Stationary distribution, extinction and density function of a stochastic prey-predator system with general anti-predator behavior and fear effect

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.

Automated summary

This paper examines a stochastic prey-predator system with fear effect and general anti-predator behavior. A p-stochastic threshold method is proposed to tackle the impact of stochastic perturbations, and a quasi-carrying capacity is defined. The existence and uniqueness of the stationary distribution of the system are established, and the approximate expression of the probability density function is derived. The extinction of prey and predator populations is also studied.