Turing patterns in hyperbolic reaction-transport vegetation models with cross-diffusion

In this paper, the pattern formation process in arid environments on flat terrains is investigated. In particular, a class of one-dimensional hyperbolic reaction-transport vegetation model with a cross-diffusion term accounting for plant roots’ suction in the soil water diffusion feedback is considered. To characterize the emerging Turing patterns, linear stability analysis on the uniform steady states is first addressed. Then, multiple-scale weakly nonlinear analysis is performed to describe the time evolution of the pattern amplitude close to the stability threshold. Finally, to validate analytical predictions, a modified Klausmeier model which takes also into account the internal competition rate is studied. The effects of the inertial times as well as the cross-diffusion and the internal competition rate are illustrated both analytically and numerically.