Conditions for permanence and ergodicity of certain stochastic predator-prey models

Abstract

In this paper we derive sufficient conditions for the permanence and ergodicity of a stochastic predator-prey model with a Beddington-DeAngelis functional response. The conditions obtained are in fact very close to the necessary conditions. Both nondegenerate and degenerate diffusions are considered. One of the distinctive features of our results is that they enable the 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 the existing literature and matters related to other stochastic predator-prey models are also given.