  We study the effect of spatial structure, genetic drift, mutation, and
selective pressure on the evolutionary dynamics in a simplified model of
asexual organisms colonizing a new territory. Under an appropriate
coarse-graining, the evolutionary dynamics is related to the directed
percolation processes that arise in voter models, the Domany-Kinzel (DK) model,
contact process, etc. We explore the differences between linear (flat front)
expansions and the much less familiar radial (curved front) range expansions.
For the radial expansion, we develop a generalized, off-lattice DK model that
minimizes otherwise persistent lattice artifacts. With both simulations and
analytical techniques, we study the survival probability of advantageous
mutants, the spatial correlations between domains of neutral strains, and the
dynamics of populations with deleterious mutations. "Inflation" at the frontier
leads to striking differences between radial and linear expansions. For a
colony with initial radius $R_0$ expanding at velocity $v$, significant genetic
demixing, caused by local genetic drift, only occurs up to a finite time $t^* =
R_0/v$, after which portions of the colony become causally disconnected due to
the inflating perimeter of the expanding front. As a result, the effect of a
selective advantage is amplified relative to genetic drift, increasing the
survival probability of advantageous mutants. Inflation also modifies the
underlying directed percolation transition, introducing novel scaling functions
and modifications similar to a finite size effect. Finally, we consider radial
range expansions with deflating perimeters, as might arise from colonization
initiated along the shores of an island.
