Variable population size

The notation for this one gets a little more complicated, but the ideas are simpler than those you just survived. Since the population size is changing we need to specify the population size at each time step. Let $N_t$ be the population size in generation $t$. Then

$$\begin{eqnarray*} f_{t+1} &=& \left(1-\frac{1}{2N_t}\right)f_t + \frac{1}{2N_t}... ..._{i=1}^K\left(1-\frac{1}{2N_{t+i}}\right)\right)(1-f_t) \quad . \end{eqnarray*}$$

Now if the population size were constant

$$\left(\prod_{i=1}^K\left(1-\frac{1}{2N_{t+i}}\right)\right) = \left(1 - \frac{1}{2N_e^{(f)}}\right)^K \quad .$$

Dealing with products and powers is inconvenient, but if we take the logarithm of both sides of the equation we get something simpler:

$$\sum_{i=1}^K\log\left(1-\frac{1}{2N_{t+i}}\right) = K\log\left(1 - \frac{1}{2N_e^{(f)}}\right) \quad .$$

It's a well-known fact²³ that $\log(1-x) \approx -x$ when $x$ is small. So if we assume that $N_e$ and all of the $N_{t}$ are large,²⁴ then

$$\begin{eqnarray*} K\left(-\frac{1}{2N_e^{(f)}}\right) &=& \sum_{i=1}^K-\frac... ...(\frac{1}{K}\right) \sum_{i=1}^K\frac{1}{N_{t+i}}\right)^{-1} \end{eqnarray*}$$

The quantity on the right side of that last equation is a well-known quantity. It's the harmonic mean of the $N_{t}$. It's another well-known fact²⁵ that the harmonic mean of a series of numbers is always less than its arithmetic mean. This means that genetic drift may play a much more imporant role than we might have imagined, since the effective size of a population will be more influenced by times when it is small than by times when it is large.

Consider, for example, a population in which $N_1$ through $N_9$ are 1000, and $N_{10}$ is 10.

$$\begin{eqnarray*} N_e &=& \left(\left(\frac{1}{10}\right) \left(9\left(\frac{... ...\left(\frac{1}{10}\right)\right)\right)^{-1} \\ &\approx& 92 \end{eqnarray*}$$

versus an arithmetic average of 901. So the population will behave with respect to the inbreeding associated with drift like a population a tenth of its arithmetic average size.