Hopf bifurcation induced by time delay and influence of Allee effect in a diffusive predator–prey system with herd behavior and prey chemotaxis

We formulate and concern a diffusive delayed predator–prey system with herd behavior and fear effect; an Allee effect term and prey chemotaxis are both considered. We first investigate the stability of the system under the strong Allee effect and weak Allee effect without spatial diffusion or time delay. For the spatiotemporal system, the constant positive steady states and semi-trivial steady states are presented. Then, the dynamic behaviors of the diffusive system are demonstrated in detail, and the conditions of Turing instability caused by prey chemotaxis are explored. In addition, we regard the time delay as a bifurcation parameter to investigate the stability of reaction–diffusion system. The normal form theory and center manifold theorem are applied to derive the properties of Hopf bifurcation of the delayed diffusive system. Finally, series of computer simulations are given to verify the theoretical analysis and show how fear effect affects the stability of system.