Stability analysis of fractional two-dimensional reaction-diffusion model with applications in biological processes

We study the stability of a two-dimensional fractional reaction-diffusion system under the Caputo differential operator in time. Our model is based on the Grey-Scott model, a well-known coupled reaction-diffusion system that describes the interaction between two chemical species. The diffusion term captures the species special spread, while the nonlinear term in the system describes the chemical reaction, resulting in a wide range of difficult, self-organizing patterns, including spots, stripes, or spirals, depending on the parameter values. We derive conditions for local stability of the homogeneous equilibrium by linearizing the system and analyzing the eigenvalues of the Jacobian. Furthermore, we construct appropriate Lyapunov functionals to establish global asymptotic stability of the discrete model under suitable conditions. This approach seeks to provide a robust framework for analyzing complex dynamical behaviors in systems governed by fractional-order in-time reaction-diffusion systems. The numerical simulations employ the Chebyshev spectral method for spatial discretization and the $L1$ scheme for fractional time derivatives. These simulations validate the theoretical findings, demonstrating the model's ability to replicate intricate patterns often observed in reaction-diffusion systems. The results suggest that the fractional-order framework enhances the understanding of pattern formation in such systems, making this model a valuable tool for studying anomalous diffusion and non-local dynamics in biological and chemical processes.