  Both evolution and ecology have long been concerned with the impact of
variable environmental conditions on observed levels of genetic diversity
within and between species. We model the evolution of a quantitative trait
under selection that fluctuates in space and time, and derive an analytical
condition for when these fluctuations promote genetic diversification. As
ecological scenario we use a generalized island model with soft selection
within patches in which we incorporate generation overlap. We allow for
arbitrary fluctuations in the environment including spatio-temporal
correlations and any functional form of selection on the trait. Using the
concepts of invasion fitness and evolutionary branching, we derive a simple and
transparent condition for the adaptive evolution and maintenance of genetic
diversity. This condition relates the strength of selection within patches to
expectations and variances in the environmental conditions across space and
time. Our results unify, clarify, and extend a number of previous results on
the evolution and maintenance of genetic variation under fluctuating selection.
Individual-based simulations show that our results are independent of the
details of the genetic architecture and on whether reproduction is clonal or
sexual. The onset of increased genetic variance is predicted accurately also in
small populations in which alleles can go extinct due to environmental
stochasticity.
