Stationary distribution and probability density function analysis of a stochastic Microcystins degradation model with distributed delay

A stochastic Microcystins degradation model with distributed delay is studied in this paper. We first demonstrate the existence and uniqueness of a global positive solution to the stochastic system. Second, we derive a stochastic critical value R-0(s) related to the basic reproduction number R-0. By constructing suitable Lyapunov function types, we obtain the existence of an ergodic stationary distribution of the stochastic system if R-0 (s)> 1. Next, by means of the method developed to solve the general four-dimensional Fokker-Planck equation, the exact expression of the probability density function of the stochastic model around the quasi-endemic equilibrium is derived, which is the key aim of the present paper. In the analysis of statistical significance, the explicit density function can reflect all dynamical properties of a chemostat model. To validate our theoretical conclusions, we present examples and numerical simulations.