  For most organisms with viscous population structure, spatially localized
growth drives the invasive advance of a favorable mutation. We model a
two-allele competition where recurrent mutation introduces a genotype with a
rate of local propagation exceeding the resident's rate. We capture
ecologically important properties of the rare invader's stochastic dynamics by
assuming discrete individuals and neighborhood interactions. To understand how
individual-level processes may govern population patterns, we invoke the
physical theory for nucleation of spatial systems. Nucleation theory
discriminates between single-cluster and multi-cluster dynamics. A sufficiently
low mutation rate, or a sufficiently small environment, generates
single-cluster dynamics, an inherently stochastic process; a favorable mutation
advances only if the invader cluster reaches a critical radius. For this mode
of invasion we identify the probability distribution of waiting times until the
favored allele advances to competitive dominance, and we ask how the critical
cluster size varies as propagation or mortality rates vary. Increasing the
mutation rate or system size generates multi-cluster invasion, where spatial
averaging produces nearly deterministic global dynamics. For this process, an
analytical approximation from nucleation theory, called Avrami's Law, describes
the time-dependent behavior of the genotype densities with remarkable accuracy.
