Qualitative analysis for a diffusive predator-prey model with hunting cooperation and Holling type \textrm{III} functional response

The Spatio-temporal pattern induced by self-diffusion of a predator-prey model with Holling type III functional response that incorporates the hunting cooperation between predators has been investigated in this paper. For the local model without structure, stability of non-negative equilibria with or without collaborative hunting in predators is studied. For the Spatio-temporal model, we analyze the effect of hunting cooperation term on diffusion-driven Turing instability of the homogeneous positive equilibria. To get an idea about patterns formation near the Turing bifurcation, we derive and give a detailed study of the amplitude equation using the multiple-scale analysis. Our result shows that hunting cooperation plays a crucial role in determining the stability and the Turing bifurcation of the model, which is in sharp contrast to the case without cooperation in hunting. Furthermore, some numerical simulations are illustrated to visualize the complex dynamic behavior of the model.