Probability of fixation
This was derived in Kimura 1962.
[math]u(p)=\frac{1-e^{4N_esp}}{1-e^{4N_es}}[/math]
If we are considering the initial frequency of a single new mutation in the population p=1/(2Ne),
[math]u(p)_1=\frac{1-e^{4N_es\frac{1}{2N_e}}}{1-e^{4N_es}}=\frac{1-e^{2s}}{1-e^{4N_es}}[/math].
And if 4Nes is large
[math]u(p)_2\approx\frac{1-e^{2s}}{1}=1-e^{2s}[/math].
[math]e^{2s}\approx 1+2s[/math]
[math]u(p)_2 \approx 1-e^{2s} \approx 1-1+2s = 2s[/math].
This agrees with the results of Fisher 1930 and Wright 1931.
Notes
This is derived from
[math]u(p) = \frac{\int_0^p G(x)\, \mbox{d} x}{\int_0^1 G(x)\, \mbox{d} x}[/math],
equation 3 of Kimura 1962.
[math]u(p,t)[/math] is the probability of fixation of an allele at frequency p within t generations.
The change in allele frequency ([math]\delta p[/math]) over short periods of time ([math]\delta t[/math]) is
[math]u(p, t+\delta t) = \int f(p, p+\delta p; \delta t) u(p+ \delta p, t) \, \mbox{d} (\delta p)[/math],
integrating over all values of changes in allele frequency ([math]\delta p[/math]).
A mean and variance of the change in allele frequency (p) per generation are defined as
[math]M_{\delta p}=\lim_{\delta t \to 0} \frac{1}{\delta t} \int (\delta p) f(p, p+\delta p; \delta t) \, \mbox{d} (\delta p)[/math]
[math]V_{\delta p}=\lim_{\delta t \to 0} \frac{1}{\delta t} \int (\delta p)^2 f(p, p+\delta p; \delta t) \, \mbox{d} (\delta p)[/math]
The probability of fixation given sufficient time for fixation to occur is
[math]u(p)=\lim_{t \to \infty} u(p,t)[/math]
(to be continued ... I need to work through this.)
