Diffusive patterns in a predator–prey system with fear and hunting cooperation

Cooperation among species is a ubiquitous behavior to better understand the system dynamics from an ecological perspective. Hunting cooperation among predators can impose fear effects on the prey population, thereby decreasing the prey’s birth rate. Considering this fact, we propose a model that incorporates hunting cooperation among predators and the fear-induced birth reduction in the prey population. We have done the complete dynamical analysis, including boundedness of solutions, persistence of the system, existence of all equilibria and their local and global stability, existence of Hopf bifurcation and its direction and stability, and existence of saddle-node bifurcation. We analyze Hopf bifurcation with respect to the hunting cooperation parameter and saddle-node bifurcation by varying the predation rate. Moreover, we analyze the multi-stability of the system and observe that bi-stability occurs in two different scenarios. In the spatially extended system, we provide a detailed stability analysis and obtain the conditions for Turing instability. Various Turing patterns such as spots, holes, and stripes are obtained and discussed the biological significance of these patterns for the two-dimensional spatial model. We performed numerical simulations to validate our analytical results for both spatial and non-spatial models.