Selection-recombination-mutation dynamics: Gradient, limit cycle, and closed invariant curve.

Abstract

In this paper, the replicator dynamics of the two-locus two-allele system under weak mutation and weak selection is investigated in a generation-wise nonoverlapping unstructured population of individuals mating at random. Our main finding is that the dynamics is gradient-like when the point mutations at the two loci are independent. This is in stark contrast to the case of one-locus-multi-allele where the existence gradient behavior is contingent on a specific relationship between the mutation rates. When the mutations are not independent in the two-locus-two-allele system, there is the possibility of nonconvergent outcomes, like asymptotically stable oscillations, through either the Hopf bifurcation or the Neimark-Sacker bifurcation depending on the strength of the weak selection. The results can be straightforwardly extended for multilocus-two-allele systems.