Conditions for protected polymorphism

There is one case in which it's fairly easy to understand the consequences of selection, and that's when one of the two alleles is very rare. Suppose, for example, that $A_1$ is very rare, then a little algebraic trickery¹¹shows that

$$\begin{eqnarray*} x_{11}' &\approx& 0 \\ x_{12}' &\approx& \frac{x_{12}(F_{12,22} + F_{22,12})/2}{F_{22,22}} \end{eqnarray*}$$

So $A_1$ will become more frequent if

$$(F_{12,22} + F_{22,12})/2 > F_{22,22}$$ (2)

Similarly, $A_2$ will become more frequent when it's very rare when

$$(F_{11,12} + F_{12,11})/2 > F_{11,11} \quad .$$ (3)

If both equation (2) and (3) are satisfied, natural selection will tend to prevent either allele from being eliminated. We have what's known as a protected polymorphism.

Conditions (2) and (3) are fairly easy to interpret intuitively: There is a protected polymorphism if the average fecundity of matings involving a heterozygote and the ``resident'' homozygote exceeds that of matings of the resident homozygote with itself.¹²

NOTE: It's entirely possible for neither inequality to be satisfied and for their to be a stable polymorphism. In other words, depending on where a population starts selection may eliminate one allele or the other or keep both segregating in the population in a stable polymorphism.¹³