Stationary distribution and periodic solutions for stochastic Holling–Leslie predator–prey systems

• Stochastic autonomous and periodic Holling–Leslie predator–prey systems are studied.
• Stationary distribution and ergodicity of autonomous system are obtained.
• Existence of the positive periodic solution of the periodic system is obtained.
• Simulations are given for stochastic Holling–Tanner and Holling-type IV system.

The stochastic autonomous and periodic predator–prey systems with Holling and Leslie type functional response are investigated. For the autonomous system, we prove that there exists a unique stationary distribution, which is ergodic by constructing a suitable Lyapunov function under relatively small white noise. The result shows that, stationary distribution doesn’t rely on the existence and the stability of the positive equilibrium in the undisturbed system. Furthermore, for the corresponding non-autonomous system, we show that there exists a positive periodic Markov process under relatively weaker condition. Finally, numerical simulations illustrate our theoretical results.