On the persistence regime for Lotka-Volterra in randomly fluctuating environments

TL;DR

This paper investigates the long-term behavior of Lotka-Volterra systems with randomly fluctuating coefficients, clarifying conditions for species extinction or persistence and characterizing the invariant measure in persistent cases.

Contribution

It provides a detailed analysis of parameter regimes dictating extinction or persistence in stochastic Lotka-Volterra models, extending previous work on environmental switching effects.

Findings

Identifies parameter ranges leading to species extinction or coexistence.

Characterizes the support of invariant measures in persistent regimes.

Refines understanding of stochastic influences on ecological dynamics.

Abstract

In this note, we study the long time behavior of Lotka-Volterra systems whose coefficients vary randomly. Bena{\"i}m and Lobry (2015) recently established that randomly switching between two environments that are both favorable to the same species may lead to different regimes: extinction of one species or the other, or persistence of both species. Our purpose here is to describe more accurately the range of parameters leading to these regimes, and the support of the invariant probability measure in case of persistence.