Extraordinary sex ratios

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The sex ratio is the ratio of males to females in a population. In most sexually reproducing Bill Hamilton expounded Fisher's argument in his paper on “​Extraordinary sex ratios” as follows, given the assumption of equal parental. Author: Hamilton WD, Journal: Science (New York, N.Y.)[/04]. W.D. Hamilton gave the following basic explanation in his paper on ".

A sex-ratio allele is expressed in the heterogametic sex only, so that assumptions of Fisher's analysis do not apply. Sex-ratio evolution drives. The sex ratio is the ratio of males to females in a population. In most sexually reproducing Bill Hamilton expounded Fisher's argument in his paper on “​Extraordinary sex ratios” as follows, given the assumption of equal parental. Extraordinary Sex Ratios is the paper that William D. Hamilton seems most proud of if the effusive self-praise in the biographical preface can be.

Extraordinary Sex Ratios. By W. D. Hamilton. See allHide authors and affiliations. Science 28 Apr Vol. , Issue , pp. Colonies of Anelosimus eximiusin Panama had an average sex ratio of ±sd , i.e. about five females for each male. The sex ratio in egg sacs reared was. Fisher's principle is an evolutionary model that explains why the sex ratio of most species that basic explanation in his paper on "Extraordinary sex ratios", given the condition that males and females cost equal amounts to produce.

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Figure 2 displays the Allee-threshold for various parameter combinations. In Figure 2 a , where , an unbiased female ratio allows the lowest total population density before extinction due to the Allee effect ensues. When the sexes have the same mortality, unbiased sex allocation also maximizes total population density at positive equilibrium [12].

Other parameters: a : ; b : ,. Figure 2 b verifies that increasing female mortality, , for given and , expands the region where the Allee effect leads to extinction. Not surprisingly, increasing male mortality produces a parallel effect. Mortality-rate asymmetry and biased female ratios distort the shape of the thresholds in Figure 2 , but the same general patterns emerge. For a resident population, we have specified how existence of a positive equilibrium depends on the interaction of female ratio at birth and sex-specific mortalities.

We also have shown that initial conditions given existence of a positive equilibrium required to avert extinction due to the Allee effect depend on the same parameters.

A practical consequence is that we must choose initial densities for numerical integration carefully, so that when the competitive dynamics results in extinction, we can clearly identify the reason as either the Allee effect or exclusion.

To quantify how population consequences of female-ratio evolution can be affected by male mortality, we must have an ecological understanding of the two-group competition model, [Eqs. The system has nine fixed points; see File S1.

One is the trivial fixed point where all densities vanish. We can easily identify four more fixed points related to those of the single-class case; there are two symmetric pairs.

At these fixed points, competitive exclusion leaves one group extinct, and one extant. S11 ] of the group with the greater female ratio. When male mortality rates imply a type-I fixed point, the greater female ratio always excludes the lesser ratio. The four remaining fixed points again, forming two pairs by symmetry are qualitatively distinct from those discussed above.

But the male population is culturally dimorphic; father to son transmission [see Eqs. Consider a stable fixed point of this sort, when. The necessary conditions are , , and positivity of the discriminant 6. The preceding condition holds if 7 and. For mathematical details, see File S1. We refer to stable fixed points combining a single female ratio and a male cultural dimorphism as type-II fixed points.

Summarily, the model does not permit equilibrium coexistence of female ratio alleles, but can permit equilibrium diversity in cultural traits governing male mortality. Also note, as is clear from the above conditions, that of type-I and type-II fixed points only one can be stable at a time.

In Figure 3 we illustrate the flow in the mean-field dynamics for a set of parameters when both type-I and type-II fixed points exits, but in the presence of co-occurring males of the other allele, only type-II is stable. Having obtained the nine fixed points for the two-group model analytically, we approached the stability analysis numerically.

To be as thorough as possible, we performed numerical integration systematically to span a significant region of the five-dimensional parameter space.

The range and step of the parameters in our numerical scheme can be found in Table 1. Each run begins with a stationary resident population, with allele and cultural trait. If model parameters allowed a stable positive equilibrium, we chose initial densities accordingly.

We then introduce the invaders, with female-ratio allele and cultural trait. For each set of parameters in each series we performed two runs, one with infinitesimal initial density of invaders and one with high invader density. Each shows a table containing 2D plots with the results of each run; the axes of each 2D plot are values of the same two cultural parameters and , all with the same range.

Another two parameters female ratios and vary across the rows and columns of the tables the 4D plots. We produced as many tables as required by the range of the fifth parameter female mortality.

In each 2D plot, one pixel represents the final stationary densities of the female ratio alleles. This way, we can visually compare all the results simultaneously, simplifying the analysis greatly. Numerical integrations are performed for the scenario where the persistent stationary resident group 1 is invaded by group 2, initially at an infinitesimal density.

Large axes indicate common parameters in rows and columns; every tile has the same axes, scaled as indicated in the bottom right corner. Color scales use independent color channels, therefore, resident and invader densities are shown independently.

Female mortality is fixed:. Our numerical results reveal immediately that female ratios determine the outcome of invasion; a successful invader in pairwise competition has the greater female ratio.

That is, successful invasion always requires , and assures that the resident resists invasion. When the invader has the greater female ratio, it excludes the resident allele competitively. Furthermore, successful invasion by a female-ratio allele assures that the associated cultural trait with value advances from rarity. As a numerical check, we note that both infinitesimal and high invader densities always result in identical final densities.

Since the female-ratio allele is sex-linked, dependence of invasion on simply recalls Hamilton [10].

But in our model, the ecological effect of invasion depends on the culturally transmitted trait. Suppose that successful invasion excludes both the resident female ratio allele and the resident cultural trait ,. Sufficient conditions for invasion and exclusion of both resident traits further require: , ensuring that the invader attains positive stable equilibrium.

Figure 5 a shows an example of successful invasion leading to exclusion of both the resident allele and resident culture. Following introduction of the invading group, the resident density drops quickly, and the successful allele females and successful culture observed in males advance to become the new resident group. Expression S7 gives the explicit cultural constraint on female ratios guaranteeing a stable, positive equilibrium. Assuming this condition holds, the invader must, secondly, have the greater female ratio.

Panel a shows successful invasion; b shows invasion followed by extinction; c shows coexistence of resident males with the invader allele; d shows coexistence of invader males with resident allele.

The vertical dotted line indicates the time when the invader was added to the system, at density both males and females. Legends shown to the right of panel b describe data on all four panels. Common parameter:. Individual parameters: a , , , ; b , , , ; c , , , here, in the final equilibrium ; d , , , here, in the final equilibrium. Recall that Eqs. The model, however, does allow for cultural coexistence, where males of both groups co-occur, but females of only one group remain extant.

For details, see File S1. In one such scenario, resident females are excluded , but resident males, a cultural designation, persist. Necessary conditions for this type of coexistence i.

Figure 5 c displays an example where the resident culture, but not the resident allele, persists after successful invasion. The invader has the greater female ratio, and excludes the resident allele competitively. The ratio of males at dynamic equilibrium is.

Note that the competitively driven increase in female ratio produces a decrease in total population density females plus males at equilibrium [ Figure 5 c ]. By the symmetry of the equations, there also exists a type-II stable fixed point with. That is, the resident population resists invading females, but the introduced cultural trait advances from rarity. Put simply, we can exchange the resident-invader roles of the two groups, and reach the same dynamic equilibrium. Necessary conditions for this case are Here, the introduced female ratio is repelled.

However, the invading male mortality culture, introduced at infinitesimal density, advances and persists at equilibrium; see Figure 5 d. The ratio of males at this equilibrium. Figure 4 includes cases of equilibrium cultural coexistence. For example, condition 9 is visible in tiles where and ; the sharp change in color along the line indicates the condition for cultural coexistence. When this condition is not satisfied, the culture associated with the lower female birth ratio always declines to extinction.

In both cases, the fixed points found numerically are identical to the analytical fixed points for the respective equilibria: Eqs. S16 for cultural coexistence, and Eq. S11 for competitive exclusion of both allele and culture. Given the competitive advantage of increased female allocation in our model, evolution of the sex-linked trait might threaten population persistence. We observe this result in numerical experiments where the invader has both the greater female ratio and the greater male mortality rate, so that Expression 4 fails to hold.

Hence, the successful invader would not advance from rarity absent the resident group. Figure 4 shows an example of invasion to extinction; note the black region of the tile where and. For a particular mortality-rate combination, Figure 5 b depicts the time-dependent densities for a case of invasion to extinction.

The necessary conditions for invasion, see Eq. Hence the invader grows when rare and excludes the resident, but the invader cannot persist. After some time the density of the resident females reaches zero. The reduced density of females means that the production of males both resident and invader is reduced. Given this result, one can envision a stable population where immigration or mutation introduces new alleles over a lengthy time scale. If a new allele has a higher female ratio than the current resident, it will advance.

A series of allelic substitutions might increase the female ratio continuously. Our model does not prevent the female ratio from surpassing the threshold defined by Eqs. Equations 1 and 2 assume that densities mix homogeneously, a strong simplification for most organisms. Furthermore, invasion most often has a distinctly spatial character, expanding from one or more foci of introduction [31]. To consider both effects, we assumed a two-dimensional habitat with local mating and random mobility of individuals.

This elaborates our model as a reaction-diffusion system [40]. A stochastic, spatial individual-based model or its Langevin-type, stochastic reaction-diffusion analogue not addressed in this work may, in principle, lead to different behaviors [45] , [46].

For example, the region of persistence in the case of a single-group two-sex population becomes significantly narrower in a stochastic lattice-based model [12]. Successful invasion in spatial environments ordinarily requires that an initial invader cluster have some minimal size for further growth [31] , [37] , [39] , [47]. This criterion may be due to an Allee effect [47] or inherent geometrical constraints on cluster expansion [39]. For systems exhibiting the Allee effect under homogeneous mixing, one can specify this minimal cluster size as the critical radius required for spatial invasion.

Assuming radially symmetric growth, one expects , where is the diffusion coefficient [47]. For simplicity, we take as a constant across all individuals. The first goal of our spatial analysis was to confirm this scaling relationship for the critical radius when a single group is introduced in an open unoccupied habitat. For spatial invasion in an open habitat, individuals diffusing away from the perimeter of the invader cluster encounter mate densities too low for population increase, given the Allee effect i.

A small invader cluster can shrink as a result. A cluster size exceeding the critical radius generates interior densities sufficient to drive cluster expansion. The critical radius depends on both density inside the cluster and the diffusion coefficient. Therefore, calculating a critical radius demands specifying initial densities within the circular cluster. We noted that as we chose densities closer to, but exceeding, the Allee threshold of the homogenous-mixing case, the critical radius increased.

Therefore, a reasonable deterministic choice is the stationary density of the non-spatial model, which we can calculate, given the female ratio and sex-specific mortality rates [see Eq. We found the critical radius by performing a binary search, using the initial interval of. At each step, a simulation runs with a particular initial radius, until all densities at all grid points come to a stationary state where all time derivatives are less than.

In this final state either all grid points have the positive, stationary densities of the non-spatial model, or all have zero densities. Time evolution of a shrinking and a successfully growing, invading population are illustrated in Figures 6 and 7 , respectively. Population dynamics in the single-group system open habitat , where the initial radius is less than the critical radius.

Simulation time: a , b , c. Parameters: , , ,. Population dynamics in the single-group system open habitat , where the initial radius is greater than the critical radius.

In other taxa females can be the heterogametic sex. Hamilton's logic is simple, if a Y chromosomal gene can distort the sex ratio so only males are produced from the point of fertilization then the Y can increase its fitness. Of course, there's an obvious problem here: once all the females disappear the males can not replicate. The Y chromosome is a selfish gene in a classic and somewhat malevolent sense here, as its interest may result in the extinction of a lineage see meoitic drive.

But remember, evolution only sees a few steps ahead. Similar principles operate for distorters on the X and autosomal chromosomes, though because of the fact that they are passed through females the selection process is far weaker. Hamilton's simulations show that catastrophic crashes occur much faster with Y sex ratio distorters. In his survey of the literature Hamilton notes that the relative lack of Y sex ratio distorters. He also observes that generally the Y chromosome is genetically inactive ergo, sex linkage of traits.

He posits that intragenome dynamics are at work here; modifiers and inactivation has been selected for over time so that the Y can no longer make mischief. It's been cut off at the knees, so to speak.

X and autsomal distorters can be found at some frequency within the population. The whole topic of selfish genes and below individual level operation of selection i. Hamilton then moves to higher levels of organization. What about population substructure? Or, one of his favorite themes, spatial viscosity? These can all effect the outcomes of sex ratios.

Consider a species which is characterized by distinct demes with little between population gene flow. If the flow is low enough in rate then a Y distorter would result in a local extinction. Eventually the region would be reoccupied by migrants from another deme. One supposes in this sort of scenario gene flow would have been very low, but because of Hamilton's use of entomological examples these sorts of dynamics may occur.

He proposes the model of a 'host' body which migrants colonize. Now, imagine that N females settle on a host, and that their reproductive output is equal.

Imagine a host which is shared by r Type a females and n - r Type b females n being the total obviously. The sex ratio,. Now we're looking for the difference between. He notes a species where this is close to the sex ratio, but admits that the example is unrealistic and improbable. As the population of females which land on the host increases the situation verges upon Fisher's panmictic assumption! By this, Hamilton now infers that that simply means that the rate limiting step in reproduction are the number of daughters that a female can produce, since one brother can inseminate innumerable sisters.

The next section is a breezy survey of the literature on various organisms, their sex ratios and possible correlates with parameters such as inbreeding and mating systems e. Hamilton concludes:. These early papers are a little flaccid in the empirical details when set next to the theoretical superstructure. But then, there is a reason that Hamilton exalts these as his crowning theoretical achievements; they were truly simply guides and pilots which might instruct upon the highest probability of fruit in experimental or observational endeavors.

Hamilton ends his treatment with a few more baby-steps in the direction of game theory. If the male fish dies, the strongest female changes its sex to become the male for the group. All of these wrasse are born female, and only become male in this situation. Other species, like clownfish, do this in reverse, where all start out as non-reproductive males, and the largest male becomes a female, with the second-largest male maturing to become reproductive.

Traditionally, farmers have discovered that the most economically efficient community of animals will have a large number of females and a very small number of males. A herd of cows with a few bulls or a flock of hens with one rooster are the most economical sex ratios for domesticated livestock. It was found that the amount of fertilizing pollen can influence secondary sex ratio in dioecious plants. Increase in pollen amount leads to decrease in number of male plants in the progeny.

This relationship was confirmed on four plant species from three families — Rumex acetosa Polygonaceae , [16] [17] Melandrium album Cariophyllaceae , [18] [19] Cannabis sativa [20] and Humulus japonicus Cannabinaceae. In charadriiform birds, recent research has shown clearly that polyandry and sex-role reversal where males care and females compete for mates as found in phalaropes , jacanas , painted snipe and a few plover species is clearly related to a strongly male-biased adult sex ratio.

Male-biased adult sex ratios have also been shown to correlate with cooperative breeding in mammals such as alpine marmots and wild canids. It is also known that in cooperative breeders where both sexes are philopatric like the varied sittella , [28] adult sex ratios are equally or more male-biased than in those cooperative species, such as fairy-wrens , treecreepers and the noisy miner [29] where females always disperse. From Wikipedia, the free encyclopedia.

For gender balance as a socio-political issue, see Gender equality. Countries with more females than males. Countries with the same number of males and females accounting that the ratio has 3 significant figures , i. Countries with more males than females. No data. Main article: Fisher's principle. Map compiled in Bibcode : Sci Nat Commun. Bibcode : NatCo