## Abstract

Species spanning the animal kingdom have evolved extravagant and costly ornaments to attract mating partners. Zahavi's handicap principle offers an elegant explanation for this: ornaments signal individual quality, and must be costly to ensure honest signalling, making mate selection more efficient. Here, we incorporate the assumptions of the handicap principle into a mathematical model and show that they are sufficient to explain the heretofore puzzling observation of bimodally distributed ornament sizes in a variety of species.

## 1. Background

Darwin was the first to suggest that both natural and sexual selection play a role in the evolution of mating displays [1]. Natural selection is the shift in population traits based on an individual's ability to survive and gather resources, while sexual selection is the shift in population traits based on an individual's ability to mate with more or better partners. Natural selection alone cannot explain ornaments because they hinder survival and provide little to no benefit to the individual [2–4]. Darwin hypothesized that female preference for exaggerated mating displays drives the evolution of male ornamentation, but he was unable to explain why females prefer features which clearly handicap the males.

Zahavi's handicap principle attempts to resolve the paradox proposed by Darwin [5]. It argues that, because costly ornaments hinder survival, only the highest quality individuals can afford significant investment in them. Thus, the cost (often correlated with size) of an ornament truthfully advertises the quality of an individual, which makes mate selection easier. There is a large body of evidence that ornaments are indeed costly to the bearer (e.g. [6–8]), that ornaments are honest signals of quality (e.g. [9,10]), and that females prefer mates with larger ornaments (e.g. [11–13]).

A variety of theoretical approaches have been used to model the handicap principle [4,14–17]. Broad categories include game theoretical approaches (e.g. [18,19]), quantitative genetics (e.g. [20,21]) and phenotypic dynamics (e.g. [22,23]). Borrowing and expanding upon ideas from all three methods, we propose a new dynamical systems approach to understanding the evolution of ornaments within a population. Our model differs from some that search for a single evolutionarily stable strategy (ESS) (e.g. [19]) in that we do not require a unique phenotype for a particular male quality; our method allows for the possibility that an optimal *distribution* of strategies may emerge for a population—even a population of equal quality males.

Curiously, it has been observed that ornament sizes frequently have bimodal distributions, resulting in distinct small- and large- ‘morphs’ in many ornamented species (e.g. [24–26]). Figure 1 illustrates a classic example of ornament dimorphism, the horned dung beetle [24]. While in some cases researchers have identified genetic and environmental factors associated with ornament size variation (e.g. [27,28]), the splitting into two *distinct* large- and small-ornamented subpopulations (morphs) remains a contentious area of study.

Some evolutionary theories suggest that variety within the sexes may be because of varied mating strategies such as mimicry, sneaking or fighting [29,30]. However, our model suggests that the handicap principle alone may be sufficient to explain the origin of the observed ornament bimodality.

## 2. Model

With the goal of examining the quantitative implications of the handicap principle, we construct a minimal dynamical systems model for the evolution of extravagant and costly ornaments on animals. This proposed model incorporates two components of ornament evolution: an intrinsic cost of ornamentation to an individual (natural selection), and a social benefit of relatively large ornaments within a population (sexual selection). We show that on an evolutionary time scale, identically healthy animals can be forced to split into two morphs, one with large ornaments and one with small.

To express our model, we introduce the idea of a ‘reproductive potential’ *φ*. This can be thought of as similar to fitness, though our definition differs from the fitness function commonly used in the replicator equation [22,31] (we make the relationship between the two explicit in the electronic supplementary material). Over long time scales, the effect of evolution is to select for individuals with higher reproductive potential.

Consider an *individual* reproductive potential *φ*^{(ind)} of a solitary male with ornament size *a* (e.g. a deer with ornamental antlers). Some ornaments have practical as well as ornamental value (e.g. anti-predation [32,33]), but have a deleterious effect beyond a certain size. We therefore expect that there exists an optimal ornament size (possibly zero), for which individual potential is maximum, and thus take this to be a singly peaked function of ornament size. For simplicity, we assume the quadratic form^{1}
2.1Following the handicap principle, we expect the optimal ornament size *a*_{opt} = *a*_{opt}(*h*) to be an increasing function of ‘intrinsic health’ *h*—i.e. healthier individuals can afford larger ornaments. See figure 2*a* for the general shape of the individual reproductive potential function.

Next, we consider a *social* reproductive potential *φ*^{(soc)} that captures the effects of competition for partners (i.e. sexual selection). We assume social potential is an increasing function of ornament size^{2} because sexual selection often favours larger or more elaborate ornaments [12]. For simplicity, and motivated by the ubiquity of power laws in nature [34,35], we choose social potential to be a power of the difference between a male's ornament size and the average herd ornament size. To ensure monotonicity, we force the social reproductive potential to be antisymmetric about the average ornament size. The social potential is then
2.2where the positive parameter *γ* quantifies the rate at which deviations from the mean influence reproductive potential, sgn is the sign function, and represents the average ornament size in the population. Loosely speaking, the parameter *γ* tunes female choice; we take this ‘female choice’ parameter to be effectively constant because female choice may evolve on a slower time scale than male ornamentation [21]. Refer to figure 2*b* for an example of the social reproductive potential function.

Because both natural and sexual selection play a role in the evolution of ornaments [21], we take total reproductive potential to be the weighted average
2.3where *s* tunes the relative importance of competitive social effects (sexual selection) versus individual effects (natural selection). We show in the electronic supplementary material that a weighted product [36] produces identical qualitative results, so we focus on this case for simplicity of calculations. See figure 2*c*,*d* for examples of total potential functions.

Assuming that evolutionary forces optimize overall reproductive potential at a rate proportional to the marginal benefit of ornamentation, ornament sizes will follow the dynamics
2.4with time-scaling parameter *c* > 0. Note that this model does not presume that individual ornaments explicitly change size: the ‘phenotype flux’ d*a*/d*t* is simply a way of describing how the distribution of ornament sizes in a large animal population changes over long time scales as a result of selection processes.

This results in a piecewise-smooth ordinary differential equation for the ornament size flux:
2.5where *N* is the population size. Plugging (equation (2.5)) into the continuity equation yields a replicator equation for the evolution of the ornament size distribution (see the electronic supplementary material).

## 3. Results

### (a) Numerical exploration

For biologically relevant values of the social sensitivity parameter *γ*, our model predicts stratification into distinct phenotypes for a population of identically healthy individuals (i.e. individuals of identical quality). See figure 2*e*,*f* for the time evolution of ornament size for two representative values of *γ*.

For 0 < *γ* < 1, the ornament sizes stratify into large-ornament and small-ornament groups, with the majority possessing a large-ornament ‘morph.’ For 1 < *γ* < 2, the population stratifies into large- and small-ornament morphs, but the majority have small ornaments. The case *γ* ≥ 2 is not a reasonable option because we have selected a quadratic form for the local approximation of the individual potential function; any power *γ* exceeding 2 implies sexual selection is the dominant evolutionary force even for extremely large ornaments, an unreasonable assumption.

These qualitative results are consistent for all *a*_{opt} and 0 ≤ *s* < 1. While for clarity we have presented predictions of a specific minimal model, the qualitative results hold for a wide range of models. See the Discussion for the generality of model predictions.

### (b) Analytical results

As numerical integration shows that the uniform and two-morph steady states are of interest, we concentrate our analysis on these equilibria. However, it can also be shown graphically that uniform and two-morph steady states are the only possible solutions for a wide range of potential functions (see the electronic supplementary material).

#### (i) Uniform steady state

To investigate the uniform equilibrium with an identically healthy population, we set producing the single ordinary differential equation,^{3}
3.1The steady state (i.e. d*a*/d*t* = 0) is clearly *a* = *a*_{opt}. Linear stability analysis within this identical ornament manifold shows the fixed point *a* = *a*_{opt} is stable for all *γ*, but numerical simulation suggests that the uniform fixed point is only stable for *γ* ≥ 2. To resolve this apparent discrepancy, we investigate the uniform fixed point of (equation (2.5)) in the continuum limit, and evaluate stability without restriction to the uniform manifold. We are then able to find *γ*-dependence that agrees with simulations (details in the electronic supplementary material).

#### (ii) Two-morph steady state

To investigate the two-morph equilibrium, we assume all males have one of two ornament sizes *a*_{1} and *a*_{2}. Taking *x* to be the fraction of males with ornament size *a*_{1}, and *N* → ∞, the dynamical system becomes
3.2There exists one two-morph steady state (i.e. solution to d*a*_{1}/d*t* = d*a*_{2}/d*t* = 0):
3.3Figure 3*a*,*b* shows how two-morph equilibria vary with the morph fractionation *x*. Within the shaded region, the fixed point is stable. To be clear, the model predicts that a bimodal population will emerge, with the fraction *x* of the individuals within the population possessing ornaments of size *a*_{1}. We are *not* claiming that a proportion *x* of populations will evolve to ornament size *a*_{1}.

The eigenvalues for the linearized system constrained to this two-morph manifold are and . Clearly, the two-morph equilibrium is stable (within the two-morph manifold) for 0 < *γ* < 2 and unstable for *γ* > 2, when *λ*_{2} > 0. Curiously, the stability of the two-morph equilibrium does not depend on *x*, the morph fractionation. This presents an apparent problem because numerical simulation suggests that only certain ranges of *x* are stable (figure 3*c*). Similarly to the uniform fixed point analysis, we investigate the fixed points of the model in the continuum limit, and evaluate stability without restriction to any manifold. We are then able to find *x*-dependence that agrees well with simulations (figure 3*d*; details in the electronic supplementary material).

## 4. Model validation

We now revisit our simplifying assumption that all males are equally healthy. More realistically, we allow the intrinsic health *h* to be taken from some distribution (perhaps set by genetic, developmental or environmental factors). Suppose this distribution is such that the individual optimal ornament size *a*_{opt}(*h*) is normally distributed. Then, the stable two-morph steady state changes from a weighted sum of perfectly narrow Dirac delta functions to a distribution roughly resembling the sum of two Gaussians—usually a *bimodal* distribution. Marginal histograms in figure 4*a*,*b* show examples of steady states with varied intrinsic health.

These examples resemble data from many species that grow ornaments. Figure 4*c*,*d* shows two examples of real-world ornament distributions that exhibit bimodality. Note that we do not expect the exact shape of the real-world distributions to match our simulations because the measured quantities will not necessarily be linear in cost. However, bimodality will be preserved regardless of the measured quantity.

In a literature search [11,12,26,37–49], we found a number of published datasets showing size distributions of suspected ornaments; 23 were of sufficient quality for testing agreement with this model. In 13 of those datasets, we found some evidence for rejecting the hypothesis of unimodality: the data were more consistent with a mixture of two or more Gaussian distributions than with a single Gaussian. In seven datasets, we found stronger evidence: non-parametric tests rejected the hypothesis of unimodality. Note that other datasets were not inconsistent with bimodality, but small sample sizes often limited the power of statistical testing. See the electronic supplementary material for histograms and statistical tests of additional datasets.

## 5. Discussion

### (a) Implications for honest signalling

Assuming this model adequately represents the handicap principle, we may ask if ornament size really does honestly advertise quality. In other words, if a female can choose among all the males, is she able to detect the healthiest (or weakest) males simply by looking at ornament size? Again taking the optimal ornament size *a*_{opt} to be normally distributed, we examine the Kendall rank correlation between intrinsic health (as indicated by our proxy *a*_{opt}) and equilibrium ornament size.

We find that the advertising is mostly honest, at least for large enough variance in health. Both observational and experimental work supports this finding [9]. Figure 4*a*,*b* shows examples of ornament size versus intrinsic health based on our model.

### (b) Generality

It is natural to wonder about the generality of the results we have presented here. For a reasonable set of potential functions (described below), the only possible stable equilibria are multimodal distributions of ornament size. The following are the requirements for our reasonable potential functions.

(i) Individual effects dominate potential for large-ornament sizes. Specifically, This prevents ornament size from growing without bound, as can occur in model (2.5) for

*γ*≥ 2.(ii) Social effects dominate potential for at least some range of ornament sizes greater than the population mean. In other words,

for at least some range of . Failure to meet this criterion could be considered ‘false’ ornamentation, as in model (2.5) for

*γ*= 1.

Assuming the potential functions are continuous,^{4} these criteria guarantee that two or more morphs will emerge (see the electronic supplementary material for details).

One benefit of our modelling approach is that the evolutionarily optimal distribution we predict is independent of the particular mechanism(s) used to maintain phenotype diversity (which might include various combinations of genetic, epigenetic, environmental or other cues; see the electronic supplementary material for more discussion of this point). Of course, our model is not the only one that can show bimodality of traits. We have provided one simple mechanism for explaining such polymorphism, but other effects (e.g. intra-sexual selection, over-dominance, negative frequency-dependent selection) might also work in concert or be able to independently explain observed data.

## 6. Conclusion

The independent evolution of costly ornamentation across species has puzzled scientists for over a century. Several general evolutionary principles have been proposed to explain this phenomenon. Among the prominent hypotheses is the handicap principle, which posits that only the healthiest individuals can afford to grow and carry large ornaments, thereby serving as honest advertising to potential mates. We base a minimal model on this idea and find that, surprisingly, it predicts two-morph stratification of ornament size, which appears to be common in nature.

Importantly, the two morphs both have ornament sizes larger than the optimum for lone individuals. This means that the population survival potential, as indicated by the population average of individual potential , is reduced. Owing to the presence of ornaments, we conclude that the evolutionary benefits of honest advertising must outweigh the net costs of ornamentation when the displays exist in nature.

## Data accessibility

Data and code available from the Dryad Digital Repository [50]: http://dx.doi.org/10.5061/dryad.vb1pp. All ornament data used for statistical tests have been included as the electronic supplementary material.

## Authors' contributions

S.M.C. and D.M.A. developed and analysed the model, S.M.C. implemented the numerical simulations and created the database, R.I.B. and S.M.C. performed statistical tests on data. All authors gave final approval for publication.

## Competing interests

The authors have no competing interests.

## Funding

This research was supported in part by James S. McDonnell Foundation grant no. 220020230 and National Science Foundation Graduate Research Fellowship no. DGE-1324585.

## Acknowledgements

We thank Daniel Thomas, Trevor Price and Stephen Pruett-Jones for valuable discussions; Doug Emlen, Craig Packer, Markus Eichhorn, and Armin Moczek for sharing biological data; and editors Innes Cuthill and Samuel Flaxman as well as anonymous referees for useful feedback.

## Footnotes

Electronic supplementary material is available online at https://dx.doi.org/10.6084/m9.figshare.c.3575198.

↵1 This is a generic form for an arbitrary smooth peaked function approximated close to its peak.

↵2 This assumption applies most naturally to inter-sexual selection, ignoring alternative reproductive strategies associated with intra-sexual selection (e.g. cryptic males).

↵3 For

*γ*≤ 1, we set*φ*^{(soc)}= 0 before setting to avoid an undefined right-hand side of (equation (2.5)).↵4 This is a stronger requirement than necessary. Actually, we only require that the two-sided limits exist everywhere.

- Received September 6, 2016.
- Accepted November 1, 2016.

- © 2016 The Author(s)

Published by the Royal Society. All rights reserved.