Variation in offspring number

I'm just going to give you this formula. I'm not going to derive it for you.²⁶

$$N_e^{(f)} = \frac{2N - 1}{1 + \frac{V_k}{2}} \quad ,$$

where $V_k$ is the variance in number of offspring among individuals in the population. Remember I told you that the number of gametes any individual has represented in the next generation is a binomial random variable in an ideal population? Well, if the population size isn't changing, that means that $V_k = 2(1 - 1/N)$ in an ideal population.²⁷ A little algebra should convince you that in this case $N_e^{(f)} = N$. It can also be shown (with more algebra) that

• $N_e^{(f)} < N$ if $V_k > 2(1 - 1/N)$ and

• $N_e^{(f)} > N$ if $V_k < 2(1 - 1/N)$.

That last fact is pretty remarkable. Conservation biologists try to take advantage of it to decrease the loss of genetic variation in small populations, especially those that are captive bred. If you can reduce the variance in reproductive success, you can substantially increase the effective size of the population. In fact, if you could reduce $V_k$ to zero, then

$$N_e^{(f)} = 2N - 1 \quad .$$

The effective size of the population would then be almost twice its actual size.