  We demonstrate a vast expansion of the theory of evolutionary stability to
finite populations with mutation, connecting the theory of the stationary
distribution of the Moran process with the Lyapunov theory of evolutionary
stability. We define the notion of stationary stability for the Moran process
with mutation and generalizations, as well as a generalized notion of
evolutionary stability that includes mutation called an incentive stable state
(ISS) candidate. For sufficiently large populations, extrema of the stationary
distribution are ISS candidates and we give a family of Lyapunov quantities
that are locally minimized at the stationary extrema and at ISS candidates. In
various examples, including for the Moran and Wright-Fisher processes, we show
that the local maxima of the stationary distribution capture the
traditionally-defined evolutionarily stable states. The classical stability
theory of the replicator dynamic is recovered in the large population limit.
Finally we include descriptions of possible extensions to populations of
variable size and populations evolving on graphs.
