  We review and extend results for mutation, selection, genetic drift, and
migration in a one-dimensional continuous population. The population is
described by a continuous limit of the stepping stone model, which leads to the
stochastic Fisher-Kolmogorov-Petrovsky-Piscounov equation with additional terms
describing mutations. Although the stepping stone model was first proposed for
population genetics, it is closely related to "voter models" of interest in
nonequilibrium statistical mechanics. The stepping stone model can also be
regarded as an approximation to the dynamics of a thin layer of actively
growing pioneers at the frontier of a colony of microorganisms undergoing a
range expansion on a Petri dish. We find that the population tends to segregate
into monoallelic domains. This segregation slows down genetic drift and
selection because these two evolutionary forces can only act at the boundaries
between the domains; the effects of mutation, however, are not significantly
affected by the segregation. Although fixation in the neutral well-mixed (or
"zero dimensional") model occurs exponentially in time, it occurs only
algebraically fast in the one-dimensional model. If selection is weak,
selective sweeps occur exponentially fast in both well-mixed and
one-dimensional populations, but the time constants are different. We also find
an unusual sublinear increase in the variance of the spatially averaged allele
frequency with time. Although we focus on two alleles or variants, q-allele
Potts-like models of gene segregation are considered as well. We also
investigate the effects of geometry at the frontier by considering growth of
circular colonies. Our analytical results are checked with simulations, and
could be tested against recent spatial experiments on range expansions off
linear inoculations of Escherichia coli and Saccharomyces cerevisiae.
