Stability Switches and Chaos in a Diffusive Toxic Phytoplankton–Zooplankton Model with Delay

Abstract

To explore the effect of toxins produced by phytoplankton in algal blooms, a diffusive toxic phytoplankton–zooplankton system with delay is studied through detailed bifurcation analysis. The local stability of the positive equilibrium is studied by analyzing the characteristic equation. The critical values of delay at which Hopf bifurcations occur are obtained and ordered, and it is proved that the delay can induce stability switches, which means that the delay has a significant impact on both the formation and the termination of algal blooms. During the process of stability switching, double Hopf bifurcation arises. Taking the rate of toxin liberation and time delay as the bifurcation parameters, the normal form for double Hopf bifurcation is derived, from which the classification for dynamics and bifurcation sets near double Hopf bifurcation point is obtained. It is proved that the system has complex dynamics, such as periodic oscillations, quasi-periodic oscillations on two- or three-torus and even chaos. Numerical simulations are presented to support the theoretical results.