  We analyze a replicator-mutator model arising in the context of directed
evolution [23], where the selection term is modulated over time by the
mean-fitness. We combine a Cumulant Generating Function approach [13] and a
spatio-temporal rescaling related to the Avron-Herbst formula [1] to give of a
complete picture of the Cauchy problem. Besides its well-posedness, we provide
an implicit/explicit expression of the solution, and analyze its large time
behaviour. As a by product, we also solve a replicator-mutator model where the
mutation coefficient is socially determined, in the sense that it is modulated
by the mean-fitness. The latter model reveals concentration or anti
diffusion/diffusion phenomena.
