Mathematical Modeling of Dynamic Feedback Loops in Carbon Cycle Using Differential Equations

Carbon dioxide is very important during photosynthesis. However, it increases as a result of deforestation, burning of fossil fuels, cement manufacturing factories and other agricultural practices and this causes global warming. Carbon cycle as a process involves interactions among major reservoirs controlled by complex feed back relationships that are not easy to understand without a well organized mathematical framework. Some of the existing carbon cycle models are too complex while others are too simple and fail to include nonlinear feedback loops between reservoirs. This study built a model that captures the dynamic feedback interactions between the most relevant carbon reservoirs and investigate their effects on carbon partitioning and stability of the system through centuries. Carbon exchanges (sequestration, respiration, decomposition and interaction with the atmosphere) were simulated using a mathematical modeling approach based on systems of differential equations. Equilibrium and stability characteristics of the model were analyzed and the behavior of the model was examined numerically under various environmental conditions. These results demonstrated that feedback interactions are important for the regulation of carbon and in determining long-term equilibrium states. The model demonstrated that a disturbance in one of the reservoirs affects the entire system, changing the overall carbon dynamics. Conditions were identified under which the system stays in balance or changes to new states. The stability analysis showed conditions for maintaining a balance or changing to new state of the system. The study concluded that differential equation modeling provides a clear framework for understanding dynamic feedback loops in the carbon cycle. It is recommended that the model be extended to incorporate additional environmental variables and be used as a basis for further quantitative studies in climate and environmental modeling.