Stationary distribution and periodic solution for stochastic predator-prey systems with nonlinear predator harvesting

• The sufficient condition of persistence and extinction in mean of the predator is given.
• The stationary distribution and ergodicity of stochastic predator-prey system with nonlinear predator harvesting are studied.
• The existence of the positive periodic solution of the corresponding non-autonomous system with stochastic disturbance is investigated.
• The result shows that, the weaker white noise will strengthen the stability of the system, but stronger white noise will result in the extinction of the species.

In this paper, we investigate the dynamics of the stochastic autonomous and non-autonomous predator-prey systems with nonlinear predator harvesting respectively. For the autonomous system, we first give the existence of the global positive solution. Then, in the case of persistence, we prove that there exists a unique stationary distribution and it has ergodicity by constructing a suitable Lyapunov function. The result shows that, the relatively weaker white noise will strengthen the stability of the system, but the stronger white noise will result in the extinction of one or two species. Particularly, for the non-autonomous periodic system, we show that there exists at least one nontrivial positive periodic solution according to the theory of Khasminskii. Finally, numerical simulations illustrate our theoretical results.