Abstract
For a class of Cannings models we prove Haldane’s formula, pi (s_N) sim frac{2s_N}{rho ^2}, for the fixation probability of a single beneficial mutant in the limit of large population size N and in the regime of moderately strong selection, i.e. for s_N sim N^{-b} and 0< b<1/2. Here, s_N is the selective advantage of an individual carrying the beneficial type, and rho ^2 is the (asymptotic) offspring variance. Our assumptions on the reproduction mechanism allow for a coupling of the beneficial allele’s frequency process with slightly supercritical Galton–Watson processes in the early phase of fixation.
Highlights
Analysing the probability of fixation of a beneficial allele that arises from a single mutant is one of the classical problems in population genetics, see Patwa and Wahl
A rule of thumb known as Haldane’s formula states that the probability of fixation of a single mutant of beneficial type with small selective advantage s > 0 and offspring variance ρ2 in a large population of individuals, whose total number N is constant over the generations, is approximately equal to 2s/ρ2. This was formulated for the model of Wright and Fisher, in which the generation arises by a multinomial sampling from the previous one
The heuristics is that the branching process approximation should be valid until the beneficial allele has either died out or has reached a fraction of the population that is substantial enough so that the law of large numbers dictates that this fraction should rise to 1
Summary
Analysing the probability of fixation of a beneficial allele that arises from a single mutant is one of the classical problems in population genetics, see Patwa and Wahl (2008) for a historical overview. A rule of thumb known as Haldane’s formula states that the probability of fixation of a single mutant of beneficial type with small selective advantage s > 0 and offspring variance ρ2 in a large population of individuals, whose total number N is constant over the generations, is approximately equal to 2s/ρ2. This was formulated for the (prototypical) model of Wright and Fisher, in which the generation arises by a multinomial sampling from the previous one We conjecture that the Haldane asymptotics (3) is valid in this case
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