Prey-taxis as a driver of spatial patterns in predator–prey systems: From spots to spirals and chaos

This study identifies a behavior-driven instability mechanism for spatial pattern formation in ecology, demonstrating that repulsive prey-taxis, a phenomenon where predators avoid well-defended prey, can operate as a primary driver of spatiotemporal complexity through mechanisms distinct from classical diffusion-driven Turing instability. We develop a diffusive predator-prey model incorporating prey-taxis and a Crowley-Martin functional response to capture the interplay between directed movement and predator interference. Our analysis establishes the global existence, uniqueness, and boundedness of classical solutions, ensuring the model's biological well-posedness. We prove that the unique positive equilibrium is globally asymptotically stable under weak attractive prey-taxis, while deriving explicit, ecologically interpretable thresholds for instability. A key finding is that sufficiently strong repulsive prey-taxis (χ<0) induces a novel instability, triggering both Turing and Hopf bifurcations as quantified by the prey-taxis coefficient χ and the conversion rate c. In contrast, attractive prey-taxis (χ>0) exerts a consistent stabilizing effect. Extensive numerical simulations confirm these predictions and unveil a rich spectrum of patterns, ranging from stationary spots, stripes, and labyrinths to dynamic spiral waves and chaos, all of which align with observable ecological phenomena. Our results fundamentally expand the theory of biological pattern formation by establishing prey-taxis, particularly in its repulsive form, as a versatile and potent mechanism for spatial self-organization beyond the effects of pure diffusion.