Dynamics of a Toxin Producing Plankton-Fish Model with Three-Dimensional Patch and Time Delay

In this paper, a toxin producing plankton-fish model with three-dimensional patch and time delay is proposed. We assume that a small part of phytoplankton gathers to form a patch to release toxins for self-defense and introduce a discrete-time delay in fish populations due to gestation. The sufficient conditions for the stability of the solution and the existence of Hopf bifurcation and transcritical bifurcation are obtained. Furthermore, the conditions of stability and direction of Hopf bifurcation in the time delay system are derived by using normal form theory and center manifold theorem. Finally, numerical simulation demonstrates the dynamical outcome such as periodic and chaotic solutions of the system, which validate our analytical findings.