fixation time
van Herwaarden OA & van der Wal N 2002 Extinction time and age of an allele in a large finite population. Theor Popul Biol 61:311-318.
- c = (ln(8Nes) + (2 − 3h) ln(h) + (3h − 1) ln(1 − h) + γ) / (2sh(1 − h)) ... (19)
- for initial frequencies x near 1 we thus obtain the boundary layer solution for the (conditional) expected time to extinction (exit at 0):
- T1(x) = c − 2Ne (1 − x) ... (20)
- for small x
- T0(x) = c − 2Nex ... (28)
- for the case h = ½ Gale (1990) has determined an asymptotic approximation for the expected fixation time of an advantageous mutant by expanding asymptotically the exact solution for small s and large Nes
- his approximation agrees with our result for T0(1 / (2N)) for this case, except for his O(1)-term that slightly differs from ours due to ignoring a small part of the solution
- in the limit Ne → ∞, the expected extinction time of an allele does not change in case the selection is reversed
- this symmetry property is known for the special case h = ½
- see, e.g., Ewens (1979)
- Watterson (1977) obtained this symmetry for h = ½ by showing that the deterministic drift coefficient of the conditioned process is independent of the sign of s
- the symmetry for h = ½ was first noted by Maruyama (1974) by inspection of the exact solution, though for a related problem concerning the age of an allele
- in the case h = ½ the condition Ne → ∞ is not needed
- it can be shown to be necessary for h ≠ ½
- we mention a symmetry property found by Maruyama and Kimura (1974), which is different from our symmetry property
- their symmetry implies that, for x close to 1, T1(x) ≈ T0(1 − x)
- for an allele with its initial frequency close to 1 the expected extinction time remains unchanged if s is replaced by −s and h by 1 − h
- this relation T1(x) ≈ T0(1 − x) for x close to 1, for which Maruyama and Kimura (1974) do not require the condition Ne → ∞, is easily seen to be present in our asymptotic solution by comparing Eqs. (20) and (28)
