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

Models for detecting the effect of adaptation on population genomic diversity are often predicated on a single newly arisen mutation sweeping rapidly to fixation. However, a population can also adapt to a new environment by multiple mutations of similar phenotypic effect that arise in parallel, at the same locus or different loci. These mutations can each quickly reach intermediate frequency, preventing any single one from rapidly sweeping to fixation globally, leading to a "soft" sweep in the population. Here we study various models of parallel mutation in a continuous, geographically spread population adapting to a global selection pressure. The slow geographic spread of a selected allele due to limited dispersal can allow other selected alleles to arise and start to spread elsewhere in the species range. When these different selected alleles meet, their spread can slow dramatically and so initially form a geographic patchwork, a random tessellation, which could be mistaken for a signal of local adaptation. This spatial tessellation will dissipate over time due to mixing by migration, leaving a set of partial sweeps within the global population. We show that the spatial tessellation initially formed by mutational types is closely connected to Poisson process models of crystallization, which we extend. We find that the probability of parallel mutation and the spatial scale on which parallel mutation occurs are captured by a single compound parameter, a characteristic length, which reflects the expected distance a spreading allele travels before it encounters a different spreading allele. This characteristic length depends on the mutation rate, the dispersal parameter, the effective local density of individuals, and to a much lesser extent the strength of selection. While our knowledge of these parameters is poor, we argue that even in widely dispersing species, such parallel geographic sweeps may be surprisingly common. Thus, we predict that as more data become available, many more examples of intraspecies parallel adaptation will be uncovered.