Question

Derive the expression for effective population size under fluctuating N (ie., that Ne is the harmonic mean of the Ni's). Start with the equation for the accumulation of identity by descent (f) under finite population size, and employ the approximation that ln[1-(1-x)]=-(1/x), which is appropriate when x>10. Try for two generations.

Answer 1

Let us consider that small population determine, the overall population size to a large extent: all future offspring will be descendants from of these few survivors. By examining the heterozygosity, the effect of variation in population size can be determined.

H_t = (1 - 1/2N)^t H₀

loss of heterozygosity is given by

H_t/H₀ = (1 - 1/2N)^t

if N varies

H_t/H₀ = $\tiny \prod i=1 t \to 1$ (1 - 1/2N_i)

overall effective population size

$\tiny \prod i=1 t \to 1$ (1 - 1/2N_i) = (1 - 1/2Ne)^t = H_t/H₀

putting e $\tiny \approx$ 1 + x, $\left | x \right | \approx$ 0 then x = 1/2N

$\tiny \prod i=1 t \to 1$ e^-1/2N_i = e^-1/2Ne

solving N_e we get,

N_e = $\sum$ t/N_i

where t - number of discrete generations of fluctuating size. Ne - single-generation harmonic mean of population sizes. harmonic mean is strongly affected by single small values compared to arithemetic mean.