Philosophical Transactions of the Royal Society B: Biological Sciences
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Fertility signalling games: should males obey the signal?

Viktor Kovalov

Viktor Kovalov

Department of Evolutionary Biology and Environmental Studies, University of Zurich, 8057 Zurich, Switzerland

[email protected]

Contribution: Investigation, Methodology, Visualization, Writing – original draft, Writing – review & editing

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Hanna Kokko

Hanna Kokko

Department of Evolutionary Biology and Environmental Studies, University of Zurich, 8057 Zurich, Switzerland

Konrad Lorenz Institute of Ethology, University of Veterinary Medicine, 1160 Vienna, Austria

Organismal and Evolutionary Biology Research Program, Faculty of Biological and Environmental Sciences, 00790 University of Helsinki, Finland

Contribution: Conceptualization, Investigation, Methodology, Project administration, Supervision, Writing – original draft, Writing – review & editing

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    Abstract

    Game theory is frequently used to study conflicting interests between the two sexes. Males often benefit from a higher mating rate than females do. A temporal component of this conflict has rarely been modelled: females' interest in mating may depend on when females become fertile. This sets conditions for male–female coevolution, where females may develop fertility signals, and males may obey the signal, such that they only target signalling females. Modelling this temporal aspect to sexual conflict yields two equilibria: (i) a trivial equilibrium without signals and with males targeting all females, and (ii) a signalling equilibrium where all females signal before ovulation, and where either some, or all, males obey the signal. The ‘all males obey the signal’ equilibrium is more likely if we assume that discriminating males have an advantage in postcopulatory sperm competition, while in the absence of this benefit, we find the ‘some males obey the signal’ equilibrium. The history of game-theoretic models of sex differences often portrays one sex as the 'winner' and the opposite sex as the ‘loser’. From early models emphasizing ‘battle of the sexes’-style terminology, we recommend moving on to describe the situation as non-signalling equilibria having stronger unresolved sexual conflict than signalling equilibria.

    This article is part of the theme issue ‘Half a century of evolutionary games: a synthesis of theory, application and future directions’.

    1. Introduction

    Sexual reproduction entails both cooperative elements and conflict. Clearly, it is of interest for both sexes that fertilizations happen without too much delay when the female has eggs ready to be fertilized. On the other hand, conflict is rife: in many situations, optimal mating rates differ between males and females [1,2]. In species with conventional sex roles males may approach females, including active coercion [310], at a time when it is not in the female's interest to invest time or energy in matings.

    Table 1. The list of parameters and notations used in the model.

    symbol description
    S label indicating females that signal for a proportion T of the time (preceding ovulation)
    s label indicating females that never signal
    D discriminating males (males that obey the signal, i.e. only approach and mate with females that currently signal)
    d non-discriminating males (males that disobey the signal and approach and mate with all females)
    x proportion of S females
    y proportion of D males
    T length of time before ovulation during which S females signal (T < 1)
    k number of matings
    c cost of suboptimally high number of matings (when k > 1)
    τ time period after a mating during which a male is unable to mate again
    md mating rate of a female with d males
    mD mating rate of a female with D males
    F(k) fecundity of females that have mated k times
    Ad(k) Poisson distribution of k for s females mating with d males
    Ed(k) Poisson distribution of k for non-signalling S females mating with d males
    Ld(k) Poisson distribution of k for signalling S females mating with d males
    AD(k) Poisson distribution of k for signalling S females mating with D males
    WS expected fitness of S females
    Ws expected fitness of s females
    WD expected fitness of D males
    Wd expected fitness of d males

    While a considerable number of theoretical and empirical studies investigate how the conflict plays out, the temporal component of the statement above appears to have received scant attention. It is uncommon that a female reproduces absolutely continually throughout her life. Much more commonly, reproduction occurs in bouts, and mating with a specific female will give the male different prospects of paternity depending on when the mating occurs. In many systems, females accordingly signal their receptivity to males: examples include sexual swellings in primates (which cycle ‘on’ and ‘off’ depending on the oestrous cycle [11,12]), ontogenetic change of body colour in damselflies [13], pheromone production in Lepidoptera (which in many species ceases after the female has acquired sperm [14,15]), temporal development of coloured patches in fishes [16] and lizards [17], and glowing in female glow worms [18].

    One might think that prospects for significant conflict are reduced in temporal settings of female receptivity: males ought to pay attention to female signals, given that these are likely to be honest indicators of the female being near her fertility peak. However, the situation is not guaranteed to be conflict-free. Sperm storage inside a female's reproductive tract is a common trait in internally fertilizing species, with sperm often staying alive for the duration of a mating season [19]. Under a temporal separation of insemination and fertilization, it may pay for a male to ignore the signal and mate with females irrespective of their signalling status, or to begin guarding females earlier than what is ideal from a female perspective (possibly even before the female is mature [20]). In seasonal systems that include hibernation (e.g. echidnas), mating activities may also begin when females are still hibernating [21].

    A male strategy of ignoring the female signal, despite potential benefits, is not obviously evolutionarily stable. Given that females are likely to mate with several males during a mating season, sperm placed inside a female early will be subject to sperm competition later, implying that sperm precedence patterns (i.e. whether sires are predominantly the first or the last males to mate) become relevant. Strong last male sperm precedence is a very common [2225] but not ubiquitous finding [2628]. Thus, males mating with females irrespective of their signalling status risk losing paternity to males who concentrate their mating attempts closer to females' ovulation. Whether such losses translate into opportunity costs is also likely to depend on how quickly males can re-mate with new females after inseminating one (male mate choice theory in general predicts males to discriminate only when time and energy costs of each mating are non-negligible [29]).

    On the female side, the fitness consequences of ‘uninvited’ matings may be either positive or negative. Superfluous courtship, harassment or mating activities may attract the attention of predators [30] or hamper feeding or other activities related to self-maintenance [3136]. On the other hand, the risk of remaining unmated [37] is likely reduced if males approach all females regardless of the signal—though the extent of this is a priori not clear, as males who mate very indiscriminately may not have time to attend to females who are currently signalling their receptivity, and indiscriminate habits of males may therefore simply shift matings over a larger temporal range rather than guarantee fertilization specifically when females ‘request’ it.

    Given the complexity of the above situation, it is surprising that few models have investigated the coevolution of female signals of fertility and male strategies (whether to pay attention to the signal or to ignore it and approach all females instead). There is a significant body of work regarding mate guarding decisions in relation to the time until a female can be mated in situations where matings are physically only possible at moult (many crustaceans [3841]), but here the models do not ask how the males can tell the time to moult (more precisely, evolution of female signals are not modelled).

    Some papers do consider female signals, however. Nakahashi [42] assumes, in a primate-inspired model, that females go in and out of oestrus independently of each other. If they are guarded by an ‘alpha’ male, they cannot mate with any others and cannot consequently achieve multiple paternity. Paternity confusion is thought to be a benefit to females, and therefore there is sexual conflict. Multiple mating is thus assumed to be a priori beneficial to females. The model needs to introduce some cost to some signalling strategies to find its equilibria; but since female signalling does not lead to costs ‘on its own’ in the framework of this model, this is done by assuming that it is costly for females to change their strategy from an ancestral one. Our view is that this assumption may be difficult to justify over long periods of evolutionary time.

    Glover & Crowley [43] consider a more complex set-up of potential male behaviours, where males may (1) invest in being helpful (at the expense of other matings) to maximize reproductive output when chosen by a female, (2) invest little into a given mating, trying to increase number of matings, or (3) resort to coercive behaviours as a way to avoid evaluation by a female (these males also do not help with parenting). Females have a preference for helpful males. They are assumed to be able to tell the male strategies apart, even between (1) and (2); whether it is possible for females to do this in reality, i.e. assess the helping tendencies of the male already at mating, may depend on the system under consideration (small primate groups, where individuals know each other, might qualify). Coercers may be removed through policing in this model, and females are left in peace when they are unreceptive. Since the last fact is an assumption, not an outcome, of this model, it does not directly focus on the question we raised above: is a rule where males obey the signal evolutionarily stable?

    Rooker & Gavrilets [44] and its follow-up paper [45] assume that females are approached by males, and become mated, all the time, irrespective of their signals; but females that do not signal, or signal less intensely, mate with a lower-quality subset of males than those that signal. This model, once again inspired by primates, also models a risk of infanticide.

    An interesting general theme emerges from this set of models (including the references in a highly useful ‘mini-review’ in [44]). It is quite common to ask the ‘why signal?’ question, but the payoff structures in existing models do not include an option where signalling unreceptivity might make the female benefit by effectively ‘giving her a break’ from constant male attention. Costs of harassment, or more generally, costs of being at the receiving end of male attention, are typically not included in the model at all, despite ample evidence of such costs (see references above). Here we fill in this gap and show when precisely males and females can coevolve to a signalling equilibrium where females switch their fertility signals on and off, and males (either to a large extent, or completely) obey the signal and leave non-signalling females alone.

    2. The model

    We assume that females and males come in two categories each, making this a game theory model. Females are denoted s or S, and males d or D (see table 1 for a summary of notation. Type s females never signal while type S females divide their time such that they switch from a non-signalling to a signalling state some time before ovulation. The duration of the time window during which matings can happen is one unit both for s and S females. The only difference is that S females spend the last portion of this time window, of length T (T < 1), signalling. Fertilization happens at the end of the time window, with sperm usage rules that are outlined below.

    Males either seek out and only mate with S females when they signal (D males, for ‘discriminating’), or they ignore the signal (‘d’ males) and consequently attempt matings with any female they encounter (both with s and S, and if S, both when she signals and when she does not). Mating attempts are successful in the sense of them leading to a copulation, while they are not necessarily successful in yielding paternity (see below). We use the notation where the current proportion of S females is x, and the current proportion of D males is y (see table 1 for the full list of parameters and notations).

    To keep the model simple, we assume that males are equally capable of finding signalling and non-signalling females. It means that signalling does not elevate females' visibility. Signalling only helps D males to recognize signalling females as a mating partner. This allows us to focus on the key question (can signalling evolve to ensure that non-signalling times are spent without unnecessary male attention?) as signals in our model do not make a female easier to find. Similarly, for simplicity, we assume no direct costs of signal production. Signalling may impose costs on the female, however, in cases where it leads to suboptimally many males approaching the female. Both s and S females suffer if male attention is higher than needed to ensure fertilization, but all else being equal, S females are approached by more males (d and D types, as opposed to d only) when they signal—making it a non-trivial exercise to understand how females as a whole (i.e. over the whole mating window) can benefit from dividing their time between signalling and non-signalling. We assume that the number of realized matings can be used as a proxy for male attention that a female receives, and females pay a cost in terms of decreased fecundity when total attention was high during the mating window. This is achieved with a cost parameter c > 0: females that mate k times before the end of the time window are assumed to have fecundity

    F(k)={0k=0ec(k1)k1.2.1

    This function peaks at k = 1, i.e. monandry is optimal for females (F(1) = 1 regardless of the value of c). Larger values of c imply that fecundity decreases faster with an increased number of matings.

    We assume that each male can target one female every τ units of time (typically τ < 1, but τ > 1 is in principle permitted as well). We will show solutions including large values, e.g. τ = 0.3 indicates that a male can only mate approximately three times during the time that it takes a female to complete a mating time window and proceed to laying eggs (or to gestation, should the species be viviparous). Males search for females independently; therefore, there is stochastic variation in the number of males approaching each female. The stochasticity takes the form of a Poisson process due to the independence assumption. We also assume that females ovulate asynchronously, which allows us to ignore temporal fluctuations in the sizes of the target populations of females of type s and S, or the availability of d and D males.

    Each female gives paternity to just one male, but the earlier matings with other males still impact the sire's fitness, as they are the proxy for total male attention that have the potential to reduce female fecundity (equation (2.1); note that the function is the same for s and S females, but the total number of matings may differ between them). In situations where a s female has mated with d males only, the sire is chosen randomly among them. The sire's offspring production is then equivalent to the female's fecundity.

    In situations where an S female has mated with one or more males, we assume a temporal order where the matings during her signalling time, if such matings exist, outcompete earlier matings in sperm competition (last male sperm precedence): an S female only gives paternity to males that mated with her before she began signalling if she was not found by any male after the start of signalling (there is a non-zero probability for this latter duration to yield zero matings, but this probability is not very large unless τ is large, making males inefficient at finding her, and T is small, meaning that signalling time is limiting). If there are matings during her signalling time, then we investigate two distinct scenarios, which differ in the biological mechanism we envisage behind last male sperm precedence. In scenario 0, we assume that all males that mated with her during the time of her signalling are equally competitive with respect to sperm competition. In other words, in scenario 0, the fact that d males have a larger pool of females to mate with is not assumed to make them sperm-depleted, and any last male sperm precedence reflects new sperm displacing older sperm instead. By contrast, scenario 1 assumes that d males do suffer from sperm depletion and are consequently less competitive in each of their matings. To investigate the maximum effect that this can have, we assume that D males have absolute paternity advantage when they mate. To be precise, in scenario 1, a d male that has mated with a signalling female can only become the sire if she has not mated with any D males. In other words, should a female have mated with males of both types during signalling, the paternity lottery only occurs among the D males in scenario 1, but among all males (that mated with her during signalling) in scenario 0.

    (a) Mating probabilities

    How often males target females and, hence, how many matings every female will experience depend on female type, male type, and (for S females) whether or not a female has already started signalling. Females of types s and S will all be approached by d males, and this applies whether or not type S females are currently signalling (since d males simply ignore the signal). All females therefore experience a baseline mating rate of md=(1y)/τ. Here a proportion 1y of males approach all females (target size 1) at a rate of 1/τ each. A signalling female experiences additional approaches by D males. This leads to an elevated mating rate for a subset of a target (female) population: the size of this target is Tx, since a proportion x of females signal a proportion T of their time; a proportion y of males focus solely on this target, at a rate of 1/τ each, yielding an additional mating rate of mD=y/Txτ for S females while they signal.

    For s females, the mating rate of md is applied over the entire mating time window of one time unit. We assume that males search for females independently of each other, and therefore the Poisson distribution will describe the number of matings that a female experiences during the time window of 1 unit of time. Thus, the probability that any given s female will mate k times equals

    Ad(k)=mdkemdk!,2.2
    where the notation Ad is chosen to reflect that these are all the matings that happen between s females and d males.

    For S females, we need to derive the number of three different types of matings. First, there is a time period of length 1 – T during which she does not signal yet. During this time, only d males approach her, and she experiences a mating rate md (just like the s females). We denote the probability that she experiences precisely k matings of this type by Ed(k), where E stands for ‘early’ and the subscript refers to d males:

    Ed(k)=((1T)md)ke(1T)mdk!.2.3

    Second, once a S female has started signalling, she will be approached by d and D males alike. Males of type d still approach her with the same mating rate md as before the start of signalling, but this rate is now applied for T units of time. We denote the probability that she experiences precisely k matings of this type by Ld(k), where L stands for ‘late’:

    Ld(k)=(Tmd)keTmdk!.2.4

    Third, D males perceive signalling S females as a target, resulting in a per-female mating rate mD. We adopt the notation AD(k) for the probability that k matings happen to a female (once again, the letter A reflects these being all the matings between S females and D males). This rate is applied for T units of time, yielding probabilities

    AD(k)=(TmD)keTmDk!.2.5

    (b) Fitness

    We next need to proceed from the Poisson-distributed mating numbers to expected fitness of females and males.

    We start with females of type s. By the time an s female is ready to lay eggs (or to give birth), she will have only mated with d males, with a probability distribution Ad(k). Her expected fitness is

    Ws=i=0Ad(i)F(i).2.6
    The frequency of s females is 1 – x, and the frequency of males that can mate with her is 1 – y, thus on a per capita basis, males of type d will gain fitness
    Wds=1x1yWs,2.7
    via the route of mating with s-type females. There are also other routes for these males as type d males may gain paternity with S females too (see below).

    Interactions between females of type S and males of either type d or D are somewhat more complicated. Females of type S have expected fitness

    WS=i=0Ed(i)j=0Ld(j)k=0AD(k)F(i+j+k).2.8

    Note that the above includes the possibility that the female does not mate at all, which is correctly accounted for since we assume that F(0 + 0 + 0) = 0. Note that even though signalling in our model has no impact on the ease with which females are found by males, S females have a lower risk of unmatedness than s females: the former are approached by a larger pool of males (d and D, as opposed to only d males).

    While the female side is rather straightforward, the male side is more complicated because of our assumptions that the temporal structure of matings matters. Males only gain paternity if the female mated at least once, which we take into account in what follows. There are three possible outcomes (ignoring the outcome of a female remaining mateless, which adds zero paternity to anyone), of which the last one comes in two variants.

    Outcome 1: The female mates at least once before she begins signalling and does not mate afterwards.

    This outcome does not give any positive paternity expectation for D males (that do not mate with non-signalling females). For d males, the per capita paternity gain is

    WdS1=x1yLd(0)AD(0)i=1Ed(i)F(i).2.9
    Outcome 2: The female mates at least once during signalling times (regardless of how many times she has mated before she started signalling), and none of those matings are by D males.

    Like in outcome 1, there is no paternity for D males in this case. For d males, the per capita paternity gain is

    WdS2=x1yAD(0)i=0Ed(i)j=1Ld(j)F(i+j).2.10
    Outcome 3: The female mates at least once during her signalling time (regardless of how many times she has mated before her start of signalling), and at least one of those matings is with D males.

    This outcome behaves in a manner that depends on our assumptions regarding sperm competition. We have two variants of the model, of which the first assumes equal male competitiveness between d and D types (scenario 0). In this variant, d males have per capita paternity gain

    WdS3=x1yi=0Ed(i)j=0Ld(j)k=1AD(k)F(i+j+k)jj+k,2.11
    while D males gain
    WDS3=xyi=0Ed(i)j=0Ld(j)k=1AD(k)F(i+j+k)kj+k.2.12

    Note that the last term, which computes the expected paternity among the males that mated with the female during signalling, only includes j and k, not i, since i counts the matings that will not lead to any paternity (latter males enjoy the advantage of sperm precedence).

    The second variant of the model assumes that D males have an advantage (scenario 1). In this case, outcome 3 never yields paternity for d males, while D males' expected paternity gain is

    WDS3=xyi=0Ed(i)j=0Ld(j)k=1AD(k)F(i+j+k).2.13
    As a whole, we then can compute d and D male fitness, respectively, as
    Wd=Wds+WdS1+WdS2+WdS3(forscenario0)2.14a
    Wd=Wds+WdS1+WdS2(forscenario1)2.14b
    andWD=WDS3.2.15

    3. Results

    For each combination of current proportions x (of females who spend some time signalling) and y (of males who obey the signal and ignore non-signalling females), we can ask if selection favours D over d (i.e. is WD > Wd) and/or if selection favours S over s (i.e. is WS > Ws). If both inequalities are true, then signalling coevolves with males obeying the signal; these are indicated as white areas in figures 1 and 2. In black areas, the opposite is true: type s females have higher fitness than S females, and d males have higher fitness than D males. In between, it is also possible that signalling increases in frequency, while obeying the signal does not (olive grey in figures 1 and 2), or vice versa (males evolve to increase the proportion of D males, while the proportion of signalling females decreases; this happens in rose-coloured regions in figures 1 and 2).

    Figure 1.

    Figure 1. Results of scenario 0 with d and D males being equally competitive in situations involving sperm competition. X-axis—proportion of S females, y-axis—proportion of D males. Values for τ (scaling how often males can mate; small τ implies frequent matings) and c (fecundity cost for suboptimally high mating frequency) are as indicated. Value of T = 0.05 throughout. Colours separate regions where the currently winning strategies (that are predicted to increase in frequency) are {s,d} (black, down-left arrows), {s,D} (rose-coloured, up-left arrows), {S,d} (olive-green, down-right arrows) and {S,D} (white, up-right arrows). Red circles indicate signalling equilibria when x = 1 and further evolution of y is not happening. (Online version in colour.)

    Figure 2.

    Figure 2. Like figure 1, but with scenario 1, where D males have a competitive advantage in situations involving sperm competition. (Online version in colour.)

    For an equilibrium where females signal and males obey the signal to be stable, one needs to see a white-coloured area near x = y = 1 (top right corners of each of the plots in figures 1 and 2). In scenario 0 where we assume that d and D males are equally competitive with respect to sperm competition, this never happens (figure 1 shows results for only one value of T, but results change very little with T (see electronic supplementary material 1 for more details about T variation)). Instead, the pattern can be summarized as follows. If τ is small, i.e. if males can mate with many females in a quick sequence, then males are selected to ignore the female signal (arrows pointing downwards in figure 1). As long as there are some D males present (y > 0), mating rates of S females are above those of s females, and since superfluous matings are costly and males are capable of high mating rates, females typically mate more often than is optimal for them. Since s females mate less often than S females, the latter suffer more from suboptimally high mating rates and the proportion of S females declines continually. This process occurs throughout the black areas in figure 1.

    The situation is different if τ is larger, so that males are not very efficient at locating females to mate with (or, alternatively, it takes them a long time to replenish sperm supplies). When τ = 0.15 and c is large, then a system near the origin x = y = 0 does not easily move away towards signalling. Here females are selected to avoid signalling (whether {x,y} is in the black or the rose-coloured region), and even if males were in the rose-coloured region where obeying the signal pays off, the system as a whole moves towards x = 0 and, since lowest values of x mean very poor fitness for D males that try to focus on mating with the very rare signalling females, the trivial equilibrium x = y = 0 cannot be easily escaped.

    However, when τ is very large (rightmost columns of figure 1), then an alternative equilibrium appears where the {x,y} pair is in the white region if y is small, and in the olive-green region when it is large. This implies that S females have better fitness than s females, while males are selected to obey the signal if the current proportion of discriminating males is small (i.e. small y implies that WD > Wd). The opposite is true at larger proportions (large y implies that Wd > WD). This means that there is negative frequency dependence on male behaviour and positive selection on female signalling; the system ends up in an equilibrium where all females signal and some males obey the signal while others do not (all cases where olive-green occurs above white, when viewed from the vertical end of each figure at x = 1). Reaching this mixed equilibrium requires, in most cases, overcoming an invasion barrier: the white region, which allows both D and S to increase in frequency simultaneously, does not usually extend to near x = y = 0, though it can do so if the costs of multiple mating are very small.

    The situation is quite different in scenario 1, where we assume discriminating D males to experience benefits of sperm competition. The biological justification of this is that d males ‘waste’ a lot of sperm in matings that occur far away (in time) from the female's fertility, and they are more likely to be sperm-depleted at any point in time. Now, whenever τ is large and/or costs of multiple mating are small (low c), the area near x = y = 1 is white (figure 2). This means that the system is capable of coevolving to an equilibrium where all females signal their fertility when they are close to their fertility peak, and all males discriminate, i.e. leave females in peace when they are not signalling.

    However, the black and rose-coloured regions still exist in figure 2, meaning that giving D males a competitive advantage does not automatically move the system to its evolutionarily stable point where all males are D and all females are S. Instead, especially when males are capable of multiple mating (low τ) and when costs of multiple mating are significant (high c), the system may remain at x = y = 0 where signalling does not evolve, and males approach any female. Since at this equilibrium females mate suboptimally much, there is unresolved sexual conflict that males can be said to have ‘won’. Whenever the area around x = y = 0 only contains black and rose-coloured regions, there is no easy way to move towards the signalling equilibrium, even if this equilibrium, should it have arisen, is stable. In some cases (high τ and low c), however, there is a white region extending to the origin. Here we can predict an easier evolution towards a signalling equilibrium, especially if evolutionary movement along the x-axis is speedier than along the y. Note that a game-theoretic treatment is agnostic about the genetic architecture of traits; in reality, it may be that evolvability is higher in females or males, or vice versa, e.g. because of more additive genetic variation underlying the relevant behavioural traits in one of the sexes, or a higher mutation rate. If such asymmetries make the evolution of female signalling easier than the male behavioural responses to the signals, then it is easier to move away from the trivial equilibrium x = y = 0 towards x = y = 1 where females signal, and males obey the signal.

    4. Discussion

    It is clearly possible to find alternative stable states in our model: no females signal and no male obeys the signal (the s–d equilibrium), all females signal and all males obey the signal (the S–D equilibrium), or also, if we give discriminating males no advantage in sperm competition, a mixed equilibrium where all females signal and some males obey the signal while others do not. This shows that females, as a group, can achieve what a priori appears difficult: even though a female unilaterally switching to signalling increases the total male attention that a female receives (since during signalling both types of males, not only the indiscriminate ones, approach the female), the system as a whole can shift to a state where more males become discriminating, and therefore females are left ‘in peace’ when they indicate unreceptivity by not signalling.

    Whether the evolutionary equilibrium features all or only some males obeying the signal, it is important to explain how signal-obeying males can increase in frequency. An important parameter in our model is τ, modulating the total mating rate that a male can achieve. If males can mate in quick succession with many females (low τ), it is difficult to evolve any discrimination. If replenishment of sperm is so quick that indiscriminating males never suffer in sperm competition situations, then there are problems in the spread of discrimination even in cases with high τ. In that case, we do not find equilibria where all males discriminate, but only approximately half of them do.

    These findings reflect a common rule in the male mate choice literature: if matings are relatively cost-free for males in terms of time or energy (which also requires that sperm replenishes easily), then it is difficult to evolve male behaviours favouring a subset of females because there is little opportunity cost paid when approaching the less profitable females. If subset-favouring males exist, this simply intensifies competition among that one part of a population [46,47], which quickly selects against males that forego their mating chances among all females. In ‘traditional’ male mate choice models, the subpopulations of females differ in their fecundity or other aspects of female quality [48]. In our case, the differences arise in two ways: first, there is temporal structure because of last male sperm precedence (females close to their fertility peak offer better chances that the current mating is the one that leads to fertilization, and signalling females offer information that they are close to their fertility peak), but this benefit reduces with an increasing frequency of discriminating males because this increases the number of matings in which signalling females participate. This reduces female fitness as male attention is assumed to be costly, but it reduces male fitness even more: costs paid by females also translate into reduced fitness for the ultimate sire, and the additional cost on the male side is that the per-male chances of becoming the sire are reduced when females mate multiply.

    We assumed the signalling females indicate that they are close to ovulating in the near future (‘honesty’ in this sense is an assumption we make, as there is no reason why females would benefit from being dishonest and signalling early but not late during the modelled time window). Whether this means that we have modelled a ‘best case scenario’ for discrimination to spread is open to debate. On the one hand, we assume last male sperm precedence, which means that targeting signalling females yields more benefits than a situation that lacks last male sperm precedence. On the other hand, we have implicitly assumed good viability of stored sperm: if non-discriminating males mate with females early in the mating time window, and she does not mate again later, there is no fertilization failure preventing the early matings from fertilizing the female's eggs. Future work could obviously take a more detailed look at the temporal dynamics that determine siring chances of sperm placed in a female's reproductive tract at different times.

    It is equally instructive to think of the game from a female perspective. It is clear that for a female to benefit from signalling, this should have an impact on her mating dynamics, with a ‘wish’ to be left in peace when matings are not yet needed, combined with male attention when required (i.e. close to ovulation). But if the D strategy is not yet prevalent among males, signalling does not achieve this goal. It is therefore unclear why signalling should start spreading from rare among females. In the absence of female signallers, D males do not mate and gain no paternity, so the coevolution towards the S–D equilibrium does not start at all. Under most realistic parameter settings (where males are capable of mating in quick succession), the required frequency of D, before S females can start benefitting from their attempt to manage male attention, is quite high. The logic here reflects the fact that opportunity costs paid by males are low in this situation (which favours indiscriminate behaviour), and when most males do not discriminate, females cannot benefit from a ‘quiet’ period before they start signalling. Simultaneously, there needs to be a substantial frequency of S females before D males can elevate their fitness above d males. This, too, is logical: without S females in their signalling state, D males cannot mate at all, as they pay no attention to non-signallers.

    Once established, however, the S–D equilibrium can be stable. When we gave D males the benefit of higher sperm competitiveness, we did not find mixed equilibria where some males pay no attention to the signal while others do. In principle, one could imagine that even if all females are S, and the resident male strategy is D, some males might ‘try their luck’ and mate with them during the non-signalling period that precedes signalling. This would correspond to a rare d male attempting to gain fitness when the resident strategy is a combination of S and D. However, in scenario 1, a rare d male will have zero fitness with all females that subsequently mate (with D males, as these predominate), and assuming a mating system where unmatedness remains rare, d has very low fitness. As a whole, therefore, the system now features positive feedback, instead of negative frequency-dependent selection that would be required for mixed equilibria to persist.

    If signals of fertility often need to overcome an invasion barrier to be selected for, we have a ‘chicken-and-egg’ problem: in the absence of female signals males do not pay attention to them, in the absence of male behaviour where non-signalling females are left in peace, females do not begin developing signals. These chicken-and-egg situations are not unique to female signals of fertility. Invasion barriers are a common feature in models where males advertise their quality, if there are costs involved [49]. We have not attempted to model the ways in which the invasion barrier can be initially overcome, but it is possible that signals evolve from cues, i.e. some trait that covaries with closeness to ovulation (or more generally the presentation of fertilizable eggs) via some obvious physical correlation; a quintessential example is eggs inside female fish that may be directly perceptible.

    Also, it is of course possible that female signals evolve to make them more visible. We did not include this feature in our model in order to keep the focus on whether a signal can evolve to control male behaviour even if the males are perfectly capable of locating non-signallers. Consider, for example, primate signals of ovulation: males can locate and approach females even if the signal is not present, but the question is whether they profit from doing so. It will be an interesting avenue for further work to see if visibility makes signalling equilibria easier to reach from low frequencies. This may be the case especially if one assumption we make is also changed: we presumed that the number of matings is a good proxy for the total harassment that a female has received, and matings therefore correlated negatively with female fitness except for females that remain unmated (the risk of which is non-trivial at low male ability to mate multiply, i.e. high value of τ). A worthwhile avenue for future work would be to separate the negative effects of harassment from the (potentially positive [50]) effects of multiple mating. One might envisage a temporal structure where matings far away from ovulation are harmful to female fitness, while this turns to a benefit close to ovulation; highly visible signals employed for a short time might then be the outcome.

    As a whole, our model complements earlier fertility signalling models [4244,51] by adding an explicit sexual conflict angle. There are often two alternative equilibria: one where females could be said to have won (in the nuanced sense of [51]) because their signalling ensures reduced or no male attention when superfluous matings would harm their fitness, and another one where no signalling occurs and males simply mate with all females that they encounter.

    The non-signalling equilibrium is typically stable whenever male mating capacities are high. At this equilibrium, sexual conflict and suboptimally high mating rates (for females) remain strong. Does this mean that males can be said to have ‘won’ in this situation? Early ideas about game theory in biology often phrased the situation as one sex winning the game [52,53]. However, it is clear that males do not actually gain more offspring than females produce for them [54], and when all males mate many times, most matings do not lead to them siring young in our model. Moreover, our model also predicts population fitness to be lowered when females frequently mate suboptimally (see [55] for a general discussion), and this automatically harms male reproduction too [56]. Instead of one sex winning and the other one losing, our modelling exercise shows that it may be better to classify equilibria as involving strong unresolved conflict or less strong conflict; in our case, these relate to the absence and presence of fertility signals, respectively.

    Data accessibility

    The code and all the data generated by the model will become available upon the publication of the manuscript.

    The MATLAB script with the model code, scripts for generating data and data itself are uploaded on Zenodo: https://zenodo.org/record/7032493#.Y8-5BC9XbBI.

    The data are provided in the electronic supplementary material [57].

    Authors' contributions

    V.K.: investigation, methodology, visualization, writing—original draft and writing—review and editing; H.K.: conceptualization, investigation, methodology, project administration, supervision, writing—original draft and writing—review and editing.

    Both authors gave final approval for publication and agreed to be held accountable for the work performed therein.

    Conflict of interest declaration

    The authors declare no competing interests.

    Funding

    The project was funded by University of Zurich.

    Acknowledgements

    Authors also thank two anonymous reviewers for their valuable and constructive comments that improved the manuscript.

    Footnotes

    One contribution of 18 to a theme issue ‘Half a century of evolutionary games: a synthesis of theory, application and future directions’.

    Present address: Institute of Organismic and Molecular Evolution, Johannes Gutenberg University, 55128 Mainz, Germany.

    Electronic supplementary material is available online at https://doi.org/10.6084/m9.figshare.c.6423951.

    Published by the Royal Society. All rights reserved.

    References