Role of Allee Effect, Hunting Cooperation, and Dispersal to Prey–Predator Model

Abstract

In prey–predator interaction, many factors, such as the fear effect, Allee effect, cooperative hunting, and group behavior, can influence the population dynamics. Hence, studying these factors in prey–predator makes the model more realistic. In this paper, we have proposed the prey–predator model having herd and Allee effect in prey population, where predators follow hunting cooperation. We have employed temporal analysis to examine the role of the Allee effect and hunting cooperation. Furthermore, we have extended the analysis to spatiotemporal analysis to examine the role of dispersal and the type of spatial structure formed by the population due to random movement. We first discuss the proposed model’s existence and positivity, then the stability of the existing equilibrium points through Routh–Hurwitz criteria. The temporal analysis is carried out through Hopf-bifurcation at the coexistence equilibrium point by considering the Allee threshold ([Formula: see text]), hunting cooperation ([Formula: see text]), and attack rate ([Formula: see text]) as controlled parameters. With the addition of diffusion to the model, we examine the spatial model stability and derive the Turing instability condition, which will give rise to various Turing patterns. Finally, numerical simulations are performed to validate the analytical results. The theoretical study and numerical simulation results demonstrate that the Allee effect, hunting cooperation, and diffusion coefficient are sensitive parameters to the model’s stability.