A method to deal with the critical case in stochastic population dynamics

TL;DR

This paper introduces a method to analyze the critical case in stochastic population models where the growth rate is zero, showing that the population converges to extinction in time average, with applications to various models.

Contribution

It provides a general method to handle the critical case in stochastic population dynamics, which is rarely addressed in prior research.

Findings

The method proves convergence to extinction in the critical case.

Applications include stochastic differential equations and piecewise deterministic Markov processes.

Demonstrates extinction behavior in prey-predator and epidemiological models.

Abstract

In numerous papers, the behaviour of stochastic population models is investigated through the sign of a real quantity which is the growth rate of the population near the extinction set. In many cases, it is proven that when this growth rate is positive, the process is persistent in the long run, while if it is negative, the process converges to extinction. However, the critical case when the growth rate is null is rarely treated. The aim of this paper is to provide a method that can be applied in many situations to prove that in the critical case, the process congerves in temporal average to extinction. A number of applications are given, for Stochastic Differential Equations and Piecewise Deterministic Markov Processes modelling prey-predator, epidemilogical or structured population dynamics.