Turing–Hopf bifurcation of a delayed diffusive predator–prey system with chemotaxis and fear effect

In this paper, we consider the impacts of the chemotaxis and delay on the dynamics of a diffusive predator–prey system with fear effect under the Neumann boundary conditions. Regarding chemotaxis coefficient and delay as bifurcation parameters, the existence of the codimension-two Turing–Hopf bifurcation is studied by analyzing the associated characteristic equation. We deduce that chemotaxis-driven Turing bifurcation and discrete delay-driven Hopf bifurcation can occur simultaneously. Finally, spatially homogeneous periodic oscillation, spatial patterns and spatiotemporal patterns appear near Turing–Hopf bifurcation by numerical simulations.