  If deleterious mutations near a fitness maximum in a spatially distributed
population are sufficiently frequent or detrimental, the population can undergo
a fitness collapse, similarly to the Muller's ratchet effect in well-mixed
populations. Recent studies of one-dimensional habitats (e.g., the frontier of
a two-dimensional range expansion) have shown that the onset of the fitness
collapse is described by a directed percolation phase transition with its
associated critical exponents. We consider population fitness collapse in
three-dimensional range expansions with both inflating and fixed-size frontiers
(applicable to, e.g., expanding and treadmilling spherical tumors,
respectively). We find that the onset of fitness collapse in these two cases
obeys different scaling laws, and that competition between species at the
frontier leads to a deviation from directed percolation scaling. As in
two-dimensional range expansions, inflating frontiers modify the critical
behavior by causally disconnecting well-separated portions of the population.
