Dynamic analysis of a stochastic toxin-mediated predator-prey model in aquatic environments

This paper considers a stochastic aquatic toxin-mediated predator-prey model. Our main aim is to investigate the effects of environmental toxins and white noise on the population dynamics of this model. First, we prove that there is a global positive solution for any given initial value. Then we show the existence of a unique stationary distribution applying the Lyapunov function method and establish extinction conditions using the invariant density of a one-dimensional diffusion process. In addition, by solving the corresponding Fokker-Plauck equation, we obtain the explicit probability density function around the quasi-stable equilibrium of the transformed stochastic system without pollution. Finally, we use numerical tests to illustrate the main results and reveal some interesting biological phenomena, such as small white noise can reduce the risk of extinction, and the same toxin concentration has different effects on predators and their prey, etc.