Turing Pattern Formation in a Semiarid Vegetation Model with Fractional-in-Space Diffusion

A fractional power of the Laplacian is introduced to a reaction–diffusion system to describe water’s anomalous diffusion in a semiarid vegetation model. Our linear stability analysis shows that the wavenumber of Turing pattern increases with the superdiffusive exponent. A weakly nonlinear analysis yields a system of amplitude equations, and the analysis of these amplitude equations shows that the spatial patterns are asymptotic stable due to the supercritical Turing bifurcation. Numerical simulations exhibit a bistable regime composed of hexagons and stripes, which confirm our analytical results. Moreover, the characteristic length of the emergent spatial pattern is consistent with the scale of vegetation patterns observed in field studies.