Bifurcation Analysis of a Predator-Prey Model with Allee Effect and Fear Effect in Prey and Hunting Cooperation in Predator

Studying predator-prey interactions with the help of mathematical modeling is helpful in understanding the mechanism, making informed decisions about their management and conservation, and estimating the behavior and dynamics of ecological systems. In this paper, we have explored the dynamics of a predator-prey model with the fear effect and Allee effect in prey and hunting cooperation in predator species. First, we prove the well-posedness of the system, which is followed by results related to the existence and stability of biologically feasible equilibrium points for the weak Allee effect as well as for the strong Allee effect. We have shown that the model exhibits a variety of bifurcations, including saddle-node, transcritical, and Hopf-bifurcation. Numerical simulation reveals that the conversion rate of predator plays an important role in the stability switching. It is observed that fear has both stabilizing and destabilizing effects depending on hunting cooperation and conversion efficiency of predators. The system experiences persistence or bistability for the weak Allee effect, whereas bistability and tristability are observed in the presence of strong Allee effect. We found that as the hunting cooperation increases, the density of prey species reduces. On the other hand, moderate cooperation is beneficial with the strong Allee effect, but the high level of hunting cooperation may force one or both species to become extinct. However, for the weak Allee effect, when the conversion rate of predators is greater than the mortality rate, system cannot collapse even with the high amount of hunting cooperation. Therefore, the theoretical and numerical results of our paper may provide some useful biological insight to estimate the behavior of species and may have a potential impact on population management.