Plankton-Oxygen Dynamics in the Context of Climate Change: A Fractional Model with A Probability Density Function Approach

Abstract

Analyze how climate change affects marine oxygen production by modeling plankton–oxygen dynamics with a fractional-order nonlinear system and establishing rigorous conditions for the model’s well-posedness.We formulate a three-dimensional system d^α x(t)\/dt^α=Ax(t)+f(x(t)), where A is a diagonal matrix of order 3 and f is nonlinear. We (i) rigorously state the model, (ii) derive a Lipschitz constant for f under suitable assumptions, and (iii) prove existence, uniqueness, and continuous dependence on initial data using a fractional formula with a probability density kernel and a generalized Grönwall inequality.Under stated conditions, f satisfies a computable Lipschitz bound that yields existence and uniqueness of solutions for the fractional system. The solutions depend continuously on initial conditions, establishing well-posedness of the plankton–oxygen model.Introduces a fractional, PDF-kernel–based framework for plankton–oxygen dynamics and provides clean, general proofs of well-posedness via a generalized Grönwall approach, capturing memory effects that classical integer-order models miss.The results justify numerical simulation and sensitivity analyses of fractional marine-ecosystem models, providing a sound base for testing mitigation or management strategies affecting oxygen dynamics.Stronger theory for oxygen-cycle modeling can support evidence-based policies aimed at protecting marine ecosystems under global warming.