  In evolutionary dynamics, a key measure of a mutant trait's success is the
probability that it takes over the population given some initial
mutant-appearance distribution. This "fixation probability" is difficult to
compute in general, as it depends on the mutation's effect on the organism as
well as the population's spatial structure, mating patterns, and other factors.
In this study, we consider weak selection, which means that the mutation's
effect on the organism is small. We obtain a weak-selection perturbation
expansion of a mutant's fixation probability, from an arbitrary initial
configuration of mutant and resident types. Our results apply to a broad class
of stochastic evolutionary models, in which the size and spatial structure are
arbitrary (but fixed). The problem of whether selection favors a given trait is
thereby reduced from exponential to polynomial complexity in the population
size, when selection is weak. We conclude by applying these methods to obtain
new results for evolutionary dynamics on graphs.
