Modeling of fractional order DPG model insight global warming and pollution effect on desertification for control mechanism.

This study presents a novel fractional-order mathematical model that investigates the dynamic interplay between dust pollutants, plant biomass, and global warming, referred to as the DPG system. This research introduces a fractional-order formulation that incorporates ecological memory and long-range interactions, providing a more realistic representation of desertification dynamics than classical integer-order models. It also establishes a comprehensive analytical numerical framework designed to capture system feedback, assess instability patterns, and evaluate the effectiveness of chaos control mechanisms. The model utilizes Caputo derivatives to capture the memory effects inherent in ecological and atmospheric processes. Key parameters such as dust emission, plant decay, and global warming feedback mechanisms are integrated into a nonlinear system of differential equations. Analytical evaluations ensure the existence, uniqueness, and generalized Hyers-Ulam-Rassias stability of the proposed system. Sensitivity analysis identifies the parameters that have the most significant influence on desertification risk. Furthermore, a Newton polynomial-based numerical scheme is constructed to efficiently simulate system behavior under varying fractional orders. Chaos control strategies are implemented to stabilize the system near critical equilibrium points. Numerical simulations reveal that lower fractional orders dampen dust accumulation, slow plant biomass regeneration, and delay global warming trends, highlighting the efficacy of fractional modeling in capturing real-world environmental inertia and feedback. This research provides a robust analytical and computational foundation for understanding ecosystem resilience in the face of both anthropogenic and climatic stressors.