Stability and Hopf bifurcation of a delayed diffusive phytoplankton-zooplankton-fish model with refuge and two functional responses

In our paper, a delayed diffusive phytoplankton-zooplankton-fish model with a refuge and Crowley-Martin and Holling II functional responses is established. First, for the model without delay and diffusion, we not only analyze the existence and stability of equilibria, but also discuss the occurrence of Hopf bifurcation by choosing the refuge proportion of phytoplankton as the bifurcation parameter. Then, for the model with delay, we set some sufficient conditions to demonstrate the existence of Hopf bifurcation caused by delay; we also discuss the direction of Hopf bifurcation and the stability of the bifurcation of the periodic solution by using the center manifold and normal form theories. Next, for a reaction-diffusion model with delay, we show the existence and properties of Hopf bifurcation. Finally, we use Matlab software for numerical simulation to prove the previous theoretical results.