  The spatial structure of an evolving population affects which mutations
become fixed. Some structures amplify selection, increasing the likelihood that
beneficial mutations become fixed while deleterious mutations do not. Other
structures suppress selection, reducing the effect of fitness differences and
increasing the role of random chance. This phenomenon can be modeled by
representing spatial structure as a graph, with individuals occupying vertices.
Births and deaths occur stochastically, according to a specified update rule.
We study death-Birth updating: An individual is chosen to die and then its
neighbors compete to reproduce into the vacant spot. Previous numerical
experiments suggested that amplifiers of selection for this process are either
rare or nonexistent. We introduce a perturbative method for this problem for
weak selection regime, meaning that mutations have small fitness effects. We
show that fixation probability under weak selection can be calculated in terms
of the coalescence times of random walks. This result leads naturally to a new
definition of effective population size. Using this and other methods, we
uncover the first known examples of transient amplifiers of selection (graphs
that amplify selection for a particular range of fitness values) for the
death-Birth process. We also exhibit new families of "reducers of fixation",
which decrease the fixation probability of all mutations, whether beneficial or
deleterious.
