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lotka-volterra.stan
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lotka-volterra.stan
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functions {
real[] dz_dt(real t, // time
real[] z, // system state {prey, predator}
real[] theta, // parameters
real[] x_r, // unused data
int[] x_i) {
real u = z[1];
real v = z[2];
real alpha = theta[1];
real beta = theta[2];
real gamma = theta[3];
real delta = theta[4];
real du_dt = (alpha - beta * v) * u;
real dv_dt = (-gamma + delta * u) * v;
return { du_dt, dv_dt };
}
}
data {
int<lower = 0> N; // number of measurement times
real ts[N]; // measurement times > 0
real y_init[2]; // initial measured populations
real<lower = 0> y[N, 2]; // measured populations
}
parameters {
real<lower = 0> theta[4]; // { alpha, beta, gamma, delta }
real<lower = 0> z_init[2]; // initial population
real<lower = 0> sigma[2]; // measurement errors
}
transformed parameters {
real z[N, 2]
= integrate_ode_rk45(dz_dt, z_init, 0, ts, theta,
rep_array(0.0, 0), rep_array(0, 0),
1e-5, 1e-3, 5e2);
}
model {
theta[{1, 3}] ~ normal(1, 0.5);
theta[{2, 4}] ~ normal(0.05, 0.05);
sigma ~ lognormal(-1, 1);
z_init ~ lognormal(log(10), 1);
for (k in 1:2) {
y_init[k] ~ lognormal(log(z_init[k]), sigma[k]);
y[ , k] ~ lognormal(log(z[, k]), sigma[k]);
}
}
generated quantities {
real y_init_rep[2];
real y_rep[N, 2];
for (k in 1:2) {
y_init_rep[k] = lognormal_rng(log(z_init[k]), sigma[k]);
for (n in 1:N)
y_rep[n, k] = lognormal_rng(log(z[n, k]), sigma[k]);
}
}