Effect of Non-local Grazing on Dry-land Vegetation Dynamics

The paper derives the same linearization matrix L(k) and dispersion relation, establishes stability of the bare state for all k, defines and computes the Turing prediction region, and reports no Hopf-onset in ecologically relevant regimes; all of this aligns with the candidate solution’s steps and formulas. The model adds mild analytic refinements (e.g., T(k) ≤ T(0) under G′ ≤ 0; a monotonicity argument for aT(σ)) that are consistent with, though not proved in, the paper. No core contradictions were found. Key overlaps include the symbol L(k) and dispersion relation (equations (19)–(22)), stability of B via diagonal negativity at neq=0, the explicit transforms for the cut-off Gaussian and uniform kernels, the definition and use of the Turing prediction region, and the observation that as σ increases the behavior approaches the uniform-kernel case; all of these elements match the candidate solution’s derivations and procedure. Citations: L(k) and dispersion relation (, ), stability of B (homogeneous and heterogeneous) (), kernel Fourier transforms and w=L convention (, ), Turing prediction region and lack of Hopf at onset (, , ), convergence toward uniform kernel as σ grows and empirical decrease of aT with σ (), and wavelength selection near aT ().