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Pattern formation at the ecosystem level is a rapidly growing area of spatial ecology. The best studied example is vegetation stripes running along contours in semi-arid regions. Theoretical models are a widely used tool for studying these banded vegetation patterns, and one important model is the system of advection-diffusion equations proposed by Klausmeier. The present study is part of a series of papers whose objective is a comprehensive understanding of patterned solutions of the Klausmeier model. The author focuses on the region of parameter space in which the propagation speed of the patterns is close to its maximum possible value. Exploiting the large value of one of the model parameters, a leading order approximation is obtained for the maximum propagation speed, and the author undertakes a detailed investigation of the parameter region in which there are patterns with speeds close to this maximum.