Fractional modelling of oxygen-plankton interactions under climate effects through proportional Hadamard-Caputo operators

This study introduces a novel fractional framework to model the dynamics of oxygen, phytoplankton, and zooplankton interactions under the influence of climate change. By employing the proportional Hadamard–Caputo derivative, the proposed system reflects the effects of nonlocal memory in ecological processes. We establish the existence and uniqueness of the solution via the nonlinear Leray–Schauder alternative and Banach contraction principle. Additionally, we present both Ulam–Hyers and extended Ulam–Hyers stability results for the proposed model. Using the predictor-corrector Adams method, numerical simulations are conducted to show the effective role of the fractional memory and warming rates on oxygen enrichment and the improvement of planktonic dynamics. The findings illustrate how fractional parameters control marine ecosystem responses to thermal effects while providing a better understanding of the long-term ecological impacts from climate variability. To model long–range effects on a logarithmic time scale, we use the proportional Hadamard–Caputo derivative. Its tuning that is proportional offers variable memory, encompasses the classical case in a limiting manner, and accommodates a stable Adams predictor–corrector scheme for our simulations.