Dynamic Behavior of Caputo Fractional-Order Model of Forest Biomass, Human Population, and Atmospheric Carbon Dioxide

This study aims to analyze the dynamical model of CO₂ concentration, human population, and forest biomass. Human activities and land-use changes in forested areas play an important role as the primary contributors to the increase in CO₂ emissions, which drive global warming. The inclusion of fractional-order derivatives is considered to examine the long-term memory effects on the interactions within the CO₂ concentration model. Theoretical results such as the existence, uniqueness, positivity, and boundedness of solutions, the local and global stability behavior of equilibrium points, and the existence of a Hopf bifurcation are explored. Furthermore, key parameters, including the deforestation rate and memory order, are investigated to determine their influence on the solution behavior of the CO₂ concentration model. A fractional-order numerical scheme is employed to illustrate various scenarios, validating the theoretical findings. The results show that fractional-order changes affect the dynamic behavior of the model. In addition, increased deforestation rates can increase CO₂ concentrations and reduce human population in the long term.