(untitled) (Journal of Nonlinear Modeling and Analysis, 2025)
The investigation of predator cooperative hunting and prey refuge is crucial for understanding ecological dynamics. In recent years, the role of cooperative hunting or prey refuge in predator-prey systems has received much attention from researchers. However, the study on the combined effects of predator cooperation and predator-dependent prey refuge in the predation system has not yet been investigated. Therefore, in this paper, we propose a prey-predator model with both the factors of predator-dependent prey refuge and cooperative hunting. The positivity and boundedness of the system’s solutions are investigated, followed by the existence and local stability of the equilibrium points. Sufficient conditions for the existence of Hopf bifurcation of the system are obtained. The direction of Hopf bifurcation in the system is investigated by using the center manifold theorem and normal form method. From the analysis of the model, we find that the dependence coefficient m of the prey refuge ratio on the number of predators may be responsible for the stability of the system. The results also indicate that suitable competition coefficients s and cooperative hunting coefficients α between predators may enable the species to coexist in the long run. Furthermore, we observe two limit cycles of the system when the parameters satisfy certain conditions. Finally, the dynamical behavior of the model is performed through intriguing numerical simulations.