A METHOD TO DEAL WITH THE CRITICAL CASE IN STOCHASTIC POPULATION DYNAMICS

摘要

In numerous papers, the behavior 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 population is persistent in the long run, while if it is negative, the population goes extinct. 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 converges in temporal average to the extinction set. A number of applications are given for stochastic differential equations and piecewise deterministic Markov processes modeling prey-predator, epidemiological or structured population dynamics.