Inbreeding effective size

You may also remember that we can think of genetic drift as analogous to inbreeding. The probability of identity by descent within populations changes in a predictable way in relation to population size, namely

$$f_{t+1} = \frac{1}{2N} + \left(1 - \frac{1}{2N}\right)f_t \quad.$$

So another way we can make our actual population equivalent to an ideal population is to make them equivalent with respect to how $f$ changes from generation to generation. We do this by calculating how the inbreeding coefficient changes from one generation to the next in our actual population, figuring out what size an ideal population would have to be to show the same change between generations, and pretending that our actual population is the same size at the ideal one. So suppose $\hat f_t$ and $\hat f_{t+1}$ are the actual inbreeding coefficients we'd have in our population at generation $t$ and $t+1$, respectively. Then

$$\begin{eqnarray*} \hat f_{t+1} &=& \frac{1}{2N_e^{(f)}} + \left(1 - \frac{1}{... ...f)} &=& \frac{1 - \hat f_t}{2(\hat f_{t+1} - \hat f_t)} \quad . \end{eqnarray*}$$

In many applications it's convenient to assume that $\hat f_t = 0$. In that case the calculation gets a lot simpler:

$$N_e^{(f)} = \frac{1}{2\hat f_{t+1}} \quad .$$

We also don't lose anything by doing so, because $N_e^{(f)}$ depends only on how much $f$ changes from one generation to the next, not on its actual magnitude.