Efficacy of stochastic and fractional approaches in modeling air, forest and water systems in India

Abstract This study investigates the coupled dynamics of population growth, industrialization, forest-based carbon sequestration, air quality and river water pH levels in India using a unified mathematical framework validated with real-world data. Two complementary modeling approaches are employed: a fractional-order deterministic model to capture memory effects, and a stochastic model to account for environmental variability. The mathematical well-posedness of both systems is established, and numerical simulations show strong agreement with observed data and are used to predict key environmental indicators. The results reveal that increased forest coverage, together with regulated population and industrial growth, significantly improves air quality and raises the average minimum pH levels of major rivers, indicating reduced water acidity. Consequently, forest conservation and expansion emerge as vital strategies for sustainable development and environmental stability. A comparative evaluation with a long short-term memory (LSTM) recurrent neural network further supports the predictive capability of the proposed models.