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ORIGINAL ARTICLE
doi:10.1111/j.1558-5646.2007.00302.x
THE MAINTENANCE OF HERITABLE VARIATION
THROUGH SOCIAL COMPETITION
W. Edwin Harris,1,2 Alan J. McKane,3,4 and Jason B. Wolf1,5
1 Faculty
of Life Sciences, The University of Manchester, Michael Smith Building, Oxford Road, Manchester M13 9PT, UK
2 E-mail:
3 Theory
[email protected]
Group, School of Physics and Astronomy, The University of Manchester, Oxford Road, Manchester M13 9PL, UK
4 E-mail:
[email protected]
5 E-mail:
[email protected]
Received June 18, 2007
Accepted October 16, 2007
The paradoxical persistence of heritable variation for fitness-related traits is an evolutionary conundrum that remains a preeminent
problem in evolutionary biology. Here we describe a simple mechanism in which social competition results in the evolutionary
maintenance of heritable variation for fitness related traits. We demonstrate this mechanism using a genetic model with two
primary assumptions: the expression of a trait depends upon success in social competition for limited resources; and competitive
success of a genotype depends on the genotypes that it competes against. We find that such social competition generates heritable
(additive) genetic variation for “competition-dependent” traits. This heritable variation is not eroded by continuous directional
selection because, rather than leading to fixation of favored alleles, selection leads instead to allele frequency cycling due to the
concerted coevolution of the social environment with the effects of alleles. Our results provide a mechanism for the maintenance
of heritable variation in natural populations and suggest an area for research into the importance of competition in the genetic
architecture of fitness related traits.
KEY WORDS:
Condition-dependence, genetic variation, quantitative genetics, selection, social evolution genetics, selection.
Explaining the maintenance of heritable genetic variation for
fitness-related traits remains a fundamental unresolved problem in
evolutionary biology (e.g., Bürger 2000; Rice 2004). Such variation is paradoxical because it should be quickly eroded by selection, leading to low levels of standing heritable variation (discussed in Mousseau and Roff 1987; Bulmer 1989). Despite this,
empirical patterns reveal that traits closely related to fitness often exhibit substantial amounts of heritable variation in natural
populations (e.g., Mousseau and Roff 1987) and therefore can
evolve (but see Houle 1992; Burt 1995). Among explanations for
the maintenance of genetic variation in natural populations are:
mutation–selection balance (Lande 1975), heterozygote advantage (or heterosis, e.g., Berger 1976), fertility selection (Feldman
et al. 1983), environmental or temporal heterogeneity (Tomkins
et al. 2004; Byers 2005), epistasis (Wolf et al. 2000), multilo
C
337
cus frequency-dependent selection (Clarke 1979; Slatkin 1979),
and others (Bürger 2000). Although all of these explanations have
some theoretical or empirical support, none has emerged as a general solution (see Bürger 2000; Rice 2004 for reviews). More importantly, mechanisms that have been shown to maintain allelic
variation do not always provide an explanation for the maintenance of heritable variation for traits at equilibrium (i.e., additive
genetic variation), especially in the face of ongoing directional
selection (Rice 2004).
Here we present a simple model demonstrating that social
(intraspecific) competition (sensu West-Eberhard 1983) can lead
to the long-term maintenance of heritable genetic variation for
fitness-related traits, even under continuous directional selection.
We illustrate this using a simple model in which the expression
of a fitness-related trait is determined by the relative success of a
C 2007 The Society for the Study of Evolution.
2007 The Author(s). Journal compilation Evolution 62-2: 337–347
W. EDWIN HARRIS ET AL.
given genotype in competition with other genotypes in a population. We assume that the pattern of competitive success of genotypes is nontransitive, such that no single genotype outcompetes
all other genotypes (e.g., Weisbrot 1966). We assume that simple
directional selection acts on this “competition-dependent trait,” as
may be the case for many condition-dependent traits; for example
those involved in mate choice and sexual selection (Bonduriansky
and Rowe 2005). We begin the presentation of the model with
a biological rationale for our assumptions and then discuss the
implications of our results.
The Model
MODEL STRUCTURE
RATIONALE
Our model is built upon two main assumptions: (1) that competition among genotypes is nontransitive, meaning that the competitive success of a genotype depends entirely on the genotype
it competes against and (2) that expression of a fitness-related
trait depends upon success in social competition. Support for the
first assumption is twofold. First, it has long been known that
the outcome of competitive interactions can be strongly dependent upon nonadditive genotypic interactions among competitors
(e.g., Lewontin 1955; Parsons 1958; Roy 1960; Park et al. 1964;
Weisbrot 1966; Wade 2000), where the competitiveness of a genotype may be largely determined by the genotypes of its competitors. Second, we assume selection should act to eliminate most
genetic variation relating to a linear hierarchy of competitors (because the most successful competitive alleles conferring success
in linear hierarchies should become fixed: West-Eberhard 1979),
leaving only complex (i.e., nonlinear and nontransitive) competitive interactions to persist in evolutionary time (excepting special
circumstances, e.g., Byers 2005).
Our assumption that competition influences trait expression
is generally well supported by available empirical evidence. The
effect of competition has often been measured through its affect
on the expression of some trait (e.g., Griffing 1989; Tonsor 1989;
Stevens et al. 1995; Wolf et al. 1998; Bonduriansky and Rowe
2005; Muir 2005), which implies that the expression of such a
trait is “competition-dependent.” More generally, variation in the
resource pool available to an individual may be a major determinant of overall condition (Rower and Houle 1996) and, therefore,
success in competition for limited resources could be an important
source of variation in “condition-dependent” traits. The idea that
condition dependence mediates the expression of fitness-related
traits has been most well explored in the context of sexual selection
(e.g., see Sheldon 1997; Merila 1999; David et al. 2000), where it is
hypothesized that sexually selected traits are costly and therefore
are honest indicators of male genetic quality (Andersson 1994).
Such condition-dependent expression has been documented in a
number of systems (e.g., see Rowe and Houle 1996; David et al.
338
2000; Cotton et al. 2004a), but the generality of such results have
been questioned and there remains a lack of evidence that such
variation is heritable (Cotton et al. 2004b). Thus, given the available data it is logical to assume that competitive success may often
be a major determinant of resource availability and condition in
natural populations and, as a result, will affect the phenotypes
of individuals. Consequently, the expression of traits affected by
competitive success should directly reflect the relative competitive ability of individuals, implying that genetic variation for
competitive success should underlie expression of competition
or condition-dependent traits.
EVOLUTION FEBRUARY 2008
Here we develop a single-locus, three-allele model to examine the
genetics and evolution of complex competitive interactions. This
is meant to represent the simplest model of competitive interactions between genotypes. Although simpler, two-allele models
are constrained because two-allele systems have limited evolutionary “degrees of freedom” (i.e., they have a single “evolutionary degree of freedom” with one allele frequency parameter, see
Gomulkiewicz and Kirkpatrick 1992). In contrast, a three-allele
system has a critical extra evolutionary degree of freedom that
allows more complex behavior and makes it a better approximation of natural systems. Therefore, we expect our model to approximate the dynamic behavior of more complex systems and,
although we do not allow for mutation in this model, we suggest
that a model that includes mutational input of alleles may show
similar evolutionary dynamics.
We assume random mating in a diploid population is large
enough to ignore drift and other stochastic processes (we discuss
the potential influence of drift below in the Discussion). We designate the three alleles at the locus affecting competitive interactions
(locus A) as A 1 , A 2 , and A 3 , which have frequencies p 1 , p 2 , and
p 3 , respectively. The frequencies of diploid genotypes prior to selection conform to Hardy–Weinberg proportions, where diploid
genotype frequencies can be calculated from the trinomial expansion of (p 1 + p 2 + p 3 )2 . The frequencies of the diploid genotypes
are designated f ij , with the subscripts designating the two alleles
that make up the genotype (i.e., subscripts corresponding to: 1 =
A 1 , 2 = A 2 , and 3 = A 3 ), where f ij = f ji . To simplify the presentation of the model throughout, parameters are defined for all nine
diploid ordered genotypes despite the fact that we assume that
parent of origin of alleles is unimportant.
Genetics of competition
We assume that locus A determines an individual’s success in
competitive interactions with other genotypes and that success in
competition determines the resources available to the individual.
We denote the phenotypic values associated with the success of a
genotype in competition with another genotype as x ijkl , where the
SOCIAL COMPETITION MAINTAINS HERITABILE VARIATION
phenotypic value is a measure of the amount of resources gained
by a genotype A i A j competing with genotype Ak Al and where
primes indicate that the second genotype is that of a competitor
(NB., the value of x ijkl is the success of the A i A j genotype, whereas
the A k A l genotype would have the value x klij from this interaction). The phenotypic values of this “competitive success” trait
(x ijkl ,) are defined as the amount of resources gained or lost in
competition.
We assume that competition conforms to a “zero-sum” game
(Maynard Smith 1982), where individuals compete for limited resources and divide the total among themselves (i.e., the competitive success of the two interacting genotypes must be equal and of
opposite sign, where x ijkl + x klij = 0,). We assume that the outcome
of these competitive interactions is nontransitive such that the
competitive success of a genotype depends on the genotype of its
competitor. We implement this model of competition by assuming
an antitransitive competitive hierarchy. This fitness assumption is
akin to that of the game theory model “rock–scissors–paper,” (see
Maynard Smith 1982), where A 1 is competitively superior to A 2 ,
A 2 is superior to A 3 , and A 3 is superior to A 1 , however here we
explicitly examine the population genetic dynamics of this system
and assess the effect of such a pattern of competition on heritable
variation. Although competition between genotypes is nontransitive, we assume that the competitive success of a genotype is
determined by the additive, independent effects of the alleles that
constitutes it (i.e., we assume no dominance). In the most general case, we assume that the relative amount of resources won
and lost in competition between each pair of alleles may differ,
making the game of rock–scissors–paper a special case in which
all competitive outcomes result in the same gain or loss of resources (see Maynard Smith 1982; Hofbauer and Sigmund 1998).
We denote the phenotypic value of the “winner” +r ij and the
“loser” −r ji , where r ij (where the first subscript denotes the focal genotype and the second denotes the competitor’s genotype)
is a parameter that measures the proportional difference in resources acquired by competitors and r ij is assumed to equal −r ji .
We denote the success of each allele against every other allele
with the parameters r 12 , r 13 , and r 23 . For example, the success of
A 1 versus A 2 is r 12 whereas the success of A 2 versus A 1 would be
−r 12 . When competition is between genotypes containing identical alleles we assume that resources are divided equally (i.e., x ijkl =
0 for two individuals of the same genotype). The values of r ij can
be considered as the “payoffs” from competition, and the full set of
competitive success of each allele or genotype against each other
allele or genotype may be thought of as a payoff matrix (e.g., see
Hofbauer and Sigmund 1998).
Because competitive success is determined by the average of
the independent effects of the alleles that make up a genotype, the
success of a homozygote competing against another homozygote
is equal to the success defined above for an allele. For example,
in competitive interactions between the homozygotes A 1 A 1 and
A2 A2 , A 1 A 1 would gain +r 12 resources, whereas A2 A2 would
lose −r 12 resources (i.e., x 1122 = +r 12 and x 2211 = −r 12 ). This is
because each A 1 allele “beats” each A2 allele, giving the winner
(r 12 + r 12 )/2 resources whereas the loser loses this same amount.
Competition between heterozygotes is more complex because the
payoff from competition between each pair of alleles is different
and, consequently, the net payoff for competition between genotypes is the average of the four pairwise allelic payoff values. For
example, success of A 1 A 2 against A1 A3 (x 1213 ) is calculated as
the average of the pairwise success of each allele in one individual against each allele in the other: [(A 1 vs. A1 ) + (A 1 vs. A3 )
+ (A 2 vs. A1 ) + (A 2 vs. A3 )]/4 = (0 − r 13 − r 12 + r 23 )/4 =
x 1213 . In the case in which all values of r ij are the same, that is, all
r ij = r, this value would correspond to −1/4 r, but when considering more complex scenarios in which competition differs between
different alleles, the success of A 1 A 2 against A1 A3 (x 1213 ) would
simply have the value (r 13 − r 12 + r 23 )/4. Although alternative
assumptions about the pattern of competitive success of genotypes
are possible, we found that the model results are not particularly
sensitive to assumptions about how we construct the success of
diploid genotypes given a pattern of competitive success of alleles,
including assumptions about dominance effects of alleles.
We assume that individuals interact at random, such that the
frequencies with which each genotype interacts with each other
genotype are determined solely by the frequencies of the genotypes. We designate the frequencies of interactions between genotypes F ijkl (following x ijkl above). Under the assumption of random
interactions these frequencies are equal to the products of the frequencies of the genotypes of competitors (i.e., F ijkl = f ij f kl ).
The expected phenotypic value of success in competition for
each genotype is calculated by averaging across its success in
competitive interactions (x ijkl ) with all other genotypes (the results
of the model are not sensitive to this assumption and the results
are essentially the same with different assumptions, e.g., assuming
pairwise interactions of genotypes). Thus, the average success of
the genotype ij is taken as x ij ·· , where the dot subscripts denote
averaging over all values of a subscript. We denote these average
phenotypic values for each genotype X ij (which are equal to x ij ·· ),
and are calculated as:
3
3
3 3 Xi j =
xijkl Fijkl
Fijkl .
(1)
k=1 l=1
k=1 l=1
Note that, although the values of x ijkl are not frequency dependent,
because they are assigned directly to a genotype in competition
with another genotype, the values of X ij are dependent on allele
frequencies as they are averaged over all competitive interactions
of a given genotype. For example, X 11 simplifies to (r 12 p 2 −
r 13 p 3 ), which shows that the competitive success of the A 1 A 1
genotype is determined by the relative abundance (p 2 ) of the A 2
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339
W. EDWIN HARRIS ET AL.
allele (which A 1 wins against) versus the frequency (p 3 ) of the
A 3 allele (which A 1 loses to) each weighted by the proportion
of resources to be won or lost in that interaction (r 12 and r 13 ,
respectively).
our assumed model, Templeton 1987). The average effect of each
allele can be calculated as the average phenotype associated with
the allele measured as a deviation from the population mean (cf.
Crow and Kimura 1970) (i = 3j=1 ci j f i j − c̄):
Expression of competition-dependent traits
We assume that the phenotypic value of a competition-dependent
trait (denoted c ij for the genotype ij) is directly determined by
the outcome of competition for resources (which determines the
resource pool available to an individual). We assume resources
affect the expression of the competition-dependent trait during
development such that the expected phenotypic value of a genotype is defined as
1 = 12 (r12 p2 − r13 p3 )
ci j = c + X ij ,
(2)
where defines the relationship between resources gained or
lost through competition and the expression of the competitiondependent trait and c gives the mean expression independent of
the effect of competition.
Genetic variances
The mean value of the competition-dependent trait (c̄) is calculated as the sum of the expected phenotypic value of each genotype
3 3
multiplied by its frequency (i.e., c̄ = i=1
j=1 ci j f i j ). Because
we assume that competition is a zero-sum game c̄= c regardless of allele frequencies or the relative effects of competition,
(i.e., values of r ij ). Consequently, the mean trait value for the
competition-dependent trait does not evolve as allele frequencies
change.
The total genetic variance for the competition-dependent
trait (V G ) is the sum of the squared deviations of the average
phenotypic values of the genotypes from the mean multiplied
by the frequency of the corresponding genotype (i.e., V G =
3 3
2
i=1
j=1 (ci j − c̄) f i j ). The total phenotypic variance of the
competition-dependent trait (V P ) depends on assumptions about
the number of competitive interactions experienced by genotypes
and the nonsocial environmental variance. Therefore, because the
model imposes no assumptions about the number of interactions
experienced by genotypes, the relative importance of random environmental sources of variation or the relative contribution of the
competition-dependent effects compared to other sources of trait
variation, we necessarily focus on the expected pattern (rather than
the amount) of heritable (i.e., additive) genetic variation. Consequently, we leave the question of the relative importance of sources
of variation as an empirical and biological problem that requires
further investigation.
To calculate the additive genetic variance we first calculate
the average effects of the three alleles (technically, we calculate
the average excess, which is equivalent to the average effect under
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EVOLUTION FEBRUARY 2008
2 = 12 (r23 p3 − r12 p1 )
3 =
1
(r13 p1
2
(3)
− r23 p2 )
The additive genetic variance (V a ) is calculated as twice the variance in average effects of the alleles (Crow and Kimura 1970)
[Va = 2( 21 p 1 + 22 p 2 + 23 p 3 )]:


p1 (r12 p2 − r13 p3 )2 +




p2 (r23 p3 − r12 p1 )2 +  .
Va = 12 2 
(4)




2
p3 (r13 p1 − r23 p2 )
Although we do not present the results here, we have also
derived the additive genetic variance using parent–offspring regression and find that the results of the two approaches are qualitatively the same (the two do not yield identical results because
allele frequencies differ between the parental and offspring generations). Here we use the analytically simpler approach of twice
the variance in average effects.
Selection and evolution
We assume positive directional selection on the competitiondependent trait, where fitness is a linear function of expression
of the competition-dependent trait (c ij ), with larger trait values
having higher fitness. Such directional selection could occur for
many reasons, for example when females prefer to mate with males
that have larger trait values (Andersson 1994). Thus, we define
the fitness of a particular genotype (w ij ) as its phenotypic value
for the competition-dependent trait (c ij ) multiplied by the strength
of selection (s) (with s being > 0 under directional selection for
larger values):
wi j = w + sci j ,
(5)
where w is expected fitness independent of the trait values of
individuals. Given the linear relationship between trait values and
fitness, the strength of selection acting at the level of the locus
(rather than the level of the trait) is s . That is, translates
selection on the trait into selection on the alleles, and therefore, s
is always multiplied by to determine the strength of selection
on alleles.
To calculate the evolutionary change in allele frequencies we
first calculate population mean fitness (w̄) from the fitness values
of each genotype weighted by its frequency
SOCIAL COMPETITION MAINTAINS HERITABILE VARIATION
w̄ =
3
3 wij f ij
i=1 j=1
(6)
= w + sc .
Thus, equation (6) demonstrates that population mean fitness does
not depend on allele frequencies.
We model the change in allele frequencies in continuous time
as the allele frequency at one point in time minus the expected
fitness of the allele in each genotype weighted by the frequencies
of the different genotypes, divided by population mean fitness.
The rate of evolution in continuous time is determined by the
value of the constants in the expression for allele frequency change
(s /w̄). This constant is the relative strength of selection on the
locus (i.e., s scaled by mean fitness,w̄) and determines the unit
of time of this system (i.e., the strength of selection is taken per
unit time). This constant (s /w̄) is denoted by . By defining the
changes in allele frequencies per unit time we can produce a set of
equations that define the evolutionary change in the frequencies
of the three alleles through time
dp1
= p1 (r12 p2 − r13 p3 )
dt
dp2
= p2 (r23 p3 − r12 p1 )
dt
dp3
= p3 (r13 p1 − r23 p2 ).
dt
Genetic variance of competition-dependent traits
Our model demonstrates that, despite assuming a nonadditive
model of competition among alleles with variable strengths of
competition between pairs of alleles, competition ultimately contributes to additive differences between the effects of the alleles at
the level of the competition-dependent phenotype (i.e., making the
average effects of alleles nonzero). This can be seen in Figure 1,
where we show the average effect of the A 1 allele as a function
of the frequencies of the three alleles (illustrated for the case in
which r 12 = r 13 = r 23 ). We see that the average effect changes
from positive to negative depending on allele frequencies. Similar plots for the average effects of the other two alleles show this
same pattern (eq. 3), but the magnitudes and signs of the effects
differ for the three alleles at nearly all allele frequencies (having
the same average effect only when all three are at the neutrally
stable equilibrium).
(7)
These equations yield four equilibria. There are the three “trivial”
equilibria in which one of the alleles is fixed (i.e., where p 1 , p 2 , or
p 3 = 1) and one neutral equilibrium point where the three alleles
have frequencies p 1 = r 23 /(r 12 + r 13 + r 23 ), p 2 = r 13 /(r 12 +
r 13 + r 23 ), and p 3 = r 12 /(r 12 + r 13 + r 23 ), respectively (see
the Appendix for details). We also find that pr123 pr213 pr312 is a
conserved quantity (i.e., does not change as the system evolves; see
the Appendix for details). Thus, we find that the system exhibits
neutrally stable allele frequency cycles around a neutrally stable
equilibrium point (see the stability analysis in the Appendix). This
stability is not a consequence of a cost or payoff for competing
against oneself (because we incorporate no such effect)—the latter
is known to produce cycling in some game theory versions of
rock–scissors–paper (Maynard Smith 1982). Note also that the
four equilibria correspond to the points where the additive genetic
variance is zero (being nonzero for all other allele frequencies,
see Fig. 2).
MODEL RESULTS
To explore the implications of the model we divide discussion
of our results into two conceptual sections. In the first section,
we discuss why the nontransitive pattern of competition results in
heritable genetic variation; in the second section, we discuss how
this variation is maintained through evolutionary time.
The average effect of the A 1 allele ( 1 ) on the
competition-dependent trait is shown as a function of allele frequencies (the average effect is equal to the average excess, which
is calculated as the mean phenotype of an allele measured as a deviation from the population mean). Shown is the case in which the
strength of competition is equal between all pairs of alleles, such
that r 12 = r 13 = r 23 . Each point in the space defined by the three
Figure 1.
axes of the ternary (triangular) plot corresponds to a particular set
of frequencies for the three alleles (the space is constrained such
that p 1 + p 2 + p 3 = 1). The elevation of the surface above the
ternary defines the average effect of the allele at a particular set
of allele frequencies in the ternary. The scale for the average effect
is arbitrary and the surface is simply defined in terms of whether
the allele has a positive or negative effect on the expression of the
trait.
EVOLUTION FEBRUARY 2008
341
W. EDWIN HARRIS ET AL.
Figure 2. Additive genetic variance for the competitiondependent trait as a function of allele frequencies for the case
in which the strength of competition equal between all pairs of alleles such that r 12 = r 13 = r 23 . As in Figure 1, allele frequencies are
defined by the ternary with the elevation of the surface giving the
additive genetic variance corresponding to a particular frequency
of the three alleles.
The differences between the average effects of alleles contribute to the additive genetic variance. This is demonstrated by
equation (4) and illustrated in Figure 2 (which shows the additive
genetic variance for the competition-dependent trait as a function
of allele frequencies, again illustrated for the case in which r 12 =
r 13 = r 23 ). The additive genetic variance contributes to parent–
offspring resemblance because alleles conferring large trait values
in one generation also confer large values in subsequent generations due to temporal viscosity in allele frequencies (i.e., rare
alleles in one generation are also rare in the following generation;
see Figs. 1 and 3).
Maintenance of genetic variation
Despite assuming constant directional selection on the
competition-dependent trait we find that heritable genetic variation persists indefinitely. Directional selection on competitiondependent traits is translated into selection on alleles affecting
competitive ability through the genotype–phenotype relationship.
Here, the genotype–phenotype relationship is frequency dependent because the effect of an allele on competitive success depends
on the frequencies of the other alleles that it competes against
(Fig. 1). This frequency-dependent genotype–phenotype relationship leads to evolutionary cycling of allele frequencies. These evo-
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EVOLUTION FEBRUARY 2008
Figure 3. Evolutionary dynamics of allele frequencies due to directional selection on the competition-dependent trait for two patterns of competition between alleles. In both cases we show three
neutrally stable cycles with arrowheads indicating the direction of
allele frequency change. (A) the case in which the strength of competition is equal between all pairs of alleles, where r 12 = r 13 = r 23 ,
(B) the case in which the strength of competition between alleles
is variable, with r 12 = 1.0; r 13 = 0.75; r 23 = 0.5.
lutionary dynamics are shown in Figure 3, which illustrates how
allele frequencies change through time due solely to directional
selection on the competition-dependent trait. Figure 3A shows the
special case in which r 12 = r 13 = r 23 and the cycling is symmetrical around the neutrally stable equilibrium point (which corresponds to the point where all three alleles are at equal frequency),
SOCIAL COMPETITION MAINTAINS HERITABILE VARIATION
which is analogous to the evolutionary dynamics of rock–scissors–
paper (Hofbauer and Sigmund 1998). Figure 3B shows the case in
which the relative competitiveness among alleles is asymmetrical
(Fig. 3B corresponds to r 12 = 1; r 13 = 0.75; r 23 = 0.5), which
results in asymmetric cycles. Indeed, for all values where all r ij
values are greater than zero, allele frequencies cycle indefinitely
despite constant directional selection. Although populations are
cycling in allele frequency space (Fig. 3), additive genetic variance is nonzero (Fig. 2), illustrating that the maintenance of allelic
variation equates with the maintenance of heritable variation.
This allele frequency cycling is different from “classic” models of frequency dependence (e.g., Clarke 1979; Christiansen
1988; Asmussen and Basnayake 1990; Sinervo and Lively 1996)
because we assume that selection on the trait is directional and independent of allele frequencies. Here frequency dependence arises
as an emergent property of competition-dependent trait expression. For example, we find that the A 1 allele has a positive effect
on the competition-dependent trait when the A 2 allele is common,
a negative effect when the A 3 allele is common, and a neutral effect when the A 1 allele is common (Fig. 1). Frequency-dependent
evolution arises from these effects due to the concerted coevolution of the effects of the three alleles with allele frequencies.
For example, when the A 1 allele is rare, selection favors the A 2
allele because it outcompetes the A 3 allele without losing to the
rare A 1 allele. Selection will then lead to the A 2 allele becoming
common, which in turn will favor the A 1 allele, which outcompetes the A 2 allele. Once the A 1 allele becomes common selection
then favors the A 3 allele, which outcompetes the A 1 allele and
so on. As a consequence of these dynamics, it is generally the
rare alleles that confer high competitive success and large values
for competition-dependent traits whereas common alleles generally confer low competitive success and small trait value. Thus,
the system generates a rare allele advantage for almost all allele
frequency space, which is expected to actively maintain allelic
variation, preventing fixation of alleles due to both selection and
genetic drift (see below).
Discussion
We show that additive genetic variation for a trait that reflects
success in competition for limited resources persists indefinitely
under conditions in which competitive success is nontransitive.
Furthermore, this is true even under situations in which the relative competitive success between genotypes is variable. This
“competition-dependent” genetic variation differs from “ordinary
genetic variation” because loci involved in competition create
the social environment experienced by conspecifics whereas, at
the same time, the effects of alleles at these loci themselves depend on the current social environment provided by conspecifics.
This reciprocal relationship between the effects of the alleles and
the social environment that they create alters the nature of the
genotype–phenotype relationship and, thereby, fundamentally alters the evolutionary dynamics of competition-dependent traits.
Nontransitive competition results in additive differences between the effects of alleles (i.e., different average effects of alleles; see Fig. 1) because the effect of an allele is determined by
averaging across competitive social environments experienced by
the allele, essentially “testing” the alleles against the current genetic background in a population. Consequently, alleles that do
well against the current competitive genetic background in a population will have positive effects on the competition-dependent
trait whereas alleles that compete poorly against the current background will have negative effects on the expression of such traits.
Thus, the testing of alleles against the current social environment
creates additive differences between the effects of alleles leading
to heritable (additive) genetic variation.
The extent to which selection acts to remove linear components of dominance hierarchies (where a single allele wins),
leaving nonlinear components, will determine how common such
a scenario might be. There is evidence, both theoretical and empirical, the latter arising from research on competitive diallels (i.e.,
genotype-by-genotype competitive interactions), suggesting significant variation among genotypes to be nonadditive in the competitive environment (e.g., Pèrez-Tomè and Toro 1982; Asmussen
and Basnayake 1990; De Miranda et al. 1991; Colegrave 1993;
Bürger and Gimelfarb 2004). However, the degree to which such
interactions are prevalent in natural populations remains an empirical problem and the generality of whether competitive hierarchies evolve to be transitive or nontransitive in nature is not well
known. For example, there is experimental evidence that threeway genetic competitive interactions can be significant and complex (e.g., Castro et al. 1985; Hemmat and Eggleston 1989), and
may exhibit negative frequency dependence (Adell et al. 1989). In
contrast, there is a much larger body of literature examining competitive transitivity in interspecific interactions, which has demonstrated that such a pattern of nontransitivity may be relatively
common, but it is obviously unclear the extent to which patterns
from interspecific competition are mirrored at the intraspecific
level. Thus, more data are required to ascertain the importance of
complex genotypic interactions for determining the outcome of
competition.
The reciprocal relationship between the effects of alleles and
the social competitive environment leads to concerted coevolution
of the two. This maintains variation because it leads to neutrally
stable allele frequency cycles, where the orbit in allele frequency
space is determined by the starting point (see Fig. 3). The evolutionary dynamics suggest that drift will randomly push allele
frequencies into different orbits, but will not directly lead to loss
of variation as a consequence. Indeed, the system is expected to
buffer the loss of allelic variation by preventing the drift fixation
EVOLUTION FEBRUARY 2008
343
W. EDWIN HARRIS ET AL.
of alleles through a process akin to negative-frequency dependence, where drift fixation is prevented because the cycles are
deflected away from regions of allele frequency space in which
a single allele would be fixed (see the stability analysis in the
Appendix). However, because drift is a stochastic process, drift
fixation is, of course, still a possible outcome, but its likelihood
should be reduced due to this frequency-dependent process. Although the evolutionary dynamics should push the system away
from the fixation of an allele due to drift in the three-allele system,
selection is not expected to push the system away from the loss
of an allele due to drift as the system orbits in allele-frequency
space. In the three-allele space shown in Figure 3, this process
is visualized as the population randomly drifting between orbits;
cycling continues because selection pushes the system away from
the corners (where drift fixation would occur). Loss of an allele
due to drift will occur if, as the system randomly drifts between
orbits, it collides with one of the axes. Once an allele is lost due
to drift, we would be left with a two-allele stochastic system with
one allele being favored over the other and the outcome being
determined by the interplay of drift and selection.
The type of cyclical dynamics seen in the system we describe
are rare in simple population genetic models and almost unknown
in models with directional selection (Bürger 2000). Here, competitive social interactions give rise to such cyclical dynamics
because of the coevolution of the effects of alleles with the social environment. That is, as allele frequencies change through
time so does the average competitive environment experienced
by genotypes and, as a result, they change the effects of alleles.
Thus, the fitness of individual alleles changes gradually leading to
slow cycling of allele frequencies and alleles favored by selection
slowly change as the success of alleles in the contemporary social
environment evolve. As a result, we find that the genotype underlying the largest (and fittest) phenotype changes through time due
to evolution of the social environment, despite the fact that the
mean of the competition-dependent trait remains constant. This
evolutionary “treadmill of competition” (Dickerson 1955) maintains heritable variation of the competition-dependent trait, even
under constant directional selection. This result is novel because
most evolutionary models examining the genetic consequences
of simple directional selection predict rapid loss of variation unless there is constant input of variation from mutation (see Bürger
2000). Here we find persistence of variation without mutational
input and expect that the presence of mutational input would simply further enhance genetic variation in the system. This finding is
robust in that, regardless of the relative competitive ability of each
allele against the other two, the system will cycle indefinitely as
long as the assumptions hold that resources are partitioned among
competitors and that there is a nonlinear hierarchy for competitiveness, where no allele beats all other alleles in competition (i.e.,
competitive success is nontransitive).
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Because allele frequencies change slowly, even under strong
selection, allelic effects change only a small amount from one
generation to the next. This means that fit individuals (e.g., with
large trait values), which are competitively successful in the current competitive environment, will also have fit offspring because
the competitive environment will be similar to that experienced by
parents. This also implies that a genotype conferring high fitness
in one generation may not necessarily do so in other generations
or populations that are at very different allele frequencies.
Because competition can produce heritable variation it can
play a role in processes for which such heritable trait variation is
important, such as the lek paradox (see Andersson 1994; Tomkins
et al. 2004). Condition dependence has been suggested as a means
of generating heritable variation for traits that are the target of
mate choice (Rowe and Houle 1996) and has emerged as the favored theory to resolve the lek paradox (Tomkins et al. 2004).
However, models of condition dependence do not actually make
the explicit prediction that condition-dependent trait expression
maintains heritable variation, they simply predict that conditiondependence provides a large mutational target, thereby leading
to a higher level of genetic variation at mutation–selection balance (Rowe and Houle 1996; Tomkins et al. 2004). Here, we suggest something fundamentally different in that, when conditiondependent traits rely on the outcome of competition, they will
reflect genetic quality directly because they will reflect relative
competitiveness of individuals and genetic variation for such traits
will be maintained actively by a frequency-dependent process.
Thus, mate choice decisions based on these traits will be adaptive
yet will not lead to the rapid loss of genetic variation for either the
traits themselves or for mate quality.
Despite the fact that competition can lead to real heritable
variation contributing to parent–offspring resemblance, this component of heritable variation does not contribute to the evolutionary response to selection. That is, the mean of the competitiondependent trait is not frequency dependent, and therefore, never
changes. As a result, the additive genetic variation associated with
competition-dependent expression of a trait is not variation that
contributes to realized heritability, as would be seen in the response
a population shows to selection on such traits. Likewise, despite
the presence of heritable variation that contributes to additive genetic differences in fitness, we find that population mean fitness
is not frequency dependent and therefore does not change (see eq.
6). This phenomenon was alluded to by Fisher in his discussion of
why mean fitness does not continue to increase when describing
his “Fundamental Theorem of Natural Selection” (Fisher 1930).
Fisher suggested that “As each organism increases in fitness, so
will its enemies and competitors increase in fitness; and this will
have the same effect, perhaps in a much more important degree,
in impairing the environment, from the point of view of each
organism concerned.” (Fisher 1930, pp 41–42). In other words,
SOCIAL COMPETITION MAINTAINS HERITABILE VARIATION
the competitive environment evolves in a way that offsets any
potential increase in trait values and, as a result, the phenotypic
population mean never changes. This has been referred to as the
“treadmill environment” (Dickerson 1955) and is akin to an intraspecific Red Queen process (Lively 1996; Rice and Holland
1997) where the competitive environment evolves to offset any
potential increases in the competition-dependent trait (see Wolf
2003). Thus, we find the surprising outcome that competition can
lead to long-term maintenance of heritable variation for traits under constant directional selection, but those traits may never show
an actual evolutionary response to selection. We suggest that competition dependence of fitness-related traits may be common in
nature because of evidence for both genetic interactions for competitive success (e.g., Hemmat and Eggleston 1989; De Miranda
et al. 1991; Colegrave 1993) and condition-dependent expression
of traits (e.g., Bakker et al. 1999; David et al. 2000; Holzer et al.
2003). However, empirical investigation is required to confirm
this.
Finally, it is interesting to note that the phenomenon of nonadditive interactions between genotypes is somewhat akin to epistasis (i.e., interactions between loci within genotypes) and has been
referred to as “genotype-by-genotype” epistasis (Wade 2000; Wolf
2000) to reflect this similarity. In the former case, the social environment provides the genetic background that a locus experiences
(which also means that a locus can interact with itself, because it
both creates and interacts with the social environment) whereas
with the latter it is other loci within the genome that provide
the genetic background. However, despite their similarity, the two
phenomena have important differences, because under directional
selection epistasis alone maintains no additive genetic variation
(e.g., see Gimelfarb 1989; Hermisson et al. 2003). In contrast, here
we show that competition can maintain additive variation under
directional selection, even in the absence of mutational input. The
differences come about because with social competition the interactions occur between alleles in different individuals, rather than
between alleles within individuals. As a result, the unit of selection (the individual genotype) does not contain the entire genotype
determining individual fitness (because individual fitness is determined by multiple genotypes).
ACKNOWLEDGMENTS
We thank P. X. Kover, A. J. Moore, J. J. Mutic, N. J. Royle, and the Wolf lab
group for insightful discussions that helped shape this work. This work
was funded by grants from The Natural Environment Research Council (UK), The Biotechnology and Biological Sciences Research Council
(UK), and The National Science Foundation (USA).
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Appendix
EQUILIBRIUM POINTS
We begin by finding the internal equilibrium point. First we note
that, there are no equilibria points (aside from the trivial equilibria) when there are just two alleles in the system because,
in all two allele cases, one allele would be competitively dominant to the other allele, and therefore, one allele would move
to fixation (this is also indicated by the linear stability analysis below). Therefore, the equilibrium point must occur when
p 1 , p 2 , and p 3 are greater than zero. From equation (7) we
can see that, at equilibrium, r 12 p 2 = r 13 p 3 , r 23 p 3 = r 12 p 1 , and
r 13 p 1 = r 23 p 2. Therefore, p2 = (r13 /r23 ) p1 , p3 = (r12 /r23 ) p1 , and
p1 [1 + (r13 /r23 ) + (r12 /r23 )] = 1 and we find that an equilibrium
exists when
p1 =
r23
r12 + r13 + r23
p2 =
r13
r12 + r13 + r23
p3 =
r12
.
r12 + r13 + r23
and
(A1)
and
We denote these internal equilibria with an asterisk (i.e., the equilibrium frequencies are denoted pi∗ ).
Linear stability analysis of equilibrium points
To examine the stability of the four equilibrium points we start
by examining the stability of one of the trivial equilibrium points
(when one allele is fixed), which we will then extend to examine
the other two trivial equilibria. We start with the stability near the
fixed equilibrium point where p 1 is close to 1 (i.e., close to fixation
of the A 1 allele). We denote the frequencies of the other two alleles
as p 2 = m and p 3 = n, where m and n are small such that the system
is close to fixation of A 1 . The frequency of A 1 is, therefore given
as 1 − m − n. Substituting these into the expressions for allele
frequency change (eq. 7) and keeping only linear terms (because
SOCIAL COMPETITION MAINTAINS HERITABILE VARIATION
higher order terms are assumed small) we can express the change
in allele frequencies as
dm
= −mr12
dt
(A2)
dn
= nr13 ,
dt
so
m(t) = m(0)e−kr12 t
(A3)
n(t) = n(0)ekr13 t ,
The matrix in equation (A5) has purely imaginary eigenvalues that
imply neutral stability.
which makes
p1 (t) = 1 − p2 (0)e−kr12 t − p3 (0)ekr13 t .
sider the case in which the system is moved from the equilibrium
in the direction where p 1 decreases whereas the other two allele
frequencies increase by small amounts, which we denote by m and
n for the increase in frequencies of p 2 and p 3 , respectively. This
makes the three allele frequencies: p 1 = p ∗1 − m − n, p 2 = p ∗2 +
m, and p 3 = p ∗3 + n. Taking these frequencies, we can examine
how the values of m and n will change near the equilibrium point,
which can be written in matrix form as
 
  
−kp3∗ r13
−kp3∗ (r13 + r23 )
n
d   
 n 
= ∗
. (A5)

∗
dt m
kp2 r12
kp2 (r12 + r23 )
m
(A4)
This demonstrates the behavior of the system seen in Figure 3,
where the system moves away from the area close to p 1 = 1
(or, more generally, away from the point where the system cycles
closest to the corner where p 1 = 1) toward increasing values of
p 3 and decreasing values of p 2 . Similar analyses demonstrate that
the system moves away from p 2 = 1 toward increasing values
of p 1 and decreasing values of p 3 and away from p 3 = 1 toward
increasing values of p 2 and decreasing values of p 1 . In other words,
the system is deflected from the areas close to the corners of the
simplex as shown in Figure 3.
At the equilibrium point where all three alleles are present in
the population the frequencies of the alleles are given by equation
(A1). To examine the stability of that equilibrium point we con-
The conserved quantity
Here we give a simple proof that the system has the conserved
quantity, pr123 pr213 pr312 , which we label C (see also Frean and Abraham 2001) To do so, we derive an expression for the simultaneous
change in the three terms in the conserved quantity ( pr123 , pr213 ,
and pr312 ) (i.e., an expression for the change in the conserved quantity through time) to show that this is zero. The change in C is a
function of the changes in allele frequencies given in equation (7).
Differentiating C with respect to time yields
dC
= kC[r23 r12 p2 − r23 r13 p3 + r13 r23 p3
dt
− r13 r12 p1 − r12 r23 p2 + r12 r13 p1 ] = 0,
(A6)
which demonstrates that, as the system evolves, the conserved
quantity does not change.
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