Finite-Time Stability in the Discrete Sel’kov-Schnakenberg

Reaction-Diffusion Model: Analytical Analysis and Numerical

Simulations

Salam Alnabulsi1, Wael Mahmoud Mohammad Salameh2, Issam Bendib3,∗, Ahmad A. Abubaker4,

Adel Ouannas5, Abdallah Al-Husban6,7

1Department of Mathematics, Faculty of Science, University of Jordan, Amman 11942, Jordan

2Faculty of Information Technology, Abu Dhabi University, Abu Dhabi, UAE

3Laboratory of Applied Mathematics and Modeling, Department of Mathematics, Faculty of Exact Sciences,

University of Constantine 1, Constantine 25017, Algeria

4Faculty of Computer Studies, Arab Open University, Saudi Arabia

5Department of Mathematics and Computer Science, University of Larbi Ben M’hidi, Oum El Bouaghi

04000, Algeria

6Department of Mathematics, Faculty of Science and Technology, Irbid National University,

P.O. Box 2600, Irbid, Jordan

7Jadara University Research Center, Jadara University, Jordan

Emails: s.alnabulsi@ju.edu.jo; wael.salameh@adu.ac.ae; bendib.issam@doc.umc.edu.dz;

a.abubaker@arabou.edu.sa; ouannas.adel@univ-oeb.dz; dralhosban@inu.edu.jo

Abstract

This study investigates the finite-time stability (FTS) of the discrete Sel’kov-Schnakenberg reaction-diffusion

(SSRD) system, a mathematical model capturing the interplay between local reactions and spatial diffusion.

A novel discretization framework based on finite difference methods (FDM) is developed to transform the

continuous reaction-diffusion (RD) system into a discrete counterpart, enabling detailed computational analy-

sis. Sufficient conditions for FTS are derived using Lyapunov functions (LF) and eigenvalue-based methods,

ensuring precise predictions of the system’s behavior. Numerical simulations validate theoretical findings,

demonstrating the proposed methods’ practical applicability to scenarios such as chemical reactions, biolog-

ical processes, and technological systems. The influence of system parameters, boundary conditions, and

initial conditions on the dynamic behavior is systematically analyzed. This study contributes to the broader

understanding of RD systems, addressing key challenges in stability analysis and offering a computationally

efficient framework with implications for science and engineering.

Keywords: Finite-time stabilit; Reaction-diffusion systems; Sel’kov-Schnakenberg model; Finite difference

methods; Lyapunov functions