Simulation studies have demonstrated that a variety of patterns in worldwide genetic variation are compatible with the trends predicted by a serial founder model, in which populations expand outward from an initial source via a process in which new populations contain only subsets of the genetic diversity present in their parental populations. Here, we provide analytical results for key quantities under the serial founder model, deriving distributions of coalescence times for pairs of lineages sampled either from the same population or from different populations. We use these distributions to obtain expectations for coalescence times and for homozygosity and heterozygosity values. A predicted approximate linear decline in expected heterozygosity with increasing distance from the source population reproduces a pattern that has been observed both in human genetic data and in simulations. Our formulas predict that populations close to the source location have lower between-population gene identity than populations far from the source, also mirroring results obtained from data and simulations. We show that different models that produce similar declining patterns in heterozygosity generate quite distinct patterns in coalescence-time distributions and gene identity measures, thereby providing a basis for distinguishing these models. We interpret the theoretical results in relation to their implications for human population genetics.
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