Spatial Autocorrelation for Subdivided Populations with Invariant Migration Schemes

For populations with geographic substructure and selectively neutral genetic data, the short term dynamics is a balance between migration and genetic drift. Before xation of any allele, the system enters into a quasi equilibrium (QE) state. Hössjer and Ryman (2012) developed a general QE methodology for computing approximations of spatial autocorrelations between subpopulations, subpopulation di erentiation ( xation indexes) and variance e ective population sizes. In this paper we treat a class of models with translationally invariant migration and use Fourier transforms for computing these quantities. We show how the QE approach is related to other methods based on conditional kinship coe cients between subpopulations under mutationmigration-drift equilibrium. We also verify that QE autocorrelations are closely related to the expected value of Moran's autocorrelation function and treat limits of continuous spatial location (isolation by distance) and in nite gitters of subpopulations. The theory is illustrated with several examples including island models, circular and torus stepping stone models, von Mises models, hierarchical island models and Gaussian models. It is well known that the xation index contains information about the e ective number of migrants. The spatial autocorrelations is complementary and typically reveal the type of migration (local or global).