Lotka-Volterra

Literature

The following is all taken from Graham’s PhD Thesis description (and Asymptotically Exact Inference), together with the following papers:

• Adapting the ABC Distance Function
• Automatic Posterior Transformation
• Fast epsilon-free inference
• Optimization Monte Carlo

Mathematical Set-Up

Suppose that $r$ is the prey population and $f$ is the predator population. Suppose ${\theta }_1,{\theta }_2,{\theta }_3,{\theta }_4$ are parameters. The Lotka-Volterra simulator is an Euler-discretization of the following SDE

$$\begin{array}{rl}dr& =({\theta }_{1}r-{\theta }_{2}rf)dt+d{n}_r\\ df& =({\theta }_{4}rf-{\theta }_{3}f)dt+d{n}_f\end{array}$$

where $n_r$ and $n_f$ are zero-mean white noise processes with variances ${\sigma }_r^2$ and ${\sigma }_f^2$ respectively. We discretize using a time step $\delta t$ an initial state $(r_0,f_0)$ and $N_s$ time points.

Coding

from math import sqrt
import numpy as np
from numpy.random import normal

def LV_simulator(theta):
  # Settings
  r = f = 100
  dt = 1.0
  sigma_f = sigma_r = 1.0
  N = 50
  t1, t2, t3, t4 = theta
  # Store observations
  y = []
  # Generate white noise in advance
  nr = normal(loc=0.0, scale=sigma_r, size=N)
  nf = normal(loc=0.0, scale=sigma_f, size=N)
  
  for i in range(N):
    r = r + dt * (t1*r - t2*r*f) + sqrt(dt) * nr[i]
    f = f + dt * (t4*r*f - t3*f) + sqrt(dt) * nf[i]
    y.append(r)
    y.append(f)
  return np.array(y)

You can find more information at the following Google Colab notebook.

Summary Statistics

Sometimes (see Epsilon-free Inference and Graham’s PhD thesis) one can use the following sufficient statistics:

• The mean of the prey population
• The mean of the predator population
• The standard deviation of the prey population (or the log variance)
• The standard deviation of the predator population (or the log variance)
• The autocorrelation coefficient of lag 1 and 2 of the prey population
• The autocorrelation coefficient of lag 1 and 2 of the predator population
• The cross correlation coefficient between the prey and the predator population.