Dynamical analysis, bifurcations and slow-fast behavior of a predator-prey model with hunting cooperation

The main focus of this paper is to investigate the dynamical behaviors of a predator-prey model incorporating cooperative hunting among predators, which is widely observed in real ecological communities. Supposing identical temporal evolution for prey and predator populations, the present paper examines saddle-node, transcritical, Hopf, cusp, and codimension-2 Bogdanov–Takens bifurcations. In view of the realistic ecological fact that the growth rate of prey populations is considerably larger than the mortality rate of predator populations, the model is further extended to a slow-fast system. With the aid of geometric singular perturbation theory and asymptotic expansion techniques, the singular Hopf bifurcation and canard explosion are systematically analyzed. Finally, we investigate the transient dynamics. Theoretical results are then illustrated and verified by numerical simulations. The current paper improves the understanding of sophisticated predator–prey dynamical interactions within single and multi-time-scale frameworks, and it provides valuable insights for revealing population persistence and community stability in natural ecosystems.