Symmetry breaking and coarsening in spatially distributed evolutionary processes including sexual reproduction and disruptive selection

Sexual reproduction presents significant challenges to formal treatment of evolutionary processes. A starting point for systematic treatments of ecological and evolutionary phenomena has been provided by the gene-centered view of evolution which assigns effective fitness to each allele instead of each organism. The gene-centered view can be formalized as a dynamic mean-field approximation applied to genes in reproduction and selection dynamics. We show that the gene-centered view breaks down for symmetry breaking and pattern formation within a population and show that spatial distributions of organisms with local mating neighborhoods in the presence of disruptive selection give rise to such symmetry breaking and pattern formation in the genetic composition of local populations. Global dynamics follows conventional coarsening of systems with nonconserved order parameters. The results have significant implications for the ecology of genetic diversity and species formation.