The opposing requirements in an evolving population for a rapid rate of multiple gene substitution and for the maintenance of normal population size can be reconciled in a variety of ways. The ways out of the impasse suggested here invoke deviations from the usual assumption of a large continuous population with constant numbers. In a colonial population system there may be significant random fluctuations in the accidental mortality rate between different colonies; and those colonies with reduced numbers of accidental deaths could tolerate the larger number of selective deaths that go hand in hand with rapid evolution. In new daughter colonies founded by one or a few colonizing individuals from a large polymorphic ancestral population, some genes may reach complete fixation in one or a few generations, without the usual concomitant selective cost. Or, in the same setup, the favored alleles may change by chance from rare to moderately common, but not to complete fixation, during the founding of some daughter colonies; and this raises the allele frequencies above the low range, where the cost of selection is greatest, so that the cost of further selective changes is bearable.