Pattern solutions of the Klausmeier model for banded vegetation in semiarid environments V: the transition from patterns to desert

Vegetation in semideserts often self-organizes into spatial patterns. On gentle slopes, these typically consist of stripes of vegetation running parallel to the contours, separated by stripes of bare ground. The Klausmeier model is one of the oldest and most established of a number of mathematical models for this "banded vegetation." The model is a system of reaction-diffusion-advection equations. Under the standard nondimensionalization, one of its dimensionless parameters (v) reflects the relative rates of water flow downhill and plant dispersal and is therefore very large. This paper is the fifth and last in a series in which the author provides a detailed analytical understanding of the existence and form of pattern solutions (periodic travelling waves) of the Klausmeier model, to leading order as v -> infinity. The problem is a very rich one because the underlying mathematics depends fundamentally on the way in which the migration speed c scales with.. This paper concerns the case 1 - O(c) and c - o(v(1/2)) as v -> infinity. The author derives leading order expressions for the curves bounding the parameter region giving patterns, and for the pattern forms in this region. An important consequence of this is leading order formulae for the maximum and minimum rainfall levels for which patterns exist. The author demonstrates via numerical simulations that a decrease in rainfall through the minimum level for patterns causes a transition to full-blown desert that cannot be reversed by increasing the rainfall again.