# Variance in reproductive success and effective population size

Background

The effective population size $N_e$ is the size of the Wright-Fisher population that experience the same amount of drift than the population under consideration.

The higher the variance in reproductive success among individuals, the lower is the effective population. Consider for example, a case where the variance in reproductive success is so high that only one individual reproduces (selfing for example) and contribute to the next generation. In such a case the variance in reproductive success is so high that the effective population size is reduced to 1.

During a meeting I have encountered the following relationship between the population size $N$, the effective population size $Ne$ and the variance in reproductive success $V$.

$$N_e = \frac{N}{V}$$

It makes intuitive sense to me. In the extreme case where $V≈0$, there is no drift and therefore no loss in diversity, everybody contribute as much to the next generation and therefore $N_e=\infty$. By definition, in a Wright-Fisher population $N=N_e$ and therefore $V=1$ should be correct.

Questions

• Where does the equation $N_e = \frac{N}{V}$ comes from? How to derive it? Is it an approximation or an equality?

• Is it true that $V=1$ in a Wright-Fisher population?

• Here is a related post. – Remi.b Sep 8 '15 at 18:50
• If you have a copy of Ewens' "Mathematical Population Genetics", check out chapter 3.7 -- he discusses this. – Daniel Weissman Sep 10 '15 at 19:37
• Yes, I have one. I'll check this out. THank you – Remi.b Sep 11 '15 at 2:21