Conditions for Permanence and Ergodicity of Certain Stochastic Predator-Prey Models

TL;DR

This paper establishes near-necessary conditions for the long-term stability and ergodic behavior of a stochastic predator-prey model with Beddington-DeAngelis response, including convergence to a unique invariant measure.

Contribution

It provides new sufficient conditions for permanence and ergodicity, characterizes the support of the invariant measure, and addresses both non-degenerate and degenerate diffusions.

Findings

Conditions close to necessary for model stability

Convergence to a unique invariant measure in total variation norm

Characterization of the support of the invariant measure

Abstract

This work derives sufficient conditions for the permanence and ergodicity of a stochastic predator-prey model with Beddington-DeAngelis functional response. The conditions obtained in fact are very close to the necessary conditions. Both non-degenerate and degenerate diffusions are considered. One of the distinctive features of our results is that our results enables characterization of the support of a unique invariant probability measure. It proves the convergence in total variation norm of the transition probability to the invariant measure. Comparisons to existing literature and related matters to other stochastic predator-prey models are also given.