Effect of hunting cooperation on the dynamic behavior for a diffusive Holling type II predator-prey model

Abstract

In this paper, the dynamics of a diffusive Holling type II predator-prey model with hunting cooperation are studied. For the local model, the stability of nonnegative equilibria, the detailed behavior of Hopf bifurcation, saddle-node bifurcation, and Bogdanov–Takens bifurcation are investigated. It is shown that under suitable conditions the model exhibits a supercritical and orbitally asymptotically stable limit cycle when the predator cooperation in hunting rate pass through a threshold value. For the reaction-diffusion model, we analyze the diffusion-driven Turing instability of both the positive equilibria and bifurcating periodic solutions, the existence, direction and stability of Hopf bifurcations. Our result shows that the hunting cooperation plays a crucial role in determining the stability and bifurcation behavior to the model, that is, it is beneficial to the coexistence of predator and prey, it can have a destabilizing effect, it can induce the bistability phenomenon, it can undergo not only the saddle-node bifurcation but also Bogdanov–Takens bifurcation, and it can induce the rise of Turing instability, which is a strong contrast to the case without hunting cooperation. Moreover, some numerical simulations are performed to visualize the complex dynamic behavior.