Lotka Volterra Model

The Lotka-Volterra Model is a set of two non-linear differential equations that describe the population dynamics of a predator and prey. It can be extended to include more interacting species. This simulation replicates the Lotka-Volterra Model in the spatial domain, using a method called Gillespie algorithm. Blue squares represent the prey whereas red squares represent the predator.







Restart the simulation if either predator or prey go extinct


Model Description

Let \(x\) denote population of the prey and \(y\) denote population of the predator. Then, according to Lotka Volterra Model:

\[ \frac{dx}{dt} = \alpha x - \beta x y \] \[ \frac{dy}{dt} = \delta x y - \gamma y \]

\(\alpha, \beta, \gamma, \delta\) are real, positive numbers that descibe the growth, mortality and interaction between the two species. Numerical simulations of this model (with certain values of model parameters) showcase oscillatory behaviour in the populations of prey and predator. This sort of behaviour is actually observed in real-world populations, for example - snowshow hare (prey) and canada lynx (predator).


Spatial Simulation

The above model is a mean-field model. We attempt to replicate the results of this model using a bare-bones probabilistic automaton with basic interactions. Each prey reproduces at some rate, in which case it populates an empty cell next to it. Prey die with some mortality rate. Predators also die with some mortality rate. However, predators require prey in order to reproduce. If a prey is surrounded by atleast \(n\) predators, then the prey is killed and a predator is born in the same cell. Using Gillespie algorithm, these rates are converted into probabilities. The entire grid is synchronously updates in every time step.
All the probabilites, as well as the number of predators required to kill a prey, can be adjusted in the above simulation.


Note:
  1. This simulation is computationally complex and may lag on lower-end devices
  2. For the default values of probabilites, we indeed see some sort of oscillatory behaviour. However, there is a small chance that this is not the case.
  3. Prey search through Von-Neumann neighbours, whereas Predators consider all 8 neighbourhood cells

Developed by ChanRT | Fork me at GitHub