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.
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