Stationary distribution and extinction of a Lotka–Volterra model with distribute delay and nonlinear stochastic perturbations

This paper devotes to the study of a Lotka–Volterra model which has one prey and two predators with nonlinear stochastic perturbations and distributed delay. It is first proved that the autonomous system has a unique global and positive solution. Then, by constructing an appropriate stochastic Lyapunov function, we obtain the sufficient conditions which guarantee the existence of a stationary distribution of the positive solutions to the model. Furthermore, some sufficient conditions for extinction of the predator population is also established. Numerical simulations are provided to demonstrate the main results in the end.