Contrasting effects of intralocus sexual conflict on sexually antagonistic coevolution

Significance

Biologists distinguish two forms of conflict between the sexes, recognized as separate drivers of rapid evolution and diversification. Sexual conflict manifests itself as a tug-of-war between selective forces acting on genes jointly expressed by males and females, allowing neither to maximize their fitness. It also appears in mating interactions, where males and females engage in evolutionary arms races to increase their reproductive success to the detriment of their partner. We model the plausible scenario that reproductive traits are involved in both types of conflict and show that their interaction can draw the sexes into perpetual cycles of escalation, alternated by phases of conflict resolution. This result sheds light on the maintenance of sexually antagonistic variation and the complex dynamics of male–female coevolution.

Abstract

Evolutionary conflict between the sexes can induce arms races in which males evolve traits that are detrimental to the fitness of their female partners, and vice versa. This interlocus sexual conflict (IRSC) has been proposed as a cause of perpetual intersexual antagonistic coevolution with wide-ranging evolutionary consequences. However, theory suggests that the scope for perpetual coevolution is limited, if traits involved in IRSC are subject to pleiotropic constraints. Here, we consider a biologically plausible form of pleiotropy that has hitherto been ignored in treatments of IRSC and arrive at drastically different conclusions. Our analysis is based on a quantitative genetic model of sexual conflict, in which genes controlling IRSC traits have side effects in the other sex, due to incompletely sex-limited gene expression. As a result, the genes are exposed to intralocus sexual conflict (IASC), a tug-of-war between opposing male- and female-specific selection pressures. We find that the interaction between the two forms of sexual conflict has contrasting effects on antagonistic coevolution: Pleiotropic constraints stabilize the dynamics of arms races if the mating traits are close to evolutionary equilibrium but can prevent populations from ever reaching such a state. Instead, the sexes are drawn into a continuous cycle of arms races, causing the buildup of IASC, alternated by phases of IASC resolution that trigger the next arms race. These results encourage an integrative perspective on the biology of sexual conflict and generally caution against relying exclusively on equilibrium stability analysis.

The sexes have followed distinct evolutionary trajectories due to divergent selection regimes that have led to, and been exaggerated by, anisogamy (1–4). This disparity has the potential to ignite two forms of sexual conflict (5): interlocus and intralocus sexual conflict (IRSC and IASC, respectively). Both forms of conflict have been described as independent drivers of divergence and speciation (5–8) and have important implications for the rate of trait evolution, the maintenance of genetic variation, and sexual selection (9–12).

IRSC arises from a direct interaction between the sexes that increases the fitness of one sex at the expense of the other. Typically, males evolve adaptations for success in sperm competition and monopolization of females (male offense traits), which often prevents females from obtaining fitness benefits through polyandry or sperm use (13–15). Subsequently, an arms race is initiated via the evolution of female counteradaptations that reduce the fitness loss (female defense traits) (14, 16). Repeated or even perpetual cycles of counteradaptation in each sex are predicted to follow over evolutionary time, leading to the rapid evolution of reproductive traits (6, 7).

Whereas IRSC clearly manifests itself as a form of conflict in mating interactions, IASC involves a more subtle type of sexual antagonism that operates at the level of phenotype expression. Here, conflict arises because the sexes share the same genome but are nevertheless under selection to express different, sex-specific phenotypes (4, 17, 18). The resolution of IASC can be achieved via the evolution of sexual dimorphism (2, 19–21). However, the observation of negative intersexual correlations for fitness indicates that appreciable levels of IASC are maintained (22–25), both in the wild (11, 26, 27) and in laboratory populations (28–31).

Given their different modes of operation, IASC and IRSC are traditionally considered as separate forces. In fact, in typical studies of IASC, fitness is frequency-independent and determined by a univariate trait, ruling out the possibility of coevolution between offense and defense traits characteristic of IRSC. Models of IRSC, however, consider interactions between at least two phenotypic characters expressed in a mating context, where the strategy of one sex is governed by a different set of loci from the trait(s) required by the other sex to counteradapt (6). This has commonly been interpreted to imply that loci involved in IRSC have sex-limited expression (5, 7, 10, 32, 33) and are, therefore, unaffected by IASC. Nevertheless, some authors have emphasized the role of pleiotropic side effects, which may not be restricted to a single sex, in stabilizing the dynamics of intersexual antagonistic coevolution (8, 32).

For instance, if the evolution of female indifference to a male mating signal is mediated by mutations in the female’s sensory system (9, 10), then those same mutations might be expressed in males as well. If so, potential negative side effects (such as a reduced foraging efficiency) of female counteradaptations to sexual conflict are subject to selection in both sexes. As in this example, many traits involved in sexual conflict have a complex genetic basis, providing ample opportunity for pleiotropic effects between male and female traits, by which the two processes of conflict can become linked. The potential that the same loci underlie both forms of conflict is further increased by the widespread occurrence of alleles associated with IASC or IRSC throughout the genome (30, 34–36). Moreover, both IASC and IRSC are predicted to stem predominantly from reproductive traits, where the evolutionary interests of the sexes diverge the most (22), although sexually dimorphic traits may, in fact, be subject to reduced IASC due to the prior evolution of sex-specific gene regulation (12, 20).

The potential for IRSC and IASC to interact has been highlighted recently (4), where it was noted that intersexual selection acting on a trait that is genetically correlated between the sexes would often give rise to intralocus sexual conflict. It was also argued that the outcome of this interaction would depend on the opportunity for IASC resolution: IASC could persist and therefore prevent counteradaptation of the trait in response to IRSC, or IASC could be resolved, resulting in the escalation of arms races stemming from IRSC. We here develop a formal, quantitative genetic model of traits involved in inter- and intralocus sexual conflict, to verify these arguments and examine their implications for the evolution of sexual conflict. Our analysis supports the intuition that IASC can stabilize antagonistic male–female coevolution but also indicates that the consequences of interaction between the two forms of sexual conflict reach much further than anticipated. Finally, we discuss the implications of these results for the occurrence of perpetual arms races and the maintenance of sexually antagonistic variation in fitness.

The Model

Biological Assumptions.

Our analysis builds on a model of sexually antagonistic coevolution introduced by Rowe et al. in 2005 (32) (henceforth referred to as rcd05). The biological scenario considered in their study is that males and females are in conflict over the rate of mating, which is taken to be an increasing sigmoid function $\psi \left(s\right)=1/\left(1+\text{exp}\left(-s\right)\right)$ of the intensity of a mating stimulus, s. This particular formulation of the model captures the situation that mating is a contest between male offense and female defense traits, in which more extreme offense traits increase the rate of mating, whereas more extreme defense traits have the opposite effect. Biological examples of offense and defense traits include grasping and antigrasping devices, as seen in water striders (13), or traumatic insemination and counteradaptations to control its harmful effects, as found in bedbugs (16) and (hermaphroditic) land snails (14).

In rcd05, the intensity of the mating stimulus perceived by a female is taken to be a function of three evolving phenotypic traits with sex-limited expression. Specifically, $s=z_{\mathrm{♀}}\times \left(y_{\mathrm{♂}}-x_{\mathrm{♀}}\right)$ depends on the difference between a persistence trait, $y_{\mathrm{♂}}$, expressed in males, and a female resistance trait, $x_{\mathrm{♀}}$, reflecting the threshold amount of persistence required to induce mating. In addition, the perceived intensity of the mating stimulus depends on the sensitivity of the female, $z_{\mathrm{♀}}$, which quantifies how strongly she discriminates between males that differ in their level of persistence. Male sexual fitness is modeled as an increasing function of the mating rate, such that sexual selection will invariably favor males who mate at a higher frequency. In contrast, females are assumed to achieve maximal reproductive success at an intermediate mating rate ${\theta }_{\psi }$. Selection may therefore act on females to reduce their rate of mating by increasing the mating threshold or evolving insensitivity to the mating stimulus. The latter response is likely when there are no pleiotropic constraints that prevent females from adjusting their sensitivity (32). However, the sensory system underlying female mating behavior is probably important in other contexts as well, such that the maximization of female reproductive success may have negative consequences for fitness components unrelated to mating interactions. Similarly, evolving higher levels of persistence is presumably associated with increasing costs for males. To capture these effects, each of the mating traits is assumed to be subject to stabilizing natural selection for an intermediate optimum.

In this paper, the analysis of rcd05 is extended in two ways. First, if the mating characters have pleiotropic effects, then these need not necessarily be restricted to one sex. Therefore, we take into account that female resistance and sensitivity genes are expressed in males, denoting the corresponding phenotypic trait values as $x_{\mathrm{♂}}$ and $z_{\mathrm{♂}}$, respectively. Likewise, male persistence genes affect a correlated phenotypic character in females, for which the trait value is denoted as $y_{\mathrm{♀}}$. Stabilizing natural selection acts on x, y, and z in both sexes in our model. The optimum trait values and the strength of stabilizing selection are allowed to differ between males and females. Note that x, y, and z still have sex-limited effects on the mating rate (as in rcd05), because their expression in the context of intersexual interactions is contingent on the asymmetry between male and female sex roles.

As a second extension, our model also considers the dynamics of arms races in cases where mating requires complementarity or matching of male and female mating characters. This alternative mating mechanism, which has frequently been considered in models of sexual conflict (8), is modeled by defining the mating rate as a unimodal function $\psi \left(s\right)=\text{exp}\left(-s^2/2\right)$ of the mating stimulus $s=z_{\mathrm{♀}}\times \left(y_{\mathrm{♂}}-x_{\mathrm{♀}}\right)$ (which here reflects the extent to which the male differs from the female’s preferred phenotype). As in sexual selection models, $x_{\mathrm{♀}}$ can then be interpreted as a female mating preference, $y_{\mathrm{♂}}$ as a male mating trait (e.g., an ornament) on which the preference acts, and $z_{\mathrm{♀}}$ as a measure of female choosiness. For simplicity, we will continue to refer to the mating characters as threshold, persistence, and sensitivity, as in rcd05, except when we are explicitly considering complementarity-based mating (in which case we will use preference, ornament, and choosiness instead). Examples of sexually antagonistic mating systems that could be considered as complementarity-based include penis length/female reproductive tract coevolution in waterfowl (3), and male seminal protein/female receptor coevolution in fruit flies (37).

A key feature of our model is that genes involved in IRSC are subject to distinct components of selection in males and females. As a result, selection is likely to favor different optimum trait values in the two sexes, setting the stage for IASC to occur. Prolonged IASC is expected when only a small fraction of the genes are regulated in a sex-specific manner, making it more difficult for males and females to diverge toward their sex-specific optima (2, 19, 23). The strength of the phenotypic correlation between brothers and sisters in their expression of a mating trait and the corresponding correlated character provides an observable measure of the degree of sex-differential expression. Additive genetic intersexual correlation coefficients, which can be inferred from comparisons between opposite-sex relatives (11, 18), therefore play a prominent role as control parameters in our further analysis: Their effect on the rate of IASC resolution allows us to systematically vary the impact of IASC on antagonistic male–female coevolution.

Mathematical Representation.

Based on fitness functions that capture the above biological assumptions, we calculated the strength of selection acting on each of the characters and used this information to determine their rate of evolution (Materials and Methods). The evolutionary dynamics of the population average trait values is described by a multivariate breeder’s equation (38), $d\mathbf{u}/dt=\mathbf{G}\hspace{0.17em}\mathit{\beta }\left(\mathbf{u}\right)$, where $\mathbf{u}$ is a (column) vector ${\left({\overline{x}}_{\mathrm{♀}},{\overline{z}}_{\mathrm{♀}},{\overline{y}}_{\mathrm{♂}},{\overline{x}}_{\mathrm{♂}},{\overline{z}}_{\mathrm{♂}},{\overline{y}}_{\mathrm{♀}}\right)}^{\text{T}}$ containing the average trait values and $\mathbf{G}$ is the additive genetic variance–covariance matrix. This matrix depends on the intersexual correlations $r_x$, $r_y$, and $r_z$, as specified in Eq. 4 in Materials and Methods. The vector $\mathit{\beta }\left(\mathbf{u}\right)$ is the selection gradient, given by

$$\mathit{\beta }\left(\mathbf{u}\right)=\left(\begin{array}{c}a\hspace{0.17em}{\overline{z}}_{\mathrm{♀}}\hspace{0.17em}\left(\overline{\psi }-{\theta }_{\psi }\right)\hspace{0.17em}\overline{\psi }\prime -c_{x_{\mathrm{♀}}}\hspace{0.17em}\left({\overline{x}}_{\mathrm{♀}}-{\theta }_{x_{\mathrm{♀}}}\right)\\ a\hspace{0.17em}\left({\overline{x}}_{\mathrm{♀}}-{\overline{y}}_{\mathrm{♂}}\right)\hspace{0.17em}\left(\overline{\psi }-{\theta }_{\psi }\right)\hspace{0.17em}\overline{\psi }\prime -c_{z_{\mathrm{♀}}}\hspace{0.17em}\left({\overline{z}}_{\mathrm{♀}}-{\theta }_{z_{\mathrm{♀}}}\right)\\ b\hspace{0.17em}{\overline{z}}_{\mathrm{♀}}\hspace{0.17em}\overline{\psi }\prime -c_{y_{\mathrm{♂}}}\hspace{0.17em}\left({\overline{y}}_{\mathrm{♂}}-{\theta }_{y_{\mathrm{♂}}}\right)\\ -c_{x_{\mathrm{♂}}}\hspace{0.17em}\left({\overline{x}}_{\mathrm{♂}}-{\theta }_{x_{\mathrm{♂}}}\right)\\ -c_{z_{\mathrm{♂}}}\hspace{0.17em}\left({\overline{z}}_{\mathrm{♂}}-{\theta }_{z_{\mathrm{♂}}}\right)\\ -c_{y_{\mathrm{♀}}}\hspace{0.17em}\left({\overline{y}}_{\mathrm{♀}}-{\theta }_{y_{\mathrm{♀}}}\right)\end{array}\right).$$ [1]

Each element of the vector $\mathit{\beta }\left(\mathbf{u}\right)$ quantifies the marginal fitness effect of varying one of the characters by one phenotypic unit, in the context of the current population with average trait values $\mathbf{u}$. The upper three elements represent selection gradients acting on the mating traits ${\overline{x}}_{\mathrm{♀}}$, ${\overline{z}}_{\mathrm{♀}}$, and ${\overline{y}}_{\mathrm{♂}}$, which depend on the fitness effects of mating interactions. The strength of sexual selection varies with $\overline{\psi }$ and $\overline{\psi }\prime$, the values of the mating rate function and its first derivative at $\overline{s}={\overline{z}}_{\mathrm{♀}}\times \left({\overline{y}}_{\mathrm{♂}}-{\overline{x}}_{\mathrm{♀}}\right)$. In addition, the impact of mating on female and male fitness is scaled by two parameters, a and b, that quantify, respectively, the cost to females of deviating from their optimum mating rate and the strength of the association between mating rate and male reproductive success. Direct selection on the correlated characters, reflected by the lower three elements of the selection gradient, occurs only in the form of stabilizing natural selection, which also acts on the mating traits. Stabilizing natural selection is parameterized for each trait by an optimum trait value θ and a selection intensity c, which determines how much fitness decreases when a phenotype is displaced by a given amount from its viability selection optimum.

Results

The coevolution of the mating characters ${\overline{x}}_{\mathrm{♀}}$, ${\overline{z}}_{\mathrm{♀}}$, and ${\overline{y}}_{\mathrm{♂}}$ in the absence of between-sex pleiotropy has been analyzed by rcd05, and we briefly recapitulate their results before examining the interaction between IRSC and IASC. A key finding is that IRSC, acting by itself, has multiple potential evolutionary outcomes (8, 32). These include escalating arms races, the evolution of female indifference to the mating stimulus, and continual coevolution of threshold, persistence, and sensitivity. Female indifference tends to evolve when females are able to adjust the shape of their preference function without major negative side effects, enabling them to avoid large fitness costs of IRSC at evolutionary equilibrium. By contrast, evolutionary arms races, which result in a significant reduction of female fitness, occur when adaptation of the female sensory system is constrained by a lack of genetic variation or strong stabilizing selection on sensitivity in contexts other than mating. Hence, rcd05 conclude that the outcome of IRSC depends critically on the constraints and selective forces that act on the female preference function.

Evolutionary Equilibria Are Stabilized by IASC.

Following Pennell and Morrow (4), we hypothesized that IASC would restrain IRSC in cases with between-sex pleiotropic trait expression, potentially preventing the escalation of arms races. In particular, if the intersexual correlations are high and strong stabilizing selection acts on the correlated characters ${\overline{x}}_{\mathrm{♂}}$, ${\overline{z}}_{\mathrm{♂}}$, and ${\overline{y}}_{\mathrm{♀}}$, IASC is predicted to keep the mating traits fixed at an evolutionary equilibrium even if this would not be stable under the sole action of IRSC. To see why, suppose that one or several of the mating traits evolve away from the equilibrium under the influence of intersexual selection (i.e., sexual selection generated by variation in ψ). This change is associated with a correlated change of the homologous characters, causing those to deviate from their viability selection optimum. As a result, the response to sexual selection is opposed by stabilizing natural selection in the other sex, which pushes the traits back to their original values if the pleiotropic fitness effects outweigh the selective forces resulting from IRSC.

A formal equilibrium and stability analysis confirms this verbal argument (SI Appendix), demonstrating that IASC can stabilize the dynamics of IRSC in the vicinity of evolutionary equilibria that would otherwise be unstable. In such cases, IASC prevents the sexes from being engaged in an escalating arms race and allows them to sustain a stable “truce.” The main prerequisite for stabilization is that the intersexual genetic correlations and selection on the correlated characters must be sufficiently strong in the direction in phenotype space along which the arms race would have otherwise unfolded, so that stabilizing natural selection is capable of overpowering the forces generated by IRSC.

The mathematical analysis leads to two additional insights. First, neither the location nor the number of fixed points depends on the values of the intersexual genetic correlations. As a result, the evolutionary equilibria of the mating characters are determined by sexual selection and within-sex stabilizing natural selection, exactly as in rcd05, whereas the equilibrium values of the correlated characters are simply given by their respective optimum trait values ${\theta }_{x_{\mathrm{♂}}}$, ${\theta }_{z_{\mathrm{♂}}}$, and ${\theta }_{y_{\mathrm{♀}}}$. Apart from being useful to characterize the evolutionary equilibria, this insight also restricts the range of phenomena that can be associated with qualitative changes in the dynamics of IRSC due to its interaction with IASC. Specifically, from the mathematical theory of qualitative changes in dynamical systems [bifurcation theory (39)], we infer that variation of the intersexual correlations can induce no other generic local bifurcation than a so-called Poincaré–Andronov–Hopf bifurcation, given that all other options require equilibria to move relative to one another. This bifurcation is associated with the emergence or disappearance of a cycle (i.e., periodic orbit), which can either act as an alternative evolutionary attractor or restrict the attainability of certain evolutionary outcomes. Hence, if IASC induces a qualitative change in the stability of an equilibrium, the associated appearance or loss of a cycle could dramatically alter the outcome of sexual conflict.

Dynamics of Sexual Conflict Away from Equilibria.

To complement the insights offered by the local stability analysis, we studied the global evolutionary dynamics of the mating characters based on numerical solutions calculated for the two mating scenario variants of the model. Fig. 1 shows an illustrative result for the case when mating is a contest between offense and defense traits. Here, IRSC leads to continuous fluctuations in the male persistence trait and the female sensitivity and mating threshold (cf. figure 5b in ref. 32). During these evolutionary cycles, the initiation of arms races between threshold and persistence is alternated by the evolution of female indifference to the mating stimulus, inducing the threshold and persistence trait to evolve back toward their optimal trait value under natural selection. The oscillations persist in the presence of low levels of pleiotropic expression between the sexes (Fig. 1A), but their amplitude decreases if the intersexual genetic correlations are stronger. In that case, fluctuations of the mating traits induce a larger correlated selection response that is opposed more strongly by stabilizing selection in the other sex. Modest-to-high values of the intersexual correlations [still below typical empirically observed values (20)] entirely prevent the initiation of arms races, causing the trait values to converge on a stable equilibrium (Fig. 1B). These results are in line with the conclusions of the mathematical analysis: Without changing the location of the equilibrium, IASC has induced an evolutionary repellor, surrounded by a stable periodic solution, to transform into a stable fixed point that attracts the dynamics in Fig. 1B. Moreover, the qualitative change in the stability of the equilibrium is associated with the disappearance of the periodic attractor present in Fig. 1A.

Fig. 1.

IASC stabilizes sexually antagonistic coevolution of offense and defense traits. Male and female mating traits (solid lines) covary with their respective homologous character in the other sex (dashed lines) as a result of between-sex pleiotropic gene expression. The population-average trait values converge on an evolutionary cycle when the intersexual genetic correlations are low (A) but are driven toward a stable equilibrium (B) when sensitivity and threshold are subject to stronger IASC. Parameters are $a=5$, $b=0.5$, ${\theta }_{x_{\mathrm{♀}}}={\theta }_{x_{\mathrm{♂}}}=0.05$, ${\theta }_{y_{\mathrm{♀}}}={\theta }_{y_{\mathrm{♂}}}=-0.05$, ${\theta }_{z_{\mathrm{♀}}}={\theta }_{z_{\mathrm{♂}}}=0.5$, ${\theta }_{\psi }=0.2$, $c_{x_{\mathrm{♀}}}=c_{x_{\mathrm{♂}}}=0.5$, $c_{y_{\mathrm{♀}}}=c_{y_{\mathrm{♂}}}=c_{z_{\mathrm{♀}}}=c_{z_{\mathrm{♂}}}=0.1$. In A, $r_x=r_z=0.1$ and $r_y=0.2$; in B, $r_x=r_z=0.5$ and $r_y=0.2$.

More puzzling to explain are the simulation results obtained for the model variant with complementarity-based mating, illustrated in Fig. 2. Here, again, Fig. 2A shows the outcome of evolution when the intersexual genetic correlations are weak. In this case, females evolve costly choosiness, and we observe an arms race to exaggerated levels of ornamentation and corresponding costly preferences. Eventually, the sexes converge on a stable equilibrium, at which chase-away sexual selection (9, 10) favoring further exaggeration is balanced by natural selection acting in the opposite direction. Antagonistic coevolution can escalate in two directions and thereby converge on two different fixed points (shown in Fig. 2A, Upper and Lower). When the intersexual correlations are high (Fig. 2B), such that female choice is subject to strong pleiotropic constraints, arms races driven by chase-away sexual selection occur as well, but they are more effectively halted by natural selection, as a consequence of stronger pleiotropic side effects in the other sex. However, as the interlocus sexual conflict built up by the arms race is resolved, sexual selection on the female preference and male ornamentation changes direction, suddenly triggering an arms race in the opposite direction. As a result, rather than stabilizing the dynamics of IRSC, IASC prevents the coevolving sexes from reaching evolutionary equilibria, so that males and females are caught in a recurrent evolutionary cycle (thick lines in Fig. 2B).

Fig. 2.

IASC gives rise to an alternative evolutionary attractor when mating is based on trait complementarity. (A) Chase-away sexual selection drives the population toward exaggerated trait values when the intersexual genetic correlations are relatively low ($r_x=r_z=0.5$ and $r_y=0.2$). (Upper and Lower) Simulations converging on two alternative equilibria. Two additional stable equilibria exist when choosiness is allowed to be negative at equilibrium. We do not show further results for this case, because the evolutionary trajectories at positive and negative values of choosiness are nearly symmetric. Thick lines in B, Upper and Lower show simulations from the same initial conditions as in A, but at higher values of the intersexual additive genetic correlation ($r_x=r_z=0.95$ and $r_y=0.2$). Here, males and females engage in arms races, which are reversed halfway on their way toward attaining equilibrium. The cause of these sudden reversals is IASC resolution, which induces a correlated selection response in the mating traits, changing the relative positions of the sexes in their coevolutionary chase. The interaction of inter- and intralocus conflict thus keeps the sexes caught in a perpetual cycle of arms races, alternated by phases of conflict resolution. The equilibrium states attained in A are also potential endpoints of evolution in B but can only be reached from initial conditions close to the equilibria. This is shown by two example simulations (thin lines) that converge on the alternative equilibria in B, Upper and Lower. Parameters (other than the intersexual correlations) are as in Fig. 1.

At first sight, these results seem to contradict the conclusions from the mathematical analysis, which stated that stable equilibria of IRSC cannot be destabilized by IASC. However, the arms races observed in Fig. 2B never get close to the endpoints of the coevolutionary chase in Fig. 2A, so we cannot yet infer the stability properties of the equilibria. We therefore ran simulations from other initial conditions, closer to the endpoints reached by evolution in Fig. 2A. The additional simulations (thin lines in Fig. 2B, Upper and Lower) indicate that the sexes can still attain the same stable state characterized by exaggerated trait expression when the intersexual correlations are strong. In other words, no differences exist between Fig. 2A and Fig. 2B in the stability properties of the evolutionary equilibria, consistent with the analytical results. We are thus led to conclude that IASC has consequences for the dynamics of sexually antagonistic coevolution far away from equilibrium that contrast sharply with predictions derived from the local equilibrium stability analysis.

Fig. 3 shows simulations for a different parameter set, in which the evolutionary cycles observed in Fig. 2B occur in a more basic form. These illustrate that the coevolutionary dynamics is dominated alternatingly by inter- and intralocus sexual conflict, with periods of arms races that displace the sexes from their optima, and phases of intralocus sexual conflict resolution that set the stage for the next arms race to occur. Note that the resolution of IASC in these simulations is only partly mediated by the evolution of sexual dimorphism. IASC is also resolved by dynamic changes in the direction of intersexual selection and associated shifts in sex-specific fitness optima, which may temporarily align the selection gradients on correlated characters in males and females. At other times, the same process may cause sex-specific optima to diverge again, leading to the renewed buildup of IASC. Furthermore, a comparison between the panels illustrates that the destabilizing effect of between-sex pleiotropy is trait-specific: In order for IASC resolution to reverse the direction of chase-away sexual selection (which requires that $x_{\mathrm{♀}}-y_{\mathrm{♂}}$ changes sign), the correlated selection response, which pulls the traits back to their viability selection optimum, must be larger in females than in males (SI Appendix, Figs. S2–S4).

Fig. 3.

Effect of between-sex pleiotropy on the dynamics of IASC and IRSC during trait evolution. Each panel shows a simulation of evolving mean trait values (Lower; line styles as in Fig. 2) with a corresponding time plot of trait-specific indices of sexually antagonistic selection (dashed: IRSC index; solid: IASC index). Positive values of IASC and IRSC indices are indicative of sexual antagonism; negative values indicate that fitness effects are concordant between the sexes (Materials and Methods). (A) Complementarity-based mating without between-sex pleiotropy ($r_x=r_y=r_z=0$). Preference and ornament evolve in an arms race driven by IRSC, first in one then in the other direction, converging eventually on a stable equilibrium. In B, the approach to the equilibrium is destabilized by IASC, which can be seen to build up during the period that IRSC is strong ($r_x=0.8$; $r_y=r_z=0$). IASC is resolved when $x_{\mathrm{♂}}$ evolves back toward it sex-specific optimum, but this process induces a correlated change in $x_{\mathrm{♀}}$ that causes the direction of sexual selection to reverse. As a result, a new IRSC arms race is triggered, initially accelerated by concordant selection on the preference and its pleiotropic character. (C) When also male ornamentation genes have pleiotropic effects in the other sex ($r_x=0.8$; $r_y=0.5$; $r_z=0$) evolution can attain the equilibrium again. Although the overall level of between-sex pleiotropy and IASC have increased relative to B, conflict resolution has become less effective in reversing the direction of sexual selection. This is because both $x_{\mathrm{♀}}$ and $y_{\mathrm{♂}}$ are pushed toward their viability selection optimum by the correlated response to stabilizing selection on, respectively, $x_{\mathrm{♂}}$ and $y_{\mathrm{♀}}$. The difference between the two traits, which determines the direction of chase-away sexual selection, is therefore less strongly affected by IASC resolution than in B. Parameters are $a=0.4$, $b=0.1$, ${\theta }_{x_{\mathrm{♀}}}={\theta }_{x_{\mathrm{♂}}}=0$, ${\theta }_{y_{\mathrm{♂}}}={\theta }_{y_{\mathrm{♀}}}=0.05$, ${\theta }_{z_{\mathrm{♀}}}=0.95$, ${\theta }_{\psi }=0.25$, $c_{x_{\mathrm{♀}}}=0.1$, $c_{x_{\mathrm{♂}}}=c_{y_{\mathrm{♂}}}=c_{y_{\mathrm{♀}}}=c_{z_{\mathrm{♀}}}=0.05$. For clarity, trait values for the correlated characters and IASC indices are not shown if the corresponding intersexual genetic correlation is equal to zero.

Global Bifurcation Analysis.

The remaining section of the Results provides a description of how the dynamical effects of between-sex pleiotropy unfold, building on insights from dynamical-systems analysis (39). Readers unfamiliar with this subject can first read Discussion, where we concentrate on the biological implications of the results.

Because time plots are not well suited to summarize the results of simulations from multiple initial conditions, we switched to a representation of the evolutionary dynamics in phase space and reduced the number of variables to enable visualization of the phase diagrams in 3D. This was accomplished by treating $z_{\mathrm{♀}}$ as a fixed character (equivalent to assuming the absence of additive genetic variation in female choosiness) and setting $r_y=r_z=0$, such that the evolution of $y_{\mathrm{♀}}$ and $z_{\mathrm{♂}}$ is decoupled from that of the other characters. The remaining characters, $x_{\mathrm{♀}}$, $x_{\mathrm{♂}}$, and $y_{\mathrm{♂}}$ appear as variables on the axes of the phase diagrams in Fig. 4. The six panels illustrate the dynamics at different values of the intersexual correlation $r_x$, increasing from 0.6 in Fig. 4A to 0.9 in Fig. 4F. In this section, we consider only the model variant with complementarity-based mating.

Fig. 4.

Between-sex pleiotropy induces a sequence of qualitative changes in the dynamics of sexual conflict. The phase diagrams (A–F) show evolutionary trajectories starting from different initial conditions at six different values of the intersexual genetic correlation $r_x$: (A) $r_x=0.6$, (B) $r_x=0.7$, (C) $r_x=0.76$, (D) $r_x=0.8$, (E) $r_x=0.85$, and (F) $r_x=0.9$. Two stable equilibria (indicated by a red and green dot) exist throughout this range of parameter values, separated by a third, unstable equilibrium (gray dot) in the interior of the trait space. Trajectories are colored according to the attractor at which they end up; differences in line thickness are used to highlight important trajectories. Seen from the upper right, trajectories generally follow a counterclockwise motion. Unstable limit cycles (present in B–F) are indicated by black dots. Stable periodic attractors (thick closed orbits in D–F) and the trajectories leading toward them are shown in blue. Throughout, female choosiness was kept fixed at $z_{\mathrm{♀}}=1.5$, consistent with the absence of genetic variation for that trait. To further reduce the number of variables, we took $r_y=r_z=0$, so that $y_{\mathrm{♀}}$ and $z_{\mathrm{♂}}$ evolve independently of the other traits. Other parameters are $a=0.4$, $b=0.1$, ${\theta }_{x_{\mathrm{♀}}}={\theta }_{x_{\mathrm{♂}}}=0$, ${\theta }_{y_{\mathrm{♂}}}=0.05$, ${\theta }_{\psi }=0.25$, $c_{x_{\mathrm{♀}}}=0.1$, $c_{x_{\mathrm{♂}}}=c_{y_{\mathrm{♂}}}=0.05$.

The phase diagram depicted in Fig. 4A reflects qualitatively what we have seen in Fig. 2A: Starting from trait values near 0, chase-away sexual selection can occur in two different directions, leading to either one of two stable endpoints (colored red and green) at which males and females express costly mating traits. IASC is fully resolved at these equilibria, with the correlated character $x_{\mathrm{♂}}$ being at its viability selection optimum (${\theta }_{x_{\mathrm{♂}}}=0$). The two attractors are separated by a saddle point (gray), which has a stable manifold that divides the phase space into the separate basins of attraction of the red and the green equilibrium. Intersexual arms races leading toward the stable fixed points exhibit damped oscillations. These have become exaggerated in Fig. 4B ($r_x=0.7$), giving rise to a first bifurcation (i.e., a qualitative change in the phase diagram) that is marked by the appearance of an unstable limit cycle (dotted black). Trajectories leading from the saddle point toward the green equilibrium loop around the cycle, eventually to converge on the red equilibrium. The bias toward the red equilibrium is caused by a slight asymmetry in the viability selection optima for the female preference and the male ornament (${\theta }_{x_{\mathrm{♀}}}=0$, ${\theta }_{y_{\mathrm{♂}}}=0.05$). As a consequence, the difference between $x_{\mathrm{♀}}$ and $y_{\mathrm{♂}}$ is smaller in the vicinity of the green equilibrium, making it easier for intralocus sexual conflict resolution to reverse the direction of chase-away sexual selection (SI Appendix, Fig. S2). However, at slightly higher values of the intersexual genetic correlation (Fig. 4C; $r_x=0.76$), the approach to the red equilibrium is also destabilized. A second unstable limit cycle has emerged, causing recurrent reversals of the direction of sexually antagonistic coevolution until the process eventually settles down at the red equilibrium.

In Fig. 4D ($r_x=0.8$), a connection between the stable and unstable manifolds of the two limit cycles has led to the appearance of a third cycle (dark blue), which attracts trajectories in the interior of the phase space. The unstable limit cycles now prevent evolution from reaching either one of the stable equilibria, except from a restricted range of initial conditions (comparable to the case illustrated by Fig. 2B). When the intersexual correlation is increased even more, two additional bifurcations occur that further restrict the possibility of establishing a truce between the sexes. First, an unstable cycle arises (Fig. 4E; $r_x=0.85$), which acts as a source for trajectories leading to the stable fixed points. This cycle tightly surrounds the saddle point and eventually collapses onto it (Fig. 4F; $r_x=0.9$), transferring its stability properties to the interior equilibrium (gray). It is possible for evolution to reach the stable fixed points from the neighborhood of this equilibrium, following trajectories that spiral outward along the manifold on which the unstable cycle existed. However, selection is much more likely to push the population along the unstable manifold of the former saddle point, which continues to dominate the dynamics around the interior equilibrium. Evolution in this direction very quickly drives the population into the basin of attraction of the stable limit cycle, so that continual arms races are a much more likely outcome of evolution.

In this entire series of bifurcations, only the final bifurcation is associated with a change in the qualitative properties of an equilibrium (without even affecting its stability). The other qualitative changes that occurred in the phase diagram result from global bifurcations (SI Appendix, Fig. S5) that are not detected by a stability analysis of the equilibria. Moreover, we note that there has not been a single qualitative change in the evolutionary dynamics close to the stable fixed points (red and green) over the entire range of intersexual correlations (Fig. 4).

The above analysis relies on a number of simplifying assumptions, so we ran several checks to verify the robustness of the results. First, we recovered a similar sequence of bifurcations in a model with coevolving female choosiness (SI Appendix, Fig. S6), indicating that the main results are not specific to the reduced model with only three coevolving characters. Another restriction, implicit in our use of a quantitative genetics framework, is that we have assumed a fixed genetic architecture of the traits, whereas, in reality, the genetic variances and covariances are responding to selection (40). In fact, previous studies have shown that IRSC is capable of generating strong linkage equilibria within populations, leading to bimodal trait distributions and possibly speciation (8, 41). However, without high mutation rates and weak selection on females, phenotypic diversification is limited in individual-based simulations of our model (SI Appendix, Figs. S7–S12). Despite being genetically explicit (enabling the genetic architecture of traits to evolve), the individual-based simulations closely follow the quantitative genetic predictions, lending support to the robust occurrence of the instability uncovered by our analysis.

Discussion

The potential for IASC to affect trait evolution and diversification caused by IRSC has recently been emphasized (4) but never before put to the test of formal analysis. Previous models of IRSC (7, 8, 10, 32, 33), which have not included IASC, predict arms races of sexually antagonistic adaptation and counteradaptation between the sexes, possibly leading to exaggerated traits and substantial fitness losses due to sexual conflict (6). We included IASC in a quantitative genetic model of IRSC trait evolution and found that, depending on the genetic architecture of traits (i.e., their degree of sex-limited expression) and the biological mechanism of mating interactions (i.e., whether compatibility is determined by a contest or trait complementarity), IASC can either restrain or induce male–female antagonistic coevolution.

The stabilizing effect of IASC dominates near evolutionary equilibria. Here, selection is weak and trait values evolve slowly, so that there is ample time for the resolution of IASC. Because the correlated characters are close to their optimum, mutations expressed in mating interactions are also exposed to purifying selection in the other sex. A general mathematical argument confirms that this additional source of stabilizing selection thwarts the initiation of arms races in populations at evolutionary equilibrium.

Far away from equilibrium, the mating traits evolve more rapidly, allowing unresolved IASC to build up. The pleiotropic effect of mating-trait mutations is then subject to directional selection in the other sex, which can slow down antagonistic coevolution, or even reverse its direction. The latter phenomenon occurs with complementarity-based mating, when the female preference, which is ahead in the coevolutionary chase, is subject to stronger pleiotropic constraints than the male ornament, which is following behind (SI Appendix, Figs. S2–S4). Furthermore, arms-race reversals occur above a critical level of the intersexual correlation, at which the correlated selection response to IASC resolution is sufficient to pull the mean female preference toward the other side of the male trait distribution, qualitatively changing the direction of chase-away sexual selection. In this way, rather than helping the sexes to maintain a truce, IASC fuels a never-ending cycle of IRSC arms races, interrupted by phases of IASC resolution, which set the stage for the next arms race to occur.

In conclusion, whether IASC stabilizes or destabilizes IRSC arms races is determined primarily by the balance between the rate of conflict resolution and the rate at which new sexually antagonistic variation is accumulated during episodes of rapid trait evolution. The timescale of both processes is affected by the genetic architecture of conflict traits, characterized in our analysis by the additive genetic intersexual correlation between mating characters and their pleiotropic homologs in the other sex. Research that has focused solely on IASC has shown that negative intersexual correlations for fitness are maintained in populations (11, 19, 26), suggesting that the resolution of IASC may be slow (20, 22) (but see ref. 42). However, intersexual genetic correlations for individual traits are varied (19), with systematic differences existing among trait types (20). Moreover, specific information on the extent of between-sex pleiotropy for traits involved in IRSC is scarce. Quantitative estimates of the strength of interaction between IASC and IRSC therefore require further developments in our understanding of the genetic basis of the two forms of sexual conflict.

The Genetic Architecture of Sexual Conflict.

Currently, the most detailed studies of the genetics of IRSC come from seminal proteins. Sex peptide (SP) is one gene that has been recognized as a mediator of IRSC (15), and other candidate genes have also been identified, including genes that are required by females to respond to SP postmating (25, 37). Several IRSC candidate genes identified by Gioti et al. (37) were not sex-limited in their expression, creating potential for them to also mediate IASC. Similarly, Innocenti and Morrow (30) found that many of the transcripts associated with IASC were enriched in reproductive tissues, such as the female spermatheca and the male accessory gland and ejaculatory duct. Altogether, this suggests that the same loci could be involved in both IASC and IRSC.

Although reproductive traits are the primary candidates to mediate both types of sexual conflict, sexually antagonistic transcripts associated with IASC have also been identified in nonreproductive tissues (30), and several aspects of morphology and physiology that are exposed to sex-specific selection but not directly related to reproductive functions have strong genetic correlations between the sexes (20). Nonreproductive traits may also be necessary to counteradapt in an IRSC arms race. For example, in cockroaches and bed bugs, behavioral, morphological, and physiological adaptations are thought to be involved in adaptations to IRSC in both sexes (16, 33). Nonreproductive traits may therefore participate in both IRSC and IASC, although reproductive traits might generally be subject to stronger sex-specific selection.

Estimating between-sex genetic correlations is a challenge, because correlated phenotypic characters can easily be overlooked due to the diffuse nature of pleiotropy. Therefore, a promising complementary approach to genomic analyses of the genetic architecture of sexual conflict traits is to test for IASC effects using experimental manipulations of candidate genes previously identified for their role in sexually antagonistic interactions. It might also be insightful to conduct artificial selection experiments on traits involved in IRSC, to identify whether the traits have a shared genetic basis, where resulting phenotypes and their fitness effects could be quantified in each sex. A recent study by Fuchikawa and Okada (43) used this method to study whether exaggeration of male mandibles in seed beetles affected female fitness via an intersexual genetic correlation. Although they found no evidence of IASC over this trait, it provides a framework for other studies of this kind.

Mechanisms of Mating Interactions.

Apart from between-sex pleiotropy, a requirement for sustained arms-race reversals in our model is that mating compatibility is determined by trait complementarity rather than by a contest between offense and defense traits. IASC was observed to complicate the dynamics of arms races in the contest scenario as well (SI Appendix, Fig. S1), but its destabilizing effects never prevented evolution from ultimately reaching an equilibrium state. Both mating mechanisms have been considered in theoretical studies of IRSC and motivated by specific biological examples (8). rcd05 already showed that evolutionary cycles, featuring fluctuations in sensitivity (Fig. 1 and SI Appendix, Fig. S1), may occur when mating is determined by a contest. However, barring changes in sensitivity, changing the direction of chase-away sexual selection is precluded in the contest model, because male sexual fitness is an increasing function of the expression of the offense trait, irrespective of the level of female defense. A similar monotonic relationship holds for female sexual fitness as a function of the defense trait. These constraints, however, are an immediate consequence of the assumption that offense and defense are unidimensional traits.

Male and female mating behaviors are frequently determined by many traits (including behavioral, morphological, and physiological characteristics). Accordingly, intersexual arms races generally occur in multidimensional phenotype space. There, male–female coevolution can unfold in many different directions, so that the resolution of IASC may trigger sudden changes in the direction of arms races. Populations evolving in multidimensional phenotype spaces, in which mating interactions are governed by a contest between more than one offense and defense trait, may therefore show complex dynamics of sexual conflict similar to what is observed in our model for complementarity-based mating. Consistent with this hypothesis, frequency-dependent selection operating on multivariate phenotypes is known to frequently result in complex nonequilibrium dynamics or even evolutionary chaos (44).

Implications for Theory.

The evolutionary cycling observed in our model of sexual conflict is a consequence of an interaction between processes on a fast and a slow timescale, typical of systems with delayed nonlinear feedback control [relaxation oscillators (45)]. Sexually antagonistic coevolution is the fast process (7), which drags along pleiotropically correlated characters. The accumulating displacement of these characters from their optimum acts as a control variable with a sudden, switch-like effect on the direction of intersexual selection, mediated by IASC resolution. The strength of the feedback and its timescale of operation are set by the additive genetic correlations. Broadening this analogy, we speculate that similar dynamical instabilities can occur in other coevolutionary processes that are subject to pleiotropic constraints, such as in the context of host–parasite coevolution (6) or biological signaling (46). Because pleiotropy has been seen predominantly as a source of evolutionary constraint so far, further theoretical investigation into its consequences for the dynamics of evolution is warranted.

The rich dynamics of sexual conflict, uncovered by a numerical bifurcation analysis, also draws the attention to the limitations of equilibrium analyses to characterize complex evolutionary phenomena. The standard approach to analyzing phenotypic models of evolution is to locate evolutionary equilibria and then characterize their stability by studying the dynamics of evolution in the vicinity of the equilibria (47). This local analysis does not provide conclusive information about the dynamics far away from equilibria, yet its results are often extrapolated to infer a qualitative description of the global evolutionary dynamics. The present model serves as an illustration of a biological scenario where such qualitative analysis leads to the wrong conclusions. Specifically, the analytical results fail to detect the emergence of an evolutionary cycle that arises through a series of global bifurcations and serves as an alternative evolutionary attractor. How commonly this problem occurs in the analysis of biological models is an open question, especially for studies of complex trait evolution, because a systematic search for global bifurcations is not routinely included in model analyses.

Conclusion

Our findings inspire an integrative perspective on the biology of inter- and intrasexual conflict that sheds new light on several issues debated in the field. First, we show that pleiotropic constraints, in the form of correlated trait expression subject to stabilizing selection, do not necessarily restrain arms races but may rather create conditions favorable to perpetual antagonistic coevolution. The reversal of arms races by IASC resolution provides a previously unidentified mechanism for explaining the ongoing evolution of mating traits, despite the presence of stabilizing natural selection preventing unlimited trait exaggeration. Second, our simulations, which show a recurrent buildup of unresolved IASC during intersexual arms races, alternated by periods of conflict resolution, suggest that IASC may be more dynamic than has so far been recognized. This idea is consistent with the observation that closely related species show markedly different patterns of sex-biased gene expression (42). It can also help to resolve the paradox that appreciable levels of sexually antagonistic genetic variation segregate in populations, whereas sexual dimorphism is known to evolve rapidly in many cases (12, 22, 48). That is, even if the evolution of sexual dimorphism leads to a rapid loss of sexually antagonistic variation, new sexually antagonistic alleles might be introduced continually as a pleiotropic side-effect of intersexual arms races.

Materials and Methods

Fitness Functions.

Individual fitness is calculated as the product of survival and reproductive success. Reproductive success depends on the mating rate ψ in both sexes, but in qualitatively different ways: Male fitness is an increasing function of ψ, whereas female fitness is maximized at an intermediate mating rate ${\theta }_{\psi }$. Male and female survival are affected similarly by stabilizing natural selection, which acts independently on each of the phenotypic characters expressed by the individual. Hence, the fitness of a female is a function of her own phenotype $\left(x_{\mathrm{♀}},y_{\mathrm{♀}},\mathrm{and}\hspace{0.5em}{z}_{\mathrm{♀}}\right)$ and of her mating rate, which also depends on the average persistence ${\overline{y}}_{\mathrm{♂}}$ of the resident males with whom she interacts:

$$W_{\mathrm{♀}}=e^{-\frac{1}{2}a{\left(\psi \left(z_{\mathrm{♀}}\left({\overline{y}}_{\mathrm{♂}}-x_{\mathrm{♀}}\right)\right)-{\theta }_{\psi }\right)}^2}\times e^{-\frac{1}{2}\left(c_{x_{\mathrm{♀}}}{\left(x_{\mathrm{♀}}-{\theta }_{x_{\mathrm{♀}}}\right)}^2+c_{y_{\mathrm{♀}}}{\left(y_{\mathrm{♀}}-{\theta }_{y_{\mathrm{♀}}}\right)}^2+c_{z_{\mathrm{♀}}}{\left(z_{\mathrm{♀}}-{\theta }_{z_{\mathrm{♀}}}\right)}^2\right)}.$$ [2]

Likewise, the fitness of a male depends on his own traits $x_{\mathrm{♂}}$, $y_{\mathrm{♂}}$, and $z_{\mathrm{♂}}$, and on the average threshold ${\overline{x}}_{\mathrm{♀}}$ and sensitivity ${\overline{z}}_{\mathrm{♀}}$ of his mating partners:

$$W_{\mathrm{♂}}=e^{b\hspace{0.17em}\psi \left({\overline{z}}_{\mathrm{♀}}\left(y_{\mathrm{♂}}-{\overline{x}}_{\mathrm{♀}}\right)\right)}\times e^{-\frac{1}{2}\left(c_{x_{\mathrm{♂}}}{\left(x_{\mathrm{♂}}-{\theta }_{x_{\mathrm{♂}}}\right)}^2+c_{y_{\mathrm{♂}}}{\left(y_{\mathrm{♂}}-{\theta }_{y_{\mathrm{♂}}}\right)}^2+c_{z_{\mathrm{♂}}}{\left(z_{\mathrm{♂}}-{\theta }_{z_{\mathrm{♂}}}\right)}^2\right)}.$$ [3]

In these expressions, the parameters a and b scale the fitness consequences to females and males of IRSC. Moreover, ${\theta }_k$ and $c_k$ (where k can stand for any of the phenotypic characters) specify the optimal value of character k under natural selection and the stabilizing selection intensity, respectively.

Calculating the Response to Selection.

The strength and direction of selection on the phenotypic characters is quantified by the selection gradient, $\mathit{\beta }={\left({\beta }_{x_{\mathrm{♀}}},{\beta }_{z_{\mathrm{♀}}},{\beta }_{y_{\mathrm{♂}}},{\beta }_{x_{\mathrm{♂}}},{\beta }_{z_{\mathrm{♂}}},{\beta }_{y_{\mathrm{♀}}}\right)}^{\text{T}}$. Its elements are calculated directly from the fitness functions (Eqs. 2 and 3), using standard methods from evolutionary quantitative genetics (49). Specifically, depending on whether k is expressed in females or males,

$${\beta }_k={\frac{d\ln W_{\mathrm{♀}}}{d\hspace{0.17em}k}|}_{\begin{array}{c}{x}_{\mathrm{♀}}={\overline{x}}_{\mathrm{♀}}\\ y_{\mathrm{♀}}={\overline{y}}_{\mathrm{♀}}\\ z_{\mathrm{♀}}={\overline{z}}_{\mathrm{♀}}\end{array}}\hspace{0.17em}\mathit{or}\hspace{0.17em}{\frac{d\ln W_{\mathrm{♂}}}{d\hspace{0.17em}k}|}_{\begin{array}{c}{x}_{\mathrm{♂}}={\overline{x}}_{\mathrm{♂}}\\ y_{\mathrm{♂}}={\overline{y}}_{\mathrm{♂}}\\ z_{\mathrm{♂}}={\overline{z}}_{\mathrm{♂}}\end{array}}.$$

In the calculation of the selection gradients, we follow the common approach of assuming weak selection and limited phenotypic variation in the population. On the scale of a phenotypic SD, the fitness function can then be approximated by a linear function, and the selection gradients become independent of the phenotypic variances. Qualitatively different outcomes of sexual conflict occur in populations with high levels of standing genetic variation, in which case the approximations underlying Eq. 1 break down (SI Appendix, Individual-based simulations).

The population average value of each character changes in response to selection acting on the character itself, and due to selection on correlated characters. The combined effect of the direct and indirect component of the selection response is found by multiplying the selection gradient with the genetic variance–covariance matrix $\mathbf{G}$ (38). Several of the off-diagonal elements of $\mathbf{G}$ represent additive genetic covariances between a mating character and its pleiotropic character in the other sex (2). These intersexual covariances are of prime interest, because they quantify to what extent the resolution of IASC is constrained by male and female traits sharing a common genetic basis. The other off-diagonal elements measure covariance between nonhomologous characters due to pleiotropy or linkage disequilibrium. For simplicity, these elements of $\mathbf{G}$ are assumed to be negligibly small.

The number of parameters can be reduced further, if all traits are measured on a standardized scale. In that case, the additive genetic variances are equal to one and $\mathbf{G}$ takes the form of a correlation matrix:

$$\mathbf{G}=\left(\begin{array}{cccccc}1& 0& 0& r_x& 0& 0\\ 0& 1& 0& 0& r_z& 0\\ 0& 0& 1& 0& 0& r_y\\ r_x& 0& 0& 1& 0& 0\\ 0& r_z& 0& 0& 1& 0\\ 0& 0& r_y& 0& 0& 1\end{array}\right).$$ [4]

Here, $r_x$, $r_y$, and $r_z$ denote the additive genetic intersexual correlations between the expression of a mating character in one sex and its homologous pleiotropic character in the other.

Quantifying IASC and IRSC.

Following ref. 19, we consider IASC to arise when the selection gradients on genetically correlated characters in males and females point in opposite directions. The indices of IASC plotted in Fig. 3 and SI Appendix, Fig. S1 are therefore calculated as

$$I_x^{\text{IASC}}=-{\beta }_{x_{\mathrm{♀}}}\hspace{0.17em}{\beta }_{x_{\mathrm{♂}}},\hspace{0.17em}{I}_y^{\text{IASC}}=-{\beta }_{y_{\mathrm{♀}}}\hspace{0.17em}{\beta }_{y_{\mathrm{♂}}}\hspace{0.17em}\text{and}\hspace{0.17em}{I}_z^{\text{IASC}}=-{\beta }_{z_{\mathrm{♀}}}\hspace{0.17em}{\beta }_{z_{\mathrm{♂}}}.$$

Comparable indices for the strength of IRSC are calculated by multiplying the selection gradient of a mating trait with its effect on fitness in the other sex, that is:

$$\begin{array}{l}{I}_x^{\text{IRSC}}=-{\beta }_{x_{\mathrm{♀}}}\frac{d\ln W_{\mathrm{♂}}}{d\hspace{0.17em}{\overline{x}}_{\mathrm{♀}}}=b\hspace{0.17em}{\beta }_{x_{\mathrm{♀}}}\hspace{0.17em}{\overline{z}}_{\mathrm{♀}}\hspace{0.17em}\overline{\psi }\prime ,\\ I_y^{\text{IRSC}}=-{\beta }_{y_{\mathrm{♂}}}\frac{d\ln W_{\mathrm{♀}}}{d\hspace{0.17em}{\overline{y}}_{\mathrm{♂}}}=a\hspace{0.17em}{\beta }_{y_{\mathrm{♂}}}\hspace{0.17em}{\overline{z}}_{\mathrm{♀}}\hspace{0.17em}\left(\overline{\psi }-{\theta }_{\psi }\right)\hspace{0.17em}\overline{\psi }\prime ,\\ I_z^{\text{IRSC}}=-{\beta }_{z_{\mathrm{♀}}}\frac{d\ln W_{\mathrm{♂}}}{d\hspace{0.17em}{\overline{z}}_{\mathrm{♀}}}=b\hspace{0.17em}{\beta }_{z_{\mathrm{♀}}}\hspace{0.17em}\left({\overline{x}}_{\mathrm{♀}}-{\overline{y}}_{\mathrm{♂}}\right)\hspace{0.17em}\overline{\psi }\prime .\end{array}$$