Complex Patterns in a Reaction–Diffusion System with Fear and Anti-Predator Responses

The intricate relationship between temporal and spatiotemporal dynamics in a Crowley–Martin predator–prey model, enriched with fear effect and anti-predator behavior, is investigated in this study. A careful mathematical analysis is conducted to explore the feasible equilibria of the model system, followed by an examination of their stability, instability, and all possible bifurcation scenarios. Asymptotic stability, bistability, and various codimension-1 and codimension-2 bifurcations, including transcritical, saddle–node, Hopf, cusp, and Bogdanov–Takens bifurcations, are demonstrated by the model. The analytical findings and the model’s applications are validated through numerical simulations, employing a biparameter bifurcation diagram for quantitative analysis. The study also observes bubbling phenomena and a scenario leading to the density-dependent hydra effect. The diffusion effect is investigated with special attention to the system’s nonlinearity and chosen parameter values. Numerical simulations reveal the emergence of spatiotemporal patterns both within and beyond the Turing space. The evolution of diffusion-driven pattern generation, including cold spots, stripes, labyrinthines, mixtures of stripes and cold spots, and complex non-Turing patterns, is demonstrated on the plane. These spatial patterns are shown to be influenced biologically by both the fear effect and anti-predator behavior.