The complex dynamics of a stochastic toxic-phytoplankton-zooplankton model

In this paper, an analytical study of a toxin-producing phytoplankton-zooplankton model with stochastic perturbation is performed. By constructing suitable Lyapunov functions, we investigate the global stability of a positive equilibrium and give the condition of the existence of Hopf bifurcation for the deterministic plankton model. Under the perturbation of environmental noise, there is a globally positive solution to the stochastic model and it is stochastically ultimately bounded. In addition, the stochastic model is stochastically permanent under some conditions. A series of numerical simulations to illustrate these mathematical findings are presented.