Mitigating atmospheric carbon dioxide through ocean-based carbon capture technologies: a delay mathematical model

The ocean serves as the largest natural sink for atmospheric carbon dioxide (\(CO _2\)), playing a vital role in regulating global climate. Ocean-based carbon removal technologies seek to enhance this natural capacity, while shellfish farming offers a complementary nature-based pathway to sequester carbon dioxide. The success of these strategies, however, depends on effective budget allocation. In this study, we develop a nonlinear mathematical model to examine how budget allocation for ocean-based carbon removal technologies and shellfish farming, along with delays between investment and impact, influences atmospheric \(CO _2\) dynamics. The model considers that a portion of total budget is allocated for the implementation of ocean-based carbon removal technologies, while the remainder is invested in shellfish farming. The formulated model is qualitatively analyzed to determine the system’s behavior in the long run. Results show that increasing the efficacy of allocated budget in enhancing oceanic \(CO _2\) uptake and shellfish production can substantially lowers atmospheric \(CO _2\) levels. However, if the budget growth rate exceeds a critical threshold, the interior equilibrium loses stability through a Hopf-bifurcation, giving rise to limit cycle oscillations. Moreover, it is noticed that the amplitude of these oscillations reduces with increasing the efficacy of budget to enhance oceanic \(CO _2\) uptake, and above a critical level, these oscillations die out and system gets stabilized to a positive equilibrium state. Furthermore, we find that the stability of the interior equilibrium is highly sensitive to delays between budget allocation and the resulting increase in oceanic \(CO _{2}\) absorption and shellfish production. Longer delays trigger multiple stability switches, leading to complex dynamic behavior. Numerical simulations are presented to support and validate the theoretical findings, providing insights into the dynamic behavior of the proposed model.