
Allison Neal
PhD Candidate  Department of Biology  University of Vermont 
Introduction
The theory of sex ratios aims to explain how Natural Selection has shaped the ratios of males to females that we see in nature. Below is a brief introduction to two models of sex ratio theory that have been applied to malaria parasites: Fisherian Sex Ratio Theory and Hamilton's Local Mate Competition. Each section describes the predictions of the models and includes a general example to show how the sex ratio of a population can influence the fitness of individuals.
Fisherian Sex Ratio Theory "...the total reproductive value of the males in [a generation of offspring] is exactly equal to the total value of all the females, because each sex must supply half the ancestry of all future generations of the species. From this it follows that the sex ratio will so adjust itself, under the influence of Natural Selection, that the total parental expenditure incurred in respect of children of each sex, shall be equal." (Fisher, The Genetical Theory of Natural Selection, 1935) The first model of sex ratio theory is generally attributed to Fisher (1935) and is commonly referred to as "Fisherian Sex Ratio Theory", but a similar argument was proposed by Darwin in 1871, and a mathematical model of the fitness consequences of sex ratio was developed by Karl Düsing in 1884. Prediction Fisherian Sex Ratio Theory predicts that selection will lead to equal investment in male and female offspring. If sons and daughters take the same investment in resources to produce each individual offspring, the population sex ratio should be 50:50. If sons and daughters differ in their cost to produce, the population sex ratio will differ from 50:50, with equal total effort to males and females. This prediction of equal investment holds true regardless of the breeding structure of the population (e.g. monogamy vs. polygamy) or whether one gender has a greater chance of survival than the other. Fisher's Argument Explained Fisher starts by pointing out that all offspring produced by sexual reproduction have one mother and one father. Because of this, the number of offspring parented by each gender is equal and thus the fitness of each gender is equal. But what if there are more males than females? The males' collective fitness would have to be spread between more individuals that the females' collective fitness, so each individual male would have lower fitness than each individual female. For example, if a generation has 50 total offspring, males and females collectively each parent all 50 of those offspring. If the parents of those 50 offspring are chosen from 50 females and 75 males, each female mothers 1 offspring, whereas on average, each male fathers only 50/75 = 2/3. Because females have higher fitness (1 vs. 2/3), genes that cause the production of more daughters per female parent will be favored, shifting the sex ratio toward the production of additional females. If there are more females in the population, each male will have higher fitness, causing a shift the other direction. Only when there are equal proportions of each sex will the average fitness of individuals of each sex be equal. An Example Let's use the number of offspring parented (or 'grandparented') as a measure of fitness and assume that offspring sex ratio is a heritable trait. Suppose we have 7 females that have already mated (represented by the seven different shapes at the top of the figure above). Each has two children (the second row of shapes), and the sex ratio of these children ends up being 64% male. These children now mate with each other to produce the grandchildren of the 7 females. Each daughter mates once and produces 2 grandchildren. Because there are only 5 daughters, only 5 of the 9 sons will get to mate. Each son will father either 0 or 2 grandchildren, but on average, sons will father (5/9)*2=1.1 offspring (since they have a 5/9 chance of mating and producing 2 offspring). The fitness of the original 7 females can be represented by the sum of the offspring parented by each of their children (numbers in the yellow box at the bottom of the figure). Notice that the female represented by a circle had the highest fitness because she produced the most of the less common sex (females). She will pass on the genes for producing more females, which will increase the proportion of females in the population until the sexes are present in equal proportions. Equal Investment
Fisherian sex ratio theory actually predicts equal investment in males and females, not necessarily equal numbers of the two genders. I won't discuss this idea in too much detail, but consider a situation in which one gender (say males) take half the resource investment (due to production, parental care, etc.) as the other. If a mother could produce 2 "offspring units", she could produce 2 daughters, or 4 sons, or 2 sons and 1 daughter. In such a situation, Fisherian sex ratio theory predicts that populations at equilibrium would contain twice as many males as females. At this sex ratio, the fitness of males is half that of females, but this is balanced by the fact that they are also half as costly to produce. Note that the total effort expended to produce sons and daughters is equal in the population, but the population sex ratio would be biased toward the gender that is less costly to produce.
Hamilton's Local Mate Competition "It is interesting, therefore, to consider what this same model situation implies when this is the only assumption broken ...the condition of panmixia. The model breeding structure forces sons into competition with one another, so that although there may be, in general, a shortage of males, production of males does not necessarily pay off in terms of grandchildren." (Hamilton, Extraordinary Sex Ratios, 1967) Hamilton recognized that selection may favor population sex ratios with unequal investment in males and females under some conditions. For example, more females than males are expected in populations that are separated into many patches in which mating occurs mainly among highly related individuals. A number of organisms are divided this way, like fig wasps. Females fig wasps lay eggs in a fig. When the eggs hatch, only the females have wings, so they mate with the wingless males (mostly brothers) before flying off to find new figs. Malaria parasites are also divided into patches (separate infections) during mating. Because males in a patch often compete locally with their brothers in a patch for mates, this model is called Local Mate Competition. Basic Prediction and Reasoning Local Mate Competition predicts that populations with the mating structure described above will have female biased sex ratios. Producing more females leads to more grandoffspring. Consider this: if a female can produce 10 offspring that will all mate only with each other, how does she get the most grandoffspring? The answer is, by producing 9 daughters and 1 son. Only daughters directly produce grandchildren, so producing more sons than are necessary to mate with all daughters reduces the total number of offspring produced. This is always true, even in the large, randomly breeding populations that Fisherian sex ratio applies to. If everyone in the world except 1 suddenly became female, and that 1 male was miraculously able to mate with all of them, essentially everyone in the world would be capable of having babies instead of only half. And you think we have a population problem now! If femalebiased sex ratios mean more offspring, why aren't they more common? This comes back to the Fisher's (Düsing's) point about the less common gender getting more mates. Every baby in the world would carry that 1 male's genes, so obviously producing more sons would be a genetically lucrative trait. It becomes a tradeoff between offspring numbers and mating opportunities: producing more females means more total offspring, but producing more sons (up to 50% in the population) means a higher proportion of genes passed on through mating. The relative importance of these competing forces depends on how divided the population is. In undivided populations, males can mate with unrelated females. If there are more females in the population than males, mothers that produce more sons can essentially steal fitness from mothers that produce more daughters because they have more sons to compete for mates. In highly subdivided populations where each patch has offspring from only one mother, there is no need to compete for mates, so producing the most grandoffspring possible is the only important consideration. The figure above illustrates this tradeoff. Each side of the balance depicts a comparison of the same 2 genotypes: one that produces a femalebiased sex ratio (light) and one that produces equal numbers of males and females (dark). On the left, the offspring of the 2 genotypes are not allowed to mate with one another, so the light genotype has the most grandoffspring because it produces more females, which produce offspring. On the right, the offspring of the 2 genotypes are allowed to mate with one another, so the dark genotype has the most grandchildren since it produces more males to compete for mates. (Note: the semitransparent males superimposed over each daughter are the daughters' mates. Because there are twice as many females as males, each male on the right mates twice, so there are 2 light superimposed males and 6 dark ones). So, in undivided populations, equal proportions of males and females are favored by selection for the reasons highlighted by Fisher and Düsing. In very divided populations, very femalebiased sex ratios are often favored because more offspring can be produced. (This also depends how many females each male can mate with, because producing more females without enough males to mate with them doesn't increase offspring production.) In populations that are divided into patches with the offspring from a few mothers, less femalebiased sex ratios are favored because the relative importance of competition among unrelated males is greater. If you're interested in the specific sex ratios favored with different numbers of mothers per patches, keep reading! Numeric Predictions As hinted at above, the specific sex ratio predicted by Local Mate Competition theory depends on the number of mothers depositing offspring in the patch as described by equation (a). The equation gives the unbeatable proportion of males predicted for a patch, where n is the number of mothers per patch, which determines the degree of inbreeding. The predicted sex ratio never dips below a minimum value determined by male fecundity to ensure that there will be enough males to mate with all females. This minimum is shown in equation (b), where c is the number of females each male can mate with. For malaria parasites, we often use the "selfing rate" s (the probability of 2 genetically identical parasites mating, comparable to the probability of a brother and sister mating), instead of of the number of different strains (comparable to number of mothers) because the number of parasites of different strains is often not equal in malaria infections. Values of n are easily converted into values of s using the equation s = 1/n. The figure to the right shows the sex ratio (proportion male) predicted based on different selfing rates (n shown just above axis for comparison) and male fecundities.
