  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
non-overlapping 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 behaviour 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
non-convergent 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 multi-locus two-allele systems.
