Dynamic Analysis of a Diffusive Predator–Prey Model with Hunting Cooperation Functional Response and Prey-Taxis

Prey-taxis shows the tendency of predator moving toward the direction of gradient of prey density function. It is well known that it plays an important role in the study of biological populations. In this paper, we introduce prey-taxis into a diffusive predator–prey model with hunting cooperation functional response. First, we investigate the effects of prey-taxis on the stability of the positive equilibrium. The results show that there exists Turing instability when the prey-taxis is less than the critical value, and the positive equilibrium is locally asymptotically stable when prey-taxis is larger than the critical value. Then, we prove the existence of nonconstant positive steady states bifurcating from the positive equilibrium by using the bifurcation theory. Finally, our theoretical analyses are illustrated by numerical simulations.