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The coalescent in two colonies with symmetric migration

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Abstract

Kingman's coalescent process is extended to two colonies with symmetric migration. The mean waiting time until a sample of genes taken from two colonies coalesces to a common ancestor is obtained. The final step in the waiting time before the process is absorbed at 1 is observed to have an intriguing behaviour. The distribution of this final waiting time converges to the known distribution of the corresponding waiting time in the case of a single population as the migration rate tends to zero. The mean, however, does not converge. The waiting time until a sample has two common ancestors is modeled as a function of the migration rate. Finally bounds for the expected waiting time for the two colonies to have j > 1 ancestors are derived.

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Authors and Affiliations

  1. School of Applied Science, Monash University Gippsland Campus, 3842, Churchill, Vic., Australia

    R. C. Griffiths

  2. Department of Mathematics, Monash University Clayton, 3168, Vic., Australia

    H. B. Nath

Authors
  1. H. B. Nath
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  2. R. C. Griffiths
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Nath, H.B., Griffiths, R.C. The coalescent in two colonies with symmetric migration. J. Math. Biol. 31, 841–851 (1993). https://doi.org/10.1007/BF00168049

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  • DOI: https://doi.org/10.1007/BF00168049

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