Abstract
•A stochastic predator–prey model with additional food and nonlinear perturbation is studied.•We establish sufficient conditions for the existence of an ergodic stationary distribution.•The stationary distribution means that the two species can be persistent in the long run.•We obtain sufficient conditions for extinction of the predator species in two cases.
In this paper, we formulate and analyze a stochastic predator–prey model with additional food and nonlinear perturbation. Firstly, by constructing a suitable stochastic Lyapunov function, we establish sufficient conditions for the existence of an ergodic stationary distribution of the positive solution to the model. Then we obtain sufficient conditions for extinction of the predator species in two cases, one is the prey species surviving and the predator species extinction; the other is both the prey and predator species are extinct. Finally, some examples together with numerical simulations are provided to illustrate the analytical results.