Dynamical analysis of a toxin-producing phytoplankton-zooplankton model with refuge

To study the impacts of toxin produced by phytoplankton and refuges provided for phytoplankton on phytoplankton-zooplankton interactions in lakes, we establish a simple phytoplankton-zooplankton system with Holling type II response function. The existence and stability of positive equilibria are discussed. Bifurcation analyses are given by using normal form theory which reveals reasonably the mechanisms and nonlinear dynamics of the effects of toxin and refuges, including Hopf bifurcation, Bogdanov-Takens bifurcation of co-dimension 2 and 3. Numerical simulations are carried out to intuitively support our analytical results and help to explain the observed biological behaviors. Our findings finally show that both phytoplankton refuge and toxin have a significant impact on the occurring and terminating of algal blooms in freshwater lakes.