Numerical Simulations of a Fractional-Order Vegetation Pattern Model Driven by Water-Toxicity Interactions
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Vegetation patterns in arid ecosystems emerge from complex feedback between water availability and soil toxicity, yet their dynamics under fractional-order derivatives remain poorly understood. In this study, we extend the integer-order model proposed by [1] by reformulating it using fractional derivatives. The model consists of three coupled reaction-diffusion partial differential equations (PDEs) governing biomass, water, and toxicity dynamics, enabling us to investigate how water-toxicity interactions drive vegetation patterning. Through numerical simulations, we explore the formation of self-organized spatial structures, including spots, stripes, and gaps, across varying fractional derivation indices. Our results demonstrate that non-local memory effects, intrinsic to fractional operators, alter pattern formation compared to classical integer-order models. These findings advance the understanding of fractional-order dynamics in plant-soil feedback systems and highlight their ecological implications for arid ecosystems under environmental stress. By integrating fractional calculus with ecological modeling, this work provides a novel mathematical framework to predict vegetation resilience in water-limited ecosystems.