The above examples consider the absolute mean and variance in fitness for males and females, but selection is a relative process, and to account for this fact, certain adjustments are necessary. When the variance in absolute fitness, Vw, is divided by the squared average fitness, W2, we obtain VW/W, a quantity known as the variance in relative fitness, Vw, or as I, the opportunity for selection. The opportunity for selection provides a dimensionless, empirical estimate of selection's maximum strength that is comparable within and among species. Selection intensity can also be measured as the slope of the line that describes the correlation between phenotype and fitness within a population. However, to plot this relationship, it is necessary either to identify the specific trait that is under selection, or to vary the trait of presumed importance experimentally to generate the regression. Such considerations are not necessary for estimates of I, whose value places an upper limit on the change in average population fitness due to selection, regardless of its source, as well as on the change in the standardized average of every other trait. In most cases, I can be calculated directly from field records of individuals who produce offspring and those who do not. When paternity can be assured, I can be calculated from estimates of mating success alone.

This approach is especially useful for understanding the strength of selection within each sex. Here, the value of I is expressed as the ratio of the variance in offspring numbers, VO, to the squared average in offspring numbers, O2, among the members of each sex. Thus, I? = VO?/O? and I, = VO,/O, 2. Because each offspring has a mother and a father, the opportunity for selection on males, I?, and the opportunity for selection on females, I,, are linked through the sex ratio and mean fitness, which must be equal for both sexes. However, the sex difference in the variance in relative fitness, I? — I, = AI may be positive, negative, or zero. Its value determines whether and to what degree the sexes will diverge in character because fitness variance is proportional to selection intensity.

How can we express these relationships for a natural population? Rewriting eqn [4], substituting values from eqns [2] and [3] and rearranging terms, we have

When R = 1, eqn [5] shows that the variance in fitness for males, VO?, equals the variance in fitness for females, VO,, plus the quantity, O, 2 Vmates. This latter term equals the average female fitness squared, O, 2 , multiplied by the variance in mate numbers among males, Vmates (=^2 pj(R — j)2). For the above example, O, 2 Vmates = 9. This shows that the sex difference in fitness variance is due to the fitness effects of a sex difference in the variance in mate numbers, Vmates. In this case, variance in mate numbers exists among males, but not among females. Now recall that I = VW/ W2. We can obtain an analogous expression for the variance in relative fitness for males in terms of offspring numbers by dividing eqn [5] by [RO,]2, that is, by the squared average offspring number for males.

When we do this, we obtain

Or, I? = (RO)(I,) + Imates, because R equals 1/OSR (=1/RO). Thus, the opportunity for selection on males, I?, equals the opportunity for selection on females, adjusted by the sex ratio, (1/R)(I,), plus the opportunity for selection arising from differences in mate numbers among males, Imates. This expression, like eqn [5], shows the relationship between male and female fitness. However, because this relationship is now standardized by the square of mean fitness, it provides estimates of relative fitness, that is, of selection opportunities for each sex.

Contrary to PIT, which considers male-biased OSRs and sexual selection as equivalent, these expressions show that the sex ratio is only part of the total opportunity for selection. When the sex ratio equals 1 (R = 1/RO = 1), subtracting

strating that the sex difference in the opportunity for selection, that is, the opportunity for sexual selection, is indeed due to differences in mate numbers between the sexes. It also shows, paradoxically from a PIT perspective, that the effects of a biased OSR (=RO = 1/R) are strongest when Imates = 0, that is, when sexual selection due to differences in mate numbers is weak.

Inserting the values from our example into this latter equation, we see that when males and females have equal mate numbers, Imates = 0. When males vary in mate numbers, I, still equals 0, so all of the opportunity for selection on males is due to sexual selection or, I? = Imates. If VO, becomes nonzero, either because females vary in their mate numbers, or because females vary in their offspring numbers, or for both reasons, I, will increase and Imates will be eroded to a degree determined by the relative magnitudes of I? and I,. If I, > I?, the sex roles will reverse because sexual selection acts on females. However, erosion of Imates may become negligible if the variance in mate numbers among individuals in one sex becomes large and in the other sex remains small. When this occurs, sexual selection on one sex can overwhelm the effects of natural selection acting on the other sex, leading to apparent cases of sexual exploitation. But such situations may not last for long. Mating systems with strong sexual selection tend to be invaded by alternative mating strategies that reduce the variance in mate numbers within the sex in which it is large. The important empirical points are: (1) Imates appears to explain much about why sex differences exist; and (2) Imates can be estimated for any population in which the mean and variance in offspring numbers among females and the mean and variance in mate numbers among males are known.

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