Role of Fear and Its Carry-Over Effects in the Dynamics of a Spatially Extended Leslie–Gower Prey–Predator System

This study creates and analyses a Leslie–Gower prey–predator scheme which includes the fear phenomenon and how it affects prey reproduction over time and space. The prey population experiences growth suppression arising from both direct predation and predator-induced fear, the latter of which persists even in the absence of direct encounters. The predator dynamics are governed by a Leslie–Gower framework, wherein the carrying capacity is determined by the availability of prey as the primary food resource. Diffusion terms are incorporated for both prey and predator populations to represent their spatial movement and dispersal. This formulation enables the investigation of spatial heterogeneity and the emergence of ecological pattern formations within the system. We explore the presence and stability of equilibria, the circumstances under which predators and prey can coexist, and examine the impact of fear and its lingering effects on long-term dynamics. Furthermore, the study explores the influence of diffusion on spatial stability, diffusion-driven (Turing) instabilities, and the associated dispersion relations with respect to the wave number, particularly under the combined effects of fear and its carry-over effects (COE). Numerical simulations are conducted to enhance the analytical findings, revealing a diverse range of spatio-temporal dynamics, including predator extinction, stable coexistence, and the formation of spatial structures. The results underscore the critical ecological significance of fear and its persistent carry-over effects in shaping population stability and spatial organization.