Spatiotemporal dynamics in a delayed diffusive predator-prey model with cooperative hunting and group defense

In this paper, we study the spatiotemporal patterns near the Turing-Hopf bifurcation of a predator-prey model, where the prey exhibits group defense and the predators exhibit cooperative hunting. First, we investigate the local stability of the positive constant steady state for the non-delayed system. Then, for the delayed predator-prey model, we discuss the existence of Turing-Hopf bifurcation, and calculate the normal form near the Turing-Hopf singularity to explore the joint effect of the delay and diffusion on the dynamics of the system. Finally, we give some numerical simulations to demonstrate the theoretical results. The results show that the presence of delay and diffusion makes the system exhibit much more spatiotemporal dynamics, including nonconstant steady states, spatially homogeneous and inhomogeneous periodic solutions.