  When beneficial mutations are relatively common, competition between multiple
unfixed mutations can reduce the rate of fixation in well-mixed asexual
populations. We introduce a one dimensional model with a steady accumulation of
beneficial mutations. We find a transition between periodic selection and
multiple-mutation regimes. In the multiple-mutation regime, the increase of
fitness along the lattice bears a striking similarity to surface growth
phenomena, with power law growth and saturation of the interface width. We also
find significant differences compared to the well-mixed model. In our lattice
model, the transition between regimes happens at a much lower mutation rate due
to slower fixation times in one dimension. Also the rate of fixation is reduced
with increasing mutation rate due to the more intense competition, and it
saturates with large population size.
