Stability analysis of a discrete-time fractional Gray-Scott system

Abstract

This paper introduces a discrete-time fractional stability framework for the Gray-Scott reaction-diffusion system, representing a novel contribution to the field. By employing the Caputo \(\hbar\)-nabla operator, our model effectively incorporates memory effects inherent in many complex systems, providing a formulation well-suited for numerical simulation. The research objective is to analyze the stability of this discrete fractional system. The methodology involves a detailed spectral analysis to determine the local asymptotic stability of equilibrium points and the use of a discrete Lyapunov method to establish sufficient conditions for global stability. The results, validated through numerical simulations, demonstrate that the fractional order is a key parameter influencing the system’s convergence rate and dynamical behavior. This study provides a rigorous mathematical framework and contributes to a deeper understanding of non-integer order dynamics, with significant potential applications in modeling biological pattern formation and designing distributed control systems where memory effects are crucial.