Extinction and stationary distribution of stochastic predator-prey model with group defense behavior

Considering that many prey populations in nature have group defense behavior, and the relationship between predator and prey is usually affected by environmental noise, a stochastic predator-prey model with group defense behavior is established in this paper. Some dynamical properties of the model, including the existence and uniqueness of global positive solution, sufficient conditions for extinction and unique ergodic stationary distribution, are investigated by using qualitative theory of stochastic differential equations, Lyapunov function analysis method, Itô formula, etc. Furthermore, the effects of group defense behavior and environmental noise on population stability are also discussed. Finally, numerical simulations are carried out to illustrate that the effects of environmental noise on both populations are negative, the appropriate group defense level of prey can maintain the stability of the relationship between two populations, and the survival threshold is strongly influenced by the intrinsic growth rate of prey population and the intensity of environmental noise.