Persistence in fluctuating environments

TL;DR

This paper develops a mathematical framework to understand how environmental fluctuations influence the coexistence of interacting populations, revealing conditions under which noise promotes or inhibits diversity.

Contribution

It extends the nonlinear theory of permanence to stochastic models, providing a robust coexistence criterion applicable to various ecological dynamics.

Findings

Environmental noise can either promote or inhibit coexistence depending on community interactions.

Stochastic variation in mortality rates does not affect coexistence criteria in Lotka-Volterra models.

Random environmental forcing can enhance genetic diversity in exploitative systems.

Abstract

Understanding under what conditions interacting populations, whether they be plants, animals, or viral particles, coexist is a question of theoretical and practical importance in population biology. Both biotic interactions and environmental fluctuations are key factors that can facilitate or disrupt coexistence. To better understand this interplay between these deterministic and stochastic forces, we develop a mathematical theory extending the nonlinear theory of permanence for deterministic systems to stochastic difference and differential equations. Our condition for coexistence requires that there is a fixed set of weights associated with the interacting populations and this weighted combination of populations' invasion rates is positive for any (ergodic) stationary distribution associated with a subcollection of populations. Here, an invasion rate corresponds to an average per-capita growth rate along a stationary distribution. When this condition holds and there is sufficient noise in the system, we show that the populations approach a unique positive stationary distribution. Moreover, we show that our coexistence criterion is robust to small perturbations of the model functions. Using this theory, we illustrate that (i) environmental noise enhances or inhibits coexistence in communities with rock-paper-scissor dynamics depending on correlations between interspecific demographic rates, (ii) stochastic variation in mortality rates has no effect on the coexistence criteria for discrete-time Lotka-Volterra communities, and (iii) random forcing can promote genetic diversity in the presence of exploitative interactions.