Modeling and analysis of a carbon capturing system in forest plantations engineering with Mittag–Leffler positive invariant and global Mittag–Leffler properties

Fast-growing forest plantations hold significant promise for capturing carbon and are essential in the fight against global warming. In this study, we develop a model to simulate the dynamics of carbon adsorption in rapidly expanding plantations, accounting for the effects of forest burning and biomass growth through fractional operators. To ensure biological feasibility and mathematical rigor, we derive results on boundedness under the Lipschitz condition, positivity, the existence of a unique solution, and zones for feasible solutions using Banach space and fixed point theory. Additionally, the continuity of the solution and Ulam–Hyers stability near the equilibrium point are addressed, lending further support to the validity of the proposed model. The Mittag–Leffler positive invariant sets and global Mittag–Leffler attractive sets of the fractional-order system help preserve the non-local and non-singular properties of the solution. The solution is derived using Lagrange polynomials with two steps, extending the Mittag–Leffler kernel to examine the influence of fractional order and fractal dimension through simulations. Results are compared with power law and exponential kernels, demonstrating improved convergence. From a theoretical standpoint, numerical simulations indicate that the model’s solutions align with the plantation’s growth dynamics. The findings suggest that optimal management strategies, such as soil fertilization and fire prevention, are effective in avoiding biomass losses during the regeneration cycles of rapidly growing plantations.