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.

Study of Certain Stochastic Predator-Prey Models

“…Moreover, in this paper, we also consider the degenerate case B 1 (·) = B 2 (·). This paper complements our recent paper [4] by providing a number of interesting examples. We also go further than [4] by discussing a stochastic predator-prey model with regime-switching and giving some examples to illustrate its distinctive properties.…”

“…This paper complements our recent paper [4] by providing a number of interesting examples. We also go further than [4] by discussing a stochastic predator-prey model with regime-switching and giving some examples to illustrate its distinctive properties. Because of the page limitation, the proofs of the results are not given but referred to [4].…”

“…We also go further than [4] by discussing a stochastic predator-prey model with regime-switching and giving some examples to illustrate its distinctive properties. Because of the page limitation, the proofs of the results are not given but referred to [4]. The rest of the paper is arranged as follows.…”

Dynamical Behaviors of the Tumor-Immune System in a Stochastic Environment

“…is the Gamma distribution with parameter a 1 and b 1 , see details in [10]. Therefore, by the Itô formula and the strong law of large numbers, noting that the mean of Gamma distribution Now, we consider the auxiliary process ψ(t) defined by (3.1).…”

“…If 2α < σ 2 2 , it can be easily verified that lim t→∞ ψ(t) = 0 a.s. If 2α > σ 2 2 , by solving the Fokker-Planck equation (see details in [10]), the process ψ(t) has a unique stationary distribution λ(·), and obeys the Gamma distribution with parameter…”