Dynamic Properties of a Prey–Predator Food Chain Chemostat Model With Ornstein–Uhlenbeck Process

The food chain in an ecosystem is a complex, interconnected system of organisms that depend on each other and their environment. Chemostat model can be used to evaluate the stability and resilience of the food chain, as well as the response capacity of the system in the face of different disturbances and environmental changes. In this paper, we construct a prey–predator food chain chemostat model with Ornstein–Uhlenbeck processes and consider the dynamics of this stochastic model. Firstly, we prove the existence and uniqueness of the global solution. Secondly, we deduce the extinction in two cases: One is the extinction of prey and predator, and the other is the extinction of predator and the survival of prey. In addition, by constructing appropriate Lyapunov functions, we obtain the sufficient condition for the existence of stationary distribution, which means that prey and predator can coexist over a long period of time. Then, on this basis, we give the concrete expression of the density function of the distribution around the positive equilibrium point of corresponding deterministic system. Finally, numerical simulations prove the correctness of the theoretical results and show how the speed of reversion and intensity of volatility affect the food chain behavior.