SIAM Journal on Applied Mathematics, Volume 80, Issue 3, Page 1567-1589, January 2020.
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.

中文翻译：

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一种处理随机种群动态临界情况的方法

SIAM应用数学杂志，第80卷，第3期，第1567-1589页，2020年1月。
在许多论文中，随机种群模型的行为都是通过一个实际数量的符号来研究的，该数量是灭绝集合附近的人口增长率。在许多情况下，事实证明，如果该增长率为正数，则从长远来看，人口将持续存在；而如果为负数，则该人口将灭绝。但是，很少考虑增长率为零的关键情况。本文的目的是提供一种可以在许多情况下应用的方法，以证明在关键情况下，该过程在时间平均上收敛于灭绝集。给出了随机微分方程和分段确定性马尔可夫过程建模的多种应用，这些过程建模了捕食者，流行病学或结构化种群动态。