Memory-based prey-taxis and environmental stress shape spatiotemporal predator-prey dynamics

This paper investigates a diffusive predator–prey system incorporating memory-based prey-taxis, the Allee effect, and environmental stressors. After establishing global well-posedness and existence conditions for equilibria, we use prey-taxis sensitivity and memory delay as parameters to identify Turing and Hopf bifurcation thresholds. Our results show that strong memory-based prey-taxis suppresses diffusion-driven Turing patterns and restores spatial homogeneity. However, this spatial stabilization does not necessarily imply greater delay tolerance: The critical Hopf delay may decrease with taxis sensitivity, revealing a trade-off between taxis-induced spatial stabilization and delay-induced temporal oscillations. Additionally, we analyze environmental stress, revealing a predator release effect where moderate stress disproportionately suppresses predators, which indirectly leads to an increase in the prey's density. We also identify a mode-jumping phenomenon in the critical delay threshold during stress-induced transitions from ordinary differential equation (ODE) to Turing instability. Finally, the numerical simulations provide a two-parameter stability map delineating four dynamic regions: Stable homogeneous states, stationary Turing patterns, spatially nonhomogeneous periodic solutions, and complex spatiotemporal dynamics from interacting instabilities.