Fractional dynamics of socio-ecological collapse: Nonlocal memory operators and hereditary stability in coupled...

Inspired by the coupled human-Earth system framework of Anderies et al. (2023), we develop a fractional-order extension using Caputo derivatives to model memory effects in socio-ecological collapse dynamics. The model describes interactions between population growth, capital accumulation, and environmental degradation through a coupled system of fractional differential equations with component-dependent memory orders α₁,α₂,α₃. We establish existence, uniqueness, and Ulam-Hyers stability of solutions under suitable assumptions. Numerical simulations based on the L₁ approximation method compare the classical case with fractional-order regimes and show how population dynamics, capital inertia, and environmental memory influence transient behavior and degradation timing. This work extends the original coupled socio-ecological framework by incorporating hereditary dynamics and provides a mathematical tool for studying path dependence and delayed responses in human-environment systems.