modeling and analysis of deforestation and pollution dynamics induced by industrialization using the fractal-fractional atangana-baleanu derivative

This study presents a fractal-fractional model in the atangana–baleanu sense to investigate the dynamics of deforestation and pollution driven by industrialization. the model is analyzed for positivity and boundedness, and the existence and uniqueness of its solution are established using fixed-point theory. the system's equilibrium points are identified, and the threshold parameter  $mathfrak{r_0}$ is determined, with local asymptotic stability confirmed for all equilibria. sensitivity analysis highlights the key parameters influencing $mathfrak{r_0}$, while ulam–hyers stability ensures robustness of the solution. lagrangian polynomial interpolation is employed to approximate the solution, and phase portraits along with numerical simulations in matlab illustrate the model's dynamic behavior. the results demonstrate that the fractal-fractional approach provides a comprehensive framework for capturing complex environmental interactions, offering valuable insights into the effects of industrialization on deforestation and pollution.