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The coalescent and genealogical process in geographically structured population. (English) Zbl 0726.92014
Summary: We shall extend Kingman’s coalescent [J. F. C. Kingman, Stochastic Processes Appl. 13, 235-248 (1982; Zbl 0491.60076)] to the geographically structured population model with migration among colonies. It is described by a continuous-time Markov chain, which is proved to be a dual process of the diffusion process of stepping-stone model. We shall derive a system of equations for the spatial distribution of a common ancestor of sampled genes from colonies and the mean time to getting to one common ancestor. These equations are solved in three particular models; a two- population model, the island model and the one-dimensional stepping-stone model with symmetric nearest-neighbour migration.

MSC:
92D10 Genetics and epigenetics
60J27 Continuous-time Markov processes on discrete state spaces
60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
92D15 Problems related to evolution
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