Fractional-Order Model for Atmospheric $CO_{2}$ Gas Dynamics and Computational Analysis

This study proposes and analyzes a fractional-order mathematical model to explore the interactive dynamics among the human population, forest biomass, and atmospheric carbondioxide (CO2) concentrations. The underlying system is formulated through three distinct compartments: atmospheric CO2 density (X), human population (Y), and forest biomass (Z). To incorporate memory effects and non-local behaviors, the fractional derivative is defined in the Caputo sense. Crucial qualitative properties of the model, including the existence, uniqueness, and non-negativity of the solutions, are rigorously established. Furthermore, computational simulations are conducted via the Generalized Euler Method, and the corresponding trajectories are illustrated graphically to validate the theoretical findings.