Thomas LoFaro and Richard Gomulkiewicz Washington State University We consider the spread of a rare favorable allele in a "black-hole sink" population sustainable by recurrent immigration from a source population. It was shown by Gomulkiewicz et. al. that in the case of periodic population dynamics, the geometric mean fitness determines the spread of a rare favorable allele. We use ergodic theory to generalize this result to population dynamics that may or may not be periodic.