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Vegetation patterns, a natural phenomenon observed worldwide, are typically driven by spatially distributed feedback. However, the spatial colonization mechanisms of clonal plants, driven by the growth of a rhizome, are usually not considered in prototypical models. Here we propose a general equation for the vegetation density that includes all main clonal-growth features as well as the essential ingredients leading to spatial self-organization. This generic model reproduces the phase diagram of a fully detailed model of clonal growth. The relation of each term of the model with the mechanisms of clonal growth is discussed.