Prey-Predator Model with Allee Effect Incorporating Prey Refuge with Hunting Cooperation

In this paper, we consider the dynamics of a prey–predator model with logistic growth incorporating refuge in the prey and cooperation among predators population. Furthermore, multiplicative Allee effect in the prey growth is added to account from biological and mathematical perspectives. First, the existence and stability of equilibria of the model are discussed. Next, the existence of several kinds of bifurcation is provided and also studied the direction and stability of Hopf bifurcation. In addition, we study the influence of hunting cooperation on the model analytically and numerically, and find that the hunting cooperation cannot only reduce the density of prey population, but also destabilize the system dynamics irrespective of Allee effect. We choose also the impact of refuge on the model numerically, and explore that refuge stabilize the system dynamics. Moreover, the comparison between the dynamics of strong and weak Allee effect is taken into consideration.