Control strategies for fractional order soil micro plastic pollution model and preserving nutrient cycle integrity

Microplastic pollution presents a formidable challenge to agricultural ecosystems, imperiling soil quality and fertility with far-reaching implications. In this study, we pioneer an innovative approach by employing the Caputo fractal dimension and fractional order modeling framework to scrutinize the intricate dynamics of soil pollution induced by microplastics. Our model encompasses four pivotal compartments: Microplastic (P(t)), Soil Pollution (S(t)), Remediation or Recycling (R(t)), and Soil Nutrient Cycle (N(t)). Through meticulous qualitative analysis, we meticulously characterize steady states, distinguishing between locally stable, globally stable, and Hyer Ulam stable states. Notably, we identify equilibrium points manifesting as pollution-free and pollution-extinct states, laying the groundwork for effective intervention strategies. To address this pressing environmental concern, we propose two robust control strategies ( $${\mathcal{U}}_{1} (t)$$ and $${\mathcal{U}}_{2}$$ (t)) meticulously crafted to curtail soil pollution. We discern nuanced variations in Basic Reproduction Numbers through comprehensive sensitivity analyses, offering critical insights into the multifaceted dynamics of microplastic contamination in agricultural soils. Employing Newton's Polynomial two step method, we derive precise numerical approximations, facilitating nuanced simulations across varying orders $$(0\le \alpha \le 1)$$ and dimensions. Our findings not only deepen our understanding of microplastic pollution dynamics but also underscore the urgent need for proactive interventions. In conclusion, our study contributes a novel synthesis of fractal fractional order modeling and tailored control strategies, providing a sophisticated framework for mitigating microplastic pollution in agricultural soils. Beyond merely elucidating the problem, our research offers actionable insights and pragmatic solutions, thereby advancing the discourse on sustainable soil management and environmental stewardship in a rapidly evolving landscape.