Global dynamics, periodicity, and bifurcations in a May–Holling–Tanner model with hunting cooperation

The study of food chains and food webs depends fundamentally on our understanding of interspecific interactions. In this context, the present study analyzes a May–Holling–Tanner-type predator–prey model that incorporates cooperative hunting. We establish the existence of a positively invariant region and show that the solutions are permanent. Furthermore, we prove that the solutions are uniformly bounded. We analyze the properties of the extinction equilibrium point for both species using an equivalent topological system and examine their region of attraction. A separatrix curve in the phase diagram divides solutions into distinct qualitative regions, revealing strong sensitivity to initial conditions near the separatrix. The system can also undergo Hopf and Bogdanov–Takens bifurcations, which affect the dynamics of the model. By calculating the Lyapunov quantities, the existence of two limit cycles around the unique positive equilibrium point was demonstrated. In addition, the sensitivity analysis showed that increased cooperation among predators reduces the equilibrium level of the coexistence state. Numerical simulations and bifurcation diagrams confirm the analytical results.