Threshold dynamics and probability density function of a stochastic prey-predator system with general functional response and fear effect

This paper focuses on a stochastic prey-predator system with general functional response, nonlinear perturbation and fear effect. By developing some new analytical techniques, we first provide a complete classification for the asymptotic behavior of the system by two real-valued parameters λ1, λ2. It is shown that: (i) If λ1 < 0, both predator and prey populations go extinct eventually; (ii) Only the predator population will die out exponentially if λ1 > 0 and λ2 < 0; (iii) If λ1 > 0 and λ2 > 0, the existence and uniqueness of an ergodic stationary distribution of the system is derived, which reflects the long-term coexistence of prey and predator populations. For practical application, a stochastic quasi-positive equilibrium N* is defined. By solving the corresponding Fokker–Planck equation, we obtain an approximate expression of the density function of the stationary distribution around N*. Finally, some numerical examples are performed to demonstrate our theoretical results. The techniques and methods used in the paper are general, and can be successfully applied to other complex stochastic population models.