J. Math. Biol. (1993) 31:841-852 Journal of Mathematical Biology © Springer-Verlag 1993 The coalescent in two colonies with symmetric migration H. B. Nath 1, R. C. Grifliths 2 I School of Applied Science, Monash University Gippsland Campus, Churchill, Vic. 3842, Australia 2 Department of Mathematics, Monash University Clayton, Vic. 3168, Australia Received 26 February 1991; received in revised form 3 August 1992 Abstract. Kingman's coalescent process is extended to two colonies with sym- metric 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. Key words: Coalescent - Migration - Ancestral process - Island model - Popu- lation genetics 1 Introduction Genealogical processes in population genetics have been well explored in the case of a single population (Griffiths 1980, Kingman 1982, Tavar6 1984, Watterson 1984). An important process is the coalescent process (Kingman 1982) which describes the ancestral configuration of a sample of genes backwards in time. In an appropriate continuous time scale the number of ancestors of a sample of n genes is a death process backwards in time with rates /& =k(k-1)/2, k = 2, 3 .... A coalescence is said to take place at the time when two ancestors, from the ancestors of the sample, have a common ancestor. Authors who have studied the coalescent process in geographically structured populations are Notohara (1990) and Takahata (1988). Notohara has a very general geographical population that includes the two colony model as a special case. Takahata discusses simulation in a structured population. In this paper we consider the coalescent process in two colonies of equal size that reproduce according to a Moran model, and exchange individuals between them randomly by migration.