On the evolution of visual female sexual signalling

Abstract

A long-standing evolutionary puzzle surrounds female sexual signals visible around the time of ovulation. Even among just primates, why do some species have substantial sexual swellings and/or bright colorations visible around females' genital regions, while other species are like humans, with no signs of ovulation visible? What is the evolutionary purpose behind not just these signs, but also this great variation seen among species? Here, we examine the evolutionary trade-offs associated with visual ovulation signalling using agent-based modelling. Our model predicts how various factors, including male genetic heterogeneity and reproductive inequality, female physiological costs, group size, and the weighting of genetic versus non-genetic benefits coming from males, each influence the strength of ovulation signalling. Our model also predicts that increasing the impacts of infanticide will increase ovulation signalling. We use comparative primate data to show that, as predicted by our model, larger group size and higher risk of infanticide each correlate with having stronger visual ovulation signs. Overall, our work resolves some old controversies and sheds new light on the evolution of visual female sexual signalling.

1. Introduction

Two general components of sexual selection are mate competition and mate choice [1,2]. Traditionally, mate competition has been associated with male–male competition for access to females, while mate choice with female choice (for example, the bright colours, exaggerated plumage and mating displays seen among males in many different species of birds to attract female mates) [2–6]. However, there are now multiple examples of female–female competition (e.g. competition among female chimpanzees for space and food [7]) and male mate choice (e.g. males preferentially mating with females of a specific age and mating call); see Clutton-Brock [8] for a review and Servedio & Lande [9] for modelling work.

A striking example where female–female competition and male mate choice may be occurring is visual female sexual signalling in primates [10,11]. Visual signals around a female's time of ovulation are widespread across primate phylogeny [12], and such signs are thought to have evolved independently at least five times among the catarrhine primates (i.e. Old World monkeys and apes) [13]. Although much varied between species, these signals primarily occur via changes in the size, swelling, shape and/or colour of the female's perineal skin [13,14]; see figure 1 for an example.

Figure 1.

Olive baboons in Gombe National Park, Tanzania (photograph by Dr Michael Wilson). Adult female with sexual swelling (left) being groomed by an adult male (right).

Visual ovulation signals clearly have significant costs [15]. Physiologically, swellings resulting from increased water retention, increased body weight, as well as diverting fluids away from potentially more important bodily functions (especially in dry/drought conditions) [16]. Higher body weight (making mobility more challenging), noticeable swellings and/or genital colorations can increase a female's risk of predation. Sexual swellings expose the female's perineal skin to parasites, as well as to cuts and scratches [12]. Visual ovulation signs can also result in social costs, for example, if increased within-group male competition hurts the group's success in between-group conflicts, leading to a reduction in the group's territory and/or food resources, and/or increased infant deaths as a result of the between-group encounters [17–19].

Given all these costs, the evolution of visual female sexual signalling was long viewed as a puzzle. In addition, its lack thereof in humans when compared with chimpanzees was also long-debated [20–23]. Various potential benefits of having visual ovulation signs have been proposed, each emphasizing different selective forces. For example, when female–female indirect competition is at play, females can signal that they are fertile [24] or of high quality [25] to simplify fertilization or to obtain good genes for their offspring. When mating opportunities are limited at least for some males, females can also signal to obtain direct benefits such as provisioning or protection by males of ovulating females (defence against injury, predation and harassment) [12]. Other benefits may instead help females' offspring; for example, having visual ovulation signals could help females increase males' perceived probabilities of paternity [12,26] which could lead to offspring protection or paternal care [27,28]. This would be important in the presence of infanticide (see below). Many of the above selective forces and benefits could be working in tandem, depending on the exact combination of sexual selection dynamics occurring.

Earlier discussions of the evolution of visual ovulation signs were often framed in terms of two female mating tactics: ‘paternity concentration’ and ‘paternity confusion’. The former emphasizes female–female competition for better mates in order to receive benefits, whether genetic or non-genetic, to their offspring or themselves [17]. The latter emphasizes matings with multiple males. Having more than one mating partner can be helpful in the presence of genetic (or gametic, immunological, etc.) incompatibilities, post-copulatory sperm competition (cryptic female choice), male investment in offspring, etc. [6,15,29–31]. Doing so allows a female to make more than one male think he may be the father of her offspring (i.e. confuse paternity), which can be beneficial, for example, in reducing the risk of infanticide for the female's offspring [15,32].

Infanticide, the killing of a female's offspring prior to weaning, is extremely detrimental to a female's fitness [24,32–35]. However, committing an act of infanticide may benefit any male who is not the father of that particular offspring, and hence infanticide is often seen when a new male takes over a group [36]. Many primate species have prolonged lactational amenorrhoea [37], resulting in a male killing a female's offspring bringing that female back to fertility sooner by prematurely ending her lactation [38,39]. The infanticidal male can then subsequently mate with this female, meaning his offspring will be able to be born sooner than would otherwise be possible had he never committed infanticide [31]. Although infanticidal events are typically rare, they have been documented in many primate species [32] and can constitute a substantial proportion of infant mortality in some primate species [6].

Over time, neither paternity concentration nor paternity confusion obtained a consensus on being the more likely benefit [12]. Two new evolutionary hypotheses were then proposed, attempting to take into account more selective forces and help reconcile the benefits of paternity concentration with the benefits of paternity confusion. The first, the reliable indicator hypothesis, claims that female–female competition drives visual sexual signals to be reliable indicators for males of each female's quality during that competition [25]. Females are actively seeking high-quality males and, were it beneficial, high-quality females would also be able to attract more than one quality male [25].

The second, the graded signal hypothesis, explicitly recognizes the fact that visual sexual signals serve as ‘graded signals’ for females, i.e. such signals are probabilistic, increasing and decreasing in strength gradually, with ovulation often happening around the time of a female's peak signal. In doing so, a female is able to attract male(s) with good genes around the peak of her signal (i.e. when she is probably most fertile), while still attracting other, lesser quality male(s) during other parts of her cycle (when she still has some sexual signal, just not at its peak, and is probably less fertile) [12].

Although often presented as opposites, both the reliable indicator and graded signal hypotheses invoke female competition for high-quality males and both implicitly assume that females are competing for high-ranking males because those males have superior genetic quality. However, in many cercopithecine primates, male rank is a function of age, rather than of good genes [13]. Note the reliable indicator hypothesis is quite vague about what fitness benefits females might be obtaining through competition. One of the advantages of our approach—as shown later—is explicit consideration of both genetic and non-genetic components.

Substantial field and laboratory work (e.g. [40–48]), as well as phylogenetic studies (e.g. [12,49,50]), have all been done in order to help understand visual ovulation signalling in an evolutionary context. As a result of all this work, the graded signal hypothesis is now largely recognized as being the hypothesis to explain the evolution of visual female sexual signalling, while the reliable indicator hypothesis was unable to find support empirically, and has fallen out of favour [12,28,42,44,47].

Many questions, however, still remain. For example, as related to infanticide, is it more likely for infanticide to serve as selection pressure for the evolution of concealed ovulation or of visual sexual signals? It has long been thought that infanticide could be a main selection pressure for concealed ovulation to evolve (e.g. see [29,32]). A female with concealed ovulation would have the vast majority of her cycle to confuse paternity (lowering her offspring's risk of infanticide) and would not have to suffer any of the physiological costs of having visual ovulation signs [32].

Completely opposite these beliefs is, as discussed earlier, the idea that females who signal prominently are better able to attract multiple males as mating partners, which (for example) is beneficial in reducing the risk of infanticide occurring to that female's offspring by confusing paternity [15]. Supporting this hypothesis are the facts that females with peak signals often mate with multiple males [51] and that such signals are not perfectly honest; ovulation does not always occur exactly at the time of a female's peak signal [12,48]. Indeed, like this example of infanticide above, many of the proposed benefits of having visual ovulation signs have also instead been proposed as costs to females. As Austin Burt says, ‘Ironically, there is considerable overlap between the hypothesized benefits of sexual signals and those of concealed ovulation.’ [26, p. 4].

Informative experiments relating to this phenomenon are by nature rather challenging to conduct, and many ideas historically have just been presented as verbal arguments which can often be easily challenged. Despite still-unanswered questions on the role and evolution of visual female sexual signalling, even just within the context of the graded signal hypothesis, there has been relatively little theoretical work done in understanding the evolutionary trade-offs behind these signs persisting in so many different species (but see [25,52,53]). In fact, two different reviews of visual ovulation signalling appealed specifically for ‘[More] theoretical approaches [to] be undertaken’ [12, p. 241] and ‘More rigorous modelling, using mathematical tools and simulation approaches, [being] needed’ [11, p. 83].

With only two modelling attempts published after these calls (discussed later), the area is ripe for improved theoretical work. Hence, it is our aim here to use mathematical modelling to help explore the evolutionary trade-offs associated with visual female sexual signalling. Note that it is not our goal here to test specific verbal arguments against one another, for example, the reliable indicator versus graded signal hypotheses. Rather, allowing for multiple evolutionary forces (including different types of selection) known to act in the populations of our models, we are interested in determining under what conditions will ovulation signalling increase and/or decrease in a multi-male, multi-female population, assuming such signals to be graded. Compared to earlier, limited modelling efforts [25,52,53], our models explore a much wider range of evolutionary factors under more biologically realistic assumptions. In particular, we explicitly allow for infanticide, consider genetic and non-genetic benefits to females, and assume that males estimate their female mates' fertilities, rather than know them exactly.

2. Material and methods

First, we use mathematical modelling, and then we conduct statistical analyses of empirical data in order to test two of our model's predictions.

(a). Model

We consider a population of individuals living in a large number of groups each with N males and N females. Generations are discrete and non-overlapping. Next, we outline major modelling components; see appendix A for full model details.

Females. We explicitly account for the female cycle, which we split into D discrete units of time. While such units could be hours, minutes, etc., we choose to call such units ‘days’ (for example, with 29 days in one female cycle). Each female is fertile for C days of this cycle; the distribution of probabilities of fertilization across the cycle has the same shape for all females but is randomly centred (meaning cycle synchrony may only occur probabilistically). Females differ genetically in visual ovulation signs present. We assume both ovulation and maximum visual ovulation signs to happen directly in the middle of each female's C fertile days so that there exists at least some reliability of the ovulation signal. We treat visual ovulation signs, x(d) for each day d of the cycle, as overlapping curves (see fig. 11.1 of [13] for comparison). Each female is characterized by two genetically controlled, evolvable traits. Magnitude m is the maximum amount of ovulation signs a female has visible during her cycle; m is a non-negative, continuous number. Length ℓ is the number of days a female has some amount of ovulation signs visible; ℓ is a non-negative, integer number. Ovulation signs are costly; these costs are scaled by parameter c. For an example of what these ovulation signs x(d) could look like across a cycle for four females with varying values of m and ℓ see figure 2.

Figure 2.

Example ovulation signs for females with different trait values over one 29 day cycle. Different colours represent four different female cycles. Numbers beside each curve represent the rank of the male's genetic component (GC) who is most probably to mate with the female on that particular day of her cycle (assuming no reproductive stochasticity).

Males. We assume males to differ in ‘quality’, i.e. any benefit from the male to a female and/or her offspring. Variation in male quality is scaled by parameter b, which also characterizes female benefits. Such quality is explicitly split into a genetic component y_g (GC) and a non-genetic component y_ng (NGC). GC is owing to the male's genes being passed to the female's offspring, while NGC is any non-genetic benefit to the female from mating with that male. For example, NGC could be protection provided to the female herself or increased food, provisioning, access to resources, etc. (as reviewed in detail in [6]). We allow for y_g to correlate with male's rank (and hence his success in reproductive competition). (Note however that, as we show below, females can still evolve to signal ovulation even with no genetic benefits included in the model.) We also allow for correlation ρ between y_g and y_ng. For example, ρ < 0 corresponds to the case when powerful males are less interested in providing any protection/provisioning to females. ρ > 0 instead corresponds to the case when being a more powerful male implies being better at protecting/provisioning females. We do not consider evolution in males, assuming instead that male traits are at a (stochastic) evolutionary equilibrium.

Mating pairs. For every unit of time in the cycle, all males and all females in each group enter the competition for mates. For computational simplicity, we assume each individual to mate exactly once on every unit of time (e.g. day). Males of higher rank (and GC) are more likely to mate with females with stronger ovulation signs visible (higher x(d)), i.e. when the female is most likely to be most fertile. Males of lower rank (and GC) mate with females who have fewer ovulation signs visible, i.e. when the female is less likely to be most fertile. For empirical support of this assumption in primates, see [41,43,45]. However, this assumption can be relaxed via reproductive stochasticity. The degree of reproductive stochasticity in males and females is controlled by parameters ε_m ≥ 0 and ε_f ≥ 0, respectively. With ε_m = ε_f = 0, mating pairs are formed deterministically so that the highest GC male mates with the female with the most ovulation signs visible, and so on (figure 2); this is the case of perfect assortment in mating. Increasing ε_m increases stochasticity in the outcome of male competition for mating success, and thus reduces male reproductive inequality. Increasing ε_f reflects a decrease in the reliability of the ovulation signal as well as a lack of opportunity or interest for fertile females to mate with a higher GC male (e.g. as shown empirically in Amboseli baboons [47]). With large ε_m, ε_f, mating becomes random.

Reproduction. For any female's potential offspring, every male in the group has an associated probability of paternity, ranging anywhere from zero to one. This quantity is determined by both the number of times that particular male mates with the female during her cycle, as well as her probability of fertilization on the days of each mating event. Assuming additivity of cost and benefit effects for simplicity, we specify female fitness (fertility) as:

Here, 0 ≤ η ≤ 1 specifies the relative weight of the NGC; with η = 0.5, male-provided genetic and non-genetic benefits are equally important for females. Fitness is each female's relative reproductive success, i.e. the expected number of offspring surviving to the age of reproduction, normalized to keep the total population size in each generation constant. Offspring production is followed by dispersal.

Infanticide. Modelling infanticide is adapted from [24]. Any male is able to help or harm (note ‘harm’ here could simply mean not helping) an offspring prior to the infant being weaned, although some males (e.g. those of higher quality, size or strength) are able to do so more effectively. Our model does not distinguish between the threat of infanticide via an outsider male or a previously subordinate male. Infanticide's effects on offspring viability are scaled by parameters α (maximum benefit of protection from infanticide) and β (relative maximum cost of infanticide). An additional parameter 0 ≤ κ ≤ 1 determines the extent to which males take visible female ovulation signs into account when estimating their paternity in determining what actions to take regarding infanticide. κ = 0 means males estimate their probability of paternity solely by the number of matings with an offspring's mother, while κ = 1 means males instead estimate paternity exclusively on the basis of the female's visible ovulation signs during their time of mating.

Our goal here is to understand how visual female ovulation signs evolve over time. Appendix A provides further details, explicit fitness functions and full definitions of equations and parameters.

(b). Testing hypotheses against data

We were able to aggregate enough data from extant primate species living in multi-male, multi-female groups to test two of our model's predictions. All data are available in the electronic supplementary material. Note that all of our data sources are data collections from primatologists whose methods and results have already been published elsewhere; we did not do any data collection ourselves. Also note we restricted these analyses to those primate species with a polygynandrous mating system because our model explicitly considers primate individuals living in multi-male, multi-female groups and mating system is already a known correlate of visual ovulation signs in primates [49,50].

Wherever available, we aggregate data per species for: (i) visible ovulation signs, (ii) group size and (iii) risk of infanticide. Ovulation signs data were available from [15,49,50,54]. We follow [50] in grouping ‘absent’ and ‘slight’ ovulation signs visible (e.g. from the ternary classification as found in [49]) together into the one category of ‘absent’. This is owing to both the fact that the category ‘slight’ is vague (e.g. does a species whose females have no sexual swellings, but just a barely visible reddening of the perineum count as ‘absent’ or ‘slight’?) and also that it has yet to be established that males use ‘slight’ swellings as any kind of signal [50]. Average group size was aggregated from [55–58] for each species for which data were available. For infanticide risk, we used Opie et al.'s data on lactation and gestation length [59]. The average ratio of L/G: = lactation length/gestation length is a known correlate of infanticide risk, with species with higher L/G having higher risks of infanticide [37].

For the first analysis, we compare visible ovulation signs with group size. Doing so gives 65 species with both a classification of having ‘present’ or ‘absent’ visual ovulation signs and the species' average group size. We use this to compare the average group sizes for species with absent versus present visual ovulation signs. To test for significance, we control for phylogeny using a simulation-based phylogenetic ANOVA test [60,61] with a recent primate phylogeny [62].

For the second analysis, we compare visible ovulation signs with risk of infanticide. Doing so gives 48 species with both a classification of ‘present’ or ‘absent’ visual ovulation signs, and a value of L/G. We use this to compare the distributions of L/G for species with absent versus present visual ovulation signs. To include phylogeny, we fit a likelihood-based model of continuous trait evolution (see [63] for specific details of the method), again using the primate phylogeny from Fabre et al. [62]. We fit both Brownian motion and Ornstein–Uhlenbeck models using the R package OUwie [63]. In particular, we compare Ornstein–Uhlenbeck models with one versus multiple optima for L/G. In the case of only one single optimum, the trait is assumed to evolve non-neutrally, but that all species (regardless of present versus absent visible ovulation signs) are pulled towards that one single optimal value of L/G. With multiple optima (one for species with present visible ovulation signs and the other for species with absent visible ovulation signs), species with present versus absent visible ovulation signs have different average values of L/G. In all cases, the Akaike information criterion (AIC) was calculated from the likelihood to compare the fit of the models, which each made the different evolutionary assumptions. See the electronic supplementary material for a further description of this method, additional results and discussion on method choice.

3. Results

Model. Agent-based simulations of our model show that the average values of the ovulation signs magnitude trait (m) and the ovulation signs length trait (ℓ) converge to the unique equilibria m*, ℓ^*. Note in most simulations, ℓ^* ≤ 7, because we set the time of fertility C to 7 days. A more comprehensive discussion of our simulations and the effects of parameters can be found in the electronic supplementary material.

Figure 3a summarizes the effects without infanticide of the benefit b, cost c, and group size N on the equilibrium values m* and ℓ^*. As expected, increasing benefits b and decreasing costs c each result in increased m^*, ℓ^*. Our simulations also show that increasing group size N results in increased m*. This is because increasing the number of females in each group intensifies female–female competition for high GC males [6], which incentivizes having more visual ovulation signs present with which to attract such males.

Figure 3.

Effects of parameters on the average equilibria values of ovulation signs magnitude (m*, i) and ovulation signs length (ℓ^*, ii). (a) The effects of variation in male quality (b), costs of having visual ovulation signs (c) and group size (N). (b) Effects relating to infanticide (α, β, κ). α = 0 represents the case with no infanticide present (and hence when β, κ are irrelevant). (c) The effects of reproductive stochasticity in males (ε_m) and females (ε_f) and the relative weighting of NGC (η). (d) The effects of variation in male quality (b) and correlation between a male's GC and NGC (ρ). Equilibria are obtained by averaging 16 initial condition runs (with standard deviation indicated by error bars). Other parameters: (a) ε_m = 0.25, ε_f = 0.01, α = 0, γ = 1, η = 0.5, ρ = 0; (b) N = 8, b = 0.2, c = 0.2, ε_m = 0.25, ε_f = 0.01, γ = 1, τ = 2, ω = 1, η = 0.5, ρ = 0; (c) N = 8, b = 0.2, c = 0.2, α = 0, γ = 1, ρ = 0; (d) N = 16, c = 0.1, η = 0.75, ε_m = 0.25, ε_f = 0.01, α = 1, γ = 1.

Figure 3b instead summarizes the effects of parameters relating to infanticide (α, β, κ) on the equilibrium values of m* and ℓ^*. Increasing the effects of infanticide α and β, and the relative weight of female ovulation signs in males' paternity estimations κ each results in increased values of m*. Intuitively, as these effects relating to infanticide increase, it becomes in females' best interests to mate with high GC males (i.e. males more likely to have their actions be effective within their group, whether owing to strength, rank, etc. and hence be better able to help protect an infant). Increased matings then give these males increased likelihoods of being the father of any subsequent offspring that female produces, which increases the likelihood of the male attempting to protect the offspring against any possible infanticide being committed. Having strong visual ovulation signs facilitates such matings with high GC males.

Figure 3c summarizes the effects without infanticide of the relative weight of male-provided non-genetic benefits η and the two reproductive stochasticity parameters, ε_m and ε_f. Decreasing η increases the value of the ‘prize’ to a female for winning the female–female competition for good genes, thus incentivizing increased visible ovulation signs with which to attract high GC males. Note these effects of η are in line with empirical work, for example; see [6,28]. In general, the more reproductive stochasticity present, the smaller the benefit to a female of having visible ovulation signs, meaning the smaller her ovulation signal. Increasing male reproductive stochasticity (meaning decreasing the amount of reproductive inequality among males) decreases the female ovulation traits m*, ℓ^*. With more male reproductive stochasticity, strongly signalling females are less likely to get mates with high GC males, simply by chance, leading to females investing less energy in such signalling. With high reproductive stochasticity among males, high reproductive stochasticity among females effectively takes away any benefit to females of having visible ovulation signs (i.e. the benefit from winning the female-female competition for high GC males), leaving such signs only as a cost and why both m*, ℓ^* sharply decrease in the case of large male and female reproductive stochasticity.

In many situations, parameter ρ (the correlation between males' GC and NGC) does not have much of an effect (see the electronic supplementary material). However, when ρ does have an effect, increasing ρ will result in increased m*, ℓ^*. Such an effect will appear with larger benefit b, larger relative weight of non-genetic benefits η, and smaller reproductive stochasticity in females ε_f. Figure 3d illustrates one such scenario. In general, visible ovulation signs will increase when females benefit from attracting high GC males. With larger ρ, high GC males are increasingly also high in NGC, thus doubling the benefit an attracting female would receive from them. Indeed, as figure 3d illustrates, ρ intuitively has larger effects with higher b, meaning when such quality coming from high GC males is the greatest. In addition, ρ will have larger effects with larger η (as females would be able to obtain even more benefit via NGC by attracting high GC mates) and smaller ε_f (as the increasingly guaranteed ‘prize’ of winning the female–female competition would be higher in both GC and NGC).

Testing hypotheses against data. As outlined in the material and methods section, we were able to test two of our model's predictions against empirical data. First, our model predicts that species with larger group sizes will have more visual ovulation signs present. Figure 4a shows these breakdowns: species with ‘present’ and ‘absent’ visual ovulation signs have different distributions of group size. As predicted by our model, species with ‘absent’ visual ovulation signs are more likely to have smaller group sizes, while species with ‘present’ visual ovulation signs are more likely to have larger group sizes. The mean group size for species with ‘absent’ visual ovulation signs is 18.67, while the mean group size for species with ‘present’ visual ovulation signs is 37.41, a 100% increase. After testing for significance using a simulation-based phylogenetic ANOVA test [60,61], we found these differences to be significant (F = 12.868,  p = 0.002).

Figure 4.

(a) Comparing primate species' average group size by females having ‘present’ or ‘absent’ visual ovulation signs. After controlling for phylogeny, species with higher group sizes are more likely to have visual ovulation signs, and species with concealed ovulation smaller group sizes (using phyloANOVA: F = 12.868,  p = 0.002). Numbers inside the bars indicate the numbers of species contained in each category. (b) Box plots depicting the distributions of L/G for species with each ‘absent’ (33 species) and ‘present’ (15 species) ovulation signs. Including phylogeny, the model with ‘absent’ and ‘present’ having distinct means has a 62% relative probability. Each box plot depicts the minimum, maximum, median, and quartile values, while the black x's mark the means.

Second, our model predicted that species with a higher risk of infanticide (i.e. higher average L/G) would have more visual ovulation signs present. Figure 4b depicts the box plots of L/G for species with ‘absent’ and ‘present’ ovulation signs visible. We see that, as predicted, the mean L/G value for species with ‘absent’ ovulation signs is less than the mean for species with ‘present’ ovulation signs, and in general they have different distributions, in line with our model's prediction. After considering phylogeny using [63], there is still support for this correlation between infanticide risk (L/G) and visual ovulation signs (62% relative probability from model selection; see the electronic supplementary material for more details).

4. Discussion

Here, we have developed a framework for modelling the evolution of visual female sexual signalling, allowing for both possible benefits of paternity concentration and paternity confusion. In general, we have shown that increased visual ovulation signalling could have evolved for a variety of reasons. For example, our model predicts increased visual ovulation signals with increased group size (N), increased variation in male quality (b), decreased costs of having ovulation signs visible (c), increased effects of infanticide (α, β, κ), increased weight of male genetic quality (i.e. decreased η), increased male reproductive inequality (i.e. decreased ε_m), and increased correlation between male-provided genetic and non-genetic benefits (ρ). These predictions are all testable.

Overall, we see a costs trade-off between ovulation signs magnitude (m) and ovulation signs length (ℓ). In situations where strong ovulation signs magnitude is favoured (for example, when males evaluate their paternity exclusively on the basis of female signs; κ = 1), we typically see smaller ℓ^*. Intuitively, a female will only have so much energy/resources to spend on visual ovulation signalling. When strong ovulation signs magnitude is favoured, the female will invest that energy allotment on such magnitude rather than the length of her visual ovulation signal. Hence, where we see a much larger m*, we typically see a smaller ℓ^*.

Our results show the evolution of ovulation signalling can be driven by indirect genetic benefits. A major conclusion of our work is that females can evolve sexual signalling even in the absence of genetic benefits provided by males (i.e. when η = 1). As expected [64], the effects of ‘good genes’ can, however, be trumped by direct costs. Moreover, the effects of good genes in our model are much weaker than those of infanticide.

It is important to note that we never intended to formalize and then test specific verbal arguments (such as the reliable indicator hypothesis against the graded signal hypothesis, which, as we discussed above, have overlapping elements and some vagueness). Rather, our approach was to start with major evolutionary forces (including different types of selection) known to act in natural populations and then investigate the effects of various parameters—group size, infanticide, genetic versus non-genetic male benefits, etc.

Our theoretical work is helpful in clearing up past confusion. Contrary to some expectations (see the Introduction), both our modelling and empirical work support greater infanticide risk leading to increased ovulation signs rather than concealed ovulation. Another example is provided by Charles Nunn's statement regarding paternity confusion, ‘[It] is not able to explain the conspicuousness of [the visual ovulation] signal. This hypothesis also cannot explain why features of exaggerated swellings bias certainty towards certain males in the social group.’ [12, p. 238]. However, our modelling results show that the ‘conspicuousness of this signal’ does not have to be a result of the selection pressures owing to infanticide, but rather to one or more of the potential ecological costs identified here. Moreover, our model shows that the risk of infanticide can indeed explain why exaggerated swellings would bias certainty towards ‘certain males’ (i.e. more powerful/effective ones). Such males are both more likely to take over a group from a current alpha-male (and hence the risk of infanticide from them be imminent), and more likely to be stronger/better able to protect a female's infant from any possible infanticide attempt by a new alpha-male.

Our work complements the sparse earlier modelling work on visual ovulation signalling. Pagel [25] used mathematical modelling to argue that male–male competition for females will always occur and to support the reliable indicator hypothesis. Huchard et al. [52] used agent-based simulations, attempting to reconcile the reliable indicator and graded signal hypotheses. They showed the reliable indicator hypothesis by itself to be viable and also concluded that mate choice for a direct benefit like fertility could lead to an indirect benefit like good genes. Nakahashi [53] modelled the coevolution of ovulation signs and male mating behaviour, assuming that males can estimate the fertility of females with whom they mate; he did not model the effects of infanticide explicitly. Compared to these earlier efforts, our model explores a much wider range of evolutionary factors (including the effects of infanticide and genetic and non-genetic benefits to females) under more biologically realistic assumptions (e.g. by explicitly splitting the female cycle into discrete units of time and by assuming that males can only estimate the fertility of females with whom they mate, rather than know it exactly).

Although our models aim to describe any primate species with multi-male multi-female groups, in particular there has been much interest on the question of why do human females have concealed ovulation [21,22]? Is this related to some larger ‘adaptive suite’ in human evolution [38]? While our results do not support the infanticide hypothesis for the evolution of concealed ovulation in humans [29], our model does make predictions about evolutionary dynamics which lead to several new hypotheses for why concealed ovulation may have evolved in our ancestors.

In particular, such ecological causes for the disappearance of visual ovulation signals could include: (i) group size decreasing (potentially as a result of changes in ecological resource distributions, predators, etc.), (ii) within-group variation in male quality decreasing or males becoming more egalitarian [65,66], (iii) non-genetic components of male quality becoming highly beneficial compared to genetic components, (iv) physiological costs of having ovulation signs increasing (owing to increased water retention, body weight, infection/damage, predation, etc.; potentially as a result of the adaptation of bipedalism), and/or (v) the effects of infanticide decreasing. Note with regard to (i) above, visual ovulation signs are expected to disappear with the transition to monogamy since the benefits of visual ovulation signalling (i.e. attracting good and/or more mates) disappear, while the costs remain present [38,67]. This is in line with both previous phylogenetic studies on the primate mating system and visual ovulation signs [49,50], and also our model should we set N = 1.

Our approach comes with several limitations. Like many models, we assume discrete generations for mathematical simplicity. We also assume an additive fitness function. Explicitly accounting for the female cycle has made obtaining informative analytical results not feasible, leading to us rely on agent-based simulations. In our model, we did not impose female cycle synchrony. Assuming so would increase female–female competition and hence ovulation signalling. We do not consider the effects of any extended mate-guarding (i.e. mate-guarding lasting longer than our one discrete unit of time). We do not explicitly allow for variation in signal reliability, instead assuming a female's peak day of fertility and day of peak visible ovulation signs to line up. We also only consider multi-male, multi-female mating systems, excluding other mating systems from direct consideration in our model. In addition, our description of ‘male quality’ is purposefully general; we do not explicitly take into account male age, rank or any other similar known factor in primate mating.

While much work still needs to go into understanding the evolution of visual female sexual signalling, our model is able to resolve some old controversies and shed new light on this important issue. In particular, our work suggests several ecological pathways by which visual ovulation signals could have evolved or diminished, while also showing that infanticide can provide important selection pressure on the evolution of visual ovulation signs. Aside from the two predictions explored in figure 4, applying more of our theoretical insights to other specific systems/species would require further detailed information on fitness components and ecological factors, which we hope to have inspired for future empirical work.

Appendix A. Model details

Females. The female fertility function is assumed to have a triangular shape with zero probability of fertilization outside of a female's C fertile days (see the electronic supplementary material, figure S1). Given female traits m_i ≥ 0 (ovulation signs magnitude) and (ovulation signs length), the amount x_i(d) of ovulation signs visible on any day d of the cycle is defined as

A 1

for γ > 0 (note figure 2 depicts x_i(d) with γ = 2) and d measured from the start of ovulation signs being visible. Note m_i represents the peak of x_i(d) and ℓ_i the width of the non-zero portion of x_i(d). Traits m and ℓ are treated as controlled by two unlinked haploid loci. Mutation effects in m_i are randomly chosen from a normal distribution N(0, σ²), whereas mutations in ℓ_i are discrete, representing adding or subtracting exactly one day to or from a female's time of having ovulation signs visible.

Males. Male-provided genetic (y_g) and non-genetic (y_ng) benefits are randomly drawn from the bivariate normal distribution, each with mean and standard deviation b, and correlation parameter ρ; parameter characterizes mean male quality and b > 0 the extent of variation. Males are ranked according to the value of their genetic quality: y_{g, j} such that y_{g, 1} > y_{g, 2} > … for each male j. Small b implies small additional benefits to females from investing in ovulation signalling. For simplicity, the distribution of male quality in the population remains constant, e.g. as a result of mutation-selection balance.

Mating pairs. On each day, mating pairs are formed by first randomly perturbing both the male trait y_g and female trait x by adding to each an independent, normally distributed random variable with standard deviation ε_m ≥ 0 and ε_f ≥ 0, respectively. Males and females are then sorted according to these perturbed values and mating occurs between individuals of the same order.

Female fertility. In the case of no infanticide, we define female fitness (fertility) as

A 2

Here, w₀ is the baseline fitness, y_{g, i} the genetic benefit provided by the male who fertilizes female i, the average non-genetic benefit provided by all female i's mates, and the costs to a female of supporting her visual ovulation signs with , a female's average visual ovulation signs. Our choice of the cost term implies that there is a fitness trade-off between the m and ℓ traits so that having large values of both traits implies very large fitness costs.

Infanticide. Each male in a female's group can affect her fitness directly, because of infanticide, although males will differ in the effectiveness of their actions' (e.g. owing to strength, rank, size and alliances). We assume each male's effectiveness to be proportional to f(j) for each male j (sorted by males' y_g) via an exponential function: f(j) ∼ e^−ωj with parameter ω > 0 controlling the amount of disparity among males in their corresponding effectiveness within the group. Note a larger value of ω indicates more disparity, and ω = 0 equality.

A male's contribution g(p_j) to female fitness (positive via helping protect the offspring, or negative via not helping and/or harming the offspring) depends on his perceived probability of paternity p_j. Adapted from van Schaik et al. [24], we define g(p_j) = α[1 − β(1 − p_j)^τ], with parameters α, β > 0 and τ ≥ 1. Note α determines the maximum benefit a female can obtain from a male protecting her offspring from infanticide, while α(1 − β) determines the maximum cost a female can incur from a male not protecting her offspring from infanticide. The overall effect of infanticide on female fitness (that gets added to the right-hand side of equation 2) is , summing over all males j. Note whenever α = 0, there are no effects of infanticide.

A male, in general, would not know his actual paternity probability and would instead have to estimate it, when deciding on protecting an infant or committing infanticide. Consider male j mating with female i with visible ovulation signs x_i(d). We postulate that the male-estimated (‘perceived’) probability p_j of being the father of her offspring is proportional to (x_i(d))^κ where κ∈[0, 1] is the weight males put on visual ovulation signs. With multiple matings with the same female, we sum up the corresponding terms (x_i(d))^κ.

Simulations. This model was implemented independently in both C++ and Matlab with similar results. Migration occurs with only females migrating between groups, although simulations were also run with only males migrating between groups and the results were unaffected. Mutation occurs with a mutation rate per gene per generation of 10⁻³. 2N offspring are created in each group every generation, with N female and N male, determined randomly. The expected number of offspring per female is 2; the actual number is random, proportional to each female's fitness.

The agent-based model uses 16 different initial conditions in all simulations: every combination of m = 0, 0.3, 0.6, 0.9 and ℓ = 1, 2, 3, 4. Parameters used include: G = 400; T = 100, 000; D = 29; C = 7 (with probabilities of fertilization on each of these days being 0.125, 0.25, 0.375, 0.5, 0.375, 0.25, 0.125, respectively); w₀ = 1; N = 4, 8, 16; b = 0.1, 0.2, 0.4; c = 0.1, 0.2, 0.4; η = 0.25, 0.5, 0.75; ρ =−0.5, 0, 0.5; γ = 0.5, 1, 2; ; ; ε_m = 0, 0.25, 0.5; ε_f = 0, 0.01, 0.05, α = 0, 0.2, 0.6, 1; β = 1.25, 1.5, 1.75; τ = 1, 2, 4; ω = 0, 1, 2; and κ = 0, 0.5, 1. Further information on parameter values tested and each of their effects is available in the electronic supplementary material.

Data accessibility

All supporting data are available in our online electronic supplementary material.

Author's contributions

K.R. and S.G. designed research, analysed data and wrote the paper.

Competing interests

We declare we have no competing interests.

Funding

K.R. was supported by a NSF Graduate Research Fellowship. S.G. was supported by the National Institute for Mathematical and Biological Synthesis through the NSF awards EF-0832858 and DBI-1300426, with additional support from The University of Tennessee. All numerical simulations were run on the Volos cluster, which is supported by the NIH award GM56693 (S.G.).