Spatial Dynamics in a Diffusive Leslie–Gower Predator–Prey Model with Allee Effect, Harvesting, and Cooperative Hunting

In this paper, we investigate the spatiotemporal dynamics of a diffusive Leslie–Gower predator–prey model incorporating the Allee effect, prey harvesting, and cooperative hunting among predators. By introducing spatial diffusion into a previously established ordinary differential system, we formulate and analyze the corresponding reaction–diffusion model. The combined effects of nonlinear ecological interactions and diffusion generate rich dynamical behaviors that are absent in the non-spatial framework. First, we establish the global existence, uniqueness, non-negativity, and ultimate boundedness of solutions, and we derive sufficient conditions for population extinction. We then analyze the existence and local stability of constant steady states and show that diffusion can destabilize spatially homogeneous coexistence equilibria through Turing instability under suitable parameter conditions. Furthermore, by studying the associated elliptic system, we prove the existence and nonexistence of nonconstant steady states, thereby characterizing the emergence of stationary spatial patterns. To further reveal the temporal complexity induced by diffusion, we investigate Hopf bifurcation and analyze the direction and stability of the bifurcating periodic solutions. Our results demonstrate that the spatial domain size plays a crucial role in pattern selection and in the formation of spatially homogeneous and spatially nonhomogeneous periodic oscillations. Numerical simulations are presented to validate the theoretical results and to illustrate the evolution of diffusion-driven spatial patterns and oscillatory spatiotemporal structures. The results show that diffusion fundamentally enriches the qualitative dynamics of the predator–prey system: Turing instability induces stationary heterogeneous patterns, whereas Hopf bifurcation generates sustained oscillatory dynamics and periodic spatiotemporal behaviors. These findings provide new insight into the mechanisms of self-organized pattern formation and dynamic transitions in nonlinear ecological reaction–diffusion systems.