Coral_Reef_Model_Markdown.Rmd

---
title: "BIOL 381_ Coral Reef Model"
author: "Gavin Briske, Matt Futia, Liza Morse"
date: "10/14/2020"
output: pdf_document
---

```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE)
library(ggplot2)
library(reshape2)
library(dplyr)
```

**Part A**

![Individual flow diagram representing societal factors that influence the willingness of the public to participate in coral reef conservation.](C:/Users/mttft/Desktop/UVM/UVM Courses/Quant Reasoning/Coral_Reef_Model/SimpleModel_Futia.png)


![Team flow diagram representing societal factors that influence the willingness of the public to participate in coral reef conservation.](C:/Users/mttft/Desktop/UVM/UVM Courses/Quant Reasoning/Coral_Reef_Model/Case Study Flow Chart.png)

```{r}
require(deSolve)
Time = seq(0,50,0.1)
# Scenario 1: C = 0.8, P = 0.5
Pars = c(kappa = 1.014, j = 1.68, sigma = 0.5, phi = 0.2, C = 0.8, P = 0.55)

social_model <- function(Time, State, Pars) {
  with(as.list(c(State, Pars)), {
    dX <- kappa*X*(1 - X)*(-1 + j*(1 - C) - sigma*P*(1 - X) + phi*(2*X - 1))
    list(c(dX))
  })
}

State = c(X = 0.2)
out.2 = data.frame(ode(y = State, times = Time, func = social_model, parms = Pars))
out.2$Initial = 0.2

State = c(X = 0.5)
out.5 = data.frame(ode(y = State, times = Time, func = social_model, parms = Pars))
out.5$Initial = 0.5


State = c(X = 0.8)
out.8 = data.frame(ode(y = State, times = Time, func = social_model, parms = Pars))
out.8$Initial = 0.8


out = rbind(out.2, out.5, out.8)

# Scenario 2: C = 0.8, P = 5
Pars2 = c(kappa = 1.014, j = 1.68, sigma = 0.5, phi = 0.2, C = 0.8, P = 5)

social_model2 <- function(Time, State, Pars2) {
  with(as.list(c(State, Pars2)), {
    dX <- kappa*X*(1 - X)*(-1 + j*(1 - C) - sigma*P*(1 - X) + phi*(2*X - 1))
    list(c(dX))
  })
}

State = c(X = 0.2)
out.2 = data.frame(ode(y = State, times = Time, func = social_model2, parms = Pars2))
out.2$Initial = 0.2

State = c(X = 0.5)
out.5 = data.frame(ode(y = State, times = Time, func = social_model2, parms = Pars2))
out.5$Initial = 0.5


State = c(X = 0.8)
out.8 = data.frame(ode(y = State, times = Time, func = social_model2, parms = Pars2))
out.8$Initial = 0.8


out2 = rbind(out.2, out.5, out.8)

# Scenario 3: C = 0.8, P = 100
Pars3 = c(kappa = 1.014, j = 1.68, sigma = 0.5, phi = 0.2, C = 0.8, P = 100)

social_model3 <- function(Time, State, Pars3) {
  with(as.list(c(State, Pars3)), {
    dX <- kappa*X*(1 - X)*(-1 + j*(1 - C) - sigma*P*(1 - X) + phi*(2*X - 1))
    list(c(dX))
  })
}

State = c(X = 0.2)
out.2 = data.frame(ode(y = State, times = Time, func = social_model3, parms = Pars3))
out.2$Initial = 0.2

State = c(X = 0.5)
out.5 = data.frame(ode(y = State, times = Time, func = social_model3, parms = Pars3))
out.5$Initial = 0.5


State = c(X = 0.8)
out.8 = data.frame(ode(y = State, times = Time, func = social_model3, parms = Pars3))
out.8$Initial = 0.8

out3 = rbind(out.2, out.5, out.8)

# Scenario 4: C = 0.4, P = 0.5
Pars4 = c(kappa = 1.014, j = 1.68, sigma = 0.5, phi = 0.2, C = 0.4, P = 0.5)

social_model4 <- function(Time, State, Pars3) {
  with(as.list(c(State, Pars3)), {
    dX <- kappa*X*(1 - X)*(-1 + j*(1 - C) - sigma*P*(1 - X) + phi*(2*X - 1))
    list(c(dX))
  })
}

State = c(X = 0.2)
out.2 = data.frame(ode(y = State, times = Time, func = social_model4, parms = Pars4))
out.2$Initial = 0.2

State = c(X = 0.5)
out.5 = data.frame(ode(y = State, times = Time, func = social_model4, parms = Pars4))
out.5$Initial = 0.5


State = c(X = 0.8)
out.8 = data.frame(ode(y = State, times = Time, func = social_model4, parms = Pars4))
out.8$Initial = 0.8

out4 = rbind(out.2, out.5, out.8)

p1 = ggplot(out, aes(x = time, y = X)) +
  geom_line(out, mapping = aes(time, X, color = as.character(Initial)), size = 1.5) +
  labs(x = 'Time', y = "Proportion willing to participate", color = "Initial proportion") +
  theme_classic() +
  theme(legend.position = c(.8, .5), 
        axis.title.y = element_text(margin = margin(t = 0, r = 10, b = 0, l = 0)))+
  scale_x_continuous(expand = c(0,0)) +
  scale_y_continuous(breaks = seq(0,1,0.2)) +
  annotate("text", x = 6, y=1, label = "Scenario 1", size = 6) +
  theme(text = element_text(size = 20))

p2 = ggplot(out2, aes(x = time, y = X)) +
  geom_line(out2, mapping = aes(time, X, color = as.character(Initial)), size = 1.5) +
  labs(x = 'Time', y = "", color = "Initial proportion") +
  theme_classic() +
  theme(legend.position = c(.8, .5), 
        axis.title.y = element_text(margin = margin(t = 0, r = 10, b = 0, l = 0)))+
  scale_x_continuous(expand = c(0,0)) +
  scale_y_continuous(breaks = seq(0,1,0.2)) +
  annotate("text", x = 6, y=1, label = "Scenario 2", size = 6) +
  theme(text = element_text(size = 20))

p3 = ggplot(out3, aes(x = time, y = X)) +
  geom_line(out3, mapping = aes(time, X, color = as.character(Initial)), size = 1.5) +
  labs(x = 'Time', y = "Proportion willing to participate", color = "Initial proportion") +
  theme_classic() +
  theme(legend.position = c(.8, .5), 
        axis.title.y = element_text(margin = margin(t = 0, r = 10, b = 0, l = 0)))+
  scale_x_continuous(expand = c(0,0)) +
  scale_y_continuous(breaks = seq(0,1,0.2)) +
  annotate("text", x = 6, y=1, label = "Scenario 3", size = 6) +
  theme(text = element_text(size = 20))

p4 = ggplot(out4, aes(x = time, y = X)) +
  geom_line(out4, mapping = aes(time, X, color = as.character(Initial)), size = 1.5) +
  labs(x = 'Time', y = "", color = "Initial proportion") +
  theme_classic() +
  theme(legend.position = c(.8, .5), 
        axis.title.y = element_text(margin = margin(t = 0, r = 10, b = 0, l = 0)))+
  scale_x_continuous(expand = c(0,0)) +
  scale_y_continuous(breaks = seq(0,1,0.2)) +
  annotate("text", x = 6, y=1, label = "Scenario 4", size = 6) +
  theme(text = element_text(size = 20))
```


```{r, fig.show = "hold", out.width = "45%", fig.cap = "Proportion of people willing to participate in coral reef conservation over time with three different initial starting proportions (0.2, 0.5, and 0.8) across four scenarios. Scenarios differ based on live coral cover (Scenario 1, 2 and 3: 0.8; Scenario 4: 0.4) and parrotfish abundance (Scenarios 1 and 4: 0.5; Scenario 2: 5; Scenario 3: 100).", fig.align = "center"}
par(mfrow = c(2,2))
p1
p2
p3
p4
```

```{r}
Table1 = data.frame(matrix(nrow = 6, ncol = 3))
colnames(Table1) = c("Parameter", "Description", "Value")
Table1$Parameter = c("$\\kappa$", 'j', 'C', 'P', "$\\sigma$", "$\\phi$")
Table1$Description = c("Interactions with others", "Sensitivity to dead coral", "Live coral cover", "Parrotfish abundance", "Fishing pressue", "Public pressure")
Table1$Value = c("1.014/y", "1.68", "a", "b", "0.5", "0.2")
```

```{r comment = '', echo = F, results = 'asis'}
knitr::kable(Table1, caption = "Model parameter symbols, description, and estimated parameter values.")
``` 

Based on the four scenarios, long-term results from this model result in either complete or absent willingness to participate in coral reef conservation. The ultimate outcome of the model depends on amount of live coral, parrotfish abundance, and the initial proportion of individuals willing to participate; high initial support for conservation with low abundance of live coral and low parrotfish abundance results in increased support for conservation. If any of these conditions are not met, the outcome will always result in no support for conservation.

**Part B**

![Team flow diagram representing ecological and societal factors that influence coral reef conservation.](C:/Users/mttft/Desktop/UVM/UVM Courses/Quant Reasoning/Coral_Reef_Model/Eco_SocioFlowChart.png)

By including both the ecological and social components in our model, we predict the model will be more complex with the direction of the curve (i.e., positive or negative) changing more frequently. In addition, we predict coral, willingness to participate, and macroalgae are likely the most important paramaters for determining overall system dynamics since those are the three things tied together with just one interaction arrow. With the other parameters, the interactions between them can be positive or negative or they have a complicated feedback loop with unknown causes.

```{r}
coupled_model = function(Time, State, Pars) {
  with(as.list(c(State, Pars)), {
    dM =  a*M*C - (P*M)/(M+T) + gamma*M*T
    dC =  r*T*C - d*C - a*M*C
    dT = (P*M)/(M+T) - gamma*M*T - r*T*C + d*C
    dP = s*P*(1 - P/C) - sigma*P*(1 - X)
    dX = kappa*X*(1 - X)*(-1 + j*(1 - C) - sigma*P*(1 - X) + phi*(2*X - 1))
    return(list(c(dM,dC,dT,dP,dX)))  
  })
}

pars = c(a = 0.1, gamma = 0.8, r = 1.0, d = 0.44, s = 0.49, sigma = 0.5, kappa= 1.014, j=1.68, sigma = 0.5, phi = 0.2)
times = seq(0, 100, by = 0.1)

# Scenario 1: M = 0.05, C = 0.9, T = 0.05, P = 0.4, X = 0.5
yini1 = c(M = 0.05, C = 0.9, T = 0.05, P = 0.4, X = 0.5)
out_total1 = data.frame(ode(yini1, times, coupled_model, pars))
final1 = melt(out_total1, id.vars = "time")
final_eco1 = filter(final1, variable != "X")
final_cons1 = filter(final1, variable == "X")

# Scenario 2: M = 0.05, C = 0.3, T = 0.05, P = 0.4, X = 0.5
yini2 = c(M = 0.05, C = 0.3, T = 0.05, P = 0.4, X = 0.5)
out_total2 = data.frame(ode(yini2, times, coupled_model, pars))
final2 = melt(out_total2, id.vars = "time")
final_eco2 = filter(final2, variable != "X")
final_cons2 = filter(final2, variable == "X")

# Scenario 3: M = 0.05, C = 0.9, T = 0.3, P = 0.4, X = 0.5
yini3 = c(M = 0.05, C = 0.9, T = 0.3, P = 0.4, X = 0.5)
out_total3 = data.frame(ode(yini3, times, coupled_model, pars))
final3 = melt(out_total3, id.vars = "time")
final_eco3 = filter(final3, variable != "X")
final_cons3 = filter(final3, variable == "X")

# Scenario 4: M = 0.05, C = 0.9, T = 0.05, P = 50, X = 0.5
yini4 = c(M = 0.05, C = 0.9, T = 0.05, P = 50, X = 0.5)
out_total4 = data.frame(ode(yini4, times, coupled_model, pars))
final4 = melt(out_total4, id.vars = "time")
final_eco4 = filter(final4, variable != "X")
final_cons4 = filter(final4, variable == "X")

# Scenario 5: M = 0.05, C = 0.9, T = 0.05, P = 0.4, X = 0.8
yini5 = c(M = 0.05, C = 0.9, T = 0.05, P = 0.4, X = 0.8)
out_total5 = data.frame(ode(yini5, times, coupled_model, pars))
final5 = melt(out_total5, id.vars = "time")
final_eco5 = filter(final5, variable != "X")
final_cons5 = filter(final5, variable == "X")

# Scenario 6: M = 0.6, C = 0.9, T = 0.05, P = 0.4, X = 0.5
yini6 = c(M = 0.6, C = 0.9, T = 0.05, P = 0.4, X = 0.5)
out_total6 = data.frame(ode(yini6, times, coupled_model, pars))
final6 = melt(out_total6, id.vars = "time")
final_eco6 = filter(final6, variable != "X")
final_cons6 = filter(final6, variable == "X")


par(mfrow=c(1,2))

eco_graph = function(data,i){
  ggplot(data, aes(x = time, y = value)) +
  geom_line(data, mapping = aes(time, value, color = variable)) +
  theme_classic() +
  theme(legend.position = c(0.8, 0.75)) +
  ylab("Relative Cover") +
  scale_x_continuous(expand = c(0,0)) +
  annotate("text", x = 4, y = max(data$value), label = i, size = 6) +
  theme(text = element_text(size = 20))
}

conserve_graph = function(data, t){
  ggplot(data, aes(x = time, y = value)) +
  geom_line(data, mapping = aes(time, value, color = variable)) +
  theme_classic() +
  ylab("Fraction that are conservationists") +
  theme(legend.position = "none") +
  scale_x_continuous(expand = c(0,0)) +
  annotate("text", x = 4, y = 1, label = t, size = 6) +
  theme(text = element_text(size = 20))
} 

f1 = eco_graph(final_eco1, "A")
f2 = eco_graph(final_eco2, "B")
f3 = eco_graph(final_eco3, "C")
f4 = eco_graph(final_eco4, "D")
f5 = eco_graph(final_eco5, "E")
f6 = eco_graph(final_eco6, "F")

c1 = conserve_graph(final_cons1, "A")
c2 = conserve_graph(final_cons2, "B")
c3 = conserve_graph(final_cons3, "C")
c4 = conserve_graph(final_cons4, "D")
c5 = conserve_graph(final_cons5, "E")
c6 = conserve_graph(final_cons6, "F")

```

```{r, fig.show = "hold", out.width = "45%", fig.cap = "Model predictions for the relative cover of macroalgae (M), coral reef (C), and algae turf (T), along with parrotfish density (P) over time. Six scenarios were modeled, with each scenario adjusting one of the ecological parameters (Scenario 1: M = 0.05, C = 0.9, T = 0.05, P = 0.4, X = 0.5; Scenario 2: M = 0.05, C = 0.3, T = 0.05, P = 0.4, X = 0.5; Scenario 3: M = 0.05, C = 0.9, T = 0.3, P = 0.4, X = 0.5; Scenario 4: M = 0.05, C = 0.9, T = 0.05, P = 50, X = 0.5; Scenario 5: M = 0.05, C = 0.9, T = 0.05, P = 0.4, X = 0.8; Scenario 6: M = 0.6, C = 0.9, T = 0.05, P = 0.4, X = 0.5).", fig.align = "center"}
par(mfcol = c(3,2))
f1
f2
f3
f4
f5
f6
```

```{r, fig.show = "hold", out.width = "45%", fig.cap = "Model predictions for the proportion of the population willing to participate in conservation over time. Six scenarios were modeled, with each scenario adjusting one of the ecological parameters and corresponding to the scenarios in Figure 5. Note the change in scale for the y axis among frames.", fig.align = "center"}
par(mfcol = c(3,2))
c1
c2
c3
c4
c5
c6
```

Under most scenarios, the proportion of the population willing to participate in conservation ultimately became 1.0 (i.e., 100% of the population) while the macro algae cover increased and the cover of all other parameters approach 0 (i.e., locally extinct). Under the scenario when parrotfish density was increased, the ecological parameters had the same outcome (increased macro algae while all others decline); however, the proportion of the population willing to participate in conservation became absent (Figure 6D). Based on this information, the parrotfish density appears to have the greatest impact on coral reef conservation when taking into account the social and ecological components.

=======
dx/dt = Kx(1−x)(−1 +j(1−C) − σP(1−x) +φ(2x−1))

*Defined Variables (description in table)*

x = fraction of individuals (humans) in the population willing to participate in coral conservation
C = live coral cover 
P = Parrotfish abundance (try different values)

*Undefined Variables (description not in table)*

K = human growth rate per year
j = coral growth rate per year
σ = Parrotfish growth rate per year
φ = unknown interaction term

```{r}
#parameters
K = 1.014
j = 1.68
σ = 0.5
φ = 0.2

#equation - change in willingness to participate in coral conservaiton
change_wtp <- Kx(1−x)(−1 +j(1−C) − σP(1−x) +φ(2x−1))


```

Equilibrium points occur when dx/dt = 0. 

dx/dt = 0 at 3 points:

1) when x = 0

In this case, none of population supports coral reef conservation.

2) when (1-x) = 0 aka x = 1

In this case, all of population supports coral reef conservation.

3) When (−1 +j(1−C) − σP(1−x) +φ(2x−1)) = 0

-1 + j - jC - σP + σPx + 2φx - φ = 0

j - jC - σP + σPx + 2φx = 1

Not sure what this means...