Effect of population size in a Prey-Predator model

TL;DR

This paper investigates how the size of the population parameter  affects the long-term behavior of a stochastic predator-prey model, revealing critical thresholds for persistence and extinction.

Contribution

It introduces a stochastic predator-prey model with a population size parameter and analyzes its sensitivity to this parameter using simulations and singular perturbation theory.

Findings

Persistence occurs at =108, extinction at =107

Qualitative dynamics are highly sensitive to population size parameter

Continuous models may misrepresent stochastic population behaviors

Abstract

We consider a stochastic version of the basic predator-prey differential equation model. The model, which contains a parameter \omega which represents the number of individuals for one unit of prey -- If x denotes the quantity of prey in the differential equation model x = 1 means that there are \omega individuals in the discontinuous one -- is derived from the classical birth and death process. It is shown by the mean of simulations and explained by a mathematical analysis based on results in singular perturbation theory (the so called theory of Canards) that qualitative properties of the model like persistence or extinction are dramatically sensitive to \omega. For instance, in our example, if \omega = 107 we have extinction and if \omega = 108 we have persistence. This means that we must be very cautious when we use continuous variables in place of jump processes in dynamic population modeling even when we use stochastic differential equations in place of deterministic ones.