Appendix

I noted earlier that $\pi$ will be little affected by a population bottleneck unless it is prolonged and severe. Here's one way of thinking about it that might make that counterintuitive assertion a little clearer.

Remember that $\pi$ is defined as $\pi = \sum x_ix_j\delta_{ij}/N$. Unless one haplotype in the population happens to be very divergent from all other haplotypes in the population, the magnitude of $\pi$ will be approximately equal to the average difference between any two nucleotide sequences times the probability that two randomly chosen sequences represent different haplotypes. Thus, we can treat haplotypes as alleles and ask what happens to heterozygosity as a result of a bottleneck. Here we recall the relationship between identity by descent and drift, and we pretend that homozygosity is the same thing as identity by descent. If we do, then the heterozygosity after a bottleneck is

$$H_t = \left(1 - \frac{1}{2N_e}\right)^tH_{0} \quad.$$

So consider a really extreme case: a population reduced to one male and one female for 5 generations. $N_e=2$, so $H_5 \approx 0.24H_0$, so the population would retain roughly 24% of its original diversity even after such a bottleneck. Suppose it were less severe, say, five males and five females for 10 generations, then $N_e=10$ and $H_{10} \approx 0.6$.

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