Dynamical Analysis of a Delayed Stochastic Lotka–Volterra Competitive Model in Polluted Aquatic Environments

Abstract

A stochastic toxin-mediated Lotka–Volterra competitive model with time-delay is formulated. Our primary goal is to study the impacts of white noise, environmental toxins and time-delay on population dynamics of the model. To begin with, we demonstrate that there exists a globally positive solution with the aid of constructing Lyapunov function. Then we discuss the uniform boundedness of the pth moment and invariant measure for the solution by Krylov–Bogoliubov theorem. Moreover, persistence and extinction are significant subjects in the study of biological population systems, so we further derive the sufficient conditions for weak persistence, persistence in time average and extinction of the solution, which can serve as a theoretical basis for protecting the diversity of aquatic organisms. In addition, using exponential martingale inequality and Borel–Cantelli lemma, the asymptotic pathwise estimation of system is given. Notably, we creatively explore the probability density function of the converted model, which is based on addressing the corresponding Fokker–Planck equation. In the end, utilizing computer simulation to illuminate the dominating results and reveal the influences of the above disturbances on the aquatic ecological population, such as high concentration of toxins can result in extinction, but a certain level of toxins can promote the persistence of highly resistant species.