Untangling role of cooperative hunting among predators and herd behavior in prey with a dynamical systems approach

In the real world, group formation is ubiquitous. This group behavior is according to the species' needs; some species use it for hunting and some for protection from predators. In this paper, we proposed a prey-predator model with cooperative hunting among predators and herd behavior in prey, and we studied the temporal and spatiotemporal analysis. We first examine the system's positivity, boundedness, and then analyze the local stability criterion of equilibrium points. We perform Hopf–bifurcation analysis at feasible equilibrium points by considering s and α as the bifurcation parameters. With the incorporation of diffusion into the system, we derive the diffusion-driven instability of the prey-predator system. Then we performed a weakly nonlinear analysis and derived amplitude equations by considering self-diffusion as Turing bifurcation parameters. The stability analysis of these amplitude equations leads to the identification of various Turing patterns, such as spots, stripes, and mixed. Employing numerical simulations, we present the main types of patterns observed for parameters in the Turing domain. Our findings reveal that cooperative hunting, interspecific competition, and self-diffusion have important implications in the prey-predator system.