article · Discrete and Continuous Dynamical Systems - S
Mathematical models of population dynamics often rely on integer-order derivatives, which fail to capture memory effects and non-local interactions. This work addresses that limitation by formulating an eco-epidemiological predator-prey model using the Caputo-Fabrizio fractional derivative. The theoretical analysis confirms the existence and uniqueness of positive solutions, local stability of equilibrium points, and generalised Hyers-Ulam stability, demonstrating that the system remains robust under small perturbations. To solve the fractional differential equations, a numerical approach based on multistep Adams-Bashforth methods with step sizes from one to six was developed and tested against three equations with known exact solutions. Numerical simulations illustrate the shifting dynamics among susceptible, infected, and predator populations across different fractional orders, supporting the theoretical results and offering an improved framework for modelling complex biological interactions.
Real-world ecological and epidemiological interactions are influenced by biological history and memory, which standard equations often overlook. By integrating fractional derivatives and verifying their mathematical stability, this research provides a more reliable foundation for tracking disease spread and predator-prey dynamics over time, improving the tools available for ecological forecasting.
This work represents early-stage theoretical and numerical research. While the computational framework could eventually inform software used by ecological researchers or epidemiological modellers, the abstract does not indicate any immediate commercial application pathway or industrial testing.
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In the study of population dynamics, understanding the interactions between different species is crucial. Traditional models often rely on integer-order derivatives, which lack memory effects and non-local interactions. This research extends eco-epidemiological models by incorporating the Caputo-Fabrizio fractional derivative, providing a more accurate representation of biological processes with memory.The existence and uniqueness of positive solutions, as well as the local stability of equilibrium points in the Caputo-Fabrizio sense, are proven, ensuring the model's reliability for studying eco-epidemiological dynamics. Furthermore, the study establishes the generalized Hyers-Ulam stability for the model, ensuring robustness to small perturbations.Additionally, we introduce and analyze a novel numerical approach by constructing multistep methods of Adams-Bashforth type with step sizes ranging from 1 to 6 to solve the proposed fractional-order differential equations, enhancing the accuracy and stability of the solutions. We applied these methods to analyze three examples of fractional differential equations with known exact solutions, focusing on the dynamics for different values of the fractional order.Numerical simulations are provided to illustrate the dynamics of susceptible, infected, and predator populations, validating the theoretical findings. This comprehensive approach offers significant insights and improvements in the modeling of complex biological and epidemiological processes.
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DOI: 10.3934/dcdss.2024181
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