Sexual selection

A classic example of sexual selection is the peacock's tail. The long, elaborate tail feathers do nothing to promote survival of male peacocks, but they are very important in determining which males attract mates and which don't. If you'll recall, when we originally derived the Hardy-Weinberg principle we said that the matings occurred randomly. Sexual selection is clearly an instance of non-random mating. Let's go back to our original mating table and see how we need to modify it to accomodate sexual selection.

		Offsrping genotype
Mating	Frequency	$A_1A_1$	$A_1A_2$	$A_2A_2$
$A_1A_1 \times A_1A_1$	$x_{11}^fx_{11}^m$	1	0	0
$A_1A_2$	$x_{11}^fx_{12}^m$	$\frac{1}{2}$	$\frac{1}{2}$	0
$A_2A_2$	$x_{11}^fx_{22}^m$	0	1	0
$A_1A_2 \times A_1A_1$	$x_{12}^fx_{11}^m$	$\frac{1}{2}$	$\frac{1}{2}$	0
$A_1A_2$	$x_{12}^fx_{12}^m$	$\frac{1}{4}$	$\frac{1}{2}$	$\frac{1}{4}$
$A_1A_2$	$x_{12}^fx_{22}^m$	0	$\frac{1}{2}$	$\frac{1}{2}$
$A_2A_2 \times A_1A_1$	$x_{22}^fx_{11}^m$	0	1	0
$A_1A_2$	$x_{22}^fx_{12}^m$	0	$\frac{1}{2}$	$\frac{1}{2}$
$A_2A_2$	$x_{22}^fx_{22}^m$	0	0	1

What I've done is to assume that there is random mating in the populations among those individuals that are included in the mating pool. We'll assume that all females are mated so that $x_{ij}^f = x_{ij}$.¹¹ We'll let the relative attractiveness of the male genotypes be $a_{11}$, $a_{12}$, and $a_{22}$. Then it's not too hard to convince yourself that

$$\begin{eqnarray*} x_{11}^m &=& \frac{x_{11}a_{11}}{\bar a} \\ x_{12}^m &=& \f... ...}{\bar a} \\ x_{22}^m &=& \frac{x_{22}a_{22}}{\bar a} \quad , \end{eqnarray*}$$

where $\bar a = x_{11}a_{11} + x_{12}a_{12} + x_{22}a_{22}$. A little more algebra and you can see that

$$x_{11}' = \frac{x_{11}^2a_{11} + x_{11}x_{12}(a_{12} + a_{11})/2 + x_{12}^2a_{12}/4}{\bar a}$$

And we could derive similar equations for $x_{12}'$ and $x_{22}'$. Now you're not likely to remember this, but that equation bears a striking resemblance to one you saw earlier, equation (1). In fact, sexual selection is equivalent to a particular type of fertility selection, in terms of how genotype frequencies will change from one generation to the next. Specifically, the fertility matrix corresponding to sexual selection on a male trait is:

$$\begin{array}{cccc} & A_1A_1 & A_1A_2 & A_2A_2 \\ A_1A_... ... & a_{22} \\ A_2A_2 & a_{11} & a_{12} & a_{22} \end{array}$$

There are, of course, a couple of other things that make sexual selection interesting. First, traits that are sexually selected in males often come at a cost in viability, so there's a tradeoff between survival and reproduction that can make the dynamics complicated and interesting. Second, the evolution of a sexually selected trait involves two traits: the male characteristic that is being selected and a female preference for that trait. In fact the two tend to become associated so that the female preference evokes a sexually selected response in males, which evokes a stronger preference in females, and so on and so on. This is a process Fisher referred to as ``runaway sexual selection.''