An Analytical and Numerical Approach to Vegetation–Nutrient Dynamics Involving Symbiotic Nitrogen Fixation and the Allee Effect

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DOI: https://doi.org/10.28924/2291-8639-23-2025-243
Published: Oct 8, 2025
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Shaher Momani, Iqbal M. Batiha, Ahmed Bouchenak, Koichi Unami, An Analytical and Numerical Approach to Vegetation–Nutrient Dynamics Involving Symbiotic Nitrogen Fixation and the Allee Effect, Int. J. Anal. Appl., 23 (2025), 243.

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Shaher Momani, Iqbal M. Batiha, Ahmed Bouchenak, Koichi Unami

Abstract

This study develops a nonlinear dynamical system with three variables representing the standardized abundances of a leguminous plant, a non-leguminous plant, and a nutrient resource. The system is governed by a set of nonlinear conformable fractional differential equations without delay. We establish the unique existence of solutions, derive analytical results for special cases, and present a numerical investigation of the agroecological dynamics. The model is grounded in field experiments involving Trifolium repens (clover) and Mentha × piperita (mint) cultivated on both flat and sloped plots. Stable isotope analysis was conducted to semi-quantitatively trace the fate of nitrogen atoms, accounting for isotope fractionation, and to validate the model structure. Leaf area was quantified via image processing, and photometric analysis of soil pore water was used to determine the concentrations of clover, mint, and nitrate-nitrogen. Using repeated least squares estimation, model parameters were calibrated based on observed and normalized variable values. The results indicate a pronounced Allee effect in the interaction between clover and the soil environment.

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