fixation time

Waxman D 2011 A compact result for the time-dependent probability of fixation at a neutral locus. J Theor Biol 274:131-135.

• an important feature of the form of P_fix(τ;p) given in Eq. (3) is that the smaller the value of τ, the smaller the number of terms that need to be included in the sum for good accuracy
• this is precisely the opposite behaviour to Kimura's result in Eq. (2)

• the approximation of P_fix(τ;p) in Eq. (5) has the feature that it equals p at the time τ = π ≃ 3.14
• beyond this value of τ the approximation exceeds p, which is the largest value that P_fix(τ;p) can take
• a simple way to proceed is to use Eq. (5) for τ < 2 and Eq. (6) for τ ≥ 2

• a closely related quantity is the distribution of the random time to fixation, T_fix, given that fixation ultimately occurs
• the probability that T_fix is smaller than t is given by Prob(T_fix < t) = P_fix(τ;p) / p