Pattern Formation by Super Cross-Diffusion in a Fractional-Order Predator-Prey Model with Fear Effect

This work examines Turing instability and pattern formation in a predator-prey system incorporating fear effects and super cross-diffusion. We first establish stability criteria for equilibrium states both with and without super cross-diffusion, demonstrating that cross-diffusion can destabilize otherwise stable equilibria. Linear stability analysis subsequently derives precise conditions for Turing instability, revealing cross-diffusion as the fundamental mechanism driving spatial pattern emergence. Using the cross-diffusion coefficient as a bifurcation parameter, we employ weakly nonlinear theory to derive amplitude equations governing excited modes near the Turing bifurcation threshold. Dynamical analysis of these equations elucidates structural transitions and stability properties across diverse Turing pattern formations. Theoretical predictions are validated through comprehensive numerical simulations.