Amplitude Equation Analysis and Pattern Transitions in a Superdiffusive Predator–Prey Model with Allee Effect

This paper proposes and analyzes a superdiffusive predator–prey model incorporating Allee effect to explore parameter-driven spatial pattern formation. Combining linear stability analysis, multiscale technique, and center manifold theory, we systematically derive and analyze the stability of amplitude equations in both two- and three-dimensional spaces. Numerical simulations demonstrate that variations in three core parameters — reproductive advantage, superdiffusion exponent, and Allee coefficient — generate distinct spatial morphologies within the Turing region. A critical dimensional divergence is observed: while 2D systems follow the classical spot-stripe transition, 3D systems exhibit stable planar lamellar structures rather than spots, with BCC configurations remaining unstable. In the Turing–Hopf region, the interplay of diffusion-driven instability and temporal oscillations produces spatially organized periodic dynamics. Pure Hopf bifurcations induce complex spatiotemporal fluctuations. These results establish an integrated analytical framework linking reproductive dynamics, dispersal characteristics, and density-dependent regulation, advancing the theory of ecological pattern formation under superdiffusion constraints with implications for conservation planning and multispecies modeling.