbal extensions of prospective evolutionary ecology models such as those of character displacement and the mapping of traits onto phylogenies (Eggleton and Vane-Wright 1994; Losos 1996)(<a class="reference external" href="https://paperpile.com/c/lwDPbc/6vwA+rB1C">https://paperpile.com/c/lwDPbc/6vwA+rB1C</a>). Some prominent themes for research are adaptive radiation into ecological niches (Schluter 2000; Losos and Miles 2002)(<a class="reference external" href="https://paperpile.com/c/lwDPbc/H1J6+uLfC">https://paperpile.com/c/lwDPbc/H1J6+uLfC</a>), Outcome of species interactions (e.g., evolutionary character displacement; (Radtkey, Fallon, and Case 1997)(<a class="reference external" href="https://paperpile.com/c/lwDPbc/Qrll">https://paperpile.com/c/lwDPbc/Qrll</a>), and the manner of community assembly (R. Gillespie 2004)(<a class="reference external" href="https://paperpile.com/c/lwDPbc/GUZ0">https://paperpile.com/c/lwDPbc/GUZ0</a>). The insights from these lines of work have served to greatly clarify the role of evolution in structuring communities and molding species interactions. For example, Gillespie (R. Gillespie 2004)(<a class="reference external" href="https://paperpile.com/c/lwDPbc/GUZ0">https://paperpile.com/c/lwDPbc/GUZ0</a>) has shown that convergent evolution into similar niches has an important role in community assembly, shedding light on the long unresolved debate in community ecology about the manner of community assembly, and the existence of “assembly rules” (Weiher and Keddy 1995)(<a class="reference external" href="https://paperpile.com/c/lwDPbc/3lUy">https://paperpile.com/c/lwDPbc/3lUy</a>). On the flipside, such studies contribute a lot towards clarifying the role of ecology in speciation, a burgeoning topic in evolutionary biology (Coyne and Orr 2004)(<a class="reference external" href="https://paperpile.com/c/lwDPbc/fVZt">https://paperpile.com/c/lwDPbc/fVZt</a>).</p>
+<p><strong>(b) <em>Demographic schema</em></strong></p>
+<p><strong>Approach</strong>*.*This relatively new area involves the inference of “historical demography” combined with analyses of properties of extant populations. This has been made possible by recent attempts to bring the coalescence theory (Kingman 2000)(<a class="reference external" href="https://paperpile.com/c/lwDPbc/PMBy">https://paperpile.com/c/lwDPbc/PMBy</a>) of population genetics and phylogenetics closer together to evaluate the interrelationships between micro- and macroevolution (Wakeley, Singh, and Uyenoyama 2004)(<a class="reference external" href="https://paperpile.com/c/lwDPbc/yxzS">https://paperpile.com/c/lwDPbc/yxzS</a>). This approach also opens up the possibility of explicitly including historical population structure in analyses of current populations, a topic that has hitherto been difficult to deal with except by verbal arguments in ecology as well as evolutionary biology. As paleo-population structure is inferred purely from nucleotide polymorphism data, this approach obviously cannot infer past environmental heterogeneity explicitly unless paleo-ecological events can be dated and implicated using molecular clock approaches. Fitnesses obviously do not even enter the picture as far as the historical component is concerned. The analyses of extant populations can of course include both, fitness measures and consideration of environmental heterogeneity.</p>
+<p><strong>Research themes:</strong> <em>theory and empirical results.</em> Obviously, the past can impinge upon almost every feature of extant organisms. The themes that will be developed in this area will only be bound by the limits of the collaboration between paleobiology, phylogenetics, coalescent theory, and ecology. These are widely separated disciplines, and a coherent theory will probably take a while to develop. Research in this area could include, but need not be restricted to, all of the themes of the approaches under the non-demographic schema mentioned above. For example, recently, Flanagan et al (Flanagan et al. 2004)(<a class="reference external" href="https://paperpile.com/c/lwDPbc/GW6y">https://paperpile.com/c/lwDPbc/GW6y</a>) used such an approach to clarify the role of historical demography in the coevolution of two Müllerian comimetic butterfly species. The potential of this area of research is huge; all the questions asked in the non-demographic schema above can be better addressed by such analyses.</p>
+<p><strong>II. Microevolutionary ecology</strong></p>
+<p>This is a much larger area of research than paleo-population biology, because prospective study of extant populations is the predominant approach in population biology. This area involves the consideration of microevolutionary changes in generating ecological patterns and processes, as well as the effect of the environment on observable evolution. Two broad categories of approach can be identified here:</p>
+<p><strong>(a) <em>Non-demographic schema</em></strong></p>
+<p><strong>Approach</strong>*.*This program involves the extension of population genetics (including quantitative genetics) to realistic ecological (in the sense of the adaptive environment) scenarios, for a detailed examination of populations’ characteristics in relation to the environment. In keeping with the population genetic framework, population size only appears if at all, as <em>Ne</em>, and fitnesses are relative, or if multiple species are considered, may be juxtaposed by analyses of mean population fitnesses (generally using quantitative genetics). The most common aspect of ecological heterogeneity is spatial, while temporal variation in the abiotic environment is increasingly being studied, with rare examples of a combination of the two (Frank and Slatkin 1990)(<a class="reference external" href="https://paperpile.com/c/lwDPbc/YH5u">https://paperpile.com/c/lwDPbc/YH5u</a>).</p>
+<p><strong>Research themes:</strong> <em>theory and empirical results.</em> Most studies in ecological genetics and molecular ecology use this approach to establish the relationships between genetic variation in nature and ecological heterogeneity (Brian Charlesworth, Charlesworth, and Barton 2003)(<a class="reference external" href="https://paperpile.com/c/lwDPbc/H4zB">https://paperpile.com/c/lwDPbc/H4zB</a>), wherein the focal population’s environs can be both biotic (e.g., herbivore-plant interactions), as well as abiotic (DeSalle and Schierwater 1998; Beebee and Rowe 2004; Conner and Hartl 2004)(<a class="reference external" href="https://paperpile.com/c/lwDPbc/S0h5+1SH7+fazZ">https://paperpile.com/c/lwDPbc/S0h5+1SH7+fazZ</a>). Other themes are evolution in temporally fluctuating environments (A. Sasaki and Ellner 1997; Ellner 1996; Akira Sasaki and Ellner 1995; J. H. Gillespie 1991)(<a class="reference external" href="https://paperpile.com/c/lwDPbc/gSfW+mNWb+aTAq+xa9v">https://paperpile.com/c/lwDPbc/gSfW+mNWb+aTAq+xa9v</a>), evolution in spatio-temporally fluctuating environments (Frank and Slatkin 1990)(<a class="reference external" href="https://paperpile.com/c/lwDPbc/YH5u">https://paperpile.com/c/lwDPbc/YH5u</a>), evolutionary outcome of species interactions with such as competition and predation with or without interactions with the environment (P. a. Abrams 2000; M. Slatkin and Maynard Smith 1979; Montogemery Slatkin 1980; Agrawal 2003)(<a class="reference external" href="https://paperpile.com/c/lwDPbc/uGXo+o46W+O2ah+DBMh">https://paperpile.com/c/lwDPbc/uGXo+o46W+O2ah+DBMh</a>). A large empirical area corresponding to the last is the whole area of evolutionary ecology of plant-animal interactions, which involves the study of coevolution between populations, which does not generally consider demography explicitly, defines fitnesses largely qualitatively in terms of apparent adaptations of populations, with or without consideration of abiotic environment (Wöhrmann and Loeschcke 1984; Price and Schluter 1991; Emlen 1973; Beattie 1985; Drake 1968; Karlin and Nevo 1976; Poulin 1998)(<a class="reference external" href="https://paperpile.com/c/lwDPbc/VruJ+4cMl+E8U4+kCz4+zzc1+xIOL+j3Gh">https://paperpile.com/c/lwDPbc/VruJ+4cMl+E8U4+kCz4+zzc1+xIOL+j3Gh</a>). An important area that has recently come to the fore is the study of the role of the environment in determining evolutionary rates at the molecular level (S. S. Pawar 2005)(<a class="reference external" href="https://paperpile.com/c/lwDPbc/BCIv">https://paperpile.com/c/lwDPbc/BCIv</a>). This has raised some interesting new questions about the evolutionary mechanisms behind latitudinal diversity gradients, an issue that has traditionally been an ecological topic. In addition, work with microorganisms has also provided valuable data for evolution in temporally fluctuating (Bell and Reboud 1997; Reboud and Bell 1997; Scheiner and Yampolsky 1998; Kassen and Bell 1998)(<a class="reference external" href="https://paperpile.com/c/lwDPbc/0eyC+P2GH+HqQK+XFKD">https://paperpile.com/c/lwDPbc/0eyC+P2GH+HqQK+XFKD</a>) as well as spatially variable (Travisano, Vasi, and Lenski 1995)(<a class="reference external" href="https://paperpile.com/c/lwDPbc/Sc6a">https://paperpile.com/c/lwDPbc/Sc6a</a>) environments.</p>
+<p><strong>(b) Indirect-demographic schema</strong></p>
+<p><strong>Approach</strong>*.* This area involves the study of populations in nature using ecological genetic techniques, and inferring demographic characteristics (generally, effective population size, <em>Ne</em>) indirectly, by using population genetics theory. Fitness measures are absolute, and generally qualitative. Ecological heterogeneity, if considered, is generally spatial.</p>
+<p><strong>Research themes:</strong> <em>theory and empirical results–</em><br />
+The theoretical basis for this line of research is traditional population genetics theory, where genetic variation is used to estimate demographic characteristics such as N<em>e</em> and population structure (Hedrick 1984)(<a class="reference external" href="https://paperpile.com/c/lwDPbc/HqLP">https://paperpile.com/c/lwDPbc/HqLP</a>). As most inferences about population characteristics are indirectly inferred, this area has a restricted range of themes, and will probably be absorbed into one or more of the other categories. Most results in this area have yielded insights into the impact of natural or anthropogenic environment variation on characteristics of single populations  (Crawford, Macleod, and Mitchell 2003; Woolfit 2003)(<a class="reference external" href="https://paperpile.com/c/lwDPbc/tPSr+D6TV">https://paperpile.com/c/lwDPbc/tPSr+D6TV</a>).</p>
+<p><strong>(b) Direct-Demographic schema</strong></p>
+<p><strong>Approach</strong>. This area of research is a rapidly growing area, and includes a range of distinct research themes that have their origins in ecology or evolutionary biology. All these involve explicit consideration of demography. The most complex topics involve a three-way interaction between the environment, demographic dynamics, and evolution, perhaps capturing the essence of what has been visualized to be a unified population biology (Travis et al. 1989; Levins 2004; Lewontin et al. 2004)(<a class="reference external" href="https://paperpile.com/c/lwDPbc/bZbA+Uyrg+fPH1">https://paperpile.com/c/lwDPbc/bZbA+Uyrg+fPH1</a>). The characterization of fitness in this diverse research area is naturally variable. To a great extent, the problems that have been mentioned above with the measurement of fitness components is perhaps the most relevant to demographically-oriented objectives of these studies.</p>
+<p><strong>Research themes:</strong> <em>theory and empirical results.</em> The theory that underlies this area has been covered in parts in the previous sections, and involves demographically explicit models that combine population genetics (including quantitative genetics) with different levels of ecological heterogeneity. The themes of research are wide ranging (Brussard and Allard 1978)(<a class="reference external" href="https://paperpile.com/c/lwDPbc/okQu">https://paperpile.com/c/lwDPbc/okQu</a>) include the effects of evolutionary change on demographic characteristics and vice versa (Michael Doebeli and Koella 1994, 1995; Michael Doebeli 1996b; M. Doebeli 1996; Michael Doebeli 1996a)(<a class="reference external" href="https://paperpile.com/c/lwDPbc/2IfO+n8sU+Abha+L8fS+dbVj">https://paperpile.com/c/lwDPbc/2IfO+n8sU+Abha+L8fS+dbVj</a>), evolutionary changes in communities (Norberg et al. 2001)(<a class="reference external" href="https://paperpile.com/c/lwDPbc/t6g5">https://paperpile.com/c/lwDPbc/t6g5</a>), evolution of population stability, evolution of life history traits (B. Charlesworth 1993; Brian Charlesworth 1980; Reznick, Bryant, and Bashey 2002; Mueller and Joshi 2000; Levins 1968; A. D. Bradshaw and Shorrocks 1984; Norberg et al. 2001; M. Doebeli and de Jong 1999; Michael Doebeli 1996b; Michael Doebeli and Koella 1995)(<a class="reference external" href="https://paperpile.com/c/lwDPbc/a4Tw+VakU+09ni+69mL+WlSX+QWRf+t6g5+h1PE+Abha+n8sU">https://paperpile.com/c/lwDPbc/a4Tw+VakU+09ni+69mL+WlSX+QWRf+t6g5+h1PE+Abha+n8sU</a>), and evolution in fluctuating environments (Reznick, Bryant, and Bashey 2002; Mueller and Joshi 2000)(<a class="reference external" href="https://paperpile.com/c/lwDPbc/09ni+69mL">https://paperpile.com/c/lwDPbc/09ni+69mL</a>).<br />
+This area has had a long history of disjunction between theory and the empirical work, partly because many of the predictions of microevolutionary ecology models cannot be tested easily either because fitness components are difficult to measure, or the rate of evolution in the model organisms is too slow to be observed in reasonable time scales. However, the inclusion of new model systems, which was mentioned in section 3.1 above, has had a big impact on this situation. For example, work with <em>Drosophila</em> and <em>Poecilia (Reznick, Bryant, and Bashey 2002; Mueller and Joshi 2000)(<a class="reference external" href="https://paperpile.com/c/lwDPbc/09ni+69mL">https://paperpile.com/c/lwDPbc/09ni+69mL</a>)</em> has produced the first concrete support for some of the predictions of theory on evolution of life history traits. Work with microorganisms too has provided strong evidence for the role of evolution in molding population dynamics (Yoshida et al. 2003; Fussmann, Ellner, and Hairston 2003)(<a class="reference external" href="https://paperpile.com/c/lwDPbc/UP82+gqas">https://paperpile.com/c/lwDPbc/UP82+gqas</a>). This has important implications for the one of the longest-lived problems in ecology of pinpointing the factors driving population cycles in microtine rodents and their predators, for which evolutionary hypotheses had been proposed before, but had received little attention by ecologists (Peter Turchin 2003; Chitty 1996)(<a class="reference external" href="https://paperpile.com/c/lwDPbc/VtwM+P2dw">https://paperpile.com/c/lwDPbc/VtwM+P2dw</a>). A number of equally valuable insights on other ecological questions from work with other microorganisms (Jessup et al. 2004)(<a class="reference external" href="https://paperpile.com/c/lwDPbc/8fxM">https://paperpile.com/c/lwDPbc/8fxM</a>).</p>
+</section>
+</section>
+<section id="discussion">
+<h2>Discussion<a class="headerlink" href="#discussion" title="Link to this heading">#</a></h2>
+<section id="has-the-marriage-come-of-age">
+<h3>Has The Marriage Come Of Age?<a class="headerlink" href="#has-the-marriage-come-of-age" title="Link to this heading">#</a></h3>
+<p>So what has been the outcome of this intricate branching of evolutionary ecology? In a recent paper, Lewontin and Levins, two key figures in the program for a unified population biology, provide an assessment, which ranges from being cautiously optimistic to rather dyspeptic <a class="reference external" href="https://paperpile.com/c/lwDPbc/fPH1">(Lewontin et al. 2004)</a>. After discussing a range of topics that are at the heart of the synthesis, Levins laments that “This program for an integrated population biology remains an aspiration that has not been carried out in practice. Instead, we see a gross imbalance among these components and a continued separation of the disciplines”. This is definitely true to the extent that the mainstream of ecology and evolutionary biology do indeed proceed happily along a divergent (or parallel) path, as was observed at the beginning of this paper. However, I am inclined to be more optimistic that Levins is.<br />
+The taxonomy of a unified population biology that has been presented here is no doubt temporary, but serves the purpose of demonstrating the impressive range of topics that studies undertaking an evolutionary approach have managed to cover, both in theory, and empirical work. One factor that Lewontin and Levins do not seem to have considered, is the crucial role that the quantitative genetics models have played in providing at the very least, heuristic insights into the role of ecology in evolution, and vice versa. Moreover, there is ample evidence for the bi-directional flow of contribution, which indicates a greater balance, perhaps not yet fully appreciated, in the nature of the unification. There is no doubt that the marriage has not yet come of age. However, its rapid progress in the last two decades or so, bode good tidings for the not too distant near future.</p>
+</section>
+<section id="on-the-dullness-of-ecology">
+<h3>On the Dullness of Ecology<a class="headerlink" href="#on-the-dullness-of-ecology" title="Link to this heading">#</a></h3>
+<p>It was pointed out at the beginning that the philosophy of biology appears to suffer from a considerable disinterest in ecology, especially in comparison to evolutionary biology <a class="reference external" href="https://paperpile.com/c/lwDPbc/PICF+MDgF+Tpjb+JR1s">(Grene and Depew 2004; Sober 1993a; Hull and Ruse 1998; Hull 1974)</a>. To some extent, this is understandable, as no other work in history has shaken the teleological foundations of human culture, as Darwin’s radical theory <a class="reference external" href="https://paperpile.com/c/lwDPbc/JR1s+MDgF">(Hull 1974; Sober 1993a)</a>. Considering the fact that the role of teleology in evolutionary biology have been an important topic of discussion <a class="reference external" href="https://paperpile.com/c/lwDPbc/rVc1">(Sterelny and Griffiths 1999)</a><a class="footnote-reference brackets" href="#id206" id="id88" role="doc-noteref"><span class="fn-bracket">[</span>10<span class="fn-bracket">]</span></a>, this disinterest in the philosophy of ecology is surprising because there has been a widespread belief in ecology of the existence of optimal strategies, stability and persistence, the roots and implications of which are worth contemplating about. In this context, Hull <a class="reference external" href="https://paperpile.com/c/lwDPbc/JR1s">(Hull 1974)</a> has a definition of teleology in the biological context that applies well to ecology: “<em>prevalence of preferred states, closed feedback loops and programs, and the origin of such systems by means of selection processes</em>.” On the other hand, apart from the issue of teleology, evolutionary biology also offers a much more coherent theoretical structure compared to ecology, and has thrown up its share of interesting topics ranging from the role of biological determinism, to issues such as the meaning of species, units of evolution and selection, and so on <a class="reference external" href="https://paperpile.com/c/lwDPbc/PICF+rVc1">(Grene and Depew 2004; Sterelny and Griffiths 1999)</a>.<br />
+This paper offers another, perhaps more fundamental reason for a reconsideration of the dullness of ecology. The question of the existence of laws in biology of has been a topic that has troubled biologists and philosophers for a while, and still provides ample fodder for rumination <a class="reference external" href="https://paperpile.com/c/lwDPbc/FE1J+WSwP+tvrS+KQ8r+wyg0">(Bergandi and Blandin 1998; Kingsland 1985; Ginzburg and Colyvan 2004; Yrjö Haila and Taylor 2001; Cooper 2003)</a>. And if philosophers are interested in the laws of biology, where better to look at the rather discordant structure of ecology? Opinions about the validity of the theoretical foundations of ecology and the presence of laws vary widely, ranging from agreement about the presence of general principles at one end <a class="reference external" href="https://paperpile.com/c/lwDPbc/SoZk+ugaQ">(Berryman and Uni 2003; P. Turchin 2002)</a>, to the declaration of the theoretical redundancy of ecology to the advancement of biology on the other <a class="reference external" href="https://paperpile.com/c/lwDPbc/KQ8r">(Yrjö Haila and Taylor 2001)</a>. Writing about the relationship between ecology and evolution, Sterelny and Griffiths <a class="reference external" href="https://paperpile.com/c/lwDPbc/rVc1">(Sterelny and Griffiths 1999)</a> invoke Sober’s <a class="reference external" href="https://paperpile.com/c/lwDPbc/08Ck">(Sober 1993b)</a> distinction between “source laws” and “consequence laws”. Source laws explain the origins of fitness differences, whereas consequence laws explain the actual evolutionary outcome of these differences. Sterelny and Griffiths judge that the role of ecology is to define the source laws in evolutionary biology.<br />
+However, I claim that the results of this paper indicate that this conclusion is incorrect, given the balanced nature of the current collaboration between ecology and evolutionary biology. Ecology and evolutionary biology have clearly made reciprocal contributions to each other’s conceptual advancement. Thus While some may consider the future of ecology as a whole to lie in a stronger integration into the human social context <a class="reference external" href="https://paperpile.com/c/lwDPbc/WSwP">(Kingsland 1985)</a>(pp 65-66) <a class="reference external" href="https://paperpile.com/c/lwDPbc/mDgv+KQ8r+A4SO">(G. A. Bradshaw and Bekoff 2001; Yrjö Haila and Taylor 2001; Yrjšo Haila and Levins 1992)</a>, the progress of the evolutionary ecology synthesis suggests that while parts of ecology might indeed have a important social role, a large part of ecology might actually come to rest in an synthetic theory that comprises a more sturdy theoretical foundation than either discipline alone. Neither discipline is theoretically redundant either to each other, or to biology as a whole. While the existence of laws in biology is an important issue, it is perhaps time to consider the theoretical and philosophical implications of the ongoing reunification of the two disciplines that were born together with Darwin’s theory, which marks the birth of modern biology itself.</p>
+</section>
+</section>
+<section id="conclusion">
+<h2>Conclusion<a class="headerlink" href="#conclusion" title="Link to this heading">#</a></h2>
+<p>In this Chapter we have examined the alliance between ecology and evolutionary biology, which has been a distinct program of research for more than a century now. In particular swe will focus on the most theoretically and empirically tangible part of this alliance: the unification of population ecology and population genetics. The potential benefits of this unification are wide ranging, and it has been sought for more than half a century now. We began by highlighting the disparities between ecological and evolutionary viewpoints and reviewed aspects of their history that point at the roots of discord as well as efforts for synthesis. There are three important differences in viewpoints that population genetics and population ecology differ at: the role of processes at different time-scales, demographic considerations (including measures of fitness), and environmental heterogeneity. Based on these, a taxonomy of theoretical approaches that merge these two disciplines was presented, with focus on their conceptual approach, themes of research, and empirical advances. This allowed an evaluation of the current status of the synthesis - it is clear that the unification of population biology through an evolutionary ecology synthesis has made steady progress, but remains a work in progress. The nature of this unification also calls for a re-evaluation of the reasons for which ecology has been considered a sterile subject by philosophers. The current structure of the synthesis clearly indicates a balanced role for the two disciplines in the conceptual advancement of biology, and philosophers should perhaps pay more attention to the progress of this synthesis. This will also perhaps be a refreshing diversion from the rather bedraggled issue of the existence of laws in ecology and in biology.</p>
+</section>
+<section id="references">
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+<p>Richard C Lewontin. <em>The genetic basis of evolutionary change</em>. Columbia University Press, New York, 1974.</p>
+</div>
+<div class="citation" id="id95" role="doc-biblioentry">
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+<p><strong>missing journal in Lewontin2004</strong></p>
+</div>
+<div class="citation" id="id125" role="doc-biblioentry">
+<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id34">Lot56</a><span class="fn-bracket">]</span></span>
+<p>Alfred J Lotka. <em>Elements of mathematical biology</em>. Dover Publications, 1956.</p>
+</div>
+<div class="citation" id="id91" role="doc-biblioentry">
+<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id87">MD72</a><span class="fn-bracket">]</span></span>
+<p><strong>missing journal in MacArthur1972b</strong></p>
+</div>
+<div class="citation" id="id163" role="doc-biblioentry">
+<span class="label"><span class="fn-bracket">[</span>Mac62<span class="fn-bracket">]</span></span>
+<span class="backrefs">(<a role="doc-backlink" href="#id51">1</a>,<a role="doc-backlink" href="#id82">2</a>)</span>
+<p>R H Robert H Macarthur. Some generalized theorems of natural selection. <em>Proc. Natl. Acad. Sci. …</em>, 18(11):1893–1897, November 1962.</p>
+</div>
+<div class="citation" id="id184" role="doc-biblioentry">
+<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id40">MC66</a><span class="fn-bracket">]</span></span>
+<p>Robert H MacArthur and Joseph H Connell. <em>The biology of populations</em>. Wiley, New York, 1966.</p>
+</div>
+<div class="citation" id="id149" role="doc-biblioentry">
+<span class="label"><span class="fn-bracket">[</span>MW67<span class="fn-bracket">]</span></span>
+<span class="backrefs">(<a role="doc-backlink" href="#id35">1</a>,<a role="doc-backlink" href="#id54">2</a>,<a role="doc-backlink" href="#id85">3</a>)</span>
+<p>Robert H MacArthur and Edward O Wilson. <em>The theory of island biogeography</em>. Princeton University Press, Princeton, N.J, 1967.</p>
+</div>
+<div class="citation" id="id142" role="doc-biblioentry">
+<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id59">MS78</a><span class="fn-bracket">]</span></span>
+<p>John Maynard Smith. <em>The evolution of sex</em>. Cambridge University Press, Cambridge Eng., 1978.</p>
+</div>
+<div class="citation" id="id156" role="doc-biblioentry">
+<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id34">Odu53</a><span class="fn-bracket">]</span></span>
+<p>Eugene Pleasants Odum. <em>Fundamentals of ecology</em>. Saunders, Philadelphia, 1953.</p>
+</div>
+<div class="citation" id="id93" role="doc-biblioentry">
+<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id70">Paw03</a><span class="fn-bracket">]</span></span>
+<p>Samraat Pawar. Taxonomic chauvinism and the methodologically challenged. <em>Bioscience</em>, 53(9):861–864, 2003.</p>
+</div>
+<div class="citation" id="id120" role="doc-biblioentry">
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+<span class="backrefs">(<a role="doc-backlink" href="#id39">1</a>,<a role="doc-backlink" href="#id40">2</a>)</span>
+<p>Eric R Pianka and Herbert Parkes Riley. <em>Evolutionary ecology</em>. Dickenson Pub. Co, San Francisco, 1970.</p>
+</div>
+<div class="citation" id="id197" role="doc-biblioentry">
+<span class="label"><span class="fn-bracket">[</span>Pro01<span class="fn-bracket">]</span></span>
+<span class="backrefs">(<a role="doc-backlink" href="#id24">1</a>,<a role="doc-backlink" href="#id45">2</a>,<a role="doc-backlink" href="#id49">3</a>)</span>
+<p>William B Provine. <em>The origins of theoretical population genetics</em>. University of Chicago Press, Chicago, 2001.</p>
+</div>
+<div class="citation" id="id123" role="doc-biblioentry">
+<span class="label"><span class="fn-bracket">[</span>Ros87<span class="fn-bracket">]</span></span>
+<span class="backrefs">(<a role="doc-backlink" href="#id37">1</a>,<a role="doc-backlink" href="#id41">2</a>)</span>
+<p>M L Rosenzweig. No Title. <em>Evol. Ecol.</em>, 1:1–3, 1987.</p>
+</div>
+<div class="citation" id="id173" role="doc-biblioentry">
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+<span class="backrefs">(<a role="doc-backlink" href="#id39">1</a>,<a role="doc-backlink" href="#id85">2</a>)</span>
+<p>J Roughgarden. <em>Theory of Population Genetics and Evolutionary Ecology: an Introduction</em>. Macmillan Publishing Co., Inc., New York, 1979.</p>
+</div>
+<div class="citation" id="id143" role="doc-biblioentry">
+<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id53">Rou72</a><span class="fn-bracket">]</span></span>
+<p>Jonathan Roughgarden. Evolution of niche width. <em>Am. Nat.</em>, 106(952):683–718, November 1972.</p>
+</div>
+<div class="citation" id="id119" role="doc-biblioentry">
+<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id39">SL70</a><span class="fn-bracket">]</span></span>
+<p>K P Sammeta and R Levins. Genetics and ecology. <em>Annu. Rev. Genet.</em>, 4(4):469–488, January 1970.</p>
+</div>
+<div class="citation" id="id172" role="doc-biblioentry">
+<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id12">SZ78</a><span class="fn-bracket">]</span></span>
+<p>Francesco M Scudo and James R Ziegler. <em>The Golden age of theoretical ecology, 1923-1940: a collection of works by V. Volterra, V. A. Kostitzin, A. J. Lotka, and A. N. Kolmogoroff</em>. Springer-Verlag, Berlin, 1978.</p>
+</div>
+<div class="citation" id="id104" role="doc-biblioentry">
+<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id56">SL76</a><span class="fn-bracket">]</span></span>
+<p>M Slatkin and R Lande. Niche width in a fluctuating environment: density independent model. <em>Am. Nat.</em>, 110(971):31–55, 1976.</p>
+</div>
+<div class="citation" id="id106" role="doc-biblioentry">
+<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id53">SMS79</a><span class="fn-bracket">]</span></span>
+<p>M Slatkin and J Maynard Smith. Models of coevolution. <em>Q. Rev. Biol.</em>, 54(3):233–263, 1979.</p>
+</div>
+<div class="citation" id="id108" role="doc-biblioentry">
+<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id53">Sla70</a><span class="fn-bracket">]</span></span>
+<p>Montgomery Slatkin. Selection and polygenic characters. <em>Proc. Natl. Acad. …</em>, 66(1):87–93, 1970.</p>
+</div>
+<div class="citation" id="id107" role="doc-biblioentry">
+<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id53">Sla79</a><span class="fn-bracket">]</span></span>
+<p>Montgomery Slatkin. The evolutionary response to frequency- and density-dependent interactions. <em>Am. Nat.</em>, 114(3):384–398, 1979.</p>
+</div>
+<div class="citation" id="id109" role="doc-biblioentry">
+<span class="label"><span class="fn-bracket">[</span>Sla80<span class="fn-bracket">]</span></span>
+<span class="backrefs">(<a role="doc-backlink" href="#id50">1</a>,<a role="doc-backlink" href="#id53">2</a>)</span>
+<p>Montogemery Slatkin. Ecological Character Displacement. <em>Ecology</em>, 61(1):163, February 1980.</p>
+</div>
+<div class="citation" id="id136" role="doc-biblioentry">
+<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id1">Sob93</a><span class="fn-bracket">]</span></span>
+<p>Elliott Sober. <em>The nature of selection: evolutionary theory in philosophical focus</em>. University of Chicago Press, Chicago, 1993.</p>
+</div>
+<div class="citation" id="id121" role="doc-biblioentry">
+<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id39">Sol80</a><span class="fn-bracket">]</span></span>
+<p>Otto Thomas Solbrig. <em>Demography and evolution in plant populations</em>. University of California Press, Berkeley, 1980.</p>
+</div>
+<div class="citation" id="id176" role="doc-biblioentry">
+<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id39">SS79</a><span class="fn-bracket">]</span></span>
+<p>Otto Thomas Solbrig and Dorothy J Solbrig. <em>Introduction to population biology &amp; evolution</em>. Addison-Wesley Pub. Co, Reading, Mass, 1979.</p>
+</div>
+<div class="citation" id="id133" role="doc-biblioentry">
+<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id8">Spe64</a><span class="fn-bracket">]</span></span>
+<p>Herbert Spencer. <em>The principles of biology</em>. William and Norgate, London etc., 1864.</p>
+</div>
+<div class="citation" id="id128" role="doc-biblioentry">
+<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id10">SRML89</a><span class="fn-bracket">]</span></span>
+<p>Steven M Stanley, J Roughgarden, RM May, and SA Levin. Fossils, macroevolution, and theoretical ecology. <em>Perspectives in ecological theory</em>, pages 125–134, 1989.</p>
+</div>
+<div class="citation" id="id132" role="doc-biblioentry">
+<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id7">SS57</a><span class="fn-bracket">]</span></span>
+<p>R C Robert C Stauffer and B Y Robert C Stauffer. Haeckel, Darwin, and ecology. <em>Q. Rev. Biol.</em>, 32(2):138–144, 1957.</p>
+</div>
+<div class="citation" id="id98" role="doc-biblioentry">
+<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id61">SG99</a><span class="fn-bracket">]</span></span>
+<p>Kim Sterelny and Paul E Griffiths. <em>Sex and death: an introduction to philosophy of biology</em>. University of Chicago Press, Chicago, Ill, 1999.</p>
+</div>
+<div class="citation" id="id146" role="doc-biblioentry">
+<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id39">SP77</a><span class="fn-bracket">]</span></span>
+<p>Bernard Stonehouse and Christopher M Perrins. <em>Evolutionary ecology</em>. University Park Press, Baltimore, Md, 1977.</p>
+</div>
+<div class="citation" id="id112" role="doc-biblioentry">
+<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id39">Str84</a><span class="fn-bracket">]</span></span>
+<p>Donald R Strong. <em>Ecological communities: conceptual issues and the evidence</em>. Princeton University Press, Princeton, N.J, 1984.</p>
+</div>
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+<p>M Tokeshi. <em>Species coexistence: ecological and evolutionary perspectives</em>. Blackwell Science, Oxford, 1999.</p>
+</div>
+<div class="citation" id="id164" role="doc-biblioentry">
+<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id34">Wei65</a><span class="fn-bracket">]</span></span>
+<p>F Weiling. Lotka, a. j.: elements of mathematical biology. dover publications inc., New‐York 1956; XXX + 465 S., 72 abb., 36 tabellen und 4 übersichtstabellen im anhang, preis \$ 2,45. <em>Biom. Z.</em>, 7(3):208–208, January 1965.</p>
+</div>
+<div class="citation" id="id170" role="doc-biblioentry">
+<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id31">Wri31</a><span class="fn-bracket">]</span></span>
+<p>Sewall Wright. Evolution in Mendelian populations. <em>Bull. Math. Biol.</em>, 52(1-2):241–295, March 1931.</p>
+</div>
+<div class="citation" id="id118" role="doc-biblioentry">
+<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id39">WL84</a><span class="fn-bracket">]</span></span>
+<p>K Wöhrmann and V Loeschcke. <em>Population biology and evolution</em>. Springer-Verlag, Berlin, 1984.</p>
+</div>
+</div>
+</div>
+<section id="footnotes">
+<h3>Footnotes<a class="headerlink" href="#footnotes" title="Link to this heading">#</a></h3>
+<!-- Some key recommendations and areas for revision / updating:
+
+### **1\. New Theoretical Insights and Themes**
+
+* **Eco-Evolutionary Dynamics**: Emphasize how eco-evolutionary feedbacks integrate both ecology and evolutionary biology, showcasing research on how evolutionary changes can affect ecological interactions and vice versa (e.g., Palkovacs et al., 2018).  
+* **Adaptive Dynamics**: Expand on adaptive dynamics as a formal framework for studying evolutionary outcomes in ecological contexts, highlighting recent advancements and applications (e.g., Kisdi, 2010).  
+* **Niche Construction**: Discuss how organisms can shape their own selective environments, integrating this with ecological theory to highlight mutual impacts (Laland et al., 2016).
+
+### **2\. Updated Empirical Examples**
+
+* **Rapid Evolution in Natural Populations**: Provide examples from recent studies showing real-time evolution, such as guppy life-history evolution in response to predation (Reznick et al., 1997; Travis et al., 2014).  
+* **Microbial Evolution Experiments**: Highlight long-term experiments with *E. coli* (e.g., Lenski's experiments) that reveal evolutionary and ecological principles in action (Lenski et al., 2015).  
+* **Climate Change Impacts**: Use contemporary examples to discuss how climate change drives rapid evolutionary responses and shifts in population dynamics (e.g., Franks et al., 2018).
+
+### **3\. Contemporary Challenges and Critiques**
+
+* **Philosophical Implications**: Expand the philosophical exploration of whether ecology and evolutionary biology can be fully unified, considering recent critiques of reductionism and the debate on predictive models (e.g., Sterelny and Griffiths, 2019).  
+* **Data Integration Challenges**: Discuss the difficulties in integrating large-scale ecological data with evolutionary models due to complexity and variability in environmental interactions.
+
+### **4\. Methodological Advances**
+
+* **Genomic Tools and Phylogenetics**: Highlight the role of whole-genome sequencing in connecting phylogenetic history with ecological traits and evolutionary processes.  
+* **Computational Ecology**: Explore how computational tools, such as agent-based modeling and machine learning, have revolutionized the analysis of eco-evolutionary systems (e.g., Schreiber et al., 2021).
+
+### **5\. Additional Citations and References**
+
+* Integrate more recent references, such as:  
+  * Laland, K. N., Matthews, B., & Feldman, M. W. (2016). "An update on niche construction theory."  
+  * Palkovacs, E. P., Kinnison, M. T., Correa, C., et al. (2018). "Eco-evolutionary feedbacks in community and ecosystem ecology."  
+  * Franks, S. J., Weber, J. J., & Aitken, S. N. (2018). "Evolutionary and plastic responses to climate change in terrestrial plant populations."
+
+### **6\. Emerging Topics for Discussion**
+
+* **Evolutionary Rescue**: Introduce how populations facing environmental changes may evolve rapidly to avoid extinction, drawing on cases of evolutionary rescue.  
+* **Anthropocene Impacts**: Discuss how human-induced changes create novel selective pressures and ecological shifts, which merge evolutionary and ecological responses.  
+* **Trait-Based Approaches**: Expand on how trait-based frameworks can elucidate the intersection of evolutionary adaptation and ecological dynamics.
+
+**Include adaptive dynamics \- (Kisdi, 2010\)**
+
+ --><hr class="footnotes docutils" />
+<aside class="footnote-list brackets">
+<aside class="footnote brackets" id="id198" role="doc-footnote">
+<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id3">2</a><span class="fn-bracket">]</span></span>
+<p>The area of niche construction, which involves the study of evolutionary dynamics in situations where populations modify their own environment <a class="reference external" href="https://paperpile.com/c/lwDPbc/1C1p">(Odling-Smee, Laland, and Feldman 2003)</a>, although an interesting interface between ecology and evolution, poses a distinct set of challenges, and will not covered here.</p>
+</aside>
+<aside class="footnote brackets" id="id199" role="doc-footnote">
+<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id6">3</a><span class="fn-bracket">]</span></span>
+<p>Ideas of population growth, which formed the basis for Darwin’s theory of natural selection predate the theory of natural selection in Malthus’ treatise on populations in 1798, and the mathematical formulation of the first model of regulated (“logistical”) population growth, by Verhulst <a class="reference external" href="https://paperpile.com/c/lwDPbc/WSwP">(Kingsland 1985)</a> (pp. 64-66).</p>
+</aside>
+<aside class="footnote brackets" id="id200" role="doc-footnote">
+<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id15">4</a><span class="fn-bracket">]</span></span>
+<p>I am referring to the Biometric and the Mendelian schools <a class="reference external" href="https://paperpile.com/c/lwDPbc/L0vt">(Provine 2001)</a>; for examples, see the preface to Gause’s book <a class="reference external" href="https://paperpile.com/c/lwDPbc/2qwm">(Gause 1934)</a>, and Kingsland’s analysis <a class="reference external" href="https://paperpile.com/c/lwDPbc/WSwP">(Kingsland 1985)</a>.</p>
+</aside>
+<aside class="footnote brackets" id="id201" role="doc-footnote">
+<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id23">5</a><span class="fn-bracket">]</span></span>
+<p>Efforts to fuse genetics and ecology were also made in the Russian school of ecology by people such as V.V. Stanchinskii <a class="reference external" href="https://paperpile.com/c/lwDPbc/WSwP">(Kingsland 1985)</a>(pp. 150, 157-160) <a class="reference external" href="https://paperpile.com/c/lwDPbc/ROCY">(Mirzoyan 1996)</a>, the details of which I am unable to access.</p>
+</aside>
+<aside class="footnote brackets" id="id202" role="doc-footnote">
+<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id32">6</a><span class="fn-bracket">]</span></span>
+<p>“Cultural differences” has been invoked as a reason for lack of communication between the two disciplines by proponents of the population genetics – population ecology synthesis. For example, Roughgarden <a class="reference external" href="https://paperpile.com/c/lwDPbc/Ccks">(J. Roughgarden 1979)</a> says,<em>”The fusion of population genetics with population ecology can be compared to a prearranged marriage between partners who speak different languages. Although both families agree that the marriage is advantageous, it is somewhat difficult to achieve because of cultural differences.”</em></p>
+</aside>
+<aside class="footnote brackets" id="id203" role="doc-footnote">
+<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id43">7</a><span class="fn-bracket">]</span></span>
+<p>In Russia as well, there appears to have been developments of a similar nature <a class="reference external" href="https://paperpile.com/c/lwDPbc/IuTu">(Shvarts 1977)</a>.</p>
+</aside>
+<aside class="footnote brackets" id="id204" role="doc-footnote">
+<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id46">8</a><span class="fn-bracket">]</span></span>
+<p>In its objectives and methods, this field is now barely distinguishable from a recent off-shoot of ecology called “Molecular Ecology”, which originated in the 1990’s as a program to use molecular genetic methods to address problems in conservation biology, but then ceased to be a purely applied field, expanding and converging in its objectives and methods on Ecological Genetics. This is an interesting example of an inadvertent convergence of ecological and evolutionary fields driven by a common motivation.</p>
+</aside>
+<aside class="footnote brackets" id="id205" role="doc-footnote">
+<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id52">9</a><span class="fn-bracket">]</span></span>
+<p>The first model was actually developed much earlier, by Kostitzin in 1936 <a class="reference external" href="https://paperpile.com/c/lwDPbc/VMHP">(Christiansen, Singh, and Uyenoyama 2004)</a>.</p>
+</aside>
+<aside class="footnote brackets" id="id206" role="doc-footnote">
+<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id88">10</a><span class="fn-bracket">]</span></span>
+<p>Due to two main reasons: Adaptationism (a belief in the determinism of evolution) and theology (belief in a creator as the determining factor in biology).</p>
+</aside>
+</aside>
+</section>
+</section>
+</section>
+
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+  <div class="page-toc tocsection onthispage">
+    <i class="fa-solid fa-list"></i> Contents
+  </div>
+  <nav class="bd-toc-nav page-toc">
+    <ul class="visible nav section-nav flex-column">
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#introduction">Introduction</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-roots-of-disharmony">The Roots of Disharmony</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#contemporary-ecology-vis-a-vis-evolutionary-biology">Contemporary ecology vis-à-vis evolutionary biology</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#an-early-association-and-subsequent-divergence-1850-1950">An early association and subsequent divergence: 1850-1950</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#marriage-and-the-quest-for-harmony-1950-1985">Marriage and the quest for harmony: 1950-1985</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-marriage-of-ecology-and-evolutionary-biology-the-ups-the-downs-and-a-taxonomy">The marriage of ecology and evolutionary biology: the ups, the downs, and a taxonomy</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#the-issue-of-ecological-vs-evolutionary-time-scales">The issue of ecological vs. evolutionary time scales</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#demographic-considerations-and-measures-of-fitness">Demographic considerations and measures of fitness</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#the-role-of-environmental-heterogeneity">The role of environmental heterogeneity</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#taxonomy-of-evolutionary-ecology">Taxonomy of evolutionary ecology**</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#discussion">Discussion</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#has-the-marriage-come-of-age">Has The Marriage Come Of Age?</a></li>
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#on-the-dullness-of-ecology">On the Dullness of Ecology</a></li>
+</ul>
+</li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#conclusion">Conclusion</a></li>
+<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#references">References</a><ul class="nav section-nav flex-column">
+<li class="toc-h3 nav-item toc-entry"><a class="reference internal nav-link" href="#footnotes">Footnotes</a></li>
+</ul>
+</li>
+</ul>
+  </nav></div>
+
+</div></div>
+              
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+          </div>
+          <footer class="bd-footer-content">
+            
+<div class="bd-footer-content__inner container">
+  
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+By MulQuaBio collective!
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+  <div class="footer-item">
+    
+
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+
+  <footer class="bd-footer">
+  </footer>
+  </body>
+</html>
\ No newline at end of file
diff --git a/notebooks/Unix.html b/notebooks/Unix.html
index f333fe8d..af8f17f0 100644
--- a/notebooks/Unix.html
+++ b/notebooks/Unix.html
@@ -8,7 +8,7 @@
     <meta charset="utf-8" />
     <meta name="viewport" content="width=device-width, initial-scale=1.0" /><meta name="viewport" content="width=device-width, initial-scale=1" />
 
-    <title>UNIX and Linux &#8212; TheMulQuaBio</title>
+    <title>UNIX and Linux &#8212; MulQuaBio</title>
   
   
   
@@ -153,8 +153,8 @@
       
     
     
-    <img src="../_static/logo.png" class="logo__image only-light" alt="TheMulQuaBio - Home"/>
-    <script>document.write(`<img src="../_static/logo.png" class="logo__image only-dark" alt="TheMulQuaBio - Home"/>`);</script>
+    <img src="../_static/logo.png" class="logo__image only-light" alt="MulQuaBio - Home"/>
+    <script>document.write(`<img src="../_static/logo.png" class="logo__image only-dark" alt="MulQuaBio - Home"/>`);</script>
   
   
 </a></div>
@@ -188,23 +188,9 @@
 <li class="toctree-l1"><a class="reference internal" href="Python.html">Biological Computing in Python</a></li>
 <li class="toctree-l1"><a class="reference internal" href="R.html">Biological Computing in R</a></li>
 </ul>
-<p aria-level="2" class="caption" role="heading"><span class="caption-text">Maths for Biologists</span></p>
-<ul class="nav bd-sidenav">
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture01/01_Making_mathematical_statements.html">01 - Making mathematical statements</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture02/02_Describing_shapes_and_patterns.html">02   - Describing shapes and patterns</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture03/03_Analysing_change%2C_one_step_at_a_time.html">03   - Analysing change, one step at a time</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture04/04_From_table_arrangements_to_powerful_tools.html">04 - From table arrangements to powerful tools</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture05/05_From_table_arrangements_to_powerful_tools_part2.html">05 - From table arrangements to powerful tools - part 2</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture06/06_Analysing_change_continuously.html">06 - Analysing change, continuously</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture07/07_Analysing_change_continuously_part2.html">07 - Analysing change, continuously (Part 2)</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture08/08_Calculating_accumulated_change.html">08 - Calculating accumulated change</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/tutorials/lecture01/Tutorial_01.html">Tutorial 01</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/tutorials/lecture05/Tutorial_05.html">Tutorial 05</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/tutorials/lecture06/Tutorial_06.html">Tutorial 06</a></li>
-</ul>
 <p aria-level="2" class="caption" role="heading"><span class="caption-text">Basic Data Analyses and Statistics</span></p>
 <ul class="nav bd-sidenav">
-<li class="toctree-l1"><a class="reference internal" href="../Stats-Intro.html">Basic Data Science and Statistics: Introduction</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../intro-Stats.html">Introduction</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Data_R.html">Data Management and Visualization</a></li>
 <li class="toctree-l1"><a class="reference internal" href="ExpDesign.html">Experimental design and Data exploration</a></li>
 <li class="toctree-l1"><a class="reference internal" href="t_F_tests.html">Basic hypothesis testing: <span class="math notranslate nohighlight">\(t\)</span> and <span class="math notranslate nohighlight">\(F\)</span> tests</a></li>
@@ -224,6 +210,7 @@
 <li class="toctree-l1"><a class="reference internal" href="Appendix-JupyIntro.html">Introduction to Jupyter</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-Data-Python.html">Data analyses  with Python &amp; Jupyter</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-Maths.html">Mathematical models in Jupyter</a></li>
+<li class="toctree-l1"><a class="reference internal" href="Appendix-MathsForBiologists/MathsForBiologists.html">Maths for Biologists</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-Databases.html">Databases <span class="tocSkip"></span></a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-NLLS-Python.html">Model fitting in Python</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-MiniProj.html">The Computing Miniproject</a></li>
@@ -283,7 +270,7 @@
       
       
       
-      <li><a href="https://github.com/mhasoba/TheMulQuaBio" target="_blank"
+      <li><a href="https://github.com/MulQuaBio/MQB" target="_blank"
    class="btn btn-sm btn-source-repository-button dropdown-item"
    title="Source repository"
    data-bs-placement="left" data-bs-toggle="tooltip"
@@ -300,7 +287,7 @@
       
       
       
-      <li><a href="https://github.com/mhasoba/TheMulQuaBio/issues/new?title=Issue%20on%20page%20%2Fnotebooks/Unix.html&body=Your%20issue%20content%20here." target="_blank"
+      <li><a href="https://github.com/MulQuaBio/MQB/issues/new?title=Issue%20on%20page%20%2Fnotebooks/Unix.html&body=Your%20issue%20content%20here." target="_blank"
    class="btn btn-sm btn-source-issues-button dropdown-item"
    title="Open an issue"
    data-bs-placement="left" data-bs-toggle="tooltip"
@@ -2020,7 +2007,7 @@ <h3>Repositories and packages in Ubuntu<a class="headerlink" href="#repositories
   <div class="footer-item">
     
 <p class="component-author">
-By TheMulQuaBio collective!
+By MulQuaBio collective!
 </p>
 
   </div>
diff --git a/notebooks/anova.html b/notebooks/anova.html
index fc16b8e7..6ba422e8 100644
--- a/notebooks/anova.html
+++ b/notebooks/anova.html
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     <meta charset="utf-8" />
     <meta name="viewport" content="width=device-width, initial-scale=1.0" /><meta name="viewport" content="width=device-width, initial-scale=1" />
 
-    <title>Linear Models: Analysis of variance &#8212; TheMulQuaBio</title>
+    <title>Linear Models: Analysis of variance &#8212; MulQuaBio</title>
   
   
   
@@ -153,8 +153,8 @@
       
     
     
-    <img src="../_static/logo.png" class="logo__image only-light" alt="TheMulQuaBio - Home"/>
-    <script>document.write(`<img src="../_static/logo.png" class="logo__image only-dark" alt="TheMulQuaBio - Home"/>`);</script>
+    <img src="../_static/logo.png" class="logo__image only-light" alt="MulQuaBio - Home"/>
+    <script>document.write(`<img src="../_static/logo.png" class="logo__image only-dark" alt="MulQuaBio - Home"/>`);</script>
   
   
 </a></div>
@@ -188,23 +188,9 @@
 <li class="toctree-l1"><a class="reference internal" href="Python.html">Biological Computing in Python</a></li>
 <li class="toctree-l1"><a class="reference internal" href="R.html">Biological Computing in R</a></li>
 </ul>
-<p aria-level="2" class="caption" role="heading"><span class="caption-text">Maths for Biologists</span></p>
-<ul class="nav bd-sidenav">
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture01/01_Making_mathematical_statements.html">01 - Making mathematical statements</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture02/02_Describing_shapes_and_patterns.html">02   - Describing shapes and patterns</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture03/03_Analysing_change%2C_one_step_at_a_time.html">03   - Analysing change, one step at a time</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture04/04_From_table_arrangements_to_powerful_tools.html">04 - From table arrangements to powerful tools</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture05/05_From_table_arrangements_to_powerful_tools_part2.html">05 - From table arrangements to powerful tools - part 2</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture06/06_Analysing_change_continuously.html">06 - Analysing change, continuously</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture07/07_Analysing_change_continuously_part2.html">07 - Analysing change, continuously (Part 2)</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture08/08_Calculating_accumulated_change.html">08 - Calculating accumulated change</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/tutorials/lecture01/Tutorial_01.html">Tutorial 01</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/tutorials/lecture05/Tutorial_05.html">Tutorial 05</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/tutorials/lecture06/Tutorial_06.html">Tutorial 06</a></li>
-</ul>
 <p aria-level="2" class="caption" role="heading"><span class="caption-text">Basic Data Analyses and Statistics</span></p>
 <ul class="current nav bd-sidenav">
-<li class="toctree-l1"><a class="reference internal" href="../Stats-Intro.html">Basic Data Science and Statistics: Introduction</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../intro-Stats.html">Introduction</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Data_R.html">Data Management and Visualization</a></li>
 <li class="toctree-l1"><a class="reference internal" href="ExpDesign.html">Experimental design and Data exploration</a></li>
 <li class="toctree-l1"><a class="reference internal" href="t_F_tests.html">Basic hypothesis testing: <span class="math notranslate nohighlight">\(t\)</span> and <span class="math notranslate nohighlight">\(F\)</span> tests</a></li>
@@ -224,6 +210,7 @@
 <li class="toctree-l1"><a class="reference internal" href="Appendix-JupyIntro.html">Introduction to Jupyter</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-Data-Python.html">Data analyses  with Python &amp; Jupyter</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-Maths.html">Mathematical models in Jupyter</a></li>
+<li class="toctree-l1"><a class="reference internal" href="Appendix-MathsForBiologists/MathsForBiologists.html">Maths for Biologists</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-Databases.html">Databases <span class="tocSkip"></span></a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-NLLS-Python.html">Model fitting in Python</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-MiniProj.html">The Computing Miniproject</a></li>
@@ -283,7 +270,7 @@
       
       
       
-      <li><a href="https://github.com/mhasoba/TheMulQuaBio" target="_blank"
+      <li><a href="https://github.com/MulQuaBio/MQB" target="_blank"
    class="btn btn-sm btn-source-repository-button dropdown-item"
    title="Source repository"
    data-bs-placement="left" data-bs-toggle="tooltip"
@@ -300,7 +287,7 @@
       
       
       
-      <li><a href="https://github.com/mhasoba/TheMulQuaBio/issues/new?title=Issue%20on%20page%20%2Fnotebooks/anova.html&body=Your%20issue%20content%20here." target="_blank"
+      <li><a href="https://github.com/MulQuaBio/MQB/issues/new?title=Issue%20on%20page%20%2Fnotebooks/anova.html&body=Your%20issue%20content%20here." target="_blank"
    class="btn btn-sm btn-source-issues-button dropdown-item"
    title="Open an issue"
    data-bs-placement="left" data-bs-toggle="tooltip"
@@ -1158,7 +1145,7 @@ <h2>Saving data<a class="headerlink" href="#saving-data" title="Link to this hea
   <div class="footer-item">
     
 <p class="component-author">
-By TheMulQuaBio collective!
+By MulQuaBio collective!
 </p>
 
   </div>
diff --git a/notebooks/glm.html b/notebooks/glm.html
index 5aa42371..47dadbd3 100644
--- a/notebooks/glm.html
+++ b/notebooks/glm.html
@@ -8,7 +8,7 @@
     <meta charset="utf-8" />
     <meta name="viewport" content="width=device-width, initial-scale=1.0" /><meta name="viewport" content="width=device-width, initial-scale=1" />
 
-    <title>Generalised Linear Models &#8212; TheMulQuaBio</title>
+    <title>Generalised Linear Models &#8212; MulQuaBio</title>
   
   
   
@@ -153,8 +153,8 @@
       
     
     
-    <img src="../_static/logo.png" class="logo__image only-light" alt="TheMulQuaBio - Home"/>
-    <script>document.write(`<img src="../_static/logo.png" class="logo__image only-dark" alt="TheMulQuaBio - Home"/>`);</script>
+    <img src="../_static/logo.png" class="logo__image only-light" alt="MulQuaBio - Home"/>
+    <script>document.write(`<img src="../_static/logo.png" class="logo__image only-dark" alt="MulQuaBio - Home"/>`);</script>
   
   
 </a></div>
@@ -188,23 +188,9 @@
 <li class="toctree-l1"><a class="reference internal" href="Python.html">Biological Computing in Python</a></li>
 <li class="toctree-l1"><a class="reference internal" href="R.html">Biological Computing in R</a></li>
 </ul>
-<p aria-level="2" class="caption" role="heading"><span class="caption-text">Maths for Biologists</span></p>
-<ul class="nav bd-sidenav">
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture01/01_Making_mathematical_statements.html">01 - Making mathematical statements</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture02/02_Describing_shapes_and_patterns.html">02   - Describing shapes and patterns</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture03/03_Analysing_change%2C_one_step_at_a_time.html">03   - Analysing change, one step at a time</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture04/04_From_table_arrangements_to_powerful_tools.html">04 - From table arrangements to powerful tools</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture05/05_From_table_arrangements_to_powerful_tools_part2.html">05 - From table arrangements to powerful tools - part 2</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture06/06_Analysing_change_continuously.html">06 - Analysing change, continuously</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture07/07_Analysing_change_continuously_part2.html">07 - Analysing change, continuously (Part 2)</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture08/08_Calculating_accumulated_change.html">08 - Calculating accumulated change</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/tutorials/lecture01/Tutorial_01.html">Tutorial 01</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/tutorials/lecture05/Tutorial_05.html">Tutorial 05</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/tutorials/lecture06/Tutorial_06.html">Tutorial 06</a></li>
-</ul>
 <p aria-level="2" class="caption" role="heading"><span class="caption-text">Basic Data Analyses and Statistics</span></p>
 <ul class="nav bd-sidenav">
-<li class="toctree-l1"><a class="reference internal" href="../Stats-Intro.html">Basic Data Science and Statistics: Introduction</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../intro-Stats.html">Introduction</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Data_R.html">Data Management and Visualization</a></li>
 <li class="toctree-l1"><a class="reference internal" href="ExpDesign.html">Experimental design and Data exploration</a></li>
 <li class="toctree-l1"><a class="reference internal" href="t_F_tests.html">Basic hypothesis testing: <span class="math notranslate nohighlight">\(t\)</span> and <span class="math notranslate nohighlight">\(F\)</span> tests</a></li>
@@ -224,6 +210,7 @@
 <li class="toctree-l1"><a class="reference internal" href="Appendix-JupyIntro.html">Introduction to Jupyter</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-Data-Python.html">Data analyses  with Python &amp; Jupyter</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-Maths.html">Mathematical models in Jupyter</a></li>
+<li class="toctree-l1"><a class="reference internal" href="Appendix-MathsForBiologists/MathsForBiologists.html">Maths for Biologists</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-Databases.html">Databases <span class="tocSkip"></span></a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-NLLS-Python.html">Model fitting in Python</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-MiniProj.html">The Computing Miniproject</a></li>
@@ -283,7 +270,7 @@
       
       
       
-      <li><a href="https://github.com/mhasoba/TheMulQuaBio" target="_blank"
+      <li><a href="https://github.com/MulQuaBio/MQB" target="_blank"
    class="btn btn-sm btn-source-repository-button dropdown-item"
    title="Source repository"
    data-bs-placement="left" data-bs-toggle="tooltip"
@@ -300,7 +287,7 @@
       
       
       
-      <li><a href="https://github.com/mhasoba/TheMulQuaBio/issues/new?title=Issue%20on%20page%20%2Fnotebooks/glm.html&body=Your%20issue%20content%20here." target="_blank"
+      <li><a href="https://github.com/MulQuaBio/MQB/issues/new?title=Issue%20on%20page%20%2Fnotebooks/glm.html&body=Your%20issue%20content%20here." target="_blank"
    class="btn btn-sm btn-source-issues-button dropdown-item"
    title="Open an issue"
    data-bs-placement="left" data-bs-toggle="tooltip"
@@ -1377,7 +1364,7 @@ <h2>Halos and lawns<a class="headerlink" href="#halos-and-lawns" title="Link to
   <div class="footer-item">
     
 <p class="component-author">
-By TheMulQuaBio collective!
+By MulQuaBio collective!
 </p>
 
   </div>
diff --git a/notebooks/regress.html b/notebooks/regress.html
index ff60fff7..40548933 100644
--- a/notebooks/regress.html
+++ b/notebooks/regress.html
@@ -8,7 +8,7 @@
     <meta charset="utf-8" />
     <meta name="viewport" content="width=device-width, initial-scale=1.0" /><meta name="viewport" content="width=device-width, initial-scale=1" />
 
-    <title>Linear Models: Regression &#8212; TheMulQuaBio</title>
+    <title>Linear Models: Regression &#8212; MulQuaBio</title>
   
   
   
@@ -153,8 +153,8 @@
       
     
     
-    <img src="../_static/logo.png" class="logo__image only-light" alt="TheMulQuaBio - Home"/>
-    <script>document.write(`<img src="../_static/logo.png" class="logo__image only-dark" alt="TheMulQuaBio - Home"/>`);</script>
+    <img src="../_static/logo.png" class="logo__image only-light" alt="MulQuaBio - Home"/>
+    <script>document.write(`<img src="../_static/logo.png" class="logo__image only-dark" alt="MulQuaBio - Home"/>`);</script>
   
   
 </a></div>
@@ -188,23 +188,9 @@
 <li class="toctree-l1"><a class="reference internal" href="Python.html">Biological Computing in Python</a></li>
 <li class="toctree-l1"><a class="reference internal" href="R.html">Biological Computing in R</a></li>
 </ul>
-<p aria-level="2" class="caption" role="heading"><span class="caption-text">Maths for Biologists</span></p>
-<ul class="nav bd-sidenav">
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture01/01_Making_mathematical_statements.html">01 - Making mathematical statements</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture02/02_Describing_shapes_and_patterns.html">02   - Describing shapes and patterns</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture03/03_Analysing_change%2C_one_step_at_a_time.html">03   - Analysing change, one step at a time</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture04/04_From_table_arrangements_to_powerful_tools.html">04 - From table arrangements to powerful tools</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture05/05_From_table_arrangements_to_powerful_tools_part2.html">05 - From table arrangements to powerful tools - part 2</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture06/06_Analysing_change_continuously.html">06 - Analysing change, continuously</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture07/07_Analysing_change_continuously_part2.html">07 - Analysing change, continuously (Part 2)</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture08/08_Calculating_accumulated_change.html">08 - Calculating accumulated change</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/tutorials/lecture01/Tutorial_01.html">Tutorial 01</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/tutorials/lecture05/Tutorial_05.html">Tutorial 05</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/tutorials/lecture06/Tutorial_06.html">Tutorial 06</a></li>
-</ul>
 <p aria-level="2" class="caption" role="heading"><span class="caption-text">Basic Data Analyses and Statistics</span></p>
 <ul class="current nav bd-sidenav">
-<li class="toctree-l1"><a class="reference internal" href="../Stats-Intro.html">Basic Data Science and Statistics: Introduction</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../intro-Stats.html">Introduction</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Data_R.html">Data Management and Visualization</a></li>
 <li class="toctree-l1"><a class="reference internal" href="ExpDesign.html">Experimental design and Data exploration</a></li>
 <li class="toctree-l1"><a class="reference internal" href="t_F_tests.html">Basic hypothesis testing: <span class="math notranslate nohighlight">\(t\)</span> and <span class="math notranslate nohighlight">\(F\)</span> tests</a></li>
@@ -224,6 +210,7 @@
 <li class="toctree-l1"><a class="reference internal" href="Appendix-JupyIntro.html">Introduction to Jupyter</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-Data-Python.html">Data analyses  with Python &amp; Jupyter</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-Maths.html">Mathematical models in Jupyter</a></li>
+<li class="toctree-l1"><a class="reference internal" href="Appendix-MathsForBiologists/MathsForBiologists.html">Maths for Biologists</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-Databases.html">Databases <span class="tocSkip"></span></a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-NLLS-Python.html">Model fitting in Python</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-MiniProj.html">The Computing Miniproject</a></li>
@@ -283,7 +270,7 @@
       
       
       
-      <li><a href="https://github.com/mhasoba/TheMulQuaBio" target="_blank"
+      <li><a href="https://github.com/MulQuaBio/MQB" target="_blank"
    class="btn btn-sm btn-source-repository-button dropdown-item"
    title="Source repository"
    data-bs-placement="left" data-bs-toggle="tooltip"
@@ -300,7 +287,7 @@
       
       
       
-      <li><a href="https://github.com/mhasoba/TheMulQuaBio/issues/new?title=Issue%20on%20page%20%2Fnotebooks/regress.html&body=Your%20issue%20content%20here." target="_blank"
+      <li><a href="https://github.com/MulQuaBio/MQB/issues/new?title=Issue%20on%20page%20%2Fnotebooks/regress.html&body=Your%20issue%20content%20here." target="_blank"
    class="btn btn-sm btn-source-issues-button dropdown-item"
    title="Open an issue"
    data-bs-placement="left" data-bs-toggle="tooltip"
@@ -1188,7 +1175,7 @@ <h2>Reporting the model<a class="headerlink" href="#reporting-the-model" title="
   <div class="footer-item">
     
 <p class="component-author">
-By TheMulQuaBio collective!
+By MulQuaBio collective!
 </p>
 
   </div>
diff --git a/notebooks/t_F_tests.html b/notebooks/t_F_tests.html
index e16c227f..691d90b0 100644
--- a/notebooks/t_F_tests.html
+++ b/notebooks/t_F_tests.html
@@ -8,7 +8,7 @@
     <meta charset="utf-8" />
     <meta name="viewport" content="width=device-width, initial-scale=1.0" /><meta name="viewport" content="width=device-width, initial-scale=1" />
 
-    <title>Basic hypothesis testing: \(t\) and \(F\) tests &#8212; TheMulQuaBio</title>
+    <title>Basic hypothesis testing: \(t\) and \(F\) tests &#8212; MulQuaBio</title>
   
   
   
@@ -153,8 +153,8 @@
       
     
     
-    <img src="../_static/logo.png" class="logo__image only-light" alt="TheMulQuaBio - Home"/>
-    <script>document.write(`<img src="../_static/logo.png" class="logo__image only-dark" alt="TheMulQuaBio - Home"/>`);</script>
+    <img src="../_static/logo.png" class="logo__image only-light" alt="MulQuaBio - Home"/>
+    <script>document.write(`<img src="../_static/logo.png" class="logo__image only-dark" alt="MulQuaBio - Home"/>`);</script>
   
   
 </a></div>
@@ -188,23 +188,9 @@
 <li class="toctree-l1"><a class="reference internal" href="Python.html">Biological Computing in Python</a></li>
 <li class="toctree-l1"><a class="reference internal" href="R.html">Biological Computing in R</a></li>
 </ul>
-<p aria-level="2" class="caption" role="heading"><span class="caption-text">Maths for Biologists</span></p>
-<ul class="nav bd-sidenav">
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture01/01_Making_mathematical_statements.html">01 - Making mathematical statements</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture02/02_Describing_shapes_and_patterns.html">02   - Describing shapes and patterns</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture03/03_Analysing_change%2C_one_step_at_a_time.html">03   - Analysing change, one step at a time</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture04/04_From_table_arrangements_to_powerful_tools.html">04 - From table arrangements to powerful tools</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture05/05_From_table_arrangements_to_powerful_tools_part2.html">05 - From table arrangements to powerful tools - part 2</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture06/06_Analysing_change_continuously.html">06 - Analysing change, continuously</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture07/07_Analysing_change_continuously_part2.html">07 - Analysing change, continuously (Part 2)</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/lecture08/08_Calculating_accumulated_change.html">08 - Calculating accumulated change</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/tutorials/lecture01/Tutorial_01.html">Tutorial 01</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/tutorials/lecture05/Tutorial_05.html">Tutorial 05</a></li>
-<li class="toctree-l1"><a class="reference internal" href="../mathscourse/tutorials/lecture06/Tutorial_06.html">Tutorial 06</a></li>
-</ul>
 <p aria-level="2" class="caption" role="heading"><span class="caption-text">Basic Data Analyses and Statistics</span></p>
 <ul class="current nav bd-sidenav">
-<li class="toctree-l1"><a class="reference internal" href="../Stats-Intro.html">Basic Data Science and Statistics: Introduction</a></li>
+<li class="toctree-l1"><a class="reference internal" href="../intro-Stats.html">Introduction</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Data_R.html">Data Management and Visualization</a></li>
 <li class="toctree-l1"><a class="reference internal" href="ExpDesign.html">Experimental design and Data exploration</a></li>
 <li class="toctree-l1 current active"><a class="current reference internal" href="#">Basic hypothesis testing: <span class="math notranslate nohighlight">\(t\)</span> and <span class="math notranslate nohighlight">\(F\)</span> tests</a></li>
@@ -224,6 +210,7 @@
 <li class="toctree-l1"><a class="reference internal" href="Appendix-JupyIntro.html">Introduction to Jupyter</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-Data-Python.html">Data analyses  with Python &amp; Jupyter</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-Maths.html">Mathematical models in Jupyter</a></li>
+<li class="toctree-l1"><a class="reference internal" href="Appendix-MathsForBiologists/MathsForBiologists.html">Maths for Biologists</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-Databases.html">Databases <span class="tocSkip"></span></a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-NLLS-Python.html">Model fitting in Python</a></li>
 <li class="toctree-l1"><a class="reference internal" href="Appendix-MiniProj.html">The Computing Miniproject</a></li>
@@ -283,7 +270,7 @@
       
       
       
-      <li><a href="https://github.com/mhasoba/TheMulQuaBio" target="_blank"
+      <li><a href="https://github.com/MulQuaBio/MQB" target="_blank"
    class="btn btn-sm btn-source-repository-button dropdown-item"
    title="Source repository"
    data-bs-placement="left" data-bs-toggle="tooltip"
@@ -300,7 +287,7 @@
       
       
       
-      <li><a href="https://github.com/mhasoba/TheMulQuaBio/issues/new?title=Issue%20on%20page%20%2Fnotebooks/t_F_tests.html&body=Your%20issue%20content%20here." target="_blank"
+      <li><a href="https://github.com/MulQuaBio/MQB/issues/new?title=Issue%20on%20page%20%2Fnotebooks/t_F_tests.html&body=Your%20issue%20content%20here." target="_blank"
    class="btn btn-sm btn-source-issues-button dropdown-item"
    title="Open an issue"
    data-bs-placement="left" data-bs-toggle="tooltip"
@@ -1091,7 +1078,7 @@ <h3>Exercise<a class="headerlink" href="#id3" title="Link to this heading">#</a>
   <div class="footer-item">
     
 <p class="component-author">
-By TheMulQuaBio collective!
+By MulQuaBio collective!
 </p>
 
   </div>
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diff --git a/search.html b/search.html
index ef49aeac..9f3ba788 100644
--- a/search.html
+++ b/search.html
@@ -6,7 +6,7 @@
 
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+    <meta name="viewport" content="width=device-width, initial-scale=1.0" /><title>Search - MulQuaBio</title>
   
   
   
@@ -152,8 +152,8 @@
       
     
     
-    <img src="_static/logo.png" class="logo__image only-light" alt="TheMulQuaBio - Home"/>
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 </a></div>
@@ -187,23 +187,9 @@
 <li class="toctree-l1"><a class="reference internal" href="notebooks/Python.html">Biological Computing in Python</a></li>
 <li class="toctree-l1"><a class="reference internal" href="notebooks/R.html">Biological Computing in R</a></li>
 </ul>
-<p aria-level="2" class="caption" role="heading"><span class="caption-text">Maths for Biologists</span></p>
-<ul class="nav bd-sidenav">
-<li class="toctree-l1"><a class="reference internal" href="mathscourse/lecture01/01_Making_mathematical_statements.html">01 - Making mathematical statements</a></li>
-<li class="toctree-l1"><a class="reference internal" href="mathscourse/lecture02/02_Describing_shapes_and_patterns.html">02   - Describing shapes and patterns</a></li>
-<li class="toctree-l1"><a class="reference internal" href="mathscourse/lecture03/03_Analysing_change%2C_one_step_at_a_time.html">03   - Analysing change, one step at a time</a></li>
-<li class="toctree-l1"><a class="reference internal" href="mathscourse/lecture04/04_From_table_arrangements_to_powerful_tools.html">04 - From table arrangements to powerful tools</a></li>
-<li class="toctree-l1"><a class="reference internal" href="mathscourse/lecture05/05_From_table_arrangements_to_powerful_tools_part2.html">05 - From table arrangements to powerful tools - part 2</a></li>
-<li class="toctree-l1"><a class="reference internal" href="mathscourse/lecture06/06_Analysing_change_continuously.html">06 - Analysing change, continuously</a></li>
-<li class="toctree-l1"><a class="reference internal" href="mathscourse/lecture07/07_Analysing_change_continuously_part2.html">07 - Analysing change, continuously (Part 2)</a></li>
-<li class="toctree-l1"><a class="reference internal" href="mathscourse/lecture08/08_Calculating_accumulated_change.html">08 - Calculating accumulated change</a></li>
-<li class="toctree-l1"><a class="reference internal" href="mathscourse/tutorials/lecture01/Tutorial_01.html">Tutorial 01</a></li>
-<li class="toctree-l1"><a class="reference internal" href="mathscourse/tutorials/lecture05/Tutorial_05.html">Tutorial 05</a></li>
-<li class="toctree-l1"><a class="reference internal" href="mathscourse/tutorials/lecture06/Tutorial_06.html">Tutorial 06</a></li>
-</ul>
 <p aria-level="2" class="caption" role="heading"><span class="caption-text">Basic Data Analyses and Statistics</span></p>
 <ul class="nav bd-sidenav">
-<li class="toctree-l1"><a class="reference internal" href="Stats-Intro.html">Basic Data Science and Statistics: Introduction</a></li>
+<li class="toctree-l1"><a class="reference internal" href="intro-Stats.html">Introduction</a></li>
 <li class="toctree-l1"><a class="reference internal" href="notebooks/Data_R.html">Data Management and Visualization</a></li>
 <li class="toctree-l1"><a class="reference internal" href="notebooks/ExpDesign.html">Experimental design and Data exploration</a></li>
 <li class="toctree-l1"><a class="reference internal" href="notebooks/t_F_tests.html">Basic hypothesis testing: <span class="math notranslate nohighlight">\(t\)</span> and <span class="math notranslate nohighlight">\(F\)</span> tests</a></li>
@@ -223,6 +209,7 @@
 <li class="toctree-l1"><a class="reference internal" href="notebooks/Appendix-JupyIntro.html">Introduction to Jupyter</a></li>
 <li class="toctree-l1"><a class="reference internal" href="notebooks/Appendix-Data-Python.html">Data analyses  with Python &amp; Jupyter</a></li>
 <li class="toctree-l1"><a class="reference internal" href="notebooks/Appendix-Maths.html">Mathematical models in Jupyter</a></li>
+<li class="toctree-l1"><a class="reference internal" href="notebooks/Appendix-MathsForBiologists/MathsForBiologists.html">Maths for Biologists</a></li>
 <li class="toctree-l1"><a class="reference internal" href="notebooks/Appendix-Databases.html">Databases <span class="tocSkip"></span></a></li>
 <li class="toctree-l1"><a class="reference internal" href="notebooks/Appendix-NLLS-Python.html">Model fitting in Python</a></li>
 <li class="toctree-l1"><a class="reference internal" href="notebooks/Appendix-MiniProj.html">The Computing Miniproject</a></li>
@@ -282,7 +269,7 @@
       
       
       
-      <li><a href="https://github.com/mhasoba/TheMulQuaBio" target="_blank"
+      <li><a href="https://github.com/MulQuaBio/MQB" target="_blank"
    class="btn btn-sm btn-source-repository-button dropdown-item"
    title="Source repository"
    data-bs-placement="left" data-bs-toggle="tooltip"
@@ -299,7 +286,7 @@
       
       
       
-      <li><a href="https://github.com/mhasoba/TheMulQuaBio/issues/new?title=Issue%20on%20page%20%2Fsearch.html&body=Your%20issue%20content%20here." target="_blank"
+      <li><a href="https://github.com/MulQuaBio/MQB/issues/new?title=Issue%20on%20page%20%2Fsearch.html&body=Your%20issue%20content%20here." target="_blank"
    class="btn btn-sm btn-source-issues-button dropdown-item"
    title="Open an issue"
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@@ -421,7 +408,7 @@ <h1>Search</h1>
   <div class="footer-item">
     
 <p class="component-author">
-By TheMulQuaBio collective!
+By MulQuaBio collective!
 </p>
 
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diff --git a/searchindex.js b/searchindex.js
index 77ed7a47..7871bd81 100644
--- a/searchindex.js
+++ b/searchindex.js
@@ -1 +1 @@
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"language-kernels"]], "Latex commands": [[27, "latex-commands"]], "Launching/Running a Jupyter Notebook": [[19, "launching-running-a-jupyter-notebook"]], "Learning goals": [[3, "learning-goals"]], "Let\u2019s get started!": [[26, "let-s-get-started"]], "Likelihood and Log Likelihood": [[23, "likelihood-and-log-likelihood"]], "Likelihood profile": [[28, "likelihood-profile"]], "Likelihood surface": [[28, "likelihood-surface"]], "Limitations of \\LaTeX": [[27, "limitations-of-latex"]], "Limits": [[6, "limits"], [20, "limits"]], "Linear Models: Analysis of variance": [[37, null]], "Linear Models: Multiple explanatory variables": [[30, null]], "Linear Models: Multiple variables with interactions": [[31, null]], "Linear Models: Regression": [[39, null]], "Linear transformations": [[8, "linear-transformations"]], "Lists": [[33, "lists"], [34, "lists"]], "Local Parallelisation": [[18, "local-parallelisation"]], "Logarithmic axes": [[24, "logarithmic-axes"]], "Logarithmic function": [[4, "logarithmic-function"]], "Logistic model fit": [[22, "logistic-model-fit"]], "Looking at strains too": [[38, "looking-at-strains-too"]], "Loops": [[33, "loops"]], "Loops and List Comprehensions": [[33, "loops-and-list-comprehensions"]], "MLE for Binomial Distribution": [[23, "mle-for-binomial-distribution"]], "Magic commands": [[33, "magic-commands"]], "Manipulating arrays": [[33, "manipulating-arrays"]], "Mapping": [[24, "mapping"]], "Markdown": [[26, "markdown"]], "Marking criteria": [[21, "marking-criteria"]], "Mathematical": [[34, "mathematical"]], "Mathematical display": [[24, "mathematical-display"]], "Mathematical modelling and Stats in R": [[34, "mathematical-modelling-and-stats-in-r"]], "Mathematical models in Jupyter": [[20, null]], "Matrices and Arrays": [[34, "matrices-and-arrays"]], "Matrix algebra": [[20, "matrix-algebra"]], "Matrix operations": [[7, "matrix-operations"]], "Matrix vs Dataframe": [[34, "matrix-vs-dataframe"]], "Matrix-vector operations": [[33, "matrix-vector-operations"]], "Maximum Likelihood using optim()": [[28, "maximum-likelihood-using-optim"]], "Measuring angles": [[5, "measuring-angles"]], "Meet the UNIX shell": [[36, "meet-the-unix-shell"]], "Memory-Intensive Tasks": [[18, "memory-intensive-tasks"]], "Metacharacters vs. regular characters": [[33, "metacharacters-vs-regular-characters"]], "Method of Moments (MoM) Estimators": [[23, "method-of-moments-mom-estimators"]], "Minimize modifying raw data by hand": [[24, "minimize-modifying-raw-data-by-hand"]], "Minimum/ maximum": [[4, "minimum-maximum"]], "Missing oaks problem": [[33, "missing-oaks-problem"]], "Model 1: Mammalian genome size": [[31, "model-1-mammalian-genome-size"]], "Model 2 (ANCOVA): Body Weight in Odonata": [[31, "model-2-ancova-body-weight-in-odonata"]], "Model Fitting the Bayesian Way": [[23, null]], "Model Fitting using Maximum Likelihood": [[28, null]], "Model Fitting using Non-linear Least-squares": [[32, null]], "Model Selection": [[28, "model-selection"]], "Model criticism": [[37, "model-criticism"]], "Model diagnostics": [[39, "model-diagnostics"]], "Model fitting in Python": [[22, null]], "Model fitting script": [[21, "model-fitting-script"]], "Model predictions": [[38, "model-predictions"]], "Model simplification": [[29, null], [29, "id1"]], "Modelling complex periodic phenomena": [[5, "modelling-complex-periodic-phenomena"]], "Modules and code compartmentalization": [[33, "modules-and-code-compartmentalization"]], "Monotonicity and concavity": [[10, "monotonicity-and-concavity"]], "More examples of loops and conditionals combined": [[33, "more-examples-of-loops-and-conditionals-combined"]], "Multi-faceted plots": [[24, "multi-faceted-plots"]], "NLLS fitting using a model object": [[32, "nlls-fitting-using-a-model-object"]], "Non-Bayesian analysis": [[23, "non-bayesian-analysis"]], "Non-parametric tests": [[40, "non-parametric-tests"]], "Numerical computing in Python": [[33, "numerical-computing-in-python"]], "Numerical evaluation": [[20, "numerical-evaluation"]], "Numerical evaluation of the posterior with JAGS": [[23, "numerical-evaluation-of-the-posterior-with-jags"]], "Numerical integration using  scipy": [[33, "numerical-integration-using-scipy"]], "Numpy": [[33, "numpy"]], "Objectives": [[21, "objectives"]], "Obtaining them": [[32, "obtaining-them"]], "On to data exploration": [[24, "on-to-data-exploration"]], "One population: Exponential growth": [[20, "one-population-exponential-growth"]], "One population: Logistic Population growth": [[20, "one-population-logistic-population-growth"]], "One-sample t tests": [[40, "one-sample-t-tests"]], "Operators": [[34, "operators"]], "Other distributions": [[38, "other-distributions"]], "Other stuff": [[33, "other-stuff"]], "Other unit testing approaches": [[33, "other-unit-testing-approaches"]], "Overall": [[21, "overall"]], "Overdispersion": [[38, "overdispersion"]], "Overlaying plots": [[24, "overlaying-plots"]], "Packages": [[34, "packages"]], "Pandas dataframes": [[16, "pandas-dataframes"]], "Parallel Processing": [[18, "parallel-processing"]], "Paranoid programming: debugging with breakpoints": [[33, "paranoid-programming-debugging-with-breakpoints"]], "Particularly useful UNIX tools": [[36, "particularly-useful-unix-tools"]], "Performing the Regression analysis": [[39, "performing-the-regression-analysis"]], "Plagiarism": [[15, "plagiarism"]], "Plot annotations": [[24, "plot-annotations"]], "Plot the fits": [[23, "plot-the-fits"]], "Plot the results": [[22, "plot-the-results"]], "Plotting a matrix": [[24, "plotting-a-matrix"]], "Plotting means and confidence intervals": [[30, "plotting-means-and-confidence-intervals"]], "Plotting the fitted Regression model": [[39, "plotting-the-fitted-regression-model"]], "Plotting two dataframes together": [[24, "plotting-two-dataframes-together"]], "Polynomial functions": [[4, "polynomial-functions"]], "Population Growth": [[21, "population-growth"], [21, "id7"]], "Population growth rates": [[32, "population-growth-rates"]], "Power functions": [[4, "power-functions"]], "Practicals": [[24, "practicals"], [24, "id1"], [24, "id3"], [26, "practicals"], [27, "practicals"], [33, "practicals"], [33, "id1"], [33, "id2"], [33, "id3"], [33, "id4"], [33, "id5"], [34, "practicals"], [34, "id1"], [34, "id2"], [35, "practicals"], [36, "practicals"]], "Practicals wrap-up": [[27, "practicals-wrap-up"], [33, "practicals-wrap-up"]], "Practicals: Some RegExercises": [[33, "practicals-some-regexercises"]], "Pre-Commit Git Hook": [[26, "pre-commit-git-hook"]], "Pre-allocating arrays": [[33, "pre-allocating-arrays"]], "Pre-allocation": [[34, "pre-allocation"]], "Pre-submission practicals wrap-up": [[15, "pre-submission-practicals-wrap-up"]], "Predicted values": [[30, "predicted-values"]], "Prior Information": [[23, "prior-information"]], "Profiling": [[33, "profiling"]], "Profiling without ipython": [[33, "profiling-without-ipython"]], "Python Input/Output": [[33, "python-input-output"]], "Python Packages": [[33, "id8"]], "Python data structures": [[33, "python-data-structures"], [33, "id7"]], "Python operators": [[33, "python-operators"]], "Python packages": [[33, "python-packages"]], "Python variables": [[33, "python-variables"]], "Python versions": [[33, "python-versions"]], "Python with strings": [[33, "python-with-strings"]], "Quick plotting with qplot": [[24, "quick-plotting-with-qplot"]], "Quick profiling with timeit": [[33, "quick-profiling-with-timeit"]], "Quotation": [[14, null]], "R Data Structures": [[34, "r-data-structures"]], "R as a Programming language": [[34, "r-as-a-programming-language"]], "R basics": [[34, "r-basics"]], "R vs. Python": [[3, "r-vs-python"]], "R-squared values": [[32, "r-squared-values"]], "Rational functions": [[4, "rational-functions"]], "Readings & Resources": [[2, "readings-resources"], [21, "readings-resources"], [22, "readings-resources"], [24, "readings-resources"], [26, "readings-resources"], [27, "readings-resources"], [35, "readings-resources"], [36, "readings-resources"]], "Readings and Resources": [[16, "readings-and-resources"], [17, "readings-and-resources"], [19, "readings-and-resources"], [20, "readings-and-resources"], [32, "readings-and-resources"], [33, "readings-and-resources"], [34, "readings-and-resources"]], "Readings and Resources <a id='Readings'></a>": [[23, "readings-and-resources"], [28, "readings-and-resources"]], "Recycling": [[34, "recycling"]], "Redirection and pipes": [[36, "redirection-and-pipes"]], "Referencing and bibliography": [[27, "referencing-and-bibliography"]], "Reflection": [[5, "reflection"]], "Regex in Python": [[33, "regex-in-python"]], "Regex special sequences": [[33, "regex-special-sequences"]], "Regular expressions": [[33, "regular-expressions"]], "Regular expressions in Python": [[33, "regular-expressions-in-python"]], "Relational databases": [[17, "relational-databases"]], "Relative paths": [[34, "relative-paths"]], "Reminder about tabbing and help!": [[16, "reminder-about-tabbing-and-help"]], "Removing files": [[26, "removing-files"]], "Removing software": [[36, "removing-software"]], "Rendering special characters": [[27, "rendering-special-characters"]], "Replacing text": [[33, "replacing-text"]], "Replacing, adding or deleting elements": [[33, "replacing-adding-or-deleting-elements"]], "Report": [[21, "report"]], "Report compiling script": [[21, "report-compiling-script"]], "Reporting the model": [[38, "reporting-the-model"], [39, "reporting-the-model"]], "Repositories and packages in Ubuntu": [[36, "repositories-and-packages-in-ubuntu"]], "Reverting to a previous version": [[26, "reverting-to-a-previous-version"]], "Running Python scripts": [[33, "running-python-scripts"]], "Running R": [[33, "running-r"]], "Running R code": [[34, "running-r-code"]], "Running R in batch mode": [[34, "running-r-in-batch-mode"]], "Running a job on the Imperial HPC": [[18, "running-a-job-on-the-imperial-hpc"]], "Running an ANOVA on the fitted model": [[39, "running-an-anova-on-the-fitted-model"]], "Running code": [[19, "running-code"]], "Running diagnostics": [[23, "running-diagnostics"]], "Running git commands on a different directory": [[26, "running-git-commands-on-a-different-directory"]], "Running git commands on multiple repositories at once": [[26, "running-git-commands-on-multiple-repositories-at-once"]], "Running processes": [[33, "running-processes"]], "Running shell scripts": [[35, "running-shell-scripts"]], "SQLite": [[17, "sqlite"]], "SQLite with Python": [[17, "sqlite-with-python"]], "Safely opening files using with open()": [[33, "safely-opening-files-using-with-open"]], "Sampling from distributions in R": [[25, "sampling-from-distributions-in-r"]], "Sampling them": [[32, "sampling-them"]], "Saving data": [[37, "saving-data"]], "Saving other results": [[25, "saving-other-results"]], "Saving the exploratory graphics": [[25, "saving-the-exploratory-graphics"]], "Saving your graphics": [[24, "saving-your-graphics"]], "Saving your plots": [[24, "saving-your-plots"]], "Saving your results": [[25, "saving-your-results"]], "Scatter Plots": [[24, "scatter-plots"]], "Scatterplots": [[24, "scatterplots"]], "Scientific computing": [[33, "scientific-computing"]], "Scientific documents with \\LaTeX": [[27, null]], "Scipy stats": [[33, "scipy-stats"]], "Sequences": [[6, "sequences"]], "Serial Processing": [[18, "serial-processing"]], "Serial vs Parallel Processing": [[18, "serial-vs-parallel-processing"]], "Series": [[11, "series"], [20, "series"]], "Series expansions": [[10, "series-expansions"]], "Sets": [[33, "sets"], [34, "sets"]], "Sets and relations": [[4, "sets-and-relations"]], "Setting transparency": [[24, "setting-transparency"]], "Setting up the Bayesian Model": [[23, "setting-up-the-bayesian-model"]], "Shell scripting": [[35, null]], "Simplification": [[20, "simplification"]], "Simulating from the Binomial using R": [[23, "simulating-from-the-binomial-using-r"]], "Solving equations": [[20, "solving-equations"]], "Some (statistical) distributions": [[38, "some-statistical-distributions"]], "Some biological examples": [[20, "some-biological-examples"]], "Some conventions and syntax rules": [[35, "some-conventions-and-syntax-rules"]], "Some data wrangling principles": [[24, "some-data-wrangling-principles"]], "Some examples of variables": [[35, "some-examples-of-variables"]], "Some general tips": [[25, "some-general-tips"]], "Some guidelines, conventions and rules": [[3, "some-guidelines-conventions-and-rules"]], "Some loops examples": [[33, "some-loops-examples"]], "Some more \\LaTeX features and tips": [[27, "some-more-latex-features-and-tips"]], "Some more control flow tools": [[34, "some-more-control-flow-tools"]], "Some more examples": [[35, "some-more-examples"]], "Some more important ggplot options": [[24, "some-more-important-ggplot-options"]], "Some preliminaries": [[20, "some-preliminaries"]], "Some regex examples": [[33, "some-regex-examples"]], "Some statistical parlance": [[25, "some-statistical-parlance"]], "Some terminology": [[33, "some-terminology"]], "Some tips and tricks for NLLS fitting": [[32, "some-tips-and-tricks-for-nlls-fitting"]], "Some useful ggplot examples": [[24, "some-useful-ggplot-examples"]], "Spaces and new lines": [[27, "spaces-and-new-lines"]], "Special Characters": [[36, "special-characters"]], "Special Variables": [[35, "special-variables"]], "Special characters": [[27, "special-characters"], [35, "special-characters"]], "Specifying the growth curve": [[23, "specifying-the-growth-curve"]], "Specifying the model": [[23, "specifying-the-model"]], "Starting values": [[32, "starting-values"]], "Statistical": [[34, "statistical"]], "Statistical inference using NLLS fits": [[32, "statistical-inference-using-nlls-fits"]], "Stretching/ Compression": [[5, "stretching-compression"]], "Strings and Pasting": [[34, "strings-and-pasting"]], "Structured models": [[7, "structured-models"]], "Structured models revisited": [[8, "structured-models-revisited"]], "Submission": [[21, "submission"]], "Subplots": [[24, "subplots"]], "Suggested Workflow": [[21, "suggested-workflow"]], "Summary of the fit": [[32, "summary-of-the-fit"]], "Summary stats of the fit": [[32, "summary-stats-of-the-fit"]], "Sums and products": [[20, "sums-and-products"]], "Supercomputers": [[18, "supercomputers"]], "Sweave and knitr, and beyond": [[34, "sweave-and-knitr-and-beyond"]], "Symbolic equations": [[20, "symbolic-equations"]], "Symbolic variables": [[20, "symbolic-variables"]], "Systems of linear equations": [[7, "systems-of-linear-equations"]], "TMQB Coursework Assessment": [[15, null]], "Take a quick look at effects of certain factors": [[25, "take-a-quick-look-at-effects-of-certain-factors"]], "Teamwork using git and common branching mistakes": [[26, "teamwork-using-git-and-common-branching-mistakes"]], "Testing pairwise differences between levels": [[37, "testing-pairwise-differences-between-levels"]], "Testing/Running blocks of code": [[33, "testing-running-blocks-of-code"]], "Text cells": [[19, "text-cells"]], "The *apply family of functions": [[34, "the-apply-family-of-functions"]], "The Binomial Distribution": [[23, "the-binomial-distribution"]], "The Binomial distribution": [[38, "the-binomial-distribution"]], "The Chi-square test and count data": [[37, "the-chi-square-test-and-count-data"]], "The Chi-square test with the mammals data": [[37, "the-chi-square-test-with-the-mammals-data"]], "The Computing Miniproject": [[21, null]], "The Data": [[21, "the-data"], [21, "id2"], [21, "id5"], [23, "the-data"], [23, "id1"]], "The Dataset and Model Options": [[21, "the-dataset-and-model-options"]], "The Docstring": [[33, "the-docstring"]], "The F test": [[40, "the-f-test"]], "The F-Value (or Ratio)": [[37, "the-f-value-or-ratio"]], "The Gamma distribution": [[38, "the-gamma-distribution"]], "The Lotka-Volterra model": [[33, "the-lotka-volterra-model"]], "The Lotka-Volterra model revisited": [[33, "the-lotka-volterra-model-revisited"]], "The Models": [[21, "the-models"], [21, "miniproject-fr-models"], [21, "miniproj-tpcs-models"]], "The Poisson distribution": [[38, "the-poisson-distribution"]], "The Project": [[21, "the-project"]], "The Question": [[21, "the-question"], [21, "id1"], [21, "id4"]], "The R Workspace and Working Directory": [[34, "the-r-workspace-and-working-directory"]], "The README file": [[26, "the-readme-file"]], "The R^2": [[37, "the-r-2"]], "The Report": [[21, "the-report"]], "The Sums of Squares": [[39, "the-sums-of-squares"]], "The Thermal Performance Curve models": [[32, "the-thermal-performance-curve-models"]], "The Zen of Python and good programming practices": [[33, "the-zen-of-python-and-good-programming-practices"]], "The data": [[37, "the-data"]], "The geom argument": [[24, "the-geom-argument"]], "The main body": [[27, "the-main-body"]], "The need for speed: profiling code": [[33, "the-need-for-speed-profiling-code"]], "The posterior distribution of the mean function": [[23, "the-posterior-distribution-of-the-mean-function"]], "The process of model simplification": [[29, "the-process-of-model-simplification"]], "The scipy package": [[33, "the-scipy-package"]], "The shebang": [[33, "the-shebang"]], "The start of the document": [[27, "the-start-of-the-document"]], "The starting values problem": [[32, "the-starting-values-problem"]], "The tapply function": [[34, "the-tapply-function"]], "The thermal response model file": [[23, "the-thermal-response-model-file"], [23, "id2"]], "Thermal Performance Curves": [[21, "thermal-performance-curves"], [21, "id9"]], "To IDE or not to IDE?": [[3, "to-ide-or-not-to-ide"]], "Tracing the error trail": [[34, "tracing-the-error-trail"]], "Trade-offs": [[18, "trade-offs"]], "Traits data as an example": [[32, "traits-data-as-an-example"]], "Transformation on graphs": [[5, "transformation-on-graphs"]], "Transformations and allometric scaling": [[39, "transformations-and-allometric-scaling"]], "Translations": [[5, "translations"]], "Tree heights": [[34, "tree-heights"]], "Trigonometric circle": [[5, "trigonometric-circle"]], "Trigonometric functions": [[5, "trigonometric-functions"]], "Trigonometric identities": [[5, "trigonometric-identities"]], "Tuples": [[33, "tuples"]], "Tutorial 01": [[12, null]], "Tutorial 05": [[13, null]], "Tutorial 06": [[14, null]], "Two basic rules of experimental design and sampling": [[25, "two-basic-rules-of-experimental-design-and-sampling"]], "Two interacting populations: The Lotka-Volterra predator-prey model": [[20, "two-interacting-populations-the-lotka-volterra-predator-prey-model"]], "Two thermal performance curve models": [[23, "two-thermal-performance-curve-models"]], "Two-sample t tests": [[40, "two-sample-t-tests"]], "Type Conversions and Special Values": [[34, "type-conversions-and-special-values"]], "Types of Parallelism": [[18, "types-of-parallelism"]], "Types of parentheses": [[34, "types-of-parentheses"]], "Typesetting math": [[27, "typesetting-math"]], "UNIX and Linux": [[36, null]], "UNIX directory structure": [[36, "unix-directory-structure"]], "Un-tracking files": [[26, "un-tracking-files"]], "Unit testing": [[33, "unit-testing"], [34, "unit-testing"]], "Unit testing with doctest": [[33, "unit-testing-with-doctest"]], "Useful R commands": [[34, "useful-r-commands"]], "Useful R functions": [[34, "useful-r-functions"]], "Useful re commands": [[33, "useful-re-commands"]], "Using NLLS": [[22, "using-nlls"], [32, "using-nlls"]], "Using OLS": [[22, "using-ols"], [32, "using-ols"]], "Using Python to build workflows": [[33, "using-python-to-build-workflows"]], "Using Rscript": [[34, "using-rscript"]], "Using by": [[34, "using-by"]], "Using debug()": [[34, "using-debug"]], "Using git through a GUI": [[26, "using-git-through-a-gui"]], "Using grep": [[36, "using-grep"]], "Using next": [[34, "using-next"]], "Using os to crawl directories": [[33, "using-os-to-crawl-directories"]], "Using plyr and ddply": [[34, "using-plyr-and-ddply"]], "Using replicate": [[34, "using-replicate"]], "Using source": [[34, "using-source"]], "Using subprocess": [[33, "using-subprocess"]], "Using subprocess to run a R script": [[33, "using-subprocess-to-run-a-r-script"]], "Using the UNIX shell": [[33, "using-the-unix-shell"]], "Using tmux": [[19, "using-tmux"]], "Variable names and Tabbing": [[34, "variable-names-and-tabbing"]], "Variable scope": [[33, "variable-scope"]], "Variable types": [[34, "variable-types"]], "Variables in shell scripts": [[35, "variables-in-shell-scripts"]], "Vector decomposition": [[8, "vector-decomposition"]], "Vectorization": [[34, "vectorization"]], "Vectorization revisited": [[33, "vectorization-revisited"]], "Vectors": [[34, "vectors"]], "Version control with Git": [[26, null]], "Visualise distributions of the variables": [[25, "visualise-distributions-of-the-variables"]], "Visualizing Regression analyses": [[24, 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patterns": [[11, "describing-shapes-and-patterns"]], "Descriptive Statistics": [[17, "descriptive-statistics"]], "Descriptive statistics in R": [[17, "descriptive-statistics-in-r"]], "Detaching and Reattaching": [[9, "detaching-and-reattaching"]], "Determinant": [[11, "determinant"]], "Determining an object\u2019s type": [[28, "determining-an-object-s-type"]], "Diagnostics": [[14, "diagnostics"]], "Diagnostics of an NLLS fit": [[26, "diagnostics-of-an-nlls-fit"]], "Dictionaries": [[28, "dictionaries"]], "Differences in means with barplots": [[34, "differences-in-means-with-barplots"]], "Differentiation": [[10, "differentiation"]], "Directory and file patterns for gitignoring": [[18, "directory-and-file-patterns-for-gitignoring"]], "Discrete-time population models": [[11, "discrete-time-population-models"]], "Discussion": [[32, "discussion"]], "Discussion Questions": [[16, "discussion-questions"], [19, "discussion-questions"], [22, "discussion-questions"]], "Dissecting Consumer-Resource 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"errors-in-your-python-code"]], "Errors that cannot be debugged": [[28, "errors-that-cannot-be-debugged"]], "Estimating the population variance": [[14, "estimating-the-population-variance"]], "Examine the residuals": [[26, "examine-the-residuals"]], "Examining your data": [[6, "examining-your-data"]], "Example": [[11, null], [11, null], [11, null], [11, null], [11, null], [11, null], [11, null], [11, null], [11, null], [11, null], [11, null], [11, null], [11, null], [11, null], [11, null], [11, null], [11, null], [11, null], [11, null], [11, null], [11, null], [11, null], [11, null], [11, null], [11, null], [11, null], [11, null], [11, null], [11, null], [11, null], [11, null], [11, null], [11, null], [11, null], [11, null], [11, null], [11, null], [11, null], [11, null], [11, null], [11, null], [11, null], [11, null]], "Example: Midge Wing Length": [[14, "example-midge-wing-length"]], "Examples": [[11, null], [11, null], [11, null], [11, null], [11, null], [11, null], [11, null], [11, null], [11, null], [11, null], [11, null]], "Exercise": [[13, "exercise"], [13, "id1"], [20, "exercise"], [23, "exercise"], [36, "exercise"], [37, "exercise"], [37, "id1"], [37, "id2"], [37, "id3"]], "Exercise Set 1": [[11, "exercise-set-1"]], "Exercise Set 2": [[11, "exercise-set-2"]], "Exercise Set 3": [[11, "exercise-set-3"]], "Exercise: Prior sensitivity": [[14, "exercise-prior-sensitivity"]], "Exercise: Updating the Bayesian model": [[14, "exercise-updating-the-bayesian-model"]], "Exercises": [[26, "id5"]], "Exercises <a id='Aedes_Exercises'></a>": [[26, "id4"]], "Exercises <a id='Albatross_Exercises'></a>": [[26, "id3"]], "Exercises <a id='Allom_Exercises'></a>": [[26, "exercises"]], "Exercises <a id='MLE_Exercises'></a>": [[21, "exercises"]], "Exercises <a id='ModelSelection_Exercises'></a>": [[26, "id2"]], "Exiting a Jupyter Notebook": [[9, "exiting-a-jupyter-notebook"]], "Expand and factor": [[10, "expand-and-factor"]], "Experimental design and Data exploration": [[17, null]], "Explore further by scatter-plotting two variables": [[17, "explore-further-by-scatter-plotting-two-variables"]], "Exploring files and directories": [[33, "exploring-files-and-directories"]], "Exploring the data": [[24, "exploring-the-data"], [36, "exploring-the-data"]], "Exploring the data with boxplots": [[34, "exploring-the-data-with-boxplots"]], "Exponential functions": [[11, "exponential-functions"]], "Extracting text from webpages": [[28, "extracting-text-from-webpages"]], "Extreme values of functions": [[11, "extreme-values-of-functions"]], "FASTA exercise": [[33, "fasta-exercise"]], "Fibonacci sequence": [[11, "fibonacci-sequence"]], "Final assessment of computing coursework": [[5, "final-assessment-of-computing-coursework"]], "Final plotting and analysis script": [[12, "final-plotting-and-analysis-script"]], "Finally, sys.exit()": [[28, "finally-sys-exit"]], "Finding all matches": [[28, "finding-all-matches"]], "Finding extrema": [[11, "finding-extrema"]], "Finding files": [[33, "finding-files"]], "Finding in files": [[28, "finding-in-files"]], "First \\LaTeX example": [[20, "first-latex-example"]], "Fitting Non-linear models to the data": [[13, "fitting-non-linear-models-to-the-data"]], "Fitting a GLM": [[35, "fitting-a-glm"]], "Fitting a GLM using a quasi-poisson distribution": [[35, "fitting-a-glm-using-a-quasi-poisson-distribution"]], "Fitting a Linear model to the data": [[13, "fitting-a-linear-model-to-the-data"]], "Fitting of abundance data": [[14, "fitting-of-abundance-data"]], "Fitting the linear model": [[24, "fitting-the-linear-model"]], "Fitting the model": [[14, "fitting-the-model"], [36, "fitting-the-model"]], "Fitting the model using NLLS": [[26, "fitting-the-model-using-nlls"]], "Fitting the model with starting values": [[26, "fitting-the-model-with-starting-values"]], "Fitting the three models using NLLS": [[26, "fitting-the-three-models-using-nlls"]], "Flattening or reshaping arrays": [[28, "flattening-or-reshaping-arrays"]], "Footnotes": [[32, "footnotes"]], "For NLLS fitting": [[12, "for-nlls-fitting"]], "For manipulating strings": [[29, "for-manipulating-strings"]], "Formulae with interactions in R": [[25, "formulae-with-interactions-in-r"]], "From data exploration to statistical analysis": [[17, "from-data-exploration-to-statistical-analysis"]], "From table arrangements to powerful tools": [[11, "from-table-arrangements-to-powerful-tools"]], "From the UNIX shell, using ipython": [[28, "from-the-unix-shell-using-ipython"]], "From within the ipython shell": [[28, "from-within-the-ipython-shell"]], "Function definitions and \u201cmodules\u201d": [[28, "function-definitions-and-modules"]], "Functional Responses": [[12, "functional-responses"]], "Functional responses": [[12, "id8"]], "Functions": [[28, "functions"]], "Functions of several variables": [[11, "functions-of-several-variables"]], "Functions with conditionals": [[29, "functions-with-conditionals"]], "Functions, Modules, and Classes": [[28, "functions-modules-and-classes"]], "Fundamental mathematical operations": [[10, "fundamental-mathematical-operations"]], "Fundamental theorem of Calculus": [[11, "fundamental-theorem-of-calculus"]], "General": [[12, "general"], [18, "general"], [20, "general"], [28, "general"], [33, "general"]], "General command sequence for connecting to Remotes": [[18, "general-command-sequence-for-connecting-to-remotes"]], "General good practice for gitignoring": [[18, "general-good-practice-for-gitignoring"]], "Generalised Linear Models": [[35, null]], "Generating Random Numbers": [[29, "generating-random-numbers"]], "Generating data": [[26, "generating-data"]], "Getting Set Up on the Imperial HPC": [[8, "getting-set-up-on-the-imperial-hpc"]], "Getting help": [[28, "getting-help"]], "Getting started": [[12, "getting-started"], [29, "getting-started"]], "Getting started with Python": [[28, "getting-started-with-python"]], "Global variables": [[28, "global-variables"]], "Gompertz model": [[13, "gompertz-model"]], "Goodness of fit": [[26, "goodness-of-fit"]], "Gooey IDEs": [[2, "gooey-ides"]], "Graphics": [[9, "graphics"]], "Grouping regex patterns": [[28, "grouping-regex-patterns"]], "Groups within multiple matches": [[28, "groups-within-multiple-matches"]], "Groupwork Practical  on Tree Heights  2": [[29, "groupwork-practical-on-tree-heights-2"]], "Groupwork Practical on Align DNA sequences": [[28, "groupwork-practical-on-align-dna-sequences"]], "Groupwork Practical on Align DNA sequences 2": [[28, "groupwork-practical-on-align-dna-sequences-2"]], "Groupwork Practical on Missing oaks problem": [[28, "groupwork-practical-on-missing-oaks-problem"]], "Groupwork Practical on Tree Heights": [[29, "groupwork-practical-on-tree-heights"]], "Groupwork Practical: Autocorrelation in Florida weather": [[29, "groupwork-practical-autocorrelation-in-florida-weather"]], "Groupwork assessment": [[5, "groupwork-assessment"]], "Groupwork execution": [[5, "groupwork-execution"]], "Groupwork practical: Compare R and Python Vectorization": [[28, "groupwork-practical-compare-r-and-python-vectorization"]], "Groupwork practical: Discrete time LV Model": [[28, "groupwork-practical-discrete-time-lv-model"]], "Groupwork practical: Discrete time LV model with stochasticity": [[28, "groupwork-practical-discrete-time-lv-model-with-stochasticity"]], "Groupwork practical: Regression analysis": [[15, "groupwork-practical-regression-analysis"]], "Guidelines": [[12, "guidelines"]], "HPC Systems": [[8, "hpc-systems"]], "HPC Workflows and Use Cases": [[8, "hpc-workflows-and-use-cases"]], "Halos and lawns": [[35, "halos-and-lawns"]], "Handling Big Data in R": [[15, "handling-big-data-in-r"]], "Handling csv\u2019s": [[28, "handling-csv-s"]], "Handling directory and file paths": [[28, "handling-directory-and-file-paths"]], "Has The Marriage Come Of Age?": [[32, "has-the-marriage-come-of-age"]], "High Throughput Computing (HTC)": [[8, "high-throughput-computing-htc"]], "Higher order derivatives": [[11, "higher-order-derivatives"]], "Histograms": [[15, "histograms"]], "Histograms and density plots": [[15, "histograms-and-density-plots"]], "How Git Stores Data": [[18, "how-git-stores-data"]], "IPython": [[28, "id6"]], "Ignoring Files": [[18, "ignoring-files"]], "Implementing the Likelihood in R": [[21, "implementing-the-likelihood-in-r"]], "Importance of Size": [[22, "importance-of-size"]], "Importance of Temperature": [[22, "importance-of-temperature"]], "Importance of the return directive": [[28, "importance-of-the-return-directive"]], "Importing Modules": [[28, "importing-modules"]], "Importing and Exporting Data": [[29, "importing-and-exporting-data"]], "Improving scripts": [[30, "improving-scripts"]], "Indexing and accessing arrays": [[28, "indexing-and-accessing-arrays"]], "Inflection points": [[11, "inflection-points"]], "Installing Jupyter": [[9, "installing-jupyter"]], "Installing LaTeX": [[20, "installing-latex"]], "Installing Pandas": [[6, "installing-pandas"]], "Installing R": [[29, "installing-r"]], "Installing and running jupyter in a virtual environment": [[9, "installing-and-running-jupyter-in-a-virtual-environment"]], "Installing from downloaded *.deb files": [[33, "installing-from-downloaded-deb-files"]], "Installing from other repositories": [[33, "installing-from-other-repositories"]], "Installing from the main repository": [[33, "installing-from-the-main-repository"]], "Installing git": [[18, "installing-git"]], "Installing or removing software in Ubuntu": [[33, "installing-or-removing-software-in-ubuntu"]], "Instantaneous rate of change": [[11, "instantaneous-rate-of-change"]], "Instructions": [[18, "instructions"], [30, "instructions"], [33, "instructions"]], "Integration": [[10, "integration"]], "Integration by partial fractions": [[11, "integration-by-partial-fractions"]], "Integration by parts": [[11, "integration-by-parts"]], "Integration by substitution": [[11, "integration-by-substitution"]], "Intercept & Roots": [[11, "intercept-roots"]], "Internal Variables": [[28, "internal-variables"]], "Introduction": [[3, null], [4, null], [6, "introduction"], [7, "introduction"], [10, "introduction"], [12, "introduction"], [13, "introduction"], [14, "introduction"], [15, "introduction"], [16, "introduction"], [17, "introduction"], [18, "introduction"], [19, "introduction"], [20, "introduction"], [21, "introduction"], [22, "introduction"], [23, "introduction"], [24, "introduction"], [25, "introduction"], [26, "introduction"], [28, "introduction"], [29, "introduction"], [30, "introduction"], [32, "introduction"], [33, "introduction"], [34, "introduction"], [35, "introduction"], [36, "introduction"], [37, "introduction"]], "Introduction to High-Performance Computing (HPC)": [[8, null]], "Introduction to Jupyter": [[9, null]], "Inverse matrix": [[11, "inverse-matrix"]], "Is Florida getting warmer?": [[29, "is-florida-getting-warmer"]], "Iterator vs Iterable": [[28, "iterator-vs-iterable"]], "Job Arrays": [[8, "job-arrays"]], "Jupyter vs. IPython": [[9, "jupyter-vs-ipython"]], "Keep a metadata file for each unique dataset": [[15, "keep-a-metadata-file-for-each-unique-dataset"]], "Keep your workflow organized": [[2, "keep-your-workflow-organized"]], "Key Computer Hardware Vocabulary": [[8, "key-computer-hardware-vocabulary"]], "Key Jupyter features": [[9, "key-jupyter-features"]], "Key \\LaTeX\u00a0features": [[20, "key-latex-features"]], "Key elements of the Jupyter nb content": [[9, "key-elements-of-the-jupyter-nb-content"]], "Keyboard shortcuts": [[9, "keyboard-shortcuts"]], "Language Kernels": [[9, "language-kernels"]], "Latex commands": [[20, "latex-commands"]], "Launching/Running a Jupyter Notebook": [[9, "launching-running-a-jupyter-notebook"]], "Learning goals": [[2, "learning-goals"]], "Let\u2019s get started!": [[18, "let-s-get-started"]], "Likelihood and Log Likelihood": [[14, "likelihood-and-log-likelihood"]], "Likelihood profile": [[21, "likelihood-profile"]], "Likelihood surface": [[21, "likelihood-surface"]], "Limitations of \\LaTeX": [[20, "limitations-of-latex"]], "Limits": [[10, "limits"], [11, "limits"]], "Linear Models: Analysis of variance": [[34, null]], "Linear Models: Multiple explanatory variables": [[24, null]], "Linear Models: Multiple variables with interactions": [[25, null]], "Linear Models: Regression": [[36, null]], "Linear transformations": [[11, "linear-transformations"]], "Lists": [[28, "lists"], [29, "lists"]], "Local Parallelisation": [[8, "local-parallelisation"]], "Logarithmic axes": [[15, "logarithmic-axes"]], "Logarithmic functions": [[11, "logarithmic-functions"]], "Logistic model fit": [[13, "logistic-model-fit"]], "Looking at strains too": [[35, "looking-at-strains-too"]], "Loops": [[28, "loops"]], "Loops and List Comprehensions": [[28, "loops-and-list-comprehensions"]], "MLE for Binomial Distribution": [[14, "mle-for-binomial-distribution"]], "Magic commands": [[28, "magic-commands"]], "Making mathematical statements": [[11, "making-mathematical-statements"]], "Manipulating arrays": [[28, "manipulating-arrays"]], "Mapping": [[15, "mapping"]], "Markdown": [[18, "markdown"]], "Marking criteria": [[12, "marking-criteria"]], "Marriage and the quest for harmony: 1950-1985": [[32, "marriage-and-the-quest-for-harmony-1950-1985"]], "Mathematical": [[29, "mathematical"]], "Mathematical Modeling": [[16, "mathematical-modeling"]], "Mathematical display": [[15, "mathematical-display"]], "Mathematical modelling and Stats in R": [[29, "mathematical-modelling-and-stats-in-r"]], "Mathematical models in Jupyter": [[10, null]], "Maths for Biologists": [[11, null]], "Matrices and Arrays": [[29, "matrices-and-arrays"]], "Matrix algebra": [[10, "matrix-algebra"]], "Matrix operations": [[11, "matrix-operations"]], "Matrix vs Dataframe": [[29, "matrix-vs-dataframe"]], "Matrix-vector operations": [[28, "matrix-vector-operations"]], "Maximum Likelihood using optim()": [[21, "maximum-likelihood-using-optim"]], "Measuring angles": [[11, "measuring-angles"]], "Meet the UNIX shell": [[33, "meet-the-unix-shell"]], "Memory-Intensive Tasks": [[8, "memory-intensive-tasks"]], "Metacharacters vs. regular characters": [[28, "metacharacters-vs-regular-characters"]], "Method of Moments (MoM) Estimators": [[14, "method-of-moments-mom-estimators"]], "Minimize modifying raw data by hand": [[15, "minimize-modifying-raw-data-by-hand"]], "Minimum / Maximum": [[11, "minimum-maximum"]], "Missing oaks problem": [[28, "missing-oaks-problem"]], "Model 1: Mammalian genome size": [[25, "model-1-mammalian-genome-size"]], "Model 2 (ANCOVA): Body Weight in Odonata": [[25, "model-2-ancova-body-weight-in-odonata"]], "Model Fitting the Bayesian Way": [[14, null]], "Model Fitting using Maximum Likelihood": [[21, null]], "Model Fitting using Non-linear Least-squares": [[26, null]], "Model Selection": [[21, "model-selection"]], "Model criticism": [[34, "model-criticism"]], "Model diagnostics": [[36, "model-diagnostics"]], "Model fitting in Python": [[13, null]], "Model fitting script": [[12, "model-fitting-script"]], "Model predictions": [[35, "model-predictions"]], "Model simplification": [[23, null], [23, "id1"]], "Modelling complex periodic phenomena": [[11, "modelling-complex-periodic-phenomena"]], "Modules and code compartmentalization": [[28, "modules-and-code-compartmentalization"]], "Monotonicity and concavity": [[11, "monotonicity-and-concavity"]], "More examples of loops and conditionals combined": [[28, "more-examples-of-loops-and-conditionals-combined"]], "Multi-faceted plots": [[15, "multi-faceted-plots"]], "Mutualistic Networks": [[16, "mutualistic-networks"], [16, "id2"], [16, "id4"]], "NLLS fitting using a model object": [[26, "nlls-fitting-using-a-model-object"]], "Non-Bayesian analysis": [[14, "non-bayesian-analysis"]], "Non-parametric tests": [[37, "non-parametric-tests"]], "Numerical computing in Python": [[28, "numerical-computing-in-python"]], "Numerical evaluation": [[10, "numerical-evaluation"]], "Numerical evaluation of the posterior with JAGS": [[14, "numerical-evaluation-of-the-posterior-with-jags"]], "Numerical integration using  scipy": [[28, "numerical-integration-using-scipy"]], "Numpy": [[28, "numpy"]], "Objectives": [[12, "objectives"]], "Obtaining them": [[26, "obtaining-them"]], "On the Dullness of Ecology": [[32, "on-the-dullness-of-ecology"]], "On to data exploration": [[15, "on-to-data-exploration"]], "One population: Exponential growth": [[10, "one-population-exponential-growth"]], "One population: Logistic Population growth": [[10, "one-population-logistic-population-growth"]], "One-sample t tests": [[37, "one-sample-t-tests"]], "Operators": [[29, "operators"]], "Other distributions": [[35, "other-distributions"]], "Other stuff": [[28, "other-stuff"]], "Other unit testing approaches": [[28, "other-unit-testing-approaches"]], "Overall": [[12, "overall"]], "Overdispersion": [[35, "overdispersion"]], "Overlaying plots": [[15, "overlaying-plots"]], "Overview": [[3, "overview"]], "Packages": [[29, "packages"]], "Pandas dataframes": [[6, "pandas-dataframes"]], "Parallel Processing": [[8, "parallel-processing"]], "Paranoid programming: debugging with breakpoints": [[28, "paranoid-programming-debugging-with-breakpoints"]], "Particularly useful UNIX tools": [[33, "particularly-useful-unix-tools"]], "Performing the Regression analysis": [[36, "performing-the-regression-analysis"]], "Plagiarism": [[5, "plagiarism"]], "Plot annotations": [[15, "plot-annotations"]], "Plot the fits": [[14, "plot-the-fits"]], "Plot the results": [[13, "plot-the-results"]], "Plotting a matrix": [[15, "plotting-a-matrix"]], "Plotting means and confidence intervals": [[24, "plotting-means-and-confidence-intervals"]], "Plotting the fitted Regression model": [[36, "plotting-the-fitted-regression-model"]], "Plotting two dataframes together": [[15, "plotting-two-dataframes-together"]], "Polynomial functions": [[11, "polynomial-functions"]], "Population Growth": [[12, "population-growth"], [12, "id7"]], "Population growth rates": [[26, "population-growth-rates"]], "Power functions": [[11, "power-functions"]], "Practicals": [[15, "practicals"], [15, "id1"], [15, "id3"], [18, "practicals"], [20, "practicals"], [28, "practicals"], [28, "id1"], [28, "id2"], [28, "id3"], [28, "id4"], [28, "id5"], [29, "practicals"], [29, "id1"], [29, "id2"], [30, "practicals"], [33, "practicals"]], "Practicals wrap-up": [[20, "practicals-wrap-up"], [28, "practicals-wrap-up"]], "Practicals: Some RegExercises": [[28, "practicals-some-regexercises"]], "Pre-Commit Git Hook": [[18, "pre-commit-git-hook"]], "Pre-allocating arrays": [[28, "pre-allocating-arrays"]], "Pre-allocation": [[29, "pre-allocation"]], "Pre-requisites": [[3, "pre-requisites"]], "Pre-submission practicals wrap-up": [[5, "pre-submission-practicals-wrap-up"]], "Predicted values": [[24, "predicted-values"]], "Predicting Consumer-Resource Interactions": [[19, "predicting-consumer-resource-interactions"]], "Prior Information": [[14, "prior-information"]], "Profiling": [[28, "profiling"]], "Profiling without ipython": [[28, "profiling-without-ipython"]], "Python Input/Output": [[28, "python-input-output"]], "Python Packages": [[28, "id8"]], "Python data structures": [[28, "python-data-structures"], [28, "id7"]], "Python operators": [[28, "python-operators"]], "Python packages": [[28, "python-packages"]], "Python variables": [[28, "python-variables"]], "Python versions": [[28, "python-versions"]], "Python with strings": [[28, "python-with-strings"]], "Quick plotting with qplot": [[15, "quick-plotting-with-qplot"]], "Quick profiling with timeit": [[28, "quick-profiling-with-timeit"]], "Quotation": [[11, null]], "R Data Structures": [[29, "r-data-structures"]], "R as a Programming language": [[29, "r-as-a-programming-language"]], "R basics": [[29, "r-basics"]], "R vs. Python": [[2, "r-vs-python"]], "R-squared values": [[26, "r-squared-values"]], "Rational functions": [[11, "rational-functions"]], "Readings & Resources": [[4, "readings-resources"], [12, "readings-resources"], [13, "readings-resources"], [15, "readings-resources"], [18, "readings-resources"], [20, "readings-resources"], [30, "readings-resources"], [33, "readings-resources"]], "Readings and Resources": [[6, "readings-and-resources"], [7, "readings-and-resources"], [9, "readings-and-resources"], [10, "readings-and-resources"], [26, "readings-and-resources"], [28, "readings-and-resources"], [29, "readings-and-resources"]], "Readings and Resources <a id='Readings'></a>": [[14, "readings-and-resources"], [21, "readings-and-resources"]], "Recycling": [[29, "recycling"]], "Redirection and pipes": [[33, "redirection-and-pipes"]], "References": [[32, "references"]], "Referencing and bibliography": [[20, "referencing-and-bibliography"]], "Reflection": [[11, "reflection"]], "Regex in Python": [[28, "regex-in-python"]], "Regex special sequences": [[28, 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