Stability analysis of a discrete fractional-order lengyel-epstein reaction-diffusion system

This paper investigates the stability of a discrete-space fractional-order Lengyel–Epstein (FO-LE) reaction–diffusion system (RDS) using Caputo–nabla (CN) and dual nabla Caputo (DNC) fractional difference operators. The classical LE model is extended to incorporate temporal memory effects through fractional calculus, and the spatial domain is discretized with finite differences under periodic boundary conditions (PBCs). We establish sufficient conditions for both local Mittag–Leffler stability (LMLS) and global Mittag–Leffler stability (GMLS) of the unique positive equilibrium point (EP). Linearization and Lyapunov direct methods are employed for the LMLS analysis, while a constructed Lyapunov function (LF) combined with fractional comparison principles is used to derive GMLS criteria. Numerical simulations using an finite difference scheme confirm the theoretical findings, demonstrating convergence to the equilibrium for different fractional orders. The results illustrate the robustness of the fractional LE system and provide a foundation for further study of memory-dependent pattern formation in discrete media.