  In evolutionary processes, population structure has a substantial effect on
natural selection. Here, we analyze how motion of individuals affects constant
selection in structured populations. Motion is relevant because it leads to
changes in the distribution of types as mutations march toward fixation or
extinction. We describe motion as the swapping of individuals on graphs, and
more generally as the shuffling of individuals between reproductive updates.
Beginning with a one-dimensional graph, the cycle, we prove that motion
suppresses natural selection for death-birth updating or for any process that
combines birth-death and death-birth updating. If the rule is purely
birth-death updating, no change in fixation probability appears in the presence
of motion. We further investigate how motion affects evolution on the square
lattice and weighted graphs. In the case of weighted graphs we find that motion
can be either an amplifier or a suppressor of natural selection. In some cases,
whether it is one or the other can be a function of the relative reproductive
rate, indicating that motion is a subtle and complex attribute of evolving
populations. As a first step towards understanding less restricted types of
motion in evolutionary graph theory, we consider a similar rule on dynamic
graphs induced by a spatial flow and find qualitatively similar results
indicating that continuous motion also suppresses natural selection.
