Effective sizes for subdivided populations

Many derivations of effective population sizes have been suggested in the literature; however, few account for the breeding structure and none can readily be expanded to subdivided populations. Breeding structures influence gene correlations through their effects on the number of breeding individuals of each sex, the mean number of progeny per female, and the variance in the number of progeny produced by males and females. Additionally, hierarchical structuring in a population is determined by the number of breeding groups and the migration rates of males and females among such groups. This study derives analytical solutions for effective sizes that can be applied to subdivided populations. Parameters that encapsulate breeding structure and subdivision are utilized to derive the traditional inbreeding and variance effective sizes. Also, it is shown that effective sizes can be determined for any hierarchical level of population structure for which gene correlations can accrue. Derivations of effective sizes for the accumulation of gene correlations within breeding groups (coancestral effective size) and among breeding groups (intergroup effective size) are given. The results converge to traditional, single population measures when similar assumptions are applied. In particular, inbreeding and intergroup effective sizes are shown to be special cases of the coancestral effective size, and intergroup and variance effective sizes will be equal if the population census remains constant. Instantaneous solutions for effective sizes, at any time after gene correlation begins to accrue, are given in terms of traditional F statistics or transition equations. All effective sizes are shown to converge upon a common asymptotic value when breeding tactics and migration rates are constant. The asymptotic effective size can be expressed in terms of the fixation indices and the number of breeding groups; however, the rate of approach to the asymptote is dependent upon dispersal rates. For accurate assessment of effective sizes, initial, instantaneous or asymptotic, the expressions must be applied at the lowest levels at which migration among breeding groups is nonrandom. Thus, the expressions may be applicable to lineages within socially structured populations, fragmented populations (if random exchange of genes prevails within each population), or combinations of intra- and interpopulation discontinuities of gene flow. Failure to recognize internal structures of populations may lead to considerable overestimates of inbreeding effective size, while usually underestimating variance effective size.