Persistence and extinction for stochastic ecological models with internal and external variables

TL;DR

This paper develops mathematical criteria for understanding when species persist or go extinct in stochastic ecological models that include internal and external environmental variables, with applications to various ecological and evolutionary systems.

Contribution

It introduces stochastic Lyapunov function-based theorems providing necessary and sufficient conditions for persistence and extinction in complex stochastic ecological models.

Findings

Environmental stochasticity can promote coexistence in some models.

Fluctuations can inhibit coexistence in others like rock-paper-scissors.

Auto-correlated environmental noise can enhance persistence of populations and diseases.

Abstract

The dynamics of species' densities depend both on internal and external variables. Internal variables include frequencies of individuals exhibiting different phenotypes or living in different spatial locations. External variables include abiotic factors or non-focal species. These internal or external variables may fluctuate due to stochastic fluctuations in environmental conditions. We prove theorems for stochastic persistence and exclusion for stochastic ecological difference equations accounting for internal and external variables. Specifically, we use a stochastic analog of average Lyapunov functions to develop sufficient and necessary conditions for (i) all population densities spending little time at low densities, and (ii) population trajectories asymptotically approaching the extinction set with positive probability. For (i) and (ii), respectively, we provide quantitative estimates on the fraction of time that the system is near the extinction set, and the probability of asymptotic extinction as a function of the initial state of the system. Furthermore, we provide lower bounds for the expected time to escape neighborhoods of the extinction set. To illustrate the applicability of our results, we analyze stochastic models of evolutionary games, Lotka-Volterra dynamics, trait evolution, and spatially structured disease dynamics. Our analysis of these models demonstrates environmental stochasticity facilitates coexistence of strategies in the hawk-dove game, but inhibits coexistence in the rock-paper-scissors game and a Lotka-Volterra predator-prey model. Furthermore, environmental fluctuations with positive auto-correlations can promote persistence of evolving populations and persistence of diseases in patchy landscapes. While our results help close the gap between the persistence theories for deterministic and stochastic systems, we highlight challenges for future research.