Analysis of a stochastic Lotka–Volterra competitive system with infinite delays and Ornstein–Uhlenbeck process

Abstract

In this paper, we construct and analyze a stochastic Lotka–Volterra competitive model with the Ornstein–Uhlenbeck process and infinite delays. First, we verify the existence and uniqueness of the global solution of the system with any initial value. Then, we investigate the pth moment boundedness, asymptotic pathwise estimation, and asymptotic behavior of the solutions of the stochastic system in turn. In addition, we develop sufficient conditions for the existence of a stationary distribution of positive solutions to the stochastic system by establishing a series of suitable Lyapunov functions. Finally, by solving the corresponding six-dimensional Fokker–Planck equation, we obtain the accurate expression of the local density function of the linear system corresponding to the stochastic system.