Dynamics of a Delayed Diffusive Predator–Prey Model With Hunting Cooperation in Predator and Antipredator Behaviors in Prey

In this paper, we investigate a delayed diffusive predator–prey model incorporating hunting cooperation in predator and antipredator behaviors in prey. Firstly, we regard time delay as a bifurcation parameter and explore its impact on the stability of the positive constant equilibrium and the occurrence of Hopf bifurcation. We obtain a minimum critical value of time delay and infer that below the critical value, the positive constant equilibrium remains stable, while it becomes unstable beyond the critical value, leading to the emergence of Hopf bifurcation precisely at the critical value. Subsequently, employing the normal form theory and center manifold theorem, we compute the direction and stability of the bifurcating periodic solutions. Finally, through numerical simulations, we validate the existence of both homogeneous and nonhomogeneous periodic solutions, thereby reinforcing our theoretical findings.