Fractional-Order Discrete Competition Dynamics in Financial Institutions

While fractional-order (FO) models have been extensively studied in ecological and biological systems, their application to financial competition dynamics particularly in discrete settings with rigorous stability guarantees remains largely unexplored. This paper addresses this gap by developing a discrete FO competition model for financial institutions, extending a continuous predator–prey framework using the Grünwald–Letnikov (GL) difference operator. The proposed model captures long-term memory effects by incorporating FO differences, enabling the representation of historical dependencies in profit dynamics. Using a Lyapunov–Krasovskii (LK) functional approach, we derive sufficient conditions for global asymptotic stability (GAS) and exponential stability (ES) of nontrivial equilibrium points (EPs). These stability conditions are expressed explicitly in terms of intrinsic growth rates, market carrying capacities, competition coefficients, and memory parameters. Numerical simulations validate the theoretical results, demonstrating damped oscillatory convergence to EPs under various competitive scenarios and parameter regimes. The findings highlight the role of fractional memory as a stabilizing mechanism, with potential applications in forecasting and strategic analysis of competitive banking markets.