Modeling the Impact of Direct Air Capture on Forest Biomass and Population Dynamics

Anthropogenic carbon dioxide ( $$\:C{O}_{2}$$ ) emissions are considered a key driver of global temperature rise, which disrupts forest biomass productivity by altering plant phenology and increasing the frequency of forest fires. To combat this, carbon capture and sequestration (CCS) technologies, including direct air capture (DAC), have received significant attention. This study presents a novel mathematical model that integrates DAC with human population, forest biomass, atmospheric temperature and concentration of $$\:C{O}_{2}$$ . The model system has four non-negative equilibria, out of which, three are boundary equilibria and one is biologically feasible positive equilibrium. The stability of the system’s positive equilibrium is established through Lyapunov’s direct method. The stability conditions demonstrate that $$\:C{O}_{2}$$ depletion rate via DAC has a stabilizing effect on the climate system, whereas the anthropogenic emission rate has destabilizing impact. This shows the potential benefits and long-term impact of DAC. Further, using historical data from 1990 to 2021, the model is calibrated to match trends in global forest biomass, human population, $$\:C{O}_{2}$$ concentration, and atmospheric temperature. To tackle the issue of high operational costs associated with DAC, while also recognizing the cost associated with detrimental impact of $$\:{CO}_{2}$$ , a cost-effective strategy is proposed for deploying DAC that leverages optimal control theory. For this purpose, a cost functional is proposed involving these two components of cost. The cost functional is found to be minimal for the optimal control strategy as compared to no control as well as constant heuristic control strategies. The findings offer valuable insights into the role of DAC in stabilizing atmospheric $$\:C{O}_{2}$$ levels and provide guidance for policymakers who aim to balance environmental and economic factors in their climate change mitigation efforts.