Vegetation Pattern Formation via Energy-Balance-Constrained Modeling

TL;DR

This paper introduces an energy-balance-constrained model for vegetation pattern formation in semi-arid environments, capturing key spatial patterns and dynamics with fewer assumptions.

Contribution

It develops a novel modeling framework based on energy and water conservation principles, leading to a family of semilinear equations that explain vegetation patterns.

Findings

Model reproduces observed increase in pattern wavelength with aridity.

Vegetation bands migrate uphill on slopes, matching empirical data.

Numerical simulations confirm linear stability analysis predictions.

Abstract

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.