  Microbial populations often have complex spatial structures, with homogeneous
competition holding only at a local scale. Population structure can strongly
impact evolution, in particular by affecting the fixation probability of
mutants. Here, we propose a model of structured microbial populations on
graphs, where each node of the graph contains a well-mixed deme whose size can
fluctuate, and where migrations are independent from birth and death events. We
study analytically and numerically the mutant fixation probabilities in
different structures, in the rare migration regime. In particular, we
demonstrate that the star graph continuously transitions between amplifying and
suppressing natural selection as migration rate asymmetry is varied. This
elucidates the apparent paradox in existing constant-size models on graphs,
where the star is an amplifier or a suppressor depending on the details of the
dynamics or update rule chosen, e.g. whether each birth event precedes or
follows a death event. The celebrated amplification property of the star graph
for large populations is preserved in our model, for specific migration
asymmetry. We further demonstrate a general mapping between our model and
constant-size models on graphs, under a constraint on migration rates, which
directly stems from assuming constant size. By lifting this constraint, our
model reconciles and generalizes previous results, showing that migration rate
asymmetry is key to determining whether a given population structure amplifies
or suppresses natural selection.
