Stationary Distribution and Periodic Solution of Stochastic Toxin-Producing Phytoplankton–Zooplankton Systems

Abstract

In this paper, we investigate the dynamics of autonomous and nonautonomous stochastic toxin-producing phytoplankton–zooplankton system. For the autonomous system, we establish the sufficient conditions for the existence of the globally positive solution as well as the solution of population extinction and persistence in the mean. Furthermore, by constructing some suitable Lyapunov functions, we also prove that there exists a single stationary distribution which is ergodic, what is more important is that Lyapunov function does not depend on existence and stability of equilibrium. For the nonautonomous periodic system, we prove that there exists at least one nontrivial positive periodic solution according to the theory of Khasminskii. Finally, some numerical simulations are introduced to illustrate our theoretical results. The results show that weaker white noise and/or toxicity will strengthen the stability of system, while stronger white noise and/or toxicity will result in the extinction of one or two populations.

Dynamic Behaviors of a Stochastic Eco-Epidemiological Model for Viral Infection in the Toxin-Producing Phytoplankton and Zooplankton System

“…), the environmental fluctuation is ubiquitous in aquatic ecosystems and prominently affects the dynamics of aquatic ecosystems. In [26], the authors assumed that the growth rate of phytoplankton and the death rate of zooplankton are subjected to the Gaussian white noise, and they studied the following stochastic model:…”

“…They proved the existence of globally positive solutions of the model and derived sufficient conditions for the existence of a unique ergodic stationary distribution. There were some works about introducing the environmental noise into ecosystems (see [26][27][28][29][30][31][32][33][34][35]).…”