Dynamical Behaviors of a Stochastic Food Chain System with Ornstein-Uhlenbeck Process

Considering the important role that food chains play in ecosystems, in this paper, we study a three-species stochastic food chain model in which the growth rates and death rates are governed by Ornstein-Uhlenbeck process. The main purpose of this paper is to study the dynamic properties of the model. We first prove the existence and uniqueness of the global solution to the model. The moment boundedness and asymptotic behavior of the solution are also verified. Secondly, we get the result of extinction of the predators. Then, a sufficient criteria for the existence of the stationary distribution to the system are established by constructing a suitable Lyapunov function. Under the same conditions, it is worth noting that we further obtain the explicit formulas for the mean and the covariance of the probability density function for a linearized system around a equilibrium point. Finally, several numerical simulations are carried out to illustrate the theoretical results.

Automated summary

In this study, a three-species stochastic food chain model with Ornstein-Uhlenbeck process governing growth and death rates was investigated. The researchers proved the existence and uniqueness of the global solution, verified moment boundedness and asymptotic behavior of the solution, and established criteria for predator extinction and stationary distribution existence. Explicit formulas for mean and covariance of probability density functions were obtained for a linearized system around an equilibrium point, and numerical simulations were conducted to illustrate theoretical results.