Pattern solutions of the Klausmeier model for banded vegetation in semi-arid environments III: The transition between homoclinic solutions

Self-organised patterns of vegetation are a characteristic feature of semi-deserts. On hillsides, these typically comprise vegetation bands running parallel to the contours, separated by regions of bare ground (“tiger bush”). The present study concerns the Klausmeier mathematical model for this phenomenon [C.A. Klausmeier, Regular and irregular patterns in semiarid vegetation, Science 284 (1999) 1826–1828], which is one of the earliest and most influential of the various theoretical models for banded vegetation. The model is a system of reaction–diffusion–advection equations, and after rescaling it contains three dimensionless parameters, one of which (the slope parameter) is much larger than the other two. The present study is the third in a series of papers in which the author exploits the large value of the slope parameter to obtain leading order approximations to the parameter regions in which patterns exist, and to the form of these patterns. The boundary of the parameter region giving patterns consists in part of two loci of homoclinic solutions, that are homoclinic to different steady states. The present paper concerns behaviour for parameters close to the intersection point of these loci. The author shows that this part of parameter space divides naturally into three regions, with a different solution structure in each. In one region, the solution corresponds to a limit cycle of a reduced system of ordinary differential equations; the other two regions involve multiple matched layers. As part of the analysis, the author derives formulae for the homoclinic solution loci, and for the location of their intersection. All of the results are valid to leading order for large values of the slope parameter. The author presents a detailed numerical verification of his analytical results. The paper concludes with discussions of the ecological implications of the results, and the main outstanding mathematical questions.

► This is the third in a series of papers on the existence and form of patterns.
► Leading order approximations are obtained for large values of the slope parameter.
► The focus is on parameters near the transition between two different homoclinic loci.
► Formulae are derived for the limit cycle loci, and their intersection point.
► The author presents a detailed numerical verification of his analytical results.