Study of Certain Stochastic Predator-Prey Models

In this paper, we provide sufficient conditions for the permanence and ergodicity of a stochastic predator-prey model with Beddington-DeAngelis functional response. These sufficient conditions are sharp and close to the necessary conditions as well. In addition to the nondegenerate diffusion, degenerate cases are also treated. In the degenerate case, our results characterize the support of the associated unique invariant probability measure. Convergence in total variation of the transition probability to the invariant measures is given. We also consider a stochastic model with regime-switching. Our results are demonstrated by several examples.