Summary: The great majority of models of the population genetics of subdivided populations have made the simplifying assumption that the gene frequencies in migrant groups are deterministic. The present paper examines models which more closely mimic natural conditions, in which the gene frequencies in migrant groups are subject to stochastic effects. It is shown that some types of stochastic migration can cause dramatic changes in spatial correlations and variance. These changes depend on how the stochastic migration effects in the gene frequency recursion equations are shared among nearby subpopulations during the same generation. Only for cases where the effects are completely unshared are the equilibrium spatial and space-time correlations among adult subpopulations unaffected, but the variance is always inflated.
The analyses here use novel methods, by recasting population genetic migration-drift models as space-time autoregressive moving average (STARMA) processes. Recent theorems for STARMA processes are employed for finding the spatial correlations, and for the first time in population genetics theory the complete set of space-time correlations, for systems with general patterns of migration rates and numbers of spatial dimensions, is obtained. The space-time correlations provide a uniquely detailed description of a system, and thus form a link between observed spatial autocorrelation statistics and the underlying space-time population genetic process. STARMA theoretical processes have direct statistical analogues that can be applied for process identification, parameter estimation, model fitting, and forecasting in real systems.