Drift and migration

I just pointed out that if populations are isolated from one another they will tend to diverge from one another as a result of genetic drift. Recurrent mutation, which ``pushes'' all populations towards the same allele frequency, is one way in which that tendency can be opposed. If populations are not isolated, but exchange migrants with one another, this also will oppose the tendency for populations to become different from one another. It should be obvious that there will be a tradeoff similar to the one with mutation: the larger the populations, the less the tendency for them to diverge from one another and, therefore, the more migration will tend to make them similar. To explore how drift and migration interact we can use an approach exactly analogous to what we used for mutation.

The model of migration we'll consider is an extremely oversimplified one. It imagines that every allele brought into a population is different from any of the resident alleles.⁵ It also imagines that all populations receive the fraction of migrants. Because any immigrant allele is different, by assumption, from any resident allele we don't even have to keep track of how far apart populations are from one another, since populations close by will be no more similar to one another than populations far apart. This is Wright's island model of migration. Given these assumptions, we can write the following:

$$f_{t+1} = \left(\left(\frac{1}{2N}\right) + \left(1 - \frac{1}{2N}\right)f_t\right)(1-m)^2 \quad .$$ (3)

That might look fairly familiar. In fact, it's identical to equation (2) except that there's an $m$ in (3) instead of a $\mu$. $m$ is the migration rate, the fraction of individuals in a population that is composed of immigrants. More precisely, $m$ is the backward migration rate. It's the probability that a randomly chosen individual in this generation came from a population different from the one in which it is currently found in the preceding generation. Normally we'd think about the forward migration rate, i.e., the probability that a randomly chosen individual with go to a different population in the next generation, but backwards migration rates turn out to be more convenient to work with in most population genetic models.⁶

It shouldn't surprise you that if equations (2) and (3) are so similar the equilibrium $f$ under drift and migration is

$$\hat f = \frac{1}{4Nm + 1}$$

In fact, the two allele analog to the mutation model I presented earlier turns out to be pretty similar, too.

• If $2Nm > 1$, the stationary distribution of allele frequencies is hump-shaped, i.e., the populations tend not to diverge from one another.⁷

• If $2Nm < 1$, the stationary distribution of allele frequencies is bowl-shaped, i.e., the populations tend to diverge from one another.

Now there's a consequence of these relationships that's both surprising and odd. $N$ is the population size. $m$ is the fraction of individuals in the population that are immigrants. So $Nm$ is the number of individuals in the population that are new immigrants in any generation. That means that if populations receive more than one new immigrant every other generation, on average, they'll tend not to diverge in allele frequency from one another.⁸ It doesn't make any difference if the populations have a million individuals a piece or ten. One new immigrant every other generation is enough to keep them from diverging.

With a little more reflection, this result is less surprising than it initially seems. After all in populations of a million individuals, drift will be operating very slowly, so it doesn't a large proportion of immigrants to keep populations from diverging.⁹ In populations with only ten individuals, drift will be operating much more quickly, so it takes a large proportion of immigrants to keep populations from diverging.¹⁰