Rich dynamics of a stochastic Michaelis–Menten-type ratio-dependent predator–prey system.

Stochastic predator–prey systems with different functional responses have been studied. But the dynamics of a Michaelis–Menten-type ratio-dependent predator–prey system with stochastic perturbation is not investigated systematically. In this paper, we give the asymptotic behavior of this system. Sufficient criteria for the existence of a stationary distribution and ergodicity are obtained, which means the species are permanent. Besides, we show the situations in which the species are non-persistence. Finally, examples and simulations are carried on to verify these results. • This paper discusses the rich dynamics of a stochastic ratio-dependent predator–prey model. • Criteria for the existence of a stationary distribution are given through constructing suitable Lyapunov functions. • Sufficient conditions for the non-persistence of the species are obtained. • Our conditions and results are consistent in its corresponding deterministic model. • White noise can make conditions in our results be satisfied, and large white noise can lead the extinction of the species. [ABSTRACT FROM AUTHOR]