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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2604.03031 |
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| _version_ | 1866914443074469888 |
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| author | Topaz, Chad M. |
| author_facet | Topaz, Chad M. |
| contents | Vegetation in semi-arid environments self-organizes into striking spatial patterns -- bands, spots, labyrinths, and gaps -- with characteristic wavelengths on the order of tens to hundreds of meters. Existing reaction-diffusion models postulate nonlinearities and transport laws from qualitative physical reasoning, making it hard to distinguish essential structural features from artifacts of the chosen forms. Here we show how energy-balance and water-conservation principles can constrain the admissible model class before a specific closure is chosen. These constraints motivate a family of semilinear closures; an Euler--Lagrange representative yields a fourth-order vegetation equation coupled to quasi-steady water transport on a one-dimensional hillslope. Linear stability analysis identifies three instability mechanisms: classical water-mediated feedback, energy-balance spatial coupling, and water deflection by vegetation gradients. Their balance depends on terrain geometry. On slopes, the water-mediated coupling dominates and the model reproduces two empirical observations: pattern wavelength increases with aridity, and vegetation bands migrate uphill. On flat terrain, the energy-balance spatial coupling can drive instability independently. Numerical simulations confirm the linear predictions, and exploratory continuation reveals a narrow hysteresis region consistent with subcritical bifurcation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_03031 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Vegetation Pattern Formation via Energy-Balance-Constrained Modeling Topaz, Chad M. Pattern Formation and Solitons Vegetation in semi-arid environments self-organizes into striking spatial patterns -- bands, spots, labyrinths, and gaps -- with characteristic wavelengths on the order of tens to hundreds of meters. Existing reaction-diffusion models postulate nonlinearities and transport laws from qualitative physical reasoning, making it hard to distinguish essential structural features from artifacts of the chosen forms. Here we show how energy-balance and water-conservation principles can constrain the admissible model class before a specific closure is chosen. These constraints motivate a family of semilinear closures; an Euler--Lagrange representative yields a fourth-order vegetation equation coupled to quasi-steady water transport on a one-dimensional hillslope. Linear stability analysis identifies three instability mechanisms: classical water-mediated feedback, energy-balance spatial coupling, and water deflection by vegetation gradients. Their balance depends on terrain geometry. On slopes, the water-mediated coupling dominates and the model reproduces two empirical observations: pattern wavelength increases with aridity, and vegetation bands migrate uphill. On flat terrain, the energy-balance spatial coupling can drive instability independently. Numerical simulations confirm the linear predictions, and exploratory continuation reveals a narrow hysteresis region consistent with subcritical bifurcation. |
| title | Vegetation Pattern Formation via Energy-Balance-Constrained Modeling |
| topic | Pattern Formation and Solitons |
| url | https://arxiv.org/abs/2604.03031 |