Asymptotic behavior of a food chain model with stochastic perturbation

Abstract

In this article, asymptotic behavior of a stochastic food chain model is analyzed. First of all, we consider the dynamic of the deterministic system by analyzing the equilibria and their stability conditions. Then by Lyapunov analysis methods, we show that the system has a unique positive solution. After that, the stochastic stability and its long time behavior around the equilibriums of the deterministic system is given. Later, we obtain the stochastic system and its corresponding deterministic system have similar properties when the white noise is small. And then,we show the conditions that there is a stationary distribution of the system which implies that the system is permanent. At last, we make simulations to verify the results of our analysis.