Global well-posedness and pattern formation in a predator-prey reaction-diffusion system with prey-taxis

This paper investigates a spatiotemporal predator-prey model that incorporates the Allee effect, the fear effect, prey-taxis, and harvesting within a Beddington-DeAngelis functional framework. The model captures the combined influence of biological interactions, behavioral responses, and harvesting activities on population dynamics in a spatially heterogeneous environment. The global existence, positivity, and boundedness of classical solutions are first established under appropriate parameter conditions. The existence and local stability of homogeneous steady states are then analyzed, and the conditions for diffusion-driven instability are derived to characterize the onset of spatial patterns. Using weakly nonlinear analysis, amplitude equations are developed to describe the modulation of spatial modes near the bifurcation threshold. Numerical investigations are conducted to complement the theoretical analysis: bifurcation diagrams are employed to examine the effects of biological parameters such as the Allee threshold ($ \beta $), fear intensity ($ \gamma $), and conversion efficiency ($ \varepsilon $), while spatiotemporal simulations are performed to visualize different scenarios and demonstrate the impact of prey-taxis on pattern formation and population organization.