A general theory of coexistence and extinction for stochastic ecological communities

TL;DR

This paper develops a comprehensive mathematical framework to analyze how ecological communities coexist or go extinct under stochastic environmental influences, applicable to various models and including auxiliary factors.

Contribution

It generalizes existing results to non-compact spaces, strengthens persistence and extinction criteria, and incorporates auxiliary variables into stochastic ecological models.

Findings

Classified dynamics for one or two species in discrete time.

Analyzed Ricker, lottery, and Lotka-Volterra models.

Extended continuous time results to degenerate noise and auxiliary variables.

Abstract

We analyze a general theory for coexistence and extinction of ecological communities that are influenced by stochastic temporal environmental fluctuations. The results apply to discrete time (stochastic difference equations), continuous time (stochastic differential equations), compact and non-compact state spaces and degenerate or non-degenerate noise. In addition, we can also include in the dynamics auxiliary variables that model environmental fluctuations, population structure, eco-environmental feedbacks or other internal or external factors. We are able to significantly generalize the recent discrete time results by Benaim and Schreiber (Journal of Mathematical Biology '19) to non-compact state spaces, and we provide stronger persistence and extinction results. The continuous time results by Hening and Nguyen (Annals of Applied Probability '18) are strengthened to include degenerate noise and auxiliary variables. Using the general theory, we work out several examples. In discrete time, we classify the dynamics when there are one or two species, and look at the Ricker model, Log-normally distributed offspring models, lottery models, discrete Lotka-Volterra models as well as models of perennial and annual organisms. For the continuous time setting we explore models with a resource variable, stochastic replicator models, and three dimensional Lotka-Volterra models.