Pattern formation and modulation in a hyperbolic vegetation model for semiarid environments

Abstract

This paper presents analytical and numerical investigations of pattern formation and modulation in a one-dimensional hyperbolic extension of the Klausmeier reaction–diffusion–advection vegetation model for semiarid environments. The hyperbolicity of the model is shown to provide an additional degree of freedom which can be used to modify the dynamical response of the system away from the parabolic limit as well as to reproduce qualitatively the experimental data. Results of linear stability analysis reveal the possibility to modulate, almost separately, the main pattern features (wavelength and uphill migration speed) by acting on two independent parameters: the slope steepness and the characteristic velocity. Moreover, a possible justification of the mechanism underlying the uphill migration of the patterns is suggested in terms of the phase shift between the spatial distributions of vegetation biomass and water. The hyperbolicity of the system has also facilitated the search for a class of exact solutions, via the differential constraint method, for a hyperbolic vegetation model whose kinetic terms are more general than Klausmeier’s ones and whose applications are potentially broad.