Lyapunov functional status of the energy-balance score functional for the coupled vegetation–water system

Establish whether the score functional E[u] = ∫ (μ U − 1/2 G(U,W)^2) dX, where G(U,W) = −Γ + U − 1/2 U^2 − U W + η(Λ1 U_X + Λ2 U_{XX}), is a Lyapunov functional for the coupled vegetation–water partial differential equation system defined by the Euler–Lagrange semilinear closure U_T = μ − G q_G + ηΛ1 G_X − ηΛ2 G_{XX} with q_G = 1 − U − W and the quasi-steady water equation ∂_X[ν(1 − χU) W − 𝒟 W_X − Δ W U_X] = ρ − (1 + β U) W. In other words, prove or refute that dE/dT ≤ 0 along solutions of this coupled system (with parameters and variables as defined in the model), thereby establishing whether E acts as a Lyapunov functional.

Background

The paper introduces a constrained modeling framework for dryland vegetation patterns based on energy-balance and water-conservation principles. A semilinear Euler–Lagrange closure is adopted for vegetation, leading to a fourth-order PDE for vegetation density U coupled to a quasi-steady water field W through a conservation law.

Although the vegetation equation is derived from a variational score functional, the authors emphasize that the coupled system is not a single-field gradient flow because water is treated quasi-steadily and breaks self-adjointness—an essential feature for traveling bands. The authors therefore pose the unresolved question of whether the score functional can nevertheless serve as a Lyapunov functional for the full coupled dynamics.