Pattern selection and biomass shifts in a vegetation model with nonlocal competition and Allee activation

Abstract

We study a single-species vegetation model in which a strong Allee effect acts locally and competition acts through a nonlocal kernel, and we describe the kernel with the one-parameter family of q-Gaussians, which goes from compact support to power-law tails. The local activation introduced by the Allee effect allows a finite-wavelength instability even when the Fourier transform of the kernel is positive, so Gaussian and heavy-tailed kernels can produce patterns. We derive in closed form the coefficients of the hexagonal amplitude equations and verify them against independent numerical steady states. The quadratic coefficient $Q$ results from the balance between the curvature of the local growth and the competitive inhibition at the pattern scale, and at onset its sign depends only on the fraction of the local activation that diffusion removes at the critical wavenumber. The same coefficient controls the morphology, since spots, labyrinths and gaps appear with pairwise competition only as $Q$ changes sign, and the amplitude equations predict the choice between hexagons and stripes in every simulated state in which the cubic truncation holds. It also controls the total biomass, which increases for spots and decreases for gaps.