Fluctuating Population Size and Genetic Drift

In general the variance from a binomial process is [math]\sigma^2 = n p (1-p)[/math], where n is the number of trials and p is the probability of one of the two outcomes.

The variance for the sampling of two alleles each generation is [math]2N p (1-p)[/math], where N is the diploid population size and p is the allele frequency.

We like to work with allele frequencies rather than allele counts in a population. The variation is scaled by 2N so the underlying standard deviation is a fraction between zero and one.

[math]\frac{\sigma}{2N} = \frac{\sqrt{2N p (1-p)}}{2N}[/math]

[math]\sigma^2 = \left(\frac{\sqrt{2N p (1-p)}}{2N}\right)^2 = \frac{2N p (1-p)}{4N^2} = \frac{p (1-p)}{2N}[/math]

Imagine the population fluctuating between two sizes, N₁ and N₂, so that it spends half of its time at N₁ and half at N₂. There is an effective population size, N_e, of constant size that has the same average variance in allele frequency change each generation as the fluctuating population.

[math]\frac{p(1-p)}{2N_e}=\frac{1}{2}\times\frac{p(1-p)}{2N_1}+\frac{1}{2}\times\frac{p(1-p)}{2N_2}[/math]

We can cancel out a lot of components to get

[math]\frac{1}{N_e}=\frac{1}{2}\times\frac{1}{N_1}+\frac{1}{2}\times\frac{1}{N_2}[/math]

and solve for N_e.

[math]N_e=\frac{1}{\frac{1}{2}\times\frac{1}{N_1}+\frac{1}{2}\times\frac{1}{N_2}}[/math]

This is a harmonic mean, the inverse of the average of the inverses of the individual population sizes.

to be continued ...