Dynamical Evaluation of the Mittag-Leffler Properties-Based Marine Ecosystem Model

This manuscript proposes a novel fractional-order mathematical model, incorporating the Mittag-Leffler function and fractal fractional operators, to study the impact of global warming on marine ecosystems, particularly focusing on marine plankton and fisheries. The model accounts for the time-varying nature of environmental variables such as greenhouse gas (GHG) concentrations, atmospheric temperature, plankton biomass, and marine fishery resources. The solutions to the fractional-order system are demonstrated to be positive and bounded within the feasible region, ensuring the model’s ecological relevance. We also examine the stability of the equilibrium points using both analytical and numerical approaches. A detailed dynamical analysis reveals that the memory effects introduced by the fractional derivative significantly alter the behavior of the system. Specifically, the fractional operator’s effects are analyzed at different fractional orders and fractal dimensions, highlighting how these parameters influence the system’s stability and long-term behavior. The Mittag-Leffler positive invariant sets (MLPISs) and global Mittag-Leffler attractive sets (MLASs) are derived to further understand the system’s global dynamics. Simulations at varying fractional orders and fractal dimensions show that the model’s response to global warming is more sensitive under fractional dynamics, with larger memory effects leading to more pronounced declines in plankton and fishery resources. These findings suggest that the impact of global warming on marine ecosystems may be more severe than traditionally predicted by integer-order models, emphasizing the importance of considering fractional effects in environmental modeling. The results highlight the critical need for timely intervention to mitigate GHG emissions and preserve marine biodiversity.