Spatiotemporal Pattern Selection in a Modified Leslie–Gower Predator–Prey System with Fear Effect and Self-Diffusion

Indirect fear effects profoundly influence predator–prey dynamics by reducing prey reproduction. Whereas previous studies have investigated fear effects or self-diffusion separately in Leslie–Gower models, the novelty of this work lies in their simultaneous incorporation into a modified Leslie–Gower predator–prey system with Allee effect, leading to previously unreported bifurcations and spatiotemporal pattern selection. The temporal system exhibits up to six equilibria and undergoes a codimension-2 Bogdanov–Takens bifurcation. In the spatial extension, Turing instability is triggered when the predator diffusion coefficient exceeds a critical threshold. Using weak nonlinear multiple-scale analysis, amplitude equations are derived, and their stability analysis classifies stationary patterns into spots, stripes, and spot–stripe mixtures depending on the distance from the Turing onset. Numerical simulations confirm that low, moderate, and high predator diffusivity, respectively, favour spotted, mixed, and striped prey distributions. These results emphasise the critical role of fear-mediated indirect interactions and diffusion in driving spatial heterogeneity and ecosystem stability.