Table 2 Reformulation of the results of Equations (1) through (25) for the case of exponential growth
IndexNumberQuantityDependenciesDescription
126Embedded ImageNoneConversion of elapsed time in units of 2NA generations in a growing population of size N(t) = N(0)eαt to elapsed time in units of 2NA generations in a diploid population of constant size N = 1
227Embedded Image1Probability that k lineages remain at time t (in units of N(0) generations) when n lineages are sampled at time 0
34Embedded Image2Joint probability that T does not coalesce by time tD, and iD lineages (not including T) remain at time tD
45Embedded Image3Probability that T does not coalesce before time tD
56ℙ(Embedded Image) = 1 − ℙ(Embedded Imagec)4Probability that T coalesces before time tD
615Embedded ImageNoneProbability that T first coalesces with a lineage of type 1, given that there are i lineages of type 1, j lineages of type 2 and k lineages of type 1–2 at time tD
716Embedded ImageNoneProbability that T first coalesces with a lineage of type 2, given that there are i lineages of type 1, j lineages of type 2 and k lineages of type 1–2 at time tD
817Embedded ImageNoneProbability that T first coalesces with a lineage of type 1-2, given that there are i lineages of type 1, j lineages of type 2 and k lineages of type 1-2 at time tD
91Embedded Image6, 5, 3, 2Probability that T first coalesces with a lineage of type 1
109Embedded Image7, 3, 2Probability that T first coalesces with a lineage of type 2
1110Embedded Image8, 3, 2Probability that T first coalesces with a lineage of type 1–2
1223Embedded Image5, 2Survival function of the time until lineage T coalesces, given that T coalesces before time tD
1322Embedded Image12Expected time until T coalesces, given that T coalesces before time tD
1424Embedded Image4, 3Expected time until T coalesces with a lineage with descendants in population 1, given that T first coalesces after time tD
1521E[T1] = E[T1|Embedded Image]ℙ(Embedded Image) + E[T1|Embedded Imagec]ℙ(Embedded Imagec)14, 13, 5, 4Expected time until T coalesces with a lineage with descendants in population 1
1625Embedded Image2Expected time until T first coalesces with a lineage with descendants in population 2
  • The derivation of each expression is the same as in the case of populations of constant size, except that hn,k(t; N(0),α) is used, rather than hn,k(t; N). The quantities on which the expressions in the table depend are given in the Dependencies column. The numbers in the Dependencies column correspond to those in the Index column. The number of each equation—or its analog for the case of populations of constant size—is given in the Number column. Formulas for the case in which populations 1 and 2 have constant size are obtained by setting α1 and α2 equal to 0.