Persistence for stochastic difference equations: A mini-review

TL;DR

This mini-review discusses the conditions under which populations modeled by stochastic difference equations persist or remain bounded, highlighting recent theoretical results and applications to various ecological models.

Contribution

It provides a concise overview of recent theoretical advances in understanding persistence and boundedness in stochastic population models, with applications to classical ecological models.

Findings

Conditions for population persistence are characterized.

Applications to stochastic versions of classical models are demonstrated.

Open conjectures and future research directions are proposed.

Abstract

Understanding under what conditions populations, whether they be plants, animals, or viral particles, persist is an issue of theoretical and practical importance in population biology. Both biotic interactions and environmental fluctuations are key factors that can facilitate or disrupt persistence. One approach to examining the interplay between these deterministic and stochastic forces is the construction and analysis of stochastic difference equations $X_{t+1}=F(X_t,\xi_{t+1})$ where $X_t \in \R^k$ represents the state of the populations and $\xi_1,\xi_2,...$ is a sequence of random variables representing environmental stochasticity. In the analysis of these stochastic models, many theoretical population biologists are interested in whether the models are bounded and persistent. Here, boundedness asserts that asymptotically $X_t$ tends to remain in compact sets. In contrast, persistence requires that $X_t$ tends to be "repelled" by some "extinction set" $S_0\subset \R^k$. Here, results on both of these proprieties are reviewed for single species, multiple species, and structured population models. The results are illustrated with applications to stochastic versions of the Hassell and Ricker single species models, Ricker, Beverton-Holt, lottery models of competition, and lottery models of rock-paper-scissor games. A variety of conjectures and suggestions for future research are presented.