Evolution of sexual cooperation from sexual conflict
Maria R. Servedioa,1, John M. Powersa,b,1, Russell Landec, and Trevor D. Priced,2
a
Department of Biology, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599; bDepartment of Ecology and Evolutionary Biology, University of
California, Irvine, CA 92617; cCenter for Biodiversity Dynamics, Norwegian University of Science and Technology, N-7491 Trondheim, Norway;
and dDepartment of Ecology and Evolution, University of Chicago, Chicago, IL 60637
In many species that form pair bonds, males display to their mate
after pair formation. These displays elevate the female’s investment into the brood. This is a form of cooperation because without the display, female investment is reduced to levels that are
suboptimal for both sexes. The presence of such displays is paradoxical as in their absence the male should be able to invest extra
resources directly into offspring, to the benefit of both sexes. We
consider that the origin of these displays lies in the exploitation of
preexisting perceptual biases which increase female investment
beyond that which is optimal for her, initially resulting in a sexual
conflict. We use a combined population genetic and quantitative
genetic model to show how this conflict becomes resolved into
sexual cooperation. A cooperative outcome is most likely when
perceptual biases are under selection pressures in other contexts
(e.g., detection of predators, prey, or conspecifics), but this is not
required. Cooperation between pair members can regularly evolve
even when this provides no net advantage to the pair and when
the display itself reduces a male’s contributions to raising the
brood. The findings account for many interactions between the
sexes that have been difficult to explain in the context of sexual
selection.
|
cooperation differential allocation
sexual stimulation
| sensory bias | sexual conflict |
I
n pair-bonding species, displays between the sexes often continue after pair formation (1). Ethologists have long argued
that a main function of ongoing displays by males is to bring the
females into, and maintain them at, prime reproductive condition (2–4) as corroborated by many subsequent experiments (5).
Nevertheless, it is unclear why such displays evolve because the
pair as whole should benefit if the male were to direct his energy
into the brood, rather than display. To date, explanations have
been focused on display maintenance rather than origination and
are based on the idea that display intensity correlates with mate
quality (6, 7). Notably, the differential allocation hypothesis
states that if a female is paired with a high-quality male, she is
under selection to invest more in the brood than when paired
with a low-quality male (6–12) and that male displays are present
because they indicate male quality to the female, which she
cannot otherwise observe. Here we show how displays may regularly become established and maintained even in the absence of
differential allocation, with their role simply to bring females
into optimal reproductive condition.
Our model is predicated on the many experimental studies in
birds which show that exaggeration of a male behavior, color, or
vocalization stimulates females to increase their investment (SI
Appendix, Table S1) (13, 14). The origin of such stimulatory effects is thought to lie in perceptual biases, arising from “properties of the environment, signals and neural systems” (15). More
than 100 examples of such biases are now known from all of the
major sensory systems (sound, sight, and smell) (16). When a
male display exploits a perceptual bias that causes a female to
increase her investment in her offspring, it can provide an advantage to the male but be detrimental to the female because it
reduces her future reproductive success. This is a sexual conflict:
the male display starts to increase in the population, but females
are under selection to resist the display (17, 18). Such conflicts
www.pnas.org/cgi/doi/10.1073/pnas.1904138116
have been modeled using computational approaches (i.e., neural
nets) (19, 20). In these models a signaler, which in our case is the
male, exploits a receiver’s perceptual bias by producing displays
that stimulate the receiver to the receiver’s detriment. At some
point in the future a mutation in the receiver (female) population leads to a failure to be stimulated by the display (19).
This mutation increases, raising female mean fitness, reducing
male mean fitness, and leading to the subsequent loss of the
display. As these evolutionary adjustments continue, new biases
arise, and the process starts over. Because each sex increases its
fitness at the expense of the other, the fitness of both sexes is not
simultaneously maximized.
Here we consider a route by which such conflicts evolve into a
stable cooperative system. We follow Mesterton-Gibbons and
Dugatkin (21) in defining cooperation as an outcome that despite individual costs is good in some appropriate sense for the
members of a group and whose achievement requires collective
action. Other definitions have required cooperative acts to have
evolved specifically for the benefits they provide (22), but this
precludes behaviors that originate out of conflict, the focus of
this study. The essential idea is that selection on female investment returns her investment to its optimal level, even as
females continue to respond to the display (1). Suppose a male
display increases in the population because it causes females to
lay more eggs in their clutch (Fig. 1). Clutch size is now above the
female optimum so females that do not respond to the novel
display are favored. In one outcome, resistance to the display
evolves, as in models of sexual conflict (lower right arrow, Fig. 1).
Significance
The past 50 y have seen much research on sexual selection.
However, in many species, displays between the sexes continue long after pair formation, even if pairs have been together for years. As shown experimentally, such displays result
in cooperation between the sexes, whereby displays by one
partner affect investment into the brood by the other. How
one gets to this cooperative outcome is not understood. We
show such outcomes evolve if a novel display exploits a preexisting sensory bias that raises receiver investment. Once
established, displays are maintained because they are required
to stimulate the partner optimally. The pair bond is strengthened, and displays between the sexes accumulate over evolutionary time, even in the absence of sexual selection.
Author contributions: M.R.S., R.L., and T.D.P. designed research; M.R.S., J.M.P., R.L., and
T.D.P. performed research; and M.R.S., J.M.P., R.L., and T.D.P. wrote the paper.
The authors declare no competing interest.
This article is a PNAS Direct Submission.
Published under the PNAS license.
Data deposition: An interactive file is available at https://powers.shinyapps.io/pqreadr/,
and C code is archived on Dryad.
1
M.R.S. and J.M.P. contributed equally to this work.
2
To whom correspondence may be addressed. Email: pricet@uchicago.edu.
This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.
1073/pnas.1904138116/-/DCSupplemental.
PNAS Latest Articles | 1 of 7
EVOLUTION
Edited by James J. Bull, The University of Texas at Austin, Austin, TX, and approved September 22, 2019 (received for review March 11, 2019)
Fig. 1. Alternative outcomes when appearance of a novel male display
causes a female to invest above optimal levels. (Left) Ancestrally, females
invest optimally by laying 3 eggs in a clutch. Males do not display in an exaggerated manner to the female and have little ornamental plumage.
(Center) A male develops ornamental blue plumage associated with a display
to his mate, stimulating her to invest more in the brood (4 eggs). The display
increases the male’s fitness but lowers that of his mate. (Lower Right) In
traditional models, the engendered sexual conflict results in females evolving
to ignore the new display (males should subsequently lose the display as it
provides no benefit). (Upper Right) Instead, we consider an alternative outcome whereby investment evolves back to the female optimum even as
females respond to the display. In this case the display is retained because
without it females would produce fewer than the optimal number of eggs,
detrimental to both sexes. Illustration courtesy of Allison Johnson (artist).
However, an alternative possibility is that a clutch size above the
female optimum generates selection on females to reduce their
investment, even as they continue to respond (upper right arrow,
Fig. 1). In this second case, investment of responding females
may evolve back to what it was before a display arose. This
becomes a cooperative system that is evolutionary stable,
because a mutation that causes the male to display at a lower
rate will reduce female investment below that which is optimal
for either sex.
Results
We evaluate conditions under which the alternative outcomes in
Fig. 1 are expected, based on a mathematical model with haploid
genetics. A quantitative trait with mean zf describes basal female
investment in her clutch (all symbols are defined for easy reference in the text, in the legend to Fig. 2 and SI Appendix, Table
S2). The mean is initially allowed to evolve to its optimum, zopt,
which depends on both the fecundity benefit of the investment
(number of surviving offspring) and its viability cost to females
(cf). The viability cost is paid after reproduction, implying some
females survive to breed to the following year, resulting in
overlapping generations (Methods and SI Appendix, Fig. S1). A
novel male display and the female response to it are each
modeled by freely recombining, diallelic, single-locus modifiers,
thus combining quantitative and population genetic approaches
(23, 24). At the locus which controls a response in females, A,
there is initially a high frequency of the responder allele (A2),
corresponding to the existence of a preexisting perceptual bias
(the nonresponder allele A1 is at a low frequency). The display
locus B is initially fixed for the allele for no display, B1. We then
introduce the allele B2 that causes a male to display. The male
display comes with a viability cost to the male (sm) but elevates
investment of an A2 female partner by a value α. For example, α
may represent extra eggs laid by a responding female (Fig. 1), but
increased investment could also come through provisioning and
other positive effects she may have on her offspring (SI Appendix, Table S1). Consequently, a pair in which the male carries the
display allele and the female carries the responder allele (i.e., A2B2
pairs) have extra female investment and hence higher fecundity
than the other 3 combinations. The number of offspring surviving to
breed the next year is affected by density dependence within the
brood (mediated by parameter a), and density dependence after
fledging (b), driven by total population size.
After offspring production males and females undergo mortality determined by their investment in the brood (cm and cf,
respectively), the cost of display (sm), and a nonselective extrinsic
death rate (dm for males and df for females). Surviving pairs may
then undergo divorce, at a constant rate (v). Death and divorce
result in males and females not always being paired to the same
partner each year, which is a necessary condition for sexual
conflict. Widows, widowers, juveniles, and any formerly single
individuals (extras of the more common sex after pairing) combine to form new mated pairs, and the yearly cycle is complete
(see Methods for the full presentation of the equations). We
Fig. 2. Evolution of display, responder, and investment. (A) The responder allele, A2, starts at high frequency (0.99). A display allele B2 is introduced at low
frequency (0.01). Basal investment (green), zf , represents mean female investment without the added effects of the response to the male trait (α); dashed
green line is mean realized investment. We confirm stability of the equilibrium by introducing a perturbation (frequency of A1 = 0.01) at 10,000 y (blue
arrow). Parameters are as follows: α = 1.2, viability cost to males of displaying; sm = 0.01, scaled cost of investing in eggs for females; cf = 0.08 and for males,
cm = 0.01, nonselective density dependence in the brood before fledging; a = 0.09 and after fledging, b = 0.00007, nonselective mortality rate for both
females and males; df = dm = 0.1, divorce rate; v = 0.3, heritability of zf; h2 = 0.5, phenotypic variance of zf; σ 2 = 1, recombination rate between the display and
responder loci; r = 0.5. (B) Case in which the additional investment, α = 2.4, induces a level of investment too high for females to sustain the response, A2,
before the quantitative investment trait z can respond. The responder A2 is lost, and the display allele B2, which evolves to be close to fixation when the
responder is still present, is slowly lost due to selection against the display. Perturbation at 10,000 y (A2 reintroduced at a frequency of 0.01) confirms stability.
(C) Alternative outcomes after 100,000 y as a function of costs of female investment, cf, and investment induced, α. Perturbations occur every 10,000 y to
reintroduce at frequency of 0.001 any allele that is within that frequency of loss. The points a and b correspond to the parameters of the dynamic panels to
the left. In the yellow region, display (B2) and responder (A2) alleles are fixed. In the red region, display never increases, and the responder allele remains at
high frequency (A2 and B1 high). In the black region, cost of the induced female response causes the responder allele to be lost, followed by loss of the display
(A1 and B1 high). See https://powers.shinyapps.io/pqreadr/ (SI Appendix, Fig. S4), for a complete interactive version of both panels of this figure, where the
effect of altering the parameters can be examined and both initial and equilibrium basal clutch size are listed.
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Servedio et al.
Selection on Perceptual Biases. The model so far assumes that
perceptual biases are not subject to any selective forces other
than those which affect their response to the display. In fact, many
biases are thought to be present because they facilitate predator,
prey, or conspecific detection (15, 16, 25). A small selective advantage (e.g., a viability advantage of sp = 1%) favoring the responder allele, A2, increases the parameter space that allows the
permanent fixation of the display (compare Fig. 2C with Fig. 3C).
If both induced investment α and costs to investment cf are high,
the display, response, and to a slight extent investment may cycle,
depending on the frequency of perturbations (Fig. 3B), as follows.
First, when the display is absent, natural selection maintains the
responder allele at high frequency. Consequently, an introduced
display will increase. Next, the responder allele declines because of
high costs of investment, which overwhelm its natural selection
advantage. Once the responder allele is at a low enough frequency, the display declines as it gives little advantage to males.
When the display is at a sufficiently low frequency, a mutation
favoring the response increases under natural selection, and the
cycle is repeated.
Fecundity Selection on Males. Male displays may become estab-
lished even if they come with a cost that reduces male investment
into the current brood (parameterized by sfec), thereby directly
affecting both male and female fitness rather than reducing the
male’s survival. Outcomes across parameter space are similar
whether costs to the male affect fecundity or viability (Fig. 4). In
the fecundity case, when the display is fixed (in yellow), all males
are now investing less into the brood than they were prior to the
introduction of the display. In this case, female investment
evolves to be greater than it was before the display was present
(zf + α is higher than was zf prior to the evolution of the display).
A higher female investment when male display costs operate
through fecundity rather than viability raises the average number
of offspring fledged per pair, but this does not completely
compensate for the lower number of offspring fledged per pair
due to the presence of the display. Increased female investment
over time raises the average number of offspring fledged per
female but reduces mean female viability, resulting in a new
balance between these 2 forces, where the average number of
offspring is lower than both the viability case and the initial
conditions. For example, for the parameters of Fig. 2A, with
male display imposing a fecundity cost of 1%, rather than a
survival cost, female investment is raised by 1.3%, female mortality is increased by 0.5%, and offspring fledged per pair are
reduced by 0.16%, compared to the viability case.
Fig. 3. Natural selection on the perceptual bias. Here natural selection favors the responder allele, A2 (sp = 0.01). As in Fig. 2 the responder allele is initially at
a frequency of 0.99 and the display allele introduced at a frequency of 0.01. If close to fixation or fixed, both the display and responder alleles are perturbed
away by 0.01 for A and B and 0.001 for C, every 500 y. (A) cf = 0.04, α = 2.5. (B) cf = 0.08, α = 4. All other parameters are as in Fig. 2. (C) Colors are as in Fig. 2
with the yellow region indicating permanent fixation of the display. The colors represent a snapshot of the frequency of the display allele (green channel) and
responder allele (red channel) as they were at 10,000 y. A blue overlay indicates strong cycling, where A2 and B2 were simultaneously low (<0.2) and then
simultaneously high (>0.8). The cycling is induced by reintroduction of the display/responder alleles. Labels a and b are the positions for the dynamic runs
illustrated in the corresponding panels. It can be seen that the cycles in B are dampening; an extended run shows that there are eventually polymorphisms at
both the A and B loci with these parameters. Note that both points a and b are in the black region of Fig. 2C, so without selection on the perceptual bias the
display would not be established.
Servedio et al.
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assume random mating at pairing and strict social and genetic
monogamy, thereby removing all forms of sexual selection (i.e.,
selection associated with competition for mates).
We find that a stable equilibrium can indeed be attained
where all males display and all females respond to the display. At
equilibrium, realized investment is restored to the same value as
it was prior to the display appearing (as in Fig. 1, Top Right; in
our evaluations of long runs, initial investment and investment at
the end of the run differed only at the sixth decimal place). An
example of the evolutionary dynamics is shown in Fig. 2A, where
investment back to the optimum for females results from evolution of the quantitative investment trait (dashed green line is
realized investment, and solid green line is investment as it would
be stripped of any female responses). An example of the alternative outcome where the sexual conflict results in loss of responder and display is shown in Fig. 2B. In Fig. 2C we delineate
the role of 2 parameters in affecting these outcomes: the increase
in investment promoted by the display α and the costs to the
female of investment, cf. In the red region, the display never
increases because its costs outweigh the benefits. In the yellow
and black regions the display rises in frequency, resulting in increased costly investment by the female. In the yellow region, the
response to this selection occurs by reduction in the quantitative
investment trait, resulting in the permanent fixation of both the
display and responder alleles. In the black region, where there is
higher induced investment by the responding females, the
responding allele is quickly lost, followed by the display, as in
traditional sexual conflict models (Figs. 2B and 1, Lower Right).
The roughly vertical border between the red and yellow regions represents the point where benefits to males of extra investment by the female balance the costs of the display, such that
the display is too costly in the red region. The horizontal border
separating the yellow and black regions from the red one arises
similarly but for a subtler reason. When female costs of investment cf are low, optimal investment zopt is relatively high, and
hence, female investment is high at the initial conditions. The
consequence is that a display gives the male a proportionally
smaller benefit from the investment it induces, again countered
by its costs. The parameter space over which the display and
response are fixed (yellow) varies depending on the strength of
selection and genetic variance in the responder allele. For example, when the initial frequency of the responder allele is 0.9,
instead of the frequency of 0.99 shown in Fig. 2, introduction of a
display results in more efficient selection against responders
(initial genetic variance is higher), and the black region extends
farther to the left (SI Appendix, Fig. S2).
Fig. 4. Comparison of fecundity and viability costs to the display. The responder allele is initially at a frequency of 0.99, the display allele is at a
frequency of 0.01, and perturbations to a frequency of 0.001 occur every 500 y.
Selection coefficients are plotted on a log scale. In the yellow region, both
display and responder alleles are permanently maintained through their
effects on investment. In the red region, the display is lost. A region of cycling at the display and responder loci is indicated by the stippling. In the
black region both responder and display alleles are lost, as in the green
region, but here the display initially rises to a high frequency and is lost only
slowly. Parameters are as in Fig. 2 but with cf = 0.07, α = 2.3. (Left) Fecundity
selection on the male display. (Right) Viability selection on the male display.
Cooperation Can Arise Directly in Changing Environments. Finally,
we identified a situation in which displays promoting investment
become established under a particularly broad range of parameter space because benefits immediately accrue to the female (SI
Appendix, Fig. S3). This happens if mean female investment is
initially below the optimal level, e.g., as a consequence of a
changing environment. Then a male’s display that increases her
investment also increases her fitness. The display is immediately
cooperative, and it can be subsequently maintained at equilibrium. The fact that displays and responses may more easily become established in changing environments than in static ones
suggests they can contribute to speciation events, which may
often be associated with entry into novel environments (26). We
caution that our conclusion that establishment may be easier under these conditions is based on a broadening of the parameter
space under which we get the fixation of display, but the question
of what parameter spaces are reasonable is always an empirical
one, and we encourage further study of these conditions.
Discussion
Our models show that a simple explanation for the presence of
displays after pairing is that they are required to stimulate females to invest adequately in offspring, as has long been proposed (1–3). Here we show how such a requirement could
evolve. In words, male displays induce the female to overinvest
thereby placing selection on females to reduce their investment.
Provided a reduction can be made by evolution at loci affecting
investment, rather than loci affecting response to the display,
there is the possibility for both the male display and female response to be maintained. Under such conditions, the male display is now required to optimally stimulate the female. We find
this outcome is most likely when costs to female investment are
high, but levels of investment induced by the display are moderate. On the other hand, when both costs and the induced investment by a displaying male are high, strong selection against
the responder allele results in its loss, as in standard sexual
conflict models. We explored this process using a model which
assumes that the responder and display are determined by single
loci of large effect. Large-effect loci are realistic for some visual
display traits (27). An alternative approach would be a full
quantitative genetic model, which would require an additional
set of simplifying assumptions. Qualitatively, we expect to find
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similar outcomes, especially when a perturbation in the display
induces at least a moderate increase in investment. Specifically, in the continuous formulation, if the mean display is
perturbed, then this will induce selection on loci affecting
both investment and response, and both should evolve. To the
extent investment evolves, we expect at least some element of the
display and response to be maintained to raise investment back to
the optimal level.
A complementary explanation for the presence of displays
after pairing assumes they are present to indicate male quality. In
this idea, a female increases her investment when paired to a
male with an exaggerated display because he is likely to be of
high quality, which her offspring will inherit (6, 7). Theoretical
models demonstrate that elevated female investment when
paired to a high-quality male is expected in some situations, but
in other cases the adaptive solution is no change in investment,
or a reduction, because high-quality males may invest more,
which compensates (8–10). We suggest mate stimulation may
explain many of the experimental findings documented in SI
Appendix, Table S1, without recourse to differential investment
arguments. Nevertheless, if differential investment is important, our results provide a model for how it may evolve. Once
costly displays have become established, they are generally
under selection to become condition dependent and thereby
correlate with male quality (28).
Any evolutionary model necessarily comes with assumptions
about the spectrum of permissible mutations. It is possible to
devise scenarios whereby mutations we have not allowed would
destroy the displaying equilibrium. For example, a reader suggested that after the displaying equilibrium is reached, a mutation that causes a female to ignore the male but maintain optimal
investment should be favored because such a mutant would not
underinvest when paired with a weakly displaying male or
overinvest when paired with a strongly displaying male. It remains an empirical question as to whether such mutations, or
others along similar lines, are plausible. We consider this particular example to be unlikely because of the pleiotropic effects
implied: if a mutation causes the female to ignore the male display, we expect this to also lower her investment, as we have
modeled. In consequence the mutation would have to simultaneously increase investment to exactly compensate for the loss due
to lack of male stimulation. Even if mutations such as these do
occur, they need not be favored. For example, the female may be
less able to assess the optimal clutch from environmental cues
than from the male, or evolution subsequent to the establishment
of the display may have resulted in other selective pressures favoring its maintenance, as would be the case if differential allocation secondarily evolves.
In the model we have eliminated all forms of sexual selection,
that is, selection associated with competition for mates. In reality, both sexual selection and sexual stimulation are present in
many species. Often the same displays and traits are employed in
both cases, presumably because perceptual biases influence both
attraction and stimulation, as exemplified by several of the
studies in SI Appendix, Table S1. Given this reasoning it is possible that in sexually selected systems, females are not only
choosing displaying males but also being reproductively stimulated by viewing displays. A link between stimulation and preference provides a novel explanation for the lek paradox, which
asks how female mating preferences can be maintained in the
absence of obvious benefits to choice (29). We suggest direct
benefits may be present in the form of the reproductive stimulation display imparts to females. For example, in a controlled
experiment in the peafowl Pavo cristatus, attractive males induce
females to lay larger eggs containing more testosterone (30). The
inference is that in the absence of peafowl display they would
underinvest in their fecundity.
Servedio et al.
Methods
We consider the evolution of 3 traits, 2 of which are each controlled by a
single locus and 1 of which is polygenic, expanding on models in refs. 23 and
24. Specifically, we assume that a responder locus (A) and display locus (B)
are each autosomal and haploid and contain 2 alleles. Both are sex-limited in
their expression. The third trait is controlled by the combination of a normally distributed, haploid, continuously varying, sex-limited trait expressed
by females, zf, and the genotype of the mated pair at the responder locus (in
the female) and display locus (in the male), together producing female investment, z. Females mate with males to form a diploid zygote that undergoes free recombination between the signal and response loci and the
quantitative trait. We track numbers of single individuals and mated pairs,
assuming a life cycle with overlapping generations (the events in the life
cycle, described below, are also shown in SI Appendix, Fig. S1). Pairing is
random, breeding occurs once each year, and pairs are strictly monogamous
within a breeding season.
Life Cycle, Demographic, and Genotypic Equations. The life cycle begins right
after pairing. Numbers of single males and females of each genotype at the
responder and display loci are denoted by nmi and nfi, respectively, where the
first index in the subscript refers to the sex of the individual and the second
index, i, ranges from 1 through 4 referring respectively to the genotypes
A1B1, A1B2, A2B1, and A2B2. The number of individuals in each mated pair
consisting of a female of genotype i and a male of genotype j is designated
by Nij. Directly after pairing, only the more common sex contains unmated
individuals.
In each mated pair combination with a female responder genotype i and a
male display genotype j, female investment, z, has a normal distribution
pij(z) with phenotypic variance σ2, and heritability h2, and mean zf + αij
where zf is the basal value in females before modification. The component
zf is set at the beginning of each year based on the response to
of the mean
selection in the previous year (see below). Weak selection justifies the assumption of normality in the adult generation, when different age classes
are combined, as are the distributions of zf in pairs with different display and
response genotypes. The alternative is to track the phenotype within each
genotypic category and cohort across years, which becomes very complex.
Weak selection also justifies the assumption of constant variance. Individual
phenotypes are assumed not to change with age. The additional component
of the mean, αij, is set by the genotype at the responder and display loci
in each pairing Nij. Specifically, if the female in a pair carries the
Servedio et al.
responder allele A2 (i = 3 or 4) and the male expresses the display allele
B 2 (j = 2 or 4), then increased investment in the clutch is triggered, and
αij = α. If either of these conditions is not met, females make no additional investment, and αij = 0.
Right after the census at the time of pairing, fecundity selection occurs,
associated with offspring production. The number of offspring that survive to
recruitment, as produced by each combination of mated pairs, is
Fij = 1 − sfec,j e−bNtot
Z
pij ðzÞwðzÞdzNij ,
[1]
where Ntot is the total number of adult individuals in the population,
wðzÞ = ze−az, and sfec,j is a fecundity cost to the pair when the male displays (such that sfec,j = sfec when j is even and sfec,j = 0 otherwise). Note
that sfec is set to 0 for most of the results discussed above. Here a represents the effect of density dependence within the brood on offspring
survival, whereas b represents the density dependence of offspring
survival after fledging on the number of adults present. The fact that
females with allele A 2 increase their investment in offspring if mated to
males with allele B2 is reflected in whether the extra investment αij = α is
triggered (added to the zf expressed by female i ) for that mated pair i, j
when the integral is taken.
of offspring of each genotype and each sex is generated
The number noff
i
from the matrix F using standard equations for free recombination and
segregation in haploids. The sex ratio of offspring is set to 1:1, which is accomplished by multiplying the total number of offspring from each pairing
Fij by 1/2 in the calculation of noff
i .
After offspring production, adults in this overlapping-generation model
undergo mortality that reflects, in part, the investment they have made in
their offspring. There are several sources of mortality. Nonselective extrinsic
death occurs with a constant death rate that may differ between the sexes.
Selective death can also occur if there is an advantage to the responder allele
in another ecological context (e.g., detection of predators, and food). These
are the only sources of death on individuals that remain single (unpaired).
Specifically, the numbers of single females and males that survive are
nsfi = ð1 − df ð1 − spi ÞÞnfi and nsmj = ð1 − dm ð1 − spj ÞÞnmj , respectively, where df
and dm are the probabilities of death for each sex, and sp reflects a perceptual bias survival advantage of the responder allele A2. This latter advantage is given in both sexes; thus, spi = sp when i is 3 or 4, and spj = sp when
j is 3 or 4 (otherwise, spi or spj are equal to 0).
Selective death in the context of pairing occurs due to viability selection
against expressing the male trait (B2 in males), with selection coefficient sm,
as well as due to selection against investment such that the survival rate
due to investment is e−ck z, where k is an index for females or males (f or m)
and ck is an investment cost parameter (cf will generally be greater than cm
because females pay a cost of egg production on top of a cost of provisioning the young, but we allow these parameters to vary freely). Note
that costs of the display can be manifest as viability costs to the male, sm,
or fecundity costs to the pair, sfec. Taking all of these sources of death into
account, the number of mated pairs with both members surviving to the
following year is
Nijs = 1 − df 1 − spi
1 − dm 1 − spj
1 − smj
Z
pðzÞe−cf z e−cm z dz Nij ,
[2]
where smj = sm if j = 2 or 4 (indicating B2 is present) and smj = 0 otherwise,
spj = sp if j = 3 or 4 and spj = 0 otherwise, and spi = sp if i = 3 or 4 and spi = 0
otherwise. Similarly, the number of widows of genotype i who were mated
to males of genotype j is
nw
fij = Nij
Z
pðzÞ 1 − df 1 − spi e−cf z 1 − 1 − dm 1 − spj 1 − smj e−cm z dz,
[3]
where the ð1 − df ð1 − spi ÞÞe−cf z term accounts for survival of the female and
the 1 − ð1 − dm ð1 − spi ÞÞð1 − smj Þe−cm z term accounts for the death of the male.
Likewise, the number of widowers of genotype j who were mated to female
of genotype i is
nw
mij = Nij
Z
pðzÞ 1 − dm 1 − spj
1 − smj e−cm z 1 − 1 − df 1 − spi e−cf z dz.
[4]
The model allows for divorce, which is assumed to occur right before the next mating season irrespective of the number of offspring
fledged from a particular nest. The number of mated pairs that remain
PNAS Latest Articles | 5 of 7
EVOLUTION
We have shown how sexual conflicts evolve into permanent
sexual cooperation, a scenario previously considered impossible
(ref. 18, p. 220). We find that evolution of cooperation from
conflict is enhanced when perceptual biases are subject to selection for other reasons (sp > 0). Natural selection pressures
have been widely invoked as the cause of perceptual biases
(16), but once displays have become established, biases may
alternatively be maintained as a result of social selection
pressures, including social stimulation, as we have modeled.
These biases may be general across whole clades (31) or restricted to certain species that occupy specific environments
(16, 32). The presence of general biases does not in itself place
a strong restriction on signal design: many different signals may
stimulate the same perceptual bias (33). Indeed, multiple
components of a display may each become sequentially established in response to the same bias, with each component initially pushing the female above her optimal level of investment
that is subsequently restored by selection for lower basal investment. Ultimately, this may lead to displays whose absence
results in a failure of the female to ovulate (4, 34). Male displays that stimulate investment also set the stage for selfstimulation by the female, e.g., by using similar vocalizations
(35); through female stimulation by signals from other conspecifics, as in colonially breeding species (36); and through
feedback mechanisms, whereby females stimulate males who
then stimulate females (24). In many of these cases, the
presence of a male attending the female is required, and
hence, evolution of such displays can also lead to increases in
other cooperative behaviors, such as brood provisioning, and a
general strengthening of the pair bond (37).
after divorce is Nijr = ð1 − vÞNijs , while the number of single individuals of
sex k created by divorce from each pair of i females and j males is
ndkij = vNijs .
We can now tally up all single females and males that are available to pair
at the beginning of the next mating season:
off
ntot
+ nsfi +
fi = ni
X
j
X
d
d
tot
off
nw
nw
+ nsmj +
mij + nmij .
fij + nfij and nmj = nj
i
calculated the number of new couples with females of genotype j and males
tot
tot
tot
tot
of genotype j as Nijnew = ntot
fi nmj =ny , where y = f if nf > nm and y = m if
tot
ntot
m > nf . The total number of pairs of each genotype, at the close of the
life cycle, equals the sum of these newly paired individuals with the pairs
that remained from the previous season after selection that did not divorce,
as Nij ðt + 1Þ = Nijnew + Nijr . The more common sex also has single individuals
tot
tot
remaining in the population, such that nyi ðt + 1Þ = ntot
yi ð1 − ny =nw Þ and
tot
nwi ðt + 1Þ = 0, where y = f and w = m if ntot
f > nm and y = m and w = f if
tot
ntot
>
n
.
These
close
the
recursions
for
the
numbers
of individuals and pairs
m
f
of each genotype in the population. The genotypic recursions can be turned
into genotypic frequencies and then transformed into recursions for the
allele frequencies for the responder A2 and the display B2, and the linkage
disequilibrium between them, D, using standard equations for haploids.
Selection Differential and Evolution of zf . We calculate the selection differential on zf, which is expressed only in females. Assuming weak selection and
that selection on adults is not age dependent, as is outlined above, the selection differential is determined by taking a weighted average over paired
and single individuals of that sex. To begin these calculations, we determine
the mean after selection of the quantitative trait zf in females of genotype j,
[6]
The equation for genetic fitness for females is
[7]
In the function g fi (z) the 3 terms represent the probability that both
sexes in the mated pair survive given that the female has phenotype z,
the probability that the female survived as a widow given that she has
phenotype z, plus the number of newly recruited offspring that a female will contribute given that she has phenotype z, respectively. The
factor of 1/2 in the last term accounts for the fact that the offspring of
a female will share half of her genes in sexual haploids (as in sexual
diploids).
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6 of 7 | www.pnas.org/cgi/doi/10.1073/pnas.1904138116
PP pair s
1
Sfij Nij + nw
fij + 2Fij
i
Sf =
These include newly recruited fledglings, surviving individuals that were
single in the previous year, widows and widowers, and divorcees.
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−c z
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[5]
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s
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j
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k
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PP
p
s
s
Npq
+ nw
kpq + nkp
,
PP
Npq + nkp
q
p
[9]
q
where k = f, p = i, and q = j for females and k = m, p = j, and q = i for males,
and the multiplicative rate of increase, λ,
λ=
PP
p
q
s
1
s
Npq
+ nw
kpq + 2Fpq + nkp
,
P P
Npq + nkp
p
[10]
q
where k = f, p = i, and q = j when females are the rarer sex and k = m, p = j,
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population growth in the absence of extrapair copulations. Generation time
is then determined for each sex k as T , where T = 1 + s* =ðλ − s* Þ, and the
k
k
k
k
average across sexes T = 12ðTf + Tm Þ can be seen as an approximation of the
average turnover rate of genes in the population (38, 39). Finally, we use the
generation time T to modify the heritability h2 of the trait zf to calculate
Δzf = 12h2 Sf =T where the change in zf comes only from selection in females,
where these genes are expressed.
Numerical simulations were run by exact iterations of the recursion
equations in 2 phases, one with B1 fixed in order to calculate the optimal
investment zopt for each parameter set and the second starting at zopt and
introducing small amounts of B2, with A2 starting at a high frequency, to
replicate the scenario described in the main text. Details of these analyses,
performed using Mathematica (40), can be found in the SI Appendix and at
an interactive reader (41) (https://powers.shinyapps.io/pqreadr/); see SI Appendix, Fig. S4, for more information.
ACKNOWLEDGMENTS. We would like to thank and remember Alexander
Kenan, who made valuable contributions to the development of this project
while he was an undergraduate at University of North Carolina. We thank Nan
Lyu, Carlos Servan, David Wheatcroft, Haven Wiley, and Justin Yeh for discussion. This project was funded in part by the National Science Foundation Grant
DEB-0919018 to M.R.S. and the Norwegian Research Council’s Center of Excellence project SFF-III 223257 to R.L.
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