Measured Games · Paper 12 of 07 · Evolutionary biology

The sex ratio

After R. A. Fisher (1930); local mate competition by W. D. Hamilton (1967), tested on fig wasps by E. A. Herre (1985)

Abstract

Because every offspring has one mother and one father, the rarer sex enjoys a per-capita reproductive advantage, and selection on parental allocation drives the population to equal investment. The argument is frequency dependence applied to an allocation rather than to a behaviour. Where mating is local and brothers compete for the same mates, the same reasoning predicts a specific female bias set by the number of foundresses sharing a patch.

KeywordsFisher's principle · equal investment · local mate competition · stable allocation

1Setup

Setup

A mother has a fixed budget of reproductive effort and must divide it between sons and daughters. Ask why populations sit near equal numbers and the answer is not that equality is efficient. It is that any departure from it is self-correcting.

Every offspring in a sexual population has exactly one mother and one father. So the total reproductive output flowing through males equals that flowing through females, whatever the numbers of each. If males are rarer, the average male has more offspring than the average female, and a mother producing sons gets more grandchildren per unit of effort. That advantage disappears precisely when the investment is equal.

2The model

The model

# population produces a fraction R of sons; a mother produces r fitness through sons ∝ r / R fitness through daughters ∝ (1-r) / (1-R) # the rarer sex pays better, so the only unbeatable allocation is r* = 1/2 (equal investment, Fisher 1930) # local mate competition: n foundresses share a patch, and all mating # happens inside it before daughters disperse fitnessi = (1 - ri) + [ ri / Σrj ] · Σ(1 - rj) # a mother's sons now compete mainly with each other, so sons are worth less r* = (n - 1) / 2n n = 1 → r* = 0 n = 2 → r* = 0.25 n → ∞ → r* → 0.5

Fisher's result is the limiting case of Hamilton's. When enough foundresses share a patch, a mother's sons compete with strangers rather than with brothers and the local structure washes out. The prediction is not merely qualitative: it names a number for each level of local competition, which is what makes it testable.

Figure 1Selection on a heritable allocation

This figure needs JavaScript. The panels below are live simulations rather than images, so with scripting turned off they stay empty and the controls do nothing. Everything the figure demonstrates is also stated in the surrounding text: the model is given in full in section 2 and the result in section 3.

Population mean allocation against the prediction

Distribution of maternal strategies in the current generation

Predicted allocation against foundress number · hover to read any value

Observed mean Predicted r* Fisher, one half

Charts are focusable. With a chart focused, use the left and right arrow keys to read values along the horizontal axis.

Foundresses per patch n

Mothers

Mutation size

Starting allocation

Predicted r*—

Observed mean—

Spread—

Foundresses—

Deviation—

Generation0

Figure 1. Allocation is heritable with mutation and nothing pushes the population toward any particular value: mothers are assigned to patches, fitness is computed from who mates with whom inside the patch, and reproduction is proportional to fitness. The mean therefore converges on (n-1)/2n because selection puts it there. Start the population at an extreme allocation and it still arrives. Switch to panmixia and it goes to one half regardless of n, which is the Fisher limit.

3Results

Results

Fisher (1930) gave the argument in a paragraph and it has survived essentially unchanged. The key move is to count in units of parental investment rather than in offspring, which is why the result is about equal investment and not necessarily equal numbers: where one sex is more expensive to produce, the equilibrium numerical ratio is skewed so that total spending stays balanced.

Hamilton (1967) found the important exception. When mating happens locally among the offspring of a small number of mothers, a mother's sons compete with one another for the same females, so extra sons cannibalise each other's returns. The predicted female bias is severe when foundresses are few and relaxes toward one half as their number rises.

What makes this the cleanest case of frequency dependence on the site is that the strategy is an allocation rather than a behaviour. Nobody chooses anything, no learning is involved, and the equilibrium is reached by selection on a heritable trait. It is the same logic that produces a stable mix in the foraging model, applied one level down.

4Evidence

Evidence

Fig wasps are close to the model’s assumptions and are the system the prediction has been tested on most often. Females lay inside a fig, offspring mate among themselves before dispersing, and the number of foundresses in each fig can be counted directly, which makes the independent variable observable rather than inferred.

Herre (1985) compared predictions against data from three fig wasp species, and the paper’s contribution was to separate two effects that earlier work had run together. Local mate competition and inbreeding both push sex ratios towards females, and Herre presented a model handling their interplay. The conclusion was that foundresses adjust the sex ratio of their offspring in response to both the intensity of local mate competition and the level of inbreeding in the population. Herre (1987) extended the work to the question of how precisely the adjustment is made and what constrains it.

Werren (1980) had shown facultative adjustment in the parasitoid wasp Nasonia vitripennis, where females alter the sex ratio of a clutch according to whether a host has already been parasitised. That is the cleanest demonstration that the adjustment is a response to circumstances rather than a fixed species trait.

5Limitations

Limitations

The quantitative fit is looser than the qualitative one, and the direction of the miss is consistent. Across several species, observed brood sex ratios are more female-biased than local mate competition theory predicts, even after extensions that account for inbreeding and for the asymmetric relatedness of haplodiploid mothers to sons and daughters. The theory gets the direction and the comparative statics right and the level somewhat wrong.

Part of that gap turned out to be a measurement artefact. Several nominal fig wasp species proved to be complexes of cryptic species, so counts that appeared to record several foundresses in one fig were in some cases recording members of different species, which overestimates the local mate competition each mother actually faces. Correcting for that closes some of the discrepancy but not all of it, and results from single confirmed species still show the bias.

The mechanism is also less settled than the pattern. What cue a foundress uses to assess competition is contested, with work pointing to clutch size rather than to a direct assessment of how many others are present, and at least one manipulative study argues that what looks like adaptive adjustment in several pollinator species is instead a by-product of interference competition between foundresses.

Finally, the clean fit belongs to a specific kind of organism. Fig wasps and parasitoids satisfy the assumptions almost exactly. In vertebrates sex is usually determined chromosomally, maternal control is limited, and reported adjustments are small, inconsistent and poorly replicated. And equal investment is not equal numbers: comparing an observed head count against one half without knowing the relative cost of each sex tests the wrong quantity.

References

Fisher, R. A. (1930). The genetical theory of natural selection. Clarendon Press.
Hamilton, W. D. (1967). Extraordinary sex ratios. Science, 156(3774), 477–488. https://doi.org/10.1126/science.156.3774.477
Herre, E. A. (1985). Sex ratio adjustment in fig wasps. Science, 228(4701), 896–898. https://doi.org/10.1126/science.228.4701.896
Herre, E. A. (1987). Optimality, plasticity and selective regime in fig wasp sex ratios. Nature, 329(6140), 627–629. https://doi.org/10.1038/329627a0
West, S. A. (2009). Sex allocation. Princeton University Press.
Werren, J. H. (1980). Sex ratio adaptations to local mate competition in a parasitic wasp. Science, 208(4448), 1157–1159. https://doi.org/10.1126/science.208.4448.1157

Verification Every reference on this page has been checked against Crossref. The claims this page makes about what those sources report have also been checked against the sources, and any corrections that check produced are recorded in the project’s claims log.