Supercritical and subcritical Turing pattern formation in a hyperbolic vegetation model for flat arid environments

Patterned vegetation dynamics in flat arid environments are investigated in the framework of a hyperbolic modified Klausmeier model. In particular, this study aims at elucidating how the properties exhibited by supercritical and subcritical patterns during the transient regime are affected by the inertial times. The present work encloses linear stability analysis to deduce the threshold condition for Turing pattern formation and weakly nonlinear analysis to describe the time evolution of the pattern amplitude close to the instability threshold. In our analysis, we consider the case in which the emerging patterns do not have any spatial structure, as it is typically assumed in small domains, as well as the scenario in which patterns form sequentially and propagate over a large domain in the form of a traveling wavefront. The hyperbolic model is also integrated numerically to validate the analytical predictions and to characterize more deeply the spatio-temporal evolution of vegetation patterns in both supercritical and subcritical regime.