Dynamical behavior of a stochastic Nicholson’s blowflies model with distributed delay and degenerate diffusion

Abstract

The mathematical model with time delay is often more practical because it is subject to current and past state. What remains unclear are the details, such as how time delay and sudden environmental changes influence the dynamic behavior of systems. The purpose of this paper is to analyze the long-time behavior of a stochastic Nicholson’s blowflies model, which includes distributed delay and degenerate diffusion. The application of the Markov semigroups theory is to prove that there exists a unique stationary distribution. What’s more, the expression of probability density function around the unique positive equilibrium of the deterministic model is briefly described under a certain condition. The results of this paper can be used to find that the weaker white noise can guarantee the existence of a unique stationary distribution and the stronger mortality rate can cause the extinction of Nicholson’s blowflies. Some numerical examples are also given to explain the effect of the white noise.