evolution - Effective population size when the population sizes varies from season to season

Let's think of a species which has four generations per year and which population size changes from season to season so that the population size is 100 in summer, 200 in spring, 50 in autumn and 20 in winter for example. In such case, the effective population size $N_e$ can be calculated by:

$$N_e = \frac{n}{\sum_{i=1}^n\frac{1}{N_i}}$$

where $n$ is the number of generation per year (4 in my example) and each $N_i$ correspond to the population size in one season.

My question

Can you please provide an explanation of why this formula (based on the harmonic mean) holds true to define the effective population size?

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Answer

This is derived from studying how heterozygosity changes over time. The standard equation for change in heterozygosity ($H$) with constant population size ($N$) is:

$H_t = \left(1 - \frac{1}{2N}\right)^tH_0$

When $N$ varies between generations you use the product of this formula:

$H_t = \left(1 - \frac{1}{2N_0}\right)\left(1 - \frac{1}{2N_1}\right)...\left(1 - \frac{1}{2N_{t-1}}\right)H_0 = \prod_{i=0}^{t-1}\left(1 - \frac{1}{2N_{i}}\right)H_0$

To get the overall $N_e$ you need to find the population size that gives the corresponding decrease in heterozygocity over t generations i.e:

$\left(1 - \frac{1}{2N_{e}}\right)^t = \prod_{i=0}^{t-1}\left(1 - \frac{1}{2N_{i}}\right)$

Rearranging gives:

$N_e = \frac{1}{2\left[1-\left[\prod_{i=0}^{t-1}\left(1 -\frac{1}{2N_{i}}\right)\right]^{1/t}\right]}$

This expression can be approximated by the harmonic mean, which is easy to verify with some toy data (you get small deviations if yearly population sizes are very small). The explanation given above can also be found in Hedrick (2009, p217ff), along with some nice examples that integrates the effects on effective population size from several factors.