Traveling Stripes in the Klausmeier Model of Vegetation Pattern Formation

The Klausmeier equation is a widely studied reaction-diffusion-advection model of vegetation pattern formation on gently sloped terrain in semiarid ecosystems. We consider the case of constantly sloped terrain and study the formation of planar vegetation stripe patterns which align in the direction transverse to the slope and travel uphill. These patterns arise as solutions to an underlying traveling wave equation, which admits a separation of scales due to the fact that water flows downhill faster than the rate at which vegetation diffuses. We rigorously construct solutions corresponding to single vegetation stripes as well as long wavelength spatially periodic wave trains using geometric singular perturbation theory. Blow-up desingularization methods are needed to understand slow passage of solutions near a degenerate transcritical bifurcation. The underlying geometry of the traveling wave equation predicts relations between pattern wavelength, speed, and terrain slope.