Threshold dynamics and probability density functions of a stochastic predator-prey model with general distributed delay

Predator-prey interactions are among the most complicated interactions between biological species, in which there may be both delay digestion and stochastic fluctuation. In the literature, due to the computational complexity and the lack of available methods, few papers have directly considered the predator-prey model for the combination of general infinite time delays with stochasticity. In this sense, this work is devoted to studying the long-term qualitative behavior of a stochastic predator-prey system with general infinite distributed delay. First, we establish sufficient and necessary conditions for exponential extinction and persistence of species. Notably, not only do our results cover the predator-prey model without stochastic noises and time delays, but also reveal the impact of environmental noises on population dynamics. Moreover, there are some challenges to analyze the stationary distribution of the numbers of prey and predator under species persistence. Based on the recent work (Zhou et al., 2020) where the density function of a three-dimensional avian influenza model with non-degenerate diffusion is concerned, we further generalize the relevant theories to a five-dimensional population system with degenerate diffusion. Via solving the corresponding Fokker-Planck equations, several approximate expressions of the density function of the stochastic predator-prey system with special distribution delay kernels are obtained. Finally, we provide some numerical simulations to supplement the analytical results and study the impact of stochastic noises.