Regular vegetation patterns appear on aerial photographs as an alternation of densely and sparsely populated zones. This spatial organization of the vegetation is an endogeneous phenomenon which is not restricted to specific kinds of plants or soils; it is a characteristic landscape of many arid regions throughout the world. Simple non-local Verhulst-Fisher models and amplitude equations will be presented which allow to explain the formation of these structures in terms of an interplay between short range cooperative interactions controlling plant reproduction and long range self-inhibitory interactions originating from plant competition for environmental resources. Isotropic as well as anisotropic environmental conditions will be discussed. We show that vegetation stripes orient themselves in the direction parallel or perpendicular with respect to a direction of anisotropy depending on whether this anisotropy influences the interactions favoring or inhibiting plant reproduction; furthermore, ground curvature is not a necessary condition for the appearance of arcuate vegetation patterns. In agreement with {\it in situ} observations, we find that the width of vegetated bands increases when environmental conditions get more arid and that patterns formed of stripes orientated parallel to the direction of a slope are static, while patterns which are perpendicular to this direction exhibit an upslope motion. Moving front "interfacial instabilities" are predicted.

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1998-1999 Mathematics in Biology