Stability and Memory Effects in a Caputo Fractional Logistic Model: RK4 Validation and Predictor-Corrector Simulation

Safa Isam Ahmed

doi:10.71229/4vkr4q07

Authors

Safa Isam Ahmed Salah Al-Din General Directorate of Education

DOI:

https://doi.org/10.71229/4vkr4q07

Keywords:

Caputo Logistic model, , Carrying capacity, , Equilibrium stability, , Adams-Bashforth-Moulton

Abstract

The dynamics of the Caputo fractional logistic population model is examined in the present work by local stability analysis and numerical simulation methods. The zero equilibrium is unstable, while for all , the carrying-capacity equilibrium is locally asymptotically stable. Small deviations from the carrying capacity level follow a Mittag-Leffler relaxation law, thus analytically describing the long-memory tail around carrying capacity. The validation of the fourth-order Runge-Kutta (RK4) implementation is performed at the integer-order limit : the maximum error decreased from for to for , and the observed convergence rates are close to 4. The cases of strictly fractional ( ) are numerically treated by the Adams-Bashforth-Moulton predictor-corrector method. The initial conditions both above and below the carrying capacity are investigated ( or ). The simulations indicate two different temporal regimes, namely smaller fractional orders generate larger early transients and slower long-time relaxation. The self-convergence orders at the terminal limit are computed: 1.60, 1.75, and 1.92 for the three different values of (0.5, 0.7, and 0.9, respectively), and are found to be close to the theoretical values. The main contribution is a unified verification framework that combines classical-limit RK4 validation, fractional stability analysis, complementary initial conditions, memory-tail interpretation, and fractional self-convergence within a single reproducible benchmark for memory-driven population models.

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