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Our models for detecting the effect of adaptation on population genomicdiversity are often predicated on a single newly arisen mutation sweepingrapidly to fixation. However, a population can also adapt to a new situation bymultiple mutations of similar phenotypic effect that arise in parallel. Thesemutations can each quickly reach intermediate frequency, preventing any singleone from rapidly sweeping to fixation globally (a "soft" sweep). Here we studymodels of parallel mutation in a geographically spread population adapting to aglobal selection pressure. The slow geographic spread of a selected allele canallow other selected alleles to arise and spread elsewhere in the speciesrange. When these different selected alleles meet, their spread can slowdramatically, and so form a geographic patchwork which could be mistaken for asignal of local adaptation. This random spatial tessellation will dissipateover time due to mixing by migration, leaving a set of partial sweeps withinthe global population. We show that the spatial tessellation initially formedby mutational types is closely connected to Poisson process models ofcrystallization, which we extend. We find that the probability of parallelmutation and the spatial scale on which parallel mutation occurs is captured bya single characteristic length that reflects the expected distance a spreadingallele travels before it encounters a different spreading allele. Thischaracteristic length depends on the mutation rate, the dispersal parameter,the effective local density of individuals, and to a much lesser extent thestrength of selection. We argue that even in widely dispersing species, suchparallel geographic sweeps may be surprisingly common. Thus, we predict, asmore data becomes available, many more examples of intra-species paralleladaptation will be uncovered.

Parallel adaptation: One or many waves of advance of an advantageous allele?