Abstract
In this paper, we examine a stochastic prey-predator system with fear effect and general anti-predator behavior. To tackle the impact of stochastic perturbations, we first propose a p-stochastic threshold method to construct several necessary p-Lyapunov functions. Then by defining a quasi-carrying capacity x*, sufficient conditions are established for the existence and uniqueness of stationary distribution pi(.) of the system. By solving the Fokker-Planck equation, the approximate expression of probability density function of the distribution pi(.) around its quasi-positive equilibrium is further derived. Besides, the extinction of prey and predator populations is studied. Finally, some numerical examples are provided to verify our theoretical results and study two aspects: (i) the impact of anti-predator behavior; (ii) the effect of prey fear.