This work aims to analyze a dynamic model of CO \(_2\) concentration, forest biomass, and the coal mining area. Coal mining activities and changes in forest area play a significant role as the primary contributors to increased CO \(_2\) emissions. Theoretical results, including the existence, uniqueness, positivity, and entanglement of solutions, as well as the local and global stability behavior of the equilibrium points, and the existence of a forward bifurcation, have been explored. Next, we present the parameter estimation of the model using the differential evolution method. The parameter estimation is based on data from forest areas and CO \(_2\) concentration in Indonesia and worldwide from 1994 to 2023. The estimation results are able to predict CO \(_2\) concentration and forest biomass each year, closely aligning with the actual data. Additionally, we conducted an optimal control analysis of the model by proposing three strategies for the optimal control problem. The results show that the implementation of reforestation and the regulation of coal mines has the least additional cost-effectiveness.
