Bifurcation and pattern dynamic behavior in a fractional-order ecosystem model

This paper investigates the dynamic behavior of a fractional-order reaction-diffusion system for vegetation pattern formation, incorporating interactions between plant biomass, soil water, and salt concentration. The model utilizes Grünwald–Letnikov fractional derivatives to capture memory effects and anomalous diffusion. A novel high-order numerical scheme is developed, featuring a high-accuracy, short-memory time discretization with a nine-point finite difference method in space to enhance stability. Bifurcation analysis is performed to determine equilibrium stability and identify Turing instability thresholds. Numerical simulations illustrate the emergence of diverse vegetation patterns, highlighting how different fractional orders influence spatiotemporal dynamics. Overall, the proposed framework provides an effective computational tool for analyzing complex fractional ecological systems.