A New Fractional Predator–Prey Model for Studying the Dynamic Relationship between Financial Corruption and Society-Simulation Study

Abstract

Background: This study investigates the existence and uniqueness of solutions for a class of nonlinear fractional differential systems governed by Caputo derivatives. Such systems are effective in modeling dynamic processes with hereditary behaviors and anomalous responses, which are commonly observed in socio-economic, biological, and engineering contexts. Traditional integer-order differential models fail to capture these features, motivating the use of fractional-order formulations. Objective: The primary goal is to establish rigorous mathematical conditions ensuring the well-posedness of nonlinear fractional systems involving coupled variables. By proving existence and uniqueness, the study provides a solid theoretical foundation for analysis, simulation, and application of fractional-order models to complex systems with hereditary dynamics. Methods: The fractional differential system is transformed into an equivalent integral formulation using Riemann–Liouville fractional integral operators. Within a Banach space, two operators are defined: one is continuous and compact, while the other is contractive. The existence of solutions is established via Krasnoselskii’s fixed point theorem, and the boundedness and Lipschitz continuity of the nonlinear terms ensure uniqueness through Banach’s contraction principle. The system admits a unique solution confined to a closed, invariant subset of the function space, providing a rigorous framework for analytical and numerical investigations. Results: Analytical results confirm that the proposed fractional-order system satisfies conditions for existence and uniqueness of solutions. The integral representation demonstrates the well-posedness of the model and validates its ability to capture hereditary dynamics accurately. Operator-based analysis ensures stability and supports convergence of numerical methods, making the model practically applicable for simulations of socio-economic and biological processes. Conclusions: This work establishes a rigorous theoretical foundation for nonlinear fractional differential systems with Caputo derivatives. By addressing existence, uniqueness, and stability, it provides a reliable framework for modeling systems where past states influence current behavior. The findings facilitate both analytical and numerical studies of fractional-order dynamics, expanding the applicability of fractional calculus in diverse fields, including economics, population dynamics, control theory, and material science.