Our analysis allows us to draw several conclusions that clarify the basic evolutionary dynamics of the genetic factors influencing human male homosexuality and the related female fecundity increase, resolving a number of open questions. As a main point, we can exclude the GFMH propagation mechanisms based on overdominance (male heterozygote advantage), because none of the models (1b), (5a), (5b) account satisfactorily for the sexual-orientation asymmetries of requirement (B1). At this level of genetic analysis, we can also exclude maternal effects, including maternal genomic imprinting, as they lead too easily to GFMH extinction or fixation, against requirement (A). Only the hypothesis that the GFMH are characterized by sexually antagonistic selection (i.e. the GFMH favor one sex and disfavor the other) produces viable population genetic models (see the case (4) above) leading to the persistence of the trait at low frequencies and capable of accounting for the related pedigree asymmetries. For this reason, predictions of possible widespread diffusion of male homo- or bisexuality in human populations 
are not warranted, as stable low levels of this character are actually compatible with a broad range of parameters in population genetic models.
The fact that both the models (4a) and (4b), and only those, fit qualitatively the available empirical data not only establishes the sexually antagonistic character of this human trait, but also indicates the presence of at least one X-linked locus for the GFMH. This agrees with the relation between X-linkage of the GFMH and sexual antagonism also pointed out in 
. The best qualitative agreement with the data is obtained through model (4b) with two X-linked loci: the subtleties of the observed asymmetries therefore indicate the genetics and inheritance dynamics of the GMFH to be modulated by an X-linked switch activating a further locus on the sexual chromosome, possibly together with other autosomal components 
not identifiable through our analysis.
The two best models (4a) and (4b) allow us to draw a number of conclusions regarding the dominance for the alleles involved in the GFMH. We recall that in both models the allele A
, which resides on the X chromosome, is dominant (this is indeed the case in all the other models considered above, for otherwise the stability of polymorphism is not guaranteed). The numerical simulations show, as general trends, that in both models the dominance of B
in females (i.e. high values of u
) improves polymorphism stability, while the localization of the GFMH entirely on X-linked loci gives qualitatively better pedigree asymmetries, as can be intuitively expected, if compared to a GFMH partially residing on the autosome. In detail, for model (4a) involving an autosomal B
, the best stability is obtained when for u
1. However, this case does not produce the correct pedigree asymmetries; small or intermediate values of u
, giving almost unaffected fitness to females heterozygous for B
, are optimal to satisfy qualitatively all conditions (A)–(B1)–(B2); see Fig. 5 in Document S1
. Therefore in model (4a) the allele B
should be almost ‘recessive’ in females (recall it is always recessive in males). Also when B
is on the X chromosome (model 4b), the best conditions for polymorphism stability are given by u
1, i.e. when females who are homo- or heterozygous for B
have the same fitness. However, in this case both the GFMH loci are X-linked, and the pedigree asymmetries result to be only slightly affected by the parameter u
, so that the requirements (A)–(B1)–(B2) are qualitatively best satisfied for u
1 (see above and Fig. 6 in Document S1
). We conclude that in model (4b) the allele B
should be dominant in females (while it is still recessive in males).
We notice that model (4b) also predicts a higher concordance in sexual orientation between biological brothers than model (4a); further information on the absolute values of frequency of homosexuality, or direct gene investigation, which are unavailable at present, will help to discriminate between the two possibilities. Closer quantitative adherence to the experimental data can in principle be obtained by increasing the number of loci related to the GFMH. Such more complex modeling however should not change the basic insight provided by the simplest two-locus approach investigated here.
Our results, which exclude both overdominance and maternal effects on male offspring, also point to a likely scenario of androphilic
phenotypic expression of the GFMH, i.e., an expression that specifically increases the attraction to males in both sexes, rather than inducing a more general phenotipic feminization. Androphilia is indeed consistent in a more natural way with the sexually antagonistic hypothesis, in contrast to the hypothesis of feminizing GFMH or maternal GFMH, which are better associated to the genetic models based respectively on overdominance in males, or on genetic maternal effects, which we considered above. See the remarks on phenotypic expression in Document S1
for more details. We notice that the conclusion of an androphilic effect of the GFMH in principle allows one to make testable predictions regarding the behavior of GFMH carriers, along the lines for instance of 
Sexually antagonistic selection has been considered in the past 
, although its role in evolutionary processes has been generally underestimated; it has however recently received both theoretical and empirical attention due to its potential ubiquity in dioecious species 
. Sexually antagonistic selection is at present recognized as a powerful mechanism through which genetic variation of fitness is maintained despite sexual selection in biological populations, in insects 
, birds 
, and mammals 
, leading to population divergence and possibly speciation 
. Our findings firmly establish, with a particularly relevant example, the occurrence of sexually antagonistic characters in humans. This point of view may help shift the focus away from male homosexuality per se
: rather than concentrating on the sole aspect of the reduced male fecundity that it entails, we can place it within the more general framework of a genetic trait with gender-specific benefit, which may have evolved by increasing the fecundity of females. A consequence of this is that the entire population exhibits a high fecundity variation, and, as we show, the trait can neither disappear nor completely invade the gene pool. Indeed, the GFMH may belong to a possibly wide, but at present still poorly understood, class of sexually antagonistic characters that contribute to the maintenance of the observed genetic variation in human populations. As such characters are mostly expected to have a sex-linked component, the present treatment of the GMFH should provide basic understanding also of the dynamics of any such general sexual antagonistic traits.
While the latter are generally assumed to favor males and penalize females (but see 
), we point out a counterintuitive implication of the presence of traits which increase female fecundity, as the sexually antagonistic GFMH. shows that, at equilibrium, the proportion
of GFMH-carrying females in the population positively correlates with the proportion η of GFMH-carrying males. Both
and η affect the population's overall fecundity at equilibrium (see Document S1
). Consider now the variation Δf
of the total fecundity due to the presence of the GFMH in a population at equilibrium (with respect to the population's baseline fecundity in the absence of GFMH):
we have that when fGFMH
is constant, Δf
is a function of α and γ only. As the normalized fecundity α of GFMH-carrying females is inversely proportional to the baseline fecundity fb
of non-GFMH-carrying females, a decrease of fb
(due, for instance, to social or economic factors) results in a decrease of the total expected fitness in the population, but also in an increase of α. From (3), we find that Δf
is a positive and monotonically increasing function of both the variables α and η. This is shown in , where we also see that the higher α, for given η, the larger is Δf
. We thus have the following consequences: (a
) in a given population (α and η fixed) the presence of the GFMH always induces a positive
of the total fecundity, with respect to the baseline value in the absence of the GFMH; (b
) all else being the same, a higher proportion of homosexuals in a population indicates a comparatively higher total fecundity increment Δf
) if due to external conditions the population's baseline fecundity is falling (which results in an increase of the fecundity α of GFMH-carrying females relative to the baseline), the increment of the population's fecundity Δf
due to the presence of the GFMH becomes proportionally more pronounced, mimicking a ‘buffer effect’ on any factors inducing the total fecundity decrease.
Equilibrium frequencies and total fecundity increase due to the GFMH.