Qualitative analysis of a prey-predator model with hunting cooperation and density-dependent death rate

In the realm of ecology, the interplay between cooperative behaviours and the Allee effect represents a crucial nexus in understanding natural dynamics. These elements elucidate various mechanisms governing prey-predator intraspecific interactions. In this study, we introduce a two-dimensional prey-predator model that incorporates a strong Allee effect impacting on prey growth function and facilitating hunting cooperation among predators. Our proposed model innovatively integrates a density-dependent death rate term referred to as predator intraspecific competition, denoted as D > 0, and employs a Holling type III functional response to enhance our analysis. The primary focus of this research is to explore how hunting cooperation influences the dynamics of the model, specially whenever the intraspecific predation affecting the predator populations. Furthermore, we meticulously document the system's dynamical behaviours through an extensive bifurcation analysis, using both one-parameter and twoparameter approaches. Through the rigorous local and global bifurcation analysis, we uncover a spectrum of significant bifurcations, including saddle-node bifurcation, Hopf bifurcation, Bogdanov-Takens (BT) bifurcation, and cusp bifurcation points (CP), alongside generalized Hopf (GH) bifurcations as local bifurcations, and a homoclinic bifurcation as a global bifurcation. This comprehensive examination not only contributes to the theoretical understanding of ecological interactions but also underscores the profound implications of cooperation and density-dependent effects in shaping ecosystem dynamics.