Dynamics of a stochastic delay predator-prey model with fear effect and diffusion for prey

Abstract

We establish a stochastic delay predator-prey model with fear effect and prey diffusion, and investigate the dynamic behavior of the model. Initially, we use Itô's formula to prove the existence and uniqueness of a global positive solution and the stochastic ultimate boundedness of the system. Subsequently, we provide sufficient conditions for the extinction and persistence of prey and predator, and discuss three possible scenarios. Beside, we construct a suitable Lyapunov function and obtain the conditions for the existence of an ergodic stationary distribution. Finally, we present some numerical simulations to verify the analytical results.