Frequency-dependent selection and the evolution of assortative mating.

Frequency-dependent selection and the evolution of assortative mating.
Copyright Ó 2008 by the Genetics Society of America
DOI: 10.1534/genetics.107.084418
Frequency-Dependent Selection and the Evolution of Assortative Mating
Sarah P. Otto,*,1 Maria R. Servedio† and Scott L. Nuismer‡
*Department of Zoology, University of British Columbia, Vancouver, British Columbia V6T 1Z4, Canada,
†
Department of Biology, University of North Carolina, Chapel Hill, North Carolina 27599-3280
and ‡Department of Biological Sciences, University of Idaho, Moscow, Idaho 83844-3051
Manuscript received November 10, 2007
Accepted for publication May 24, 2008
ABSTRACT
A long-standing goal in evolutionary biology is to identify the conditions that promote the evolution of
reproductive isolation and speciation. The factors promoting sympatric speciation have been of particular
interest, both because it is notoriously difficult to prove empirically and because theoretical models have
generated conflicting results, depending on the assumptions made. Here, we analyze the conditions under
which selection favors the evolution of assortative mating, thereby reducing gene flow between sympatric
groups, using a general model of selection, which allows fitness to be frequency dependent. Our analytical
results are based on a two-locus diploid model, with one locus altering the trait under selection and the other
locus controlling the strength of assortment (a ‘‘one-allele’’ model). Examining both equilibrium and
nonequilibrium scenarios, we demonstrate that whenever heterozygotes are less fit, on average, than
homozygotes at the trait locus, indirect selection for assortative mating is generated. While costs of assortative
mating hinder the evolution of reproductive isolation, they do not prevent it unless they are sufficiently great.
Assortative mating that arises because individuals mate within groups (formed in time or space) is most
conducive to the evolution of complete assortative mating from random mating. Assortative mating based on
female preferences is more restrictive, because the resulting sexual selection can lead to loss of the trait
polymorphism and cause the relative fitness of heterozygotes to rise above homozygotes, eliminating the
force favoring assortment. When assortative mating is already prevalent, however, sexual selection can itself
cause low heterozygous fitness, promoting the evolution of complete reproductive isolation (akin to
‘‘reinforcement’’) regardless of the form of natural selection.
U
NDERSTANDING the conditions that give rise to
new species is one of the oldest and most intriguing
questions in evolutionary biology (Darwin 1859).
There is a general consensus that spatially separated
populations can diverge through time to the point
where previously separated individuals become unable
to mate and/or to produce fit progeny should they
come into contact. This divergence can be driven by
natural or sexual selection or can arise stochastically via
random genetic drift. While genetic divergence is
inevitable among isolated populations (allopatric speciation; e.g., Orr and Orr 1996), it can also arise when
individuals are arrayed across a spatial landscape without
strict barriers to migration, as long as the selective forces
leading to local adaptation and divergence are stronger
than the opposing forces of migration and recombination (parapatric speciation; e.g., Gavrilets et al. 1998,
2000; Doebeli and Dieckmann 2003). By contrast, there
is a great deal of debate about the importance of sympatric speciation, whereby divergence occurs in situ,
without any substantial degree of spatial isolation. Several
1
Corresponding author: Department of Zoology, University of British
Columbia, 6270 University Blvd., Vancouver, BC V6T 1Z4, Canada.
E-mail: otto@zoology.ubc.ca
Genetics 179: 2091–2112 (August 2008)
models demonstrate that sympatric speciation is possible
given the right combination of disruptive selection,
mating preferences, and genetic variation (e.g., Dieckmann
and Doebeli 1999; Kondrashov and Kondrashov
1999; Doebeli and Dieckmann 2000; see reviews by
Kirkpatrick and Ravigné 2002; Gavrilets 2003,
2004). The core of the debate centers on exactly where
the boundary delineating the ‘‘right’’ combination of
parameters lies. This boundary has been difficult to
determine both because of the large number of possible
parameters and alternative scenarios and because the
majority of studies of speciation in sexual populations
are numerical. Here, we develop and analyze a two-locus
diploid model of speciation, where one locus affects a
trait subject to frequency-dependent or -independent
selection and the second modifies the degree of
assortative mating with respect to the trait locus. Using
a combination of analytical techniques, we determine
exactly when speciation is possible and when it is not.
We refer the reader to recent reviews of speciation
(Turelli et al. 2001; Kirkpatrick and Ravigné 2002;
Gavrilets 2003, 2004; Coyne and Orr 2004) and provide only a brief background to place this work in context.
As described by Felsenstein (1981), there are two
classes of speciation models: ‘‘one-allele’’ and ‘‘two-allele’’
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S. P. Otto, M. R. Servedio and S. L. Nuismer
(see also Endler 1977). This classification refers to the
genetic change required to turn a randomly mating
population into two species. In one-allele models, the
spread of a single allele throughout the population is
sufficient to cause reproductive isolation. For example,
the allele might increase the tendency to remain within
particular habitats (e.g., Maynard Smith 1966; Balkau
and Feldman 1973) or the tendency to mate assortatively
with respect to a phenotype under selection [e.g.,
Maynard Smith 1966; Felsenstein’s (1981) ‘‘D’’ locus;
Dieckmann and Doebeli’s (1999) ‘‘mating character’’].
An example of such a one-allele mechanism acting to
increase the degree of assortative mating was recently
found in sympatric populations of Drosophila pseusoobscura
and D. persimilis (Ortı́z-Barrientos and Noor 2005). In
two-allele models, different alleles (say M1 and M2) must
establish in each of the nascent species for reproductive
isolation to arise. For example, if individuals mate
assortatively with respect to the M1 and M2 alleles, then
reproductive isolation will result if each allele becomes
established in a different subgroup of the population
[e.g., Udovic 1980; Felsenstein’s (1981) ‘‘A’’ locus;
Dieckmann and Doebeli’s (1999) ‘‘marker character’’;
Doebeli 2005]. Alternatively, if the M1 and M2 alleles
alter female preferences, then reproductive isolation will
result if each allele becomes established in the subgroup
containing the preferred male (e.g., Higashi et al. 1999;
Kondrashov and Kondrashov 1999; Doebeli 2005).
Speciation is more difficult in two-allele models because the two alleles must remain associated with their
subgroups, which is hampered when recombination
breaks down linkage disequilibrium between the locus
bearing the two alleles and loci responsible for the trait
differences between the subgroups. Only if selection and
assortative mating are sufficiently strong and/or linkage
between the loci sufficiently tight will speciation ensue
(Felsenstein 1981). By contrast, one-allele models are
more conducive to speciation, because they are not as
sensitive to the development of disequilibria and, hence,
to the rate of recombination (Felsenstein 1981).
In this article, we limit our attention to a one-allele
diploid model and ask under what conditions can a
modifier allele, M, spread if it increases the strength of
assortative mating. Alleles at the modifier locus ‘‘M’’ tune
the degree of assortment, which can range from zero in a
random-mating population to one with complete reproductive isolation. Exactly who mates with whom is
based on the strength of assortment (controlled by the M
locus) and by who appears similar to whom (based on a
locus A). Locus A is assumed to be polymorphic and to
affect a trait subject to natural selection; for simplicity, we
call this the trait locus. This scenario, where the trait locus
A forms the basis of assortative mating and is subject to
selection, is particularly conducive to sympatric speciation (a so-called ‘‘magic’’ trait, e.g., Gavrilets 2004;
Schneider and Bürger 2006). If separate traits controlled these functions, recombination would tend to
disassociate them, rendering speciation more difficult
(Felsenstein 1981; Gavrilets 2004). It should thus be
kept in mind that we are considering a class of models
that is most likely to lead to sympatric speciation.
Three analytical studies have recently investigated the
evolution of assortative mating, using modifier models
similar to the one investigated here (Matessi et al. 2001;
de Cara et al. 2008; Pennings et al. 2008). For brevity, we
have summarized the key differences between the
models in Table 1, providing references in the text to
related results from these studies, as appropriate. Although our study focuses only on one trait locus (unlike
de Cara et al. 2008), focusing on a single-trait locus allows
us to explore a broad array of forms of assortative mating
and to consider both strong and weak selection, modifiers of strong and weak effect, and arbitrary costs.
The main strength of this article is that we allow the
nature of selection acting on the trait locus A to be
completely general: fitnesses may be constant or frequency dependent, and selection may be directional
(favoring the spread of one allele) or balancing (maintaining a polymorphism). Frequency-dependent selection is commonly considered in speciation models
because it can, under the right circumstances, generate
disruptive selection while maintaining a polymorphism.
Frequency-dependent selection arises under a wide
variety of different circumstances: for example, when
individuals compete for resources, when predators more
readily detect common genotypes, when pathogens more
readily infect previously common genotypes, when pollinators prefer common genotypes (or unusual ones), or
when females mate preferentially with common males
(or unusual ones). Density-dependent selection can also
be approximated using a model of frequency-dependent
selection if one assumes that population size dynamics
equilibrate rapidly relative to the timescale of selection,
in which case the fitness of each genotype rapidly approaches a constant value given the current genotypic
frequencies. Many speciation models have focused on
specific causal mechanisms that give rise to frequency- or
density-dependent natural selection; such specific models are helpful in clarifying the ecological conditions that
facilitate speciation, but they are less general in scope
and can obscure the fundamental processes driving the
evolution of assortment.
As we shall see, the evolution of some amount of
assortative mating within an initially random-mating
population occurs when (a) selection is directional and
the average fitness of homozygotes is greater than
heterozygotes or (b) there is a polymorphic equilibrium
at which selection is disruptive, with heterozygotes less
fit than either homozygote. Furthermore, any costs of
assortative mating must be sufficiently weak that they do
not overpower the benefit of assortative mating that lies
in the reduced frequency of heterozygotes among
descendants. Potential costs of assortative mating include
the energetic costs of searching for appropriate mates,
Evolution of Assortative Mating
2093
TABLE 1
Comparison between current and related models
This study
Matessi et al. (2001)
Pennings et al. (2008)
de Cara et al. (2008)
No. of selected loci
Method of analysis
Form of selection on trait
One
Stability and QLEa
General
One
Stability
Gaussian competition
Arbitrary
QLE
General
Dynamics of trait allele
Equilibrium or
changing
General
One
Stability
Quadratic frequency
dependence
Equilibrium
Equilibrium
Equilibrium
p̂A ¼ 12
p̂A ¼ 12
General (focus on
Gaussianc)
General
(focus on p̂A ¼ 12Þ
Preference based
or neutralizedb
General (focus on
Gaussianc)
Present
General
Present or absent
Absent
Frequency at trait locus
Form of assortment
Preference function
Preference based
or group based
General
Sexual selection
Costs of assortment
Present or absent
General
Preference based
Preference based
or neutralizedb
General (focus on
Gaussian or
quadraticc)
Present or absent
Strong (plant model)
or absent
(neutralizedb)
These studies focus on a trait that is subject to natural selection and that forms the basis of assortative mating, the strength of
which is determined by a modifier locus.
a
QLE denotes a ‘‘quasi-linkage equilibrium’’ analysis, which assumes that genetic associations equilibrate faster than allele frequencies change. We use the term QLE even when considering genetic associations, such as the departure from Hardy–Weinberg,
that do not involve ‘‘linkage.’’
b
To eliminate sexual selection, these articles consider a ‘‘neutralized’’ model of preference-based assortative mating, where females mate preferentially but then the mating success of all genotypes is equalized (not necessarily for each sex separately, but
across both sexes).
c
With one locus, a Gaussian preference function is a particular form of matrix (3), where ð1 r2 Þ ¼ ð1 r1 Þ4 , while a quadratic
preference function sets ð1 r2 Þ ¼ ð1 r1 Þ2 .
the risk of rejecting all potential mates and remaining
unmated, the costs of mechanisms permitting perception of mate similarity, and the fitness costs of mating at a
suboptimal time or place to mate with similar individuals.
The magnitude of these costs may or may not depend on
the composition of the population; for example, search
costs should decline as the relative frequency of compatible mates increases (a ‘‘relative’’ cost), but mechanistic
costs should remain the same (a ‘‘fixed’’ cost).
Even when costs of assortment are sufficiently weak,
sexual selection complicates the picture and can prevent the evolution of strong assortment. As described
more fully below, models of assortative mating may or
may not induce sexual selection on the A locus
(Gavrilets 2003, 2004). Sexual selection raises two
distinct obstacles in models of speciation (Kirkpatrick
and Nuismer 2004). First, sexual selection can induce
directional selection at the selected loci, leading to the
loss of the trait polymorphism that is required for
assortment to evolve. And, second, sexual selection
can cause disruptive selection to become stabilizing
(in our model, altering whether homozygotes or heterozygotes are more fit), eliminating the selective benefit
of assortative mating. The reverse is also possible,
however, and sexual selection itself can induce disruptive selection and facilitate the speciation process
(Verzijden et al. 2005). We describe the conditions
under which sexual selection blocks or facilitates the
evolution of higher levels of assortative mating.
We turn now to a description of the model, followed
by the key results of two different types of analysis: a
quasi-linkage equilibrium (QLE) analysis and a local
stability analysis. Because these approaches require
different assumptions, the joint results provide a more
complete picture of how and when assortative mating
evolves in response to selection at a single gene.
MODEL
We develop a two-locus diploid model where one locus,
A, is subject to selection and determines the similarity of
potential mates and a second modifier locus, M, alters
the strength of assortative mating, r. Recombination
occurs between the two loci at rate r. The key question
that we address is whether modifier alleles altering the
level of r can invade a population. If so, we wish to know
the conditions under which high levels of assortative
mating might evolve (r 1), thereby generating substantial reproductive isolation among genotypes.
Our model is similar to that of Udovic (1980) in assuming that the A locus is subject to frequency-dependent
selection of an arbitrary nature, with fitnesses of the
2094
S. P. Otto, M. R. Servedio and S. L. Nuismer
three diploid genotypes (AA, Aa, and aa) given by the
functions
1 1 SAA ðX A Þ;
1 1 SAa ðX A Þ;
1 1 Saa ðXA Þ;
ð1Þ
where XA ¼ ffreqðAAÞ; freqðAa Þ; freqðaa Þg is the vector of genotypic frequencies at the A locus and the Si are
selection coefficients that depend on these frequencies.
The fitness functions are assumed constant over time,
so that the fitness of an individual remains the same as
long as the genotypic frequencies remain constant but
may change as the genotypic frequencies evolve. We
use Equation 1 to derive a number of results without
specifying the exact nature of the fitness functions.
We investigate the conditions under which alleles increasing the degree of assortment spread at the modifier
locus. We define assortative mating broadly as any
mechanism that makes it more likely for individuals to
mate with genotypically similar individuals. There is a
plethora of ways that such assortment can be accomplished, and we investigate two classes of models: ‘‘groupbased’’ and ‘‘preference-based.’’
Group-based model: The first class of assortment
models is based on group membership (O’Donald
1960; Felsenstein 1981). We assume that each individual is a member of a group; females mate within their
group with probability r, choosing randomly among the
males within the group, and otherwise mate with a male
chosen randomly from the entire population. Groupings
might be spatial (e.g., genotypes prefer different host
plants) or temporal (e.g., individuals release pollen or are
most active at different times of day). Grouping might
also occur by self-referent phenotype matching (Hauber
and Sherman 2001) if phenotypically similar individuals
tend to aggregate together. Specifically, we consider three
groups, whose composition is based on the genotype at
the A locus, such that individuals of genotype iP
join
3
group j with probability gi;j (Figure 1), where
j¼1
gi;j ¼ 1. The model can also be applied to the case where
only two groups form by setting gi;j ¼ 0 for one of the
three groups. Assortative mating is most efficient when
each genotype forms its own group (gAA;1 ¼ gAa;2 ¼
gaa;3 ¼ 1), which we refer to as ‘‘genotypic grouping.’’
We assume that any unmated females and all males join
a random-mating pool, which for brevity we call a ‘‘lek.’’
For example, the probability that a female of genotype
AA mates with a male of genotype Aa is
r
3
X
j¼1
gAA;j
gAa;j ðfreq of Aa malesÞ
P
i gi;j ðfreq of i malesÞ
1 ð1 rÞðfreq of Aa malesÞ;
!
ð2Þ
where i sums over the three genotypes fAA; Aa; aa g.
The first term accounts for the probability that an AA
female is in a particular group, j, and mates with an Aa
male within her group, while the last term accounts for
mating within the random-mating pool.
Figure 1.—Grouping model of assortative mating. A population is structured into groups, wherein mating occurs randomly with probability r. Assortative mating results because
different genotypes at locus A have different probabilities
of joining the different groups. Following the period of assortative mating, we assume that all unmated females mate at
random by choosing mates at times or places where each genotype is proportionately represented (e.g., in flight rather
than on a host plant, during a swarming period, or in a
lek). We assume that all individuals are a part of some group,
although they may or may not mate within their group.
Alleles at the modifier locus alter the probability that
a female mates within her group according to
rMM ;
rMm ;
rmm :
Modifier alleles that increase r strengthen the degree of
assortative mating because individuals that mate within
their group are more likely to mate with a genetically
similar individual at the A locus. In appendix a, we
consider two variants to this core model: (1) males that
mate within the group do not join the lek, and (2) the
groups and the lek form simultaneously, with individuals joining one or the other.
In the grouping model, females pay no inherent costs
for mating assortatively because each female is guaranteed an equal chance of mating, either within her group
or within the lek. To this basic model, we add two potential costs of assortative mating. One is a fixed cost, cf , paid
by females that mate within their group, which is paid regardless of the size of the group. A fixed cost might arise
if mating within the group is risky or suboptimal (e.g.,
before the optimal time in the season for mating). A
second relative cost, cr , is added that depends on the
frequency of the group. We assume that the density of
mates within a group scales with the frequency of that
group, so that females have an easier time encountering
mates in groups that are well populated. Specifically, the
fitness of a female is multiplied by a factor, 1 cr 3 r 3
ð1 frequency of her groupÞ, which falls from 1 to 1 cr 3 r as mates become scarcer (i.e., as the frequency of
Evolution of Assortative Mating
her group falls from 1 toward 0). This relative cost
represents the additional time and energy needed to find
a mate within a group containing few individuals. We
assume that the relative cost of assortment declines
linearly as the frequency of the group rises. The cost of
restricting mating to within a group might, however, be
negligible unless group size is very small. Cost functions
that decline more rapidly toward zero as the frequency of
the group rises would be more conducive to the evolution
of assortative mating than the linear cost function
explored here.
Preference-based model: In the second type of model
considered, females prefer to mate with certain males
over others, according to a preference matrix
Male genotype
0
Female genotype
aa
1
aa
B
Aa @ 1 r1
AA 1 r2
Aa
1 r1
1
1 r1
AA
refer to as her ‘‘fertility’’) is then MAA;k ¼ ð1 cr Þ 1
cr TAA;k , which is reduced below one to the extent that
the female is choosy. This cost of assortative mating is
relative; even a very picky female suffers no loss in
fertility if every male encountered is similar.
To be concrete, the overall fraction of matings between
a female of genotype i at the trait locus and k at the modifier locus and a male of genotype j at the trait locus is
ðfreq of i; k femalesÞ 3
1 r2
C
1 r1 A :
1
ð3Þ
The terms r1 and r2 measure the degree to which a
female dislikes males that differ by one allele and two
alleles, respectively. We measure the relative ability of a
female to distinguish males that differ by one vs. two
alleles, using K ¼ r1 =r2 . The r terms are assumed to be
positive (or zero) and to depend on the female’s
genotype at the modifier locus (e.g., MM females dislike
males that differ by two alleles by an amount r2;MM ). In
the text, we focus on assortative mating using the
symmetrical preference matrix (3), but the results for
a general preference matrix are analogous and are
presented in appendix b.
Each female encounters a male and chooses to mate
with him with a probability equal to the appropriate
entry in matrix (3). For example, consider an encounter
involving a female of genotype AA at the trait locus and
genotype k at the modifier locus (k ¼ MM, Mm, or mm).
The probability that the encounter is with
a male
of genotype Aa and results in mating is 1 r1;k 3
ðfreq of Aa malesÞ. Summed over all possible types of
males, the overall probability that an AA female of
modifier genotype k accepts a male during a mating
encounter is
TAA;k ¼ ð1Þ 3 ðfreq of AA malesÞ 1 ð1 r1;k Þ
3 ðfreq of Aa malesÞ 1 ð1 r2;k Þ 3 ðfreq of aa malesÞ:
ð4Þ
If a female rejects a male, she may or may not be able to
recuperate the lost mating opportunity. To account for
this potential cost, we assume that a fraction of the time,
ð1 cr Þ, a female is able to recover the fitness lost by
rejecting a dissimilar mate, and otherwise she suffers a
loss in fitness. The overall chance that a female of trait
genotype AA and modifier genotype k mates (which we
Mi;k
M
|fflfflfflffl{zfflfflfflffl}
Relative probability
that female of type i; k
has mated
3
!
Pkij 3 ðfreq of j malesÞ
Ti;k
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
;
ð5Þ
Probability male is
type j given female
has mated
is the average fertility of females,
where M
¼
M
1
2095
3 X
3
X
ðfreq of i; k femalesÞ 3 Mi;k
k¼1 i¼1
3 X
3
X
¼
ðfreq of i; k femalesÞ 3 ð1 cr 1 cr Ti;k Þ;
k¼1 i¼1
and Pijk refers to the entry in the ith row and the jth
column of matrix (3) for females of genotype k at the
modifier locus.
When relative costs are absent (cr ¼ cf ¼ 0), all females
have equal fertility. This special case has been called the
‘‘animal’’ model of assortment (Kirkpatrick and
Nuismer 2004), a reference to animals with lek-based
mating systems where the cost of searching for a different
mate is presumed negligible. In contrast, when lost
mating opportunities are never recovered (cr ¼ 1), the
fertility of females of genotype i relative to the average
, becomes Ti;k =T , which is less than one if
fertility, Mi;k =M
type i, k females reject more males than other females.
This special case was described by Moore (1979) and has
been called the ‘‘plant’’ model of assortment (Kirkpatrick and Nuismer 2004); it is appropriate for plants
that are pollen limited (or animals that are limited by
mating opportunities), such that any tendency to reject
pollen (males) directly reduces fertility.
We also allowed for a fixed cost of assortment, c f ,
which is paid by choosy females regardless of the types of
males encountered. Specifically, the fitness of a female
of genotype k at the modifier locus was multiplied by a
fixed factor, 1 c f;1 r1;k c f;2 r2;k .
A critical feature of the preference-based model is
that it induces strong sexual selection on the A locus.
The mating scheme embodied in matrix (3) selects
against rare genotypes (positive frequency-dependent
selection) because the most common females prefer
males with their own genotype. In contrast, the grouping model ensures that everybody gets an equal ‘‘kick-atthe-bucket’’ (each individual belongs to one and only
one group, and the number of receptive females per
male is the same in each group) and so induces little
sexual selection on the A locus. (Technically, some
sexual selection is induced by the grouping model if
males that mate within the group are also allowed to join
2096
S. P. Otto, M. R. Servedio and S. L. Nuismer
TABLE 2
Model variables and parameters
Terms
Model
pA ; pM
r
Si ðX A Þ
XA
Wi
Htot
gi;j
rj
(G)
(G)
r1;j ; r2;j
K ¼ r1,j/r2,j
(P)
(P)
Ti
(P)
cf
cr
R
(G)
u
(G)
mate1, mate2
DAM, DA,M
DA,A
(P)
DAM,A
Definitions
Frequency of allele A at trait locus A or allele M at modifier locus M; qi ¼ 1 pi .
Rate of recombination between loci A and M.
Strength of natural selection acting on genotype i.
Array of genotype frequencies; X A ¼ ffreqðAAÞ; freqðAaÞ; freqðaaÞg.
Total fitness of genotype i, accounting for both natural and sexual selection. Wi also depends on
the composition of the population.
Total fitness advantage of homozygotes over heterozygotes Htot ¼ WAA 1 Waa 2WAa , accounting for
natural ðHns Þ and sexual ðHss Þ selection.
The probability that genotype i joins group j (Figure 1).
Strength of assortative mating for a female of genotype j (MM, Mm, or mm); specifically, the
probability that a female chooses a mate from within her group.
Dr ¼ pM ðrMM rMm Þ 1 qM ðrMm rmm Þ measures the difference in strength
of assortment if a female carries allele M instead of m.
Strength of assortative mating for a female of genotype j (MM, Mm, or mm) as described by matrix (3).
Strength of assortative mating against males that differ by one trait allele relative to those that differ
by two in the preference-based model.
The probability that a female of genotype i accepts a male during a mating encounter, given
the current population composition and her preferences.
A fixed cost that directly selects against assortative mating in proportion to the strength of
assortative mating.
A relative cost that directly selects against assortative mating in proportion to the difficulty of finding
a preferred mate.
The rarity of males experienced by females averaged over all groups; DR measures the difference in
rarity if a female carries allele M instead of m.
The effect of mating within a group on homozygosity at locus A; Du measures the difference in
production of homozygotes if a female carries allele M instead of m.
The probability that a potential mate differs by one or two alleles at the A locus.
Linkage disequilibrium within (cis) or between (trans) homologous chromosomes.
Excess homozygosity at locus A; DDA;A measures the effect of the modifier on DA,A following a
single round of mating.
Trigenic disequilibrium measuring the association between allele M and excess
homozygosity at locus A.
Terms specific to the group-based or preference-based model are denoted in the second column by (G) or (P). The value of a
parameter x averaged over the population is denoted by x. The QLE value of a variable D is denoted by D̃.
the lek, but the induced selection is very weak unless the
modifier has a strong effect on the level of assortative
mating; in variant model 1 of appendix a, where males
that mate within the group do not join the lek, even this
slight sexual selection is eliminated.)
Recursions were developed in Mathematica (supplemental online material), on the basis of the life cycle:
natural selection, mating, recombination and gamete
production, and gamete union within mated pairs.
Allele frequencies and genetic associations were then
assessed among the offspring (the census point). These
recursions were analyzed using two approaches. We first
assumed that selection was weak and allowed genetic
associations to reach their steady-state values given the
current allele frequencies; essentially, we performed a
separation of timescales, assuming that departures from
Hardy–Weinberg and linkage disequilibria equilibrate
on a faster timescale than allele frequencies change.
This is known as a QLE analysis (Barton and Turelli
1991; Nagylaki 1993; Kirkpatrick et al. 2002). Second,
we assumed that the population had reached a polymorphic equilibrium at the A locus, at which point a
new modifier allele M was introduced. A local stability
analysis was then performed to determine the conditions under which M would spread. By combining the
two approaches—a QLE that assumes weak selection
and a stability analysis that allows strong selection but is
valid only near equilibria—we gain a more complete
picture of the forces favoring and impeding the evolution of assortative mating. All derivations are presented
in the accompanying Mathematica files, and a list of
variables and parameters is provided in Table 2.
The results for the group-based and preference-based
models were confirmed by computer simulations, which
numerically iterated the exact recursion equations.
These simulations consisted of two steps. In the first,
the allele frequencies at locus A were allowed to
equilibrate under a combination of frequency-dependent natural selection, using Si ðXA Þ ¼ ai 1 bi ðpA qA Þ
for the fitnesses in Equation 1 and assortative mating as
Evolution of Assortative Mating
determined by the ancestral genotype at the modifier
locus (mm). In the second step, the modifier allele M was
introduced in linkage equilibrium with the alleles at
locus A, and evolution proceeded until a final equilibrium was reached in the system or until the modifier was
lost or spread to fixation.
QLE RESULTS ASSUMING LOW LEVELS
OF ASSORTMENT
QLE in the group-based model of assortative mating:
We begin by assuming that selection coefficients are
small [Sij ðXA Þ ¼ O ðeÞ, where e is small], as are the initial
levels of assortment [rij ¼ O ðeÞ] and the costs of
assortment [c f ¼ O ðeÞ; c r ¼ O ðeÞ]. In this case, all
genetic associations, including linkage disequilibria
and departures from Hardy–Weinberg, rapidly reach a
steady-state value that is small, of order e. At this point,
the frequency pA of allele A changes across a generation
by an amount
DpA ¼ pA qA ðpA ðSAA ðXA Þ SAa ðXA ÞÞ
1 qA ðSAa ðXA Þ Saa ðX A ÞÞÞ 1 Oðe2 Þ; ð6Þ
where qA ¼ 1 pA . Only frequency-dependent natural
selection (1) enters into Equation 6 and not the mating
parameters (gi;j ; r), confirming that the grouping
model does not induce sexual selection on the A locus
to leading order (see appendix a). In later sections, we
report results from a QLE analysis when assortment is
already prevalent and from a stability analysis that allows
for strong selection.
Of greater relevance, the frequency of allele M
(pM ¼ 1 qM ) changes across a generation by an amount
DpM
1
¼ pM qM ðcf Dr 1 cr DRÞ
2
1 DAM ;A Htot 1 ðDAM
DpA
1 DA;M Þ
1 Oðe3 Þ: ð7Þ
pA qA
In the following paragraphs, we describe the terms in
Equation 7.
The first line of (7) reflects the costs of assortative
mating, which directly select against modifier alleles
that increase the level of assortative mating. The cost is
multiplied by a factor of 12 because the modifier is
expressed only in females and thus only females pay
the cost of assortment. Here, Dr ¼ pM ðrMM rMm Þ 1
qM ðrMm rmm Þ measures the effect of allele M on the
level of assortative mating, given the current modifier
frequency. DR is the difference in the rarity of mates
experienced by a female that carries an M allele in place
of an m allele. Under our assumption that the relative
cost to a female of mating within her group declines
linearly with the frequency of her group, the rarity of
mates experienced by females is, on average,
R¼
2097
number
of groups
X
ðfraction of females in groupi Þ
i¼1
3 ð1 fraction of males in groupi Þ:
ð8Þ
For example, if there are three groups comprising 20,
70, and 10% of the population, respectively, then
R ¼ 0:2 3 0:8 1 0:7 3 0:3 1 0:1 3 0:9 ¼ 0:46. The minimum value of R is zero and occurs when there is only
one group (all females occur in the same group as all of
the males); the maximum value of R is 23 and occurs when
all three groups are equal in size (every female is in a group
containing one-third of the males). To evaluate the change
in modifier frequency (Equation 7), we need only keep
leading-order terms within DR, and so we can calculate
the frequencies of each group without accounting for
genetic associations (for example, ‘‘freq of group1’’ ¼
pA2 gAA;1 1 2pA qA gAa;1 1 qA2 gaa;1 , see Figure 1). Doing so,
we determined that DR Dr R when assortative mating
is rare, but when assortative mating is prevalent (or with
different group structures, as in variant 2 of appendix
a), DR must be calculated from the effect of a change in
assortative mating on (8). Depending on the group
structure (i.e., on gi;j ; pA ), mates may become harder or
easier to find as assortative mating becomes more
prevalent, causing the costs of assortment to rise or fall.
The second line in (7) reflects indirect selection on
the modifier arising from genetic associations. In this
article, we use the central-moment association measures
defined in Barton and Turelli (1991). The term
DAM ;A is the genetic association between the modifier
allele M and excess homozygosity at the A locus. This
term is multiplied by Htot, which measures the degree to
which homozygotes are, on average, more fit than heterozygotes at the A locus with respect to total fitness, W,
Htot ¼ WAA 1 Waa 2WAa ;
ð9aÞ
which in turn depends on the current genotypic
frequencies. Because sexual selection is absent in the
grouping model (to leading order), Htot equals the
average fitness advantage of homozygotes over heterozygotes due to natural selection alone:
Hns ¼ SAA ðX A Þ 1 Saa ðX A Þ 2SAa ðXA Þ:
ð9bÞ
In the preference-based models of assortment, sexual
selection will also contribute to Htot by an amount Hss .
The degree to which homozygotes are more fit than
heterozygotes, Htot , plays a critical role in selecting for
assortative mating (Endler 1977). Htot is zero whenever
selection at the A locus is additive; it is positive whenever
the average fitness of homozygotes is higher than the
fitness of heterozygotes; and it is negative whenever the
average fitness of homozygotes is lower than the fitness
of heterozygotes. Finally, the terms, DAM and DA;M in
Equation 7 measure linkage disequilibrium between loci
A and M on the same chromosome (in cis) and on homologous chromosomes (in trans), respectively. More gener-
2098
S. P. Otto, M. R. Servedio and S. L. Nuismer
ally (i.e., even if the loci are unlinked), DAM and DA;M
measure the association between alleles at loci A and M
inherited from the same parent vs. different parents.
To interpret Equation 7, we need to determine the
genetic associations in terms of the parameters of the
model. We do this by setting the change in the association measures across a generation to zero and solving
for the D terms to order e given the current allele
frequencies. That is, we assume that the association
measures have reached their steady-state values expected at QLE, denoting these steady-state values as D̃.
Doing so reveals that the cis and trans linkage disequilibria D̃AM and D̃A;M are zero to this order, while
1
D̃AM ;A ¼ pM qM ðDDA;A Þ 1 Oðe2 Þ:
2
ð10Þ
DDA;A measures the effect of the modifier on the QLE
departure from Hardy–Weinberg. We can relate DDA;A
to the mating parameters by considering how mating
within a group affects the production of homozygous
offspring, relative to the parental generation:
u¼
3
X
1
j¼1
2
aaj
Aaj
Aaj
aaj
AAj
ðfreq of groupj Þ
ðfreq of groupj Þ
AAj
:
ðfreq of groupj Þ
ð11Þ
Mating among heterozygotes increases homozygosity by
a factor of 12, whereas mating among opposite homozygotes decreases homozygosity. In Equation 11, Aaj is
shorthand for the frequency of genotype Aa in group j.
Putting Equation 11 in words, the more that heterozygotes mate with one another, the more efficient
assortative mating is in converting heterozygotes into
homozygotes, and the larger u becomes. If groups form
randomly in a population at Hardy–Weinberg equilibrium, u ¼ 0. If each genotype forms its own group
(gAA;1 ¼ gAa;2 ¼ gaa;3 ¼ 1 in Figure 1), half of the heterozygotes are converted into homozygotes by each generation of assortative mating [u ¼ freqðAaÞ=2], which is
the most efficient form of assortative mating. If individuals make errors in which group they join, this effectively reduces u (toward the case of random mating) and
makes assortative mating less efficient. By altering the
tendency to mate within a group, a modifier allele M
affects the production of homozygotes by an amount,
Du, following a single round of mating at QLE. In terms
of Du, Equation 10 is equivalent to
1
Du
D̃AM ;A ¼ pM qM
1 Oðe2 Þ:
ð12Þ
2
2
(The 12 enters from the definition of DA;A , where
freqðAAÞ ¼ pA2 1 DA;A , so that freqðhomozygotesÞ ¼
pA2 1 qA2 1 2DA;A . Consequently, DA;A measures only half
the excess homozygosity caused by assortative mating.)
When assortment is rare, a modifier allele that
increases the degree of assortative mating (Dr . 0)
increases the rate of production of homozygous offspring by Du ¼ Dr u. Consequently, the modifier allele
tends to be found in individuals with higher levels of
homozygosity at the A locus (D̃AM ;A . 0 from Equation
10b). This association indirectly selects for the modifier
allele if homozygotes are more fit, on average, than
heterozygotes (Equation 7). Following up on suggestions made earlier by Dobzhansky (1940, 1941),
Endler (1977) argued that assortative mating would
evolve whenever Hns . 0. To do so, he ignored all
genetic associations except the departure from Hardy–
Weinberg at the A locus; this method does not exactly
predict the change in the modifier (the magnitude of
D̃AM ;A is not right), but the qualitative result is correct.
Whenever heterozygotes are less fit, on average, assortative mating is favored, whether or not the population
is at equilibrium. For example, if frequency-independent
selection favors the spread of a beneficial allele A,
modifiers that increase assortative mating will rise in
frequency as long as A is partially recessive (so that
Hns . 0). Conversely, disassortative mating would be
favored if A were partially dominant. Such a process is
transient, however; as soon as the A allele fixes, all
individuals belong to the same group, and mating
within a group becomes equivalent to mating at random
within the population at large. Nevertheless, the potential for assortment would persist if fixed costs are
negligible, and assortment would be revealed once
genetic variation reappears. Furthermore, the recurrent spread of partially recessive beneficial alleles
could select for increasingly high levels of assortative
mating. Although beneficial alleles are not typically
recessive, dominance varies among traits and among
alleles, and it is plausible that beneficial alleles will
tend to be partially recessive in the face of certain
environmental challenges (e.g., Anderson et al. 2003).
Selection in such environments would then promote
assortment.
If we assume that the population is at a polymorphic
equilibrium such that DpA ¼ 0 in (6), then ðSAA ðXA Þ SAa ðX A ÞÞ and ðSAa ðX A Þ Saa ðXA ÞÞ must have opposite
signs, implying either overdominance (Hns , 0) or underdominance (Hns . 0). In the absence of frequencydependent selection, the polymorphic equilibrium is
stable only with overdominance, in which case assortative mating would be selected against (Equation 7).
With frequency-dependent selection, a local stability
analysis shows that a polymorphism is possible with
underdominance as long as
0 , Hns , p̂A ðS9Aa ðX̂A Þ S9AA ðX̂A ÞÞ
1 ð1 p̂A ÞðS9aa ðX̂A Þ S9Aa ðX̂A ÞÞ;
ð13Þ
where X̂A ¼ fp̂A2 ; 2p̂A q̂A ; q̂A2 g, a caret indicates a value
at equilibrium, and S9 refers to the derivative of the
Evolution of Assortative Mating
selection coefficient with respect to pA evaluated at the
equilibrium (assuming mating is nearly random). Condition (13) requires that the fitness of individuals
bearing the A allele declines fast enough as the
frequency of A rises, relative to the fitness of individuals
bearing the a allele. In models where competition for
resources induces frequency-dependent selection (such
as that considered by Dieckmann and Doebeli 1999
and Pennings et al. 2008), condition (13) requires that
the resources are better utilized by a polymorphic
population than by a fixed population and would be
even more fully utilized if there were an excess of
homozygotes over Hardy–Weinberg expectations. With
multiple loci, de Cara et al. (2008) identify epistasis as
playing a similar role to Htot , so that indirect selection
favors assortative mating if epistasis or Htot is positive; in
either case, extreme phenotypes are fitter than intermediate ones.
Costs of assortative mating always hinder the evolution of assortative mating. If these costs are substantially
larger than the strength of natural selection acting at the
A locus, assortative mating cannot evolve. For example,
consider the case where each genotype forms its own
group [gAA;1
¼
gAa;2 ¼ gaa;3 ¼ 1, so that Du ¼ Dr
pA qA and
DR ¼ Dr pA2 1 pA2 1 2pA qA ð1 2pA qA Þ 1 qA2 1 qA2 ].
Plugging Equation 12 into Equation 7, we find that
assortment cannot evolve despite the fact that homozygotes
are fitter, on average, than heterozygotes (Hns . 0) if the
fixed cost of assortment is greater than
cf . pA qA Hns =2
ð14aÞ
or if the relative cost of assortment is greater than
cr .
Hns
:
4ð2 3pA qA Þ
ð14bÞ
These conditions are illustrated in Figure 2. Clearly, costs
cannot be too substantial for assortment to evolve.
All else equal, relative costs are less prohibitive to the
evolution of assortative mating than fixed costs, especially
as the allele frequency approaches zero or one. This is
because the relative costs become negligible if most
individuals find themselves in well-populated groups.
QLE in the preference-based model of assortative
mating: In this model, females exhibit mating preferences, which induce sexual selection. For the mating
preference scheme given by matrix (3), a QLE analysis
assuming weak selection in a population with initially
weak assortative mating shows that the frequency of
allele A changes by an amount given by Equation 6 due
to natural selection plus
1 1 cr
pA qA ðr2 ð1 2pA ÞpA qA 1 r1 ð1 2pA Þ3 Þ 1 Oðe2 Þ
2
ð15Þ
due to sexual selection, where costs of assortment are
allowed to be weak or strong. Here, an overbar is used to
2099
Figure 2.—Strong costs prohibit the evolution of assortative
mating. In the grouping model, assortative mating will evolve if
homozygotes are, on average, more fit (Hns . 0, Equation 9) and
if the costs of assortative mating are sufficiently weak (below
curves, from Equation 14). Here we assume that Aa individuals
form their own group (gAA;1 ¼ gAa;2 ¼ gaa;3 ¼ 1) and consider
costs to be either fixed (dashed curve) or relative to the difficulty
of finding a mate (solid curve). Given that Hns . 0, assortative
mating can evolve only if costs lie below the appropriate curve
and only as long as the A locus remains polymorphic. Note that
the y-axis is measured relative to Hns (Equation 9b), the degree to
which homozygotes are more fit, on average, than heterozygotes
given the current genotype frequencies.
denote an average across the population. For example,
2
2
r2;MM 1 2pM qM r2;Mm 1 qM
r2;mm . According to
r2 ¼ pM
(15), assortative mating (r1 ; r2 $ 0) selects against A
when rare (pA , 12) and favors A when common (pA . 12),
thereby generating positive frequency-dependent selection. This potentially places a break on the evolution
of assortative mating under frequency-dependent selection because a polymorphism at locus A can be sustained
only while sexual selection remains weak relative to
natural selection (Kirkpatrick and Nuismer 2004).
The force of sexual selection acting on the A locus is
twice as strong in the plant model (cr ¼ 1) as in the
animal model (cr ¼ 0); in the plant model, the rare allele
is selected against in both females and males, while in the
animal model, only males experience sexual selection.
Although the QLE results derived below do not
require that the A locus is at equilibrium, it is worth
considering the conditions under which there would be
a protected polymorphism at locus A when both sexual
and natural selection act. Specifically, we wish to know
the conditions under which pA rises when A is rare
(XA f0; 0; 1g) and falls when A is common (X A f1; 0; 0g). Adding Equation 15 to Equation 6 reveals that
a protected polymorphism is guaranteed if
r1
1 1 cr
, SAa ðf0; 0; 1gÞ Saa ðf0; 0; 1gÞ
2
and
r1
1 1 cr
, SAa ðf1; 0; 0gÞ SAA ðf1; 0; 0gÞ:
2
ð16Þ
When homozygous females at the A locus strongly
prefer similar homozygotes over heterozygotes (so that
2100
S. P. Otto, M. R. Servedio and S. L. Nuismer
r1 is large), the alternative allele at the A locus cannot
spread when rare because it finds itself in heterozygotes
that are less likely to obtain a mate. Thus, mating
preferences against individuals that differ by one allele
(r1 ) cannot evolve to very high frequency before the
fixation states become stable. Interestingly, a mating
preference against individuals that differ by two alleles
(r2 ) does not affect the conditions for a protected
polymorphism; this is because the dynamics of a rare
allele at the A locus are insensitive to preferences against
the opposite homozygote because such homozygotes
are so rare. In summary, the evolution of sexual
preferences will stabilize the fixation of alleles at the A
locus if females avoid mating with genotypes that differ
by one allele. Even when both fixation states are stable,
however, frequency-dependent natural selection could
still maintain a stable polymorphism at an intermediate
allele frequency, as long as negative frequency-dependent natural selection is strong enough to dominate
sexual selection near the polymorphic equilibrium (see
also Verzijden et al. 2005).
With preference-based assortative mating, the change
in frequency of the M modifier allele is
1
DpM ¼ pM qM ðcf;1 Dr1 1 cf;2 Dr2 1 cr mate1 Dr1 1 cr mate2 Dr2 Þ
2
DpA
1 DAM ;A Htot 1 ðDAM 1 DA;M Þ
1 Oðe3 Þ:
ð17Þ
p A qA
Again, the first line in (17) reflects direct selection
against modifier alleles arising from the costs of
assortative mating. Here, the effect of the modifier on
female preferences against
males differing
by i alleles is
given by Dri ¼ pM ri;MM ri;Mm 1 qM ri;Mm ri;mm .
As mentioned previously, cf;1 and cf;2 are the fixed costs
of being able to detect males that differ by one and two
alleles, respectively. The term cr is the relative cost to a
female of rejecting a potential mate when she is unable
to replace this lost mating opportunity. The relative cost
enters twice, first, when multiplied by the probability
that a female encounters a male
differing
by one allele
[mate1 ¼ qA2 ð2pA qA Þ 1 2pA qA qA2 1 pA2 1 pA2 ð2pA qA Þ]times
the increased probability that such males are rejected by
a female carrying the M allele (Dr1 ) and, second, when
multiplied by the probability that a female
encounters
a
male differing by two alleles [mate2 ¼ qA2 pA2 1 pA2 qA2 ]
times the increased probability that such males are
rejected by a female carrying the M allele (Dr2 ). As
expected, both fixed and relative costs hinder the
evolution of assortative mating. Indeed, with strong
costs (of order one, as in the plant model where cr ¼ 1),
the first line in (17) becomes large (of order e), the
second line becomes negligible (of order e2 ), and
modifier alleles that increase the strength of preferences (Dri . 0) always decline in frequency. In particular,
we find that assortative mating can never evolve in the
plant model because potential mates are rejected and
never replaced (as found by de Cara et al. 2008).
We next focus on the second line in Equation 17,
which measures the indirect selective forces acting on
assortative mating, assuming that the costs of assortment are small (of order e). There is one key difference
between the change in modifier frequency in the
preference-based model (Equation 17) and that observed in the grouping model (Equation 7): sexual
selection alters the relative fitness of homozygotes vs.
heterozygotes at the A locus. With both natural and
sexual selection acting, Htot ¼ Hns 1 Hss , where the
contribution due to natural selection is given by (9b)
and the contribution due to sexual selection is
1
Hss ¼ ðpA2 ð2r1 1 r2 Þ 1 2pA qA ð2r1 Þ 1 qA2 ðr2 2r1 ÞÞ:
2
ð18Þ
Preferences against genotypes differing by one allele
(r1 ) increase the fitness of homozygotes relative to
heterozygotes, causing Htot to become more positive
and facilitating the evolution of assortment. In contrast,
preferences against genotypes differing by two alleles
(r2 ) decrease the fitness of homozygotes relative to
heterozygotes, causing Htot to become less positive and
hindering the evolution of assortment. More generally,
when females dislike males of all other genotypes
(r1 . 0 and r2 . 0), sexual selection will hinder the
evolution of assortative mating (by decreasing Htot )
unless homozygotes are common and K ¼ r1 =r2 . 12.
Assuming that selection is weak relative to recombination, we again calculated the steady-state (QLE)
values of the genetic associations. Unlike the grouping
model, cis and trans linkage disequilibria are generated
in the preference-based model,
D̃AM ¼ D̃A;M ¼ pM qM ðDcÞ 1 Oðe2 Þ;
ð19aÞ
where Dc measures the effect of the modifier allele M
on DpA due to a single round of nonrandom mating (i.e.,
the effect of the modifier on sexual selection):
1
Dc ¼ pA qA pA ðDr2 pA qA 1 Dr1 ð1 2pA Þ2 Þ 1 Oðe2 Þ:
2
ð19bÞ
Unless the allele frequency is 12, a modifier increasing
the level of assortative mating (Dri . 0) induces sexual
selection favoring the more common of the two alleles
at the A locus (Dc . 0 if pA . 12 and vice versa). This in
turn generates genetic associations between the modifier and the more common allele (D̃AM . 0 and D̃A;M . 0
if pA . 12 and vice versa). These associations develop
because males carrying the common allele are more
often preferred by females (because females that also
carry the common allele are more abundant). This
coupling of preference alleles with the trait alleles that
are preferred is typical of Fisherian models of sexual
selection (e.g., Kirkpatrick 1982). According to Equation
Evolution of Assortative Mating
17, cis and trans linkage disequilibria will then favor the
spread of a modifier increasing the level of assortment
as long as the common allele is increasing in frequency.
Note that if the A locus is at equilibrium (DpA ¼ 0), the
cis and trans linkage disequilibria have no influence on
the modifier (Equation 17).
In addition to the cis and trans linkage disequilibria,
the change in the modifier (17) depends on the genetic
association, DAM ;A , between the modifier allele, M, and
excess homozygosity, which at QLE is
D̃AM ;A
1
¼ pM qM ðDDA;A Þ 1 Oðe2 Þ;
2
ð20aÞ
where
DDA;A ¼ 2pA2 qA2 ðDr2 pA qA 1 Dr1 ð1 2pA Þ2 Þ 1 Oðe2 Þ:
ð20bÞ
DDA;A again measures the effect of the modifier on the
departure from Hardy–Weinberg following a single
round of mating at QLE. A modifier allele that increases
the level of assortment (Dr1 ; Dr2 $ 0 causing DDA;A . 0)
tends to be found in individuals with higher levels of
homozygosity at the A locus (DAM ;A . 0). According to
Equation 17, this association indirectly selects for the
modifier allele if homozygotes are more fit, on average,
than heterozygotes (Htot . 0).
In summary, the results from the preference-based
model of assortative mating differ from the grouping
model in three ways. The first is that sexual selection
induces positive frequency-dependent selection on
the A locus, which makes it less likely that a polymorphism will persist (if r1 . 0). The second is that
linkage disequilibria develop that couple modifier
alleles increasing the level of assortative mating with
the common allele at the A locus, which can cause the
evolution of increased assortative mating if the common
allele is rising in frequency. The third is that sexual
selection alters the fitness of homozygotes relative
to heterozygotes. Assortative mating is favored only
while homozygotes are more fit than heterozygotes,
on average (Htot . 0); sexual selection can make this
condition harder or easier to satisfy as assortative mating evolves, depending on the relative mating success
of homozygotes and heterozygotes, which in turn
depends on the values of r1 and r2 , as well as the allele
frequency, pA .
Evolutionarily stable strategy: We can use result (17) of
this QLE analysis to determine the level of assortative
mating that can evolve before selection on subsequent
modifier alleles equals zero (to leading order), as would
occur at an evolutionarily stable strategy (ESS). For
clarity of presentation, we assume that there are no
direct costs of assortment (cf;1 ¼ cf;2 ¼ cr ¼ 0), that mating is initially random, and that the A locus is initially at
a stable equilibrium with Htot ¼ Hns . 0. When pA ¼ 12
and the population is at Hardy–Weinberg proportions,
2101
increased levels of assortative mating cause heterozygotes to become fitter because heterozygous males
appeal to the large class of heterozygous females (see
Equation 18), decreasing Htot , and decreasing selection
for assortative mating. Indeed, assortative mating is
expected to evolve only to the point at which heterozygotes have the same average fitness as homozygotes
(Htot ¼ 0). Combining (9b) and (18) and setting pA ¼ 12,
this point occurs at
r2 ¼ 4Hns
¼ 4ðSAA ðX A Þ 2SAa ðX A Þ 1 Saa ðXA ÞÞ:
ð21Þ
Equation 21 represents the evolutionarily stable level of
assortative mating when pA ¼ 12 and costs are absent.
This result is consistent with the results of Matessi
et al. (2001), who identified the ESS as n 2a [our r2 is
their ð1 yÞ4 4y, and Hns ¼ 2a with their quadratic
fitness function]. More generally, Figure 3 plots the ESS
level of assortative mating as a function of the equilibrium allele frequency. When pA 6¼ 12, however, sexual
selection may favor the common allele at the A locus so
strongly that the internal polymorphism is destabilized,
preventing assortment from evolving to the Htot ¼ 0
curve and precluding an ESS along this curve. To
determine whether there is a stable polymorphism
on the Htot ¼ 0 curve requires that the form of frequency dependence be specified and that a stability
analysis be performed. This was done in Figure 3, using the
linear form of frequency-dependent selection: Si ðX A Þ ¼
ai 1 bi ðpA qA Þ. Only when frequency-dependent selection (bi ) was sufficiently strong was there a stable
polymorphism along the Htot ¼ 0 curve; in these cases,
simulations introducing modifier alleles with small
effects confirmed that assortment levels evolved to a
point along this curve and then ceased to evolve
further.
Thus, starting at a low level of assortative mating with
weak natural selection, the level of assortment will rise
until the relative fitnesses of homozygotes and heterozygotes become equal. This requires that sexual selection favors heterozygotes, which occurs when pA is
initially near 12 and/or when females discriminate more
strongly against males that differ by two alleles than
males that differ by one allele (r1 small relative to
r2 ). Under such conditions, heterozygous males more
readily find mates than homozygous males. Because we
have assumed in this section that natural selection is
weak, the degree of assortative mating evolves only to
a small multiple of Hns (the degree to which natural
selection favors homozygotes over heterozygotes) before being counterbalanced by sexual selection (Figure
3). In other cases (pA far from 12 and r1 large relative to
r2 ), however, a balance between sexual and natural
selection is not reached, and we must expand our
analysis to consider what happens in populations once
the level of assortment becomes high.
2102
S. P. Otto, M. R. Servedio and S. L. Nuismer
unwieldy for the generic grouping model, so we focus
only on the case where each genotype forms its own
group (gAA;1 ¼ gAa;2 ¼ gaa;3 ¼ 1). The steady-state value
of DA;A then equals
D̃A;A ¼ pA qA
Figure 3.—Evolutionarily stable level of assortative mating
in the preference-based model without costs. Assortative mating
initially evolves when Hns . 0, but this process is self-limiting if
heterozygotes more readily find mates. Once the strength of
sexual selection generated by assortment causes heterozygotes and homozygotes to become equally fit (Htot ¼ 0;
curves), further increases in the level of assortative mating
are no longer favored (Equation 17). Whether or not a stable
internal polymorphism exists along these curves depends on
the form of frequency-dependent natural selection. The solid
circles show stable equilibrium points, p̂A , using a specific
model of frequency dependence: Si ðXA Þ ¼ ai 1 bi ðpA qA Þ
with aAA ¼ 0, aAa ¼ 0:067, aaa ¼ 0:1, bAa ¼ ðbAA 1 baa Þ=2,
and bAA ¼ baa . Frequency-dependent natural selection becomes weaker as baa is reduced from 1 to 0.1 in increments
of 0.1 (solid circles from left to right, without assortment
along the x-axis or with assortment along the Htot ¼ 0 curves).
Assortative mating drives the frequency of allele A away from
pA ¼ 12 (arrow with light shading, baa ¼ 0.3; arrow with dark
shading, baa ¼ 0.2; solid arrow, baa ¼ 0:1). A stable internal
equilibrium exists along the Htot ¼ 0 curve only if frequency-dependent natural selection is sufficiently strong
(baa $ 0:20 when K ¼ r1 =r
2 ¼ 1; baa $ 0:13 when K ¼ 12;
baa $ 0:07 when K ¼ 0). Otherwise, the frequency of allele
A rises as assortment evolves, until the polymorphism is lost.
Note that the y-axis is measured relative to Hns in the current
population.
QLE ALLOWING HIGH LEVELS OF ASSORTMENT
The above results suggest that higher and higher
levels of assortment should evolve in the grouping
model, that assortment will often halt at an ESS with
only a low level of assortment in the preference-based
model when costs are weak (as in the animal model),
and that random mating should prevail if costs are
sufficiently strong (as in the plant model). These results
assume that natural selection is weak and that modifier
alleles increase assortment by a small amount. They
further assume that the degree of assortment is currently low. Here we extend the QLE analysis to populations that already exhibit a substantial degree of
assortment (r is of constant order).
QLE in the group-based model with high levels of
assortment: When the rate of assortative mating is high
in the grouping model, the departure from Hardy–
Weinberg equilibrium at the A locus, DA;A , is no longer
small. Unfortunately, the QLE calculations become
r
1 OðeÞ:
2 r
ð22Þ
Whenever the steady-state value of an association
measure is of constant order, there is no guarantee that
it will be approached quickly, calling into question the
validity of a QLE analysis. Fortunately, the departure
from Hardy–Weinberg rapidly approaches (22), as long
as a substantial degree of random mating remains.
Specifically, the departure of DA;A from its QLE value
changes from generation to generation by a factor r=2,
ðD9A;A D̃A;A Þ ¼ ðDA;A D̃A;A Þr=2 1 OðeÞ;
ð23Þ
implying that the system will be close to (22) after only a
few generations. This rapid approach occurs because
the force causing DA;A to decay (random mating)
immediately returns the offspring produced by random
mating to Hardy–Weinberg proportions. Below, we
assume that the population has reached the steady-state
departure from Hardy–Weinberg given by (22). All
remaining associations rapidly approach a steady-state
value that is small (of order e) as long as the effective
rate of recombination rate is sufficiently high. (This
requires that the rate of assortative mating not be too
near one, because with r 1, most loci are homozygous
and recombination becomes ineffective at breaking
down linkage disequilibria.)
Recalculating the association measures to order e, we
find that Equation 7 continues to describe the frequency change at the modifier locus, DpM , with the term
DpA replaced by Equation 6. Furthermore, the cis and
trans linkage disequilibria D̃AM and D̃A;M remain zero to
order e, and Equation 10 continues to apply, while (12)
becomes
D̃AM ;A ¼
1
Dr u
pM qM 1 Oðe2 Þ:
4 ð1 r=2Þð1 rÞ
ð24Þ
Thus, in the grouping model with gAA;1 ¼ gAa;2 ¼ gaa;3 ¼
1 [implying u ¼ freqðAa Þ=2], there are only two changes
that must be made to the QLE prediction for DpM once
assortative mating becomes common. First, the effect
of the modifier on the efficiency of assortment, Du, in
Equation 12 becomes Dr u=ðð1 r=2Þð1 rÞÞ. Second,
the effect of the modifier on the rarity of mates, DR,
in Equation 7 must be updated; this value of DR is
somewhat complicated (see supplemental Mathematica
file) because it accounts for two effects of the modifier:
the direct effect caused by changing a female’s tendency
to mate within a group and an indirect effect caused by
changing the groups to which males and females
belong.
Evolution of Assortative Mating
As the current level of assortative mating rises, genetic
associations reach higher QLE values. In particular,
D̃AM ;A increases in magnitude, accelerating the rate at
which further assortative mating evolves. On the other
hand, the relative costs of assortative mating can
become more or less severe as the level of assortment
rises; the costs increase in severity when heterozygotes
are relatively common (i.e., when pA is near 12 and r is
small) but decrease in severity when homozygotes are
already the most common class, because further increases in homozygosity then make it easier to encounter a mate. Regardless, it can be shown that, if Hns
remains constant and if the relative costs, cr , are small
enough that modifier alleles introducing assortative
mating are able to spread within a randomly mating
population, then modifier alleles increasing assortment
will always be favored, no matter how high r gets. That
is, the costs do not rise fast enough to overwhelm
the benefits experienced by a modifier as the level of
assortment rises. Consequently, higher levels of assortment are expected to evolve unless costs are substantial,
in which case random mating is stable.
This assumes, however, that Hns is relatively constant,
but frequency-dependent natural selection can alter the
relative fitness of homozygotes vs. heterozygotes as
homozygotes become increasingly common, altering
the magnitude and even the sign of Hns . If the average
fitness of heterozygotes ever becomes equal to the
average fitness of homozygotes (Hns ¼ 0), then in the
absence of costs, the level of assortative mating will cease
to evolve (to the order of magnitude of our approximations). This phenomenon was observed in the
model of density-dependent competition studied by
Pennings et al. (2008). In their model, if homozygotes
are unable to completely utilize the resources available
to heterozygotes, the fitness of heterozygotes relative to
homozygotes rises with increasing levels of assortment
(i.e., as heterozygotes become rarer). In this case, assortment levels evolve only to the point where Hns ¼ 0
(their Equation 12), and the three genotypes are all
stably maintained within the population, with Aa heterozygotes occupying their own ecological niche (see
their Figure 1, partial-isolation regime).
QLE in the preference-based model with high levels
of assortment: A general QLE analysis is not possible for
the preference-based models because the sexual selection induced in these models is typically strong (of
constant order) once assortative mating becomes common, invalidating the QLE assumption that allele
frequencies change slowly. There are, however, three
cases where sexual selection on the A locus is weak:
when assortative mating is rare (analyzed above), when
assortative mating is nearly complete (where a QLE
analysis is not valid), and when pA ¼ 12.
Here, we briefly consider the preference-based model
in the absence of costs when pA ¼ 12, so that sexual
selection on the two alleles is exactly balanced. A QLE
2103
Figure 4.—Evolution when assortative mating is already
common. The QLE results are shown for the preferencebased model when pA ¼ 12, as a function of the current level
of assortment (vertical axis) and the relative preference
against individuals that differ by one allele, K ¼ r1 =r
2 (horizontal axis). Modifiers that increase the rate of assortative
mating decrease in frequency (DpM , 0) for the majority of
the parameter space; assortment is expected to decrease until
it reaches the ESS at r2 ¼ 4Hns (horizontal line, illustrated for
Hns ¼ 0:02). If r1 =r
2 . 12 and if assortative mating is sufficiently
common (above the thick curve), then higher rates of assortative mating are expected to evolve. If r1 is a constant multiple of r2 , then preferences will evolve either up or down (no
horizontal movement). The preference function used by Matessi et al. (2001) constrains preferences to evolve along the
dashed curve; the two points at which the QLE predicts that
DpM switches sign (the ESS at the solid star and the repelling
point at the open star) are consistent with their numerical results for weak selection (their Table 3). The graph assumes no
direct cost of assortative mating, weak selection, and modifiers of small effect. The shading at the top serves to remind
the reader that the QLE assumptions fail to hold if reproductive isolation is nearly complete.
analysis, assuming weak selection and a modifier causing only small changes to the degree of assortment, was
performed. As summarized in Figure 4, when assortative
mating is present but not very common, the level of
assortative mating decreases toward the ESS given by
Equation 21, with selection on the modifier being
strong (now of order e, rather than e2 as before). Interestingly, if assortative mating is very common [above
the thick curve in Figure 4, such that r1 . r2 =2 .
2r1 ð1 r1 Þ], then even higher levels of assortment can
evolve. This result was obtained by examining when the
sign of DpM switches according to the QLE approximation, but this switch coincides exactly with the point at
which heterozygotes become less fit than homozygotes
(Htot switches from negative to positive). This switch
occurs not because of natural selection but because of
sexual selection. Indeed, natural selection can be absent
2104
S. P. Otto, M. R. Servedio and S. L. Nuismer
(Hns ¼ 0) and yet the mating preferences would lead
to complete assortative mating above the thick curve
(Figure 4), solely because of the sexual selection
induced (as noted by Arnegard and Kondrashov
2004).
When assortative mating is rare and pA ¼ 12, the most
common females are Aa heterozygotes, who prefer to
mate with similar heterozygous males, inducing heterozygote advantage via sexual selection. If assortative
mating is sufficiently common, as might be expected
for previously allopatric populations that are already
partially reproductively isolated, most females are
homozygous and favor similar homozygous males,
inducing heterozygote disadvantage via sexual selection (above the thick curve in Figure 4). While Hss due
to sexual selection is more complicated in the case of
high levels of assortative mating (not presented), one
can already see from Equation 18 that Hss would
increase as homozygotes become more common as
long as females dislike males that differ by a single
allele at least half as much as they dislike males that
differ by two alleles (r1 =r2 . 12). Once above the thick
curve (Figure 4), sexual selection itself induces disruptive selection, facilitating the evolution of complete
reproductive isolation. The bistability of low and high
levels of assortative mating was also found by Doebeli
(1996), Matessi et al. (2001), and Pennings et al.
(2008) in similar modifier models and by Kirkpatrick
and Ravigné (2002) in a model examining the
dynamics of linkage disequilibrium between selected
loci given a certain amount of assortative mating. The
generality of this result suggests that, if preferencebased assortative mating is commonplace, the evolution of higher rates of assortative mating would be
more efficient at completing the process of speciation
(‘‘reinforcement,’’ in this case due to heterozygous
‘‘hybrids’’ having a hard time finding a mate) than
initiating it.
Exact numerical simulations of the recursion equations confirm the existence of the two curves shown
in Figure 4, using the fitness function Si ðX A Þ ¼ ai 1
bi ðpA qA Þ with the parameter values given in Figure 3
and with K ¼ r1;i =r2;i set to 0, 0.25, 0.5, 0.75, and 1.
Particularly, even when pA 6¼ 12, a modifier allele that
strengthens the degree of assortative mating increases
when r2 is small, decreases when r2 is between 4Hss
and the thick curve shown in Figure 4, and increases to
fixation when r2 is above the thick curve.
The above results ignored costs of assortment, but
they remain valid if such costs are sufficiently weak. With
strong costs, however, assortative mating is selected
against at QLE, regardless of the current level of
assortative mating. In particular, in the plant model
with cr ¼ 1, the costs of rejecting dissimilar males and
thereby losing the offspring that would have been
produced are too severe to permit the evolution of
nonrandom mating.
STABILITY ANALYSES
QLE analyses allow complex models to be analyzed
more readily, because the steady-state values of the
genetic associations are calculated in a separate step
from assessing the allele-frequency dynamics. But QLE
analyses require that selection be weak and that modifier alleles do not cause a large change in the level of
assortment; otherwise the genetic associations can depart significantly from their steady-state values. In this
section, we supplement the above results with local
stability analyses, assuming that a new modifier allele, M,
is introduced into a population at a stable equilibrium
that is polymorphic for alleles A/a and fixed for allele m.
Because these local stability analyses require the analysis
of rather complicated stability matrices, we focus on two
cases: the grouping model where Aa individuals form
their own group (gAA;1 ¼ gAa;2 ¼ gaa;3 ¼ 1 in Figure 1)
and the preference-based model, ignoring costs in both
cases. The technical details are presented in appendix c
and in the Mathematica package available as supplemental material. As we shall see, the above conclusions
remain valid qualitatively, even with strong selection and
strong modifiers.
Stability analysis in the grouping model of assortative mating: Several special cases of the genotypic
grouping model with gAA;1 ¼ gAa;2 ¼ gaa;3 ¼ 1 were considered (see appendix c). In each case, a modifier that
increases the degree of assortative mating is able
to spread when introduced into a population at an
equilibrium where homozygotes are more fit, on average, than heterozygotes, regardless of the strength
of selection or the effect of the modifier on levels of
assortment. These results are entirely consistent with
the QLE analysis.
Stability analysis in the preference-based model of
assortative mating: In appendix c, we also describe
results from stability analyses assuming p̂A ¼ 12 in the
preference-based model. We first summarize the results
when natural selection is weak. As in the QLE analysis,
lower levels of assortative mating are generally favored
except when the population mates nearly at random
(and Hns . 0) or when assortative mating is prevalent
and females dislike males that differ by one allele almost
as much as males that differ by two alleles. Indeed, the
parameter space in which modifiers that increase
assortment are favored is exactly as in Figure 4, based
on the QLE analysis. The stability analysis provides the
additional insight that modifiers causing a very large
change in the level of assortative mating are favored if
and only if modifiers causing a very small change are
also favored. Ironically, this means that it is possible for
a modifier causing complete assortment to invade a
random-mating population but not a population in
which there is already some level of assortment (between
the line at 4Hns and the thick curve in Figure 4). Numerical simulations using the fitness function Si ðXA Þ ¼
Evolution of Assortative Mating
2105
homozygotes over heterozygotes by $10%, higher levels
of assortative mating are able to invade an equilibrium
with p̂A ¼ 12, regardless of the current level of assortative
mating, as long as females dislike males that differ
by one allele at least half as much as they dislike males
that differ by two alleles (K ¼ r1;mm =r2;mm $ 12). Thus, if
natural selection in favor of homozygotes is sufficiently
strong, bistability disappears, and assortative mating can
evolve from initially low to high levels. Qualitatively
similar results were obtained from a numerical stability
analysis using a specific model of frequency-dependent
1
selection with p̂A ¼ 10
(supplemental material).
CONCLUSIONS
Figure 5.—Parameter space in which modifiers increasing
the level of assortative mating are able to invade. The results
of a local stability analysis for the preference-based model
without costs are shown. Arrows indicate whether a modifier
allele M that increases the level of assortative mating (by any
amount) is able to invade an equilibrium with m fixed and
p̂A ¼ 12. The curves specify the parameter combinations at
which DpM ¼ 0, which coincide with when Htot ¼ 0. Each
curve is associated with a particular percentage by which natural selection favors homozygotes over heterozygotes, at the
equilibrium before the M allele is introduced (solid boxes).
Only when natural selection is weak (natural selection favors
homozygotes by #1%) are the results similar to the QLE
results summarized in Figure 4. The vertical axis gives the initial level of assortment r2;mm , while the horizontal axis measures how much females dislike males that differ by one allele
relative to those that differ by two alleles, r1;mm =r2;mm .
ai 1 bi ðpA qA Þ indicate that modifiers slightly increasing assortment within a randomly mating population
spread only to a point at which r2 approximates 4Hns ,
while modifiers causing high levels of assortment (above
the thick curve in Figure 4) are able to invade and spread
to fixation.
The stability analysis also allows us to investigate cases
where natural selection is strong. Assuming that natural
selection favors homozygotes (Hns . 0), Figure 5 demonstrates that stronger natural selection facilitates the
evolution of assortative mating by increasing the parameter space in which homozygotes have an overall
higher fitness than heterozygotes under the combined
forces of natural and sexual selection (Htot . 0). This
result was also observed by Matessi et al. (2001), for the
specific case where frequency-dependent selection (1)
is a quadratic and symmetric function of the allele
frequencies. Interestingly, the strength of natural selection does not have to be inordinately high to relax
considerably the conditions under which assortative
mating can evolve. Indeed, if natural selection favors
In this article, we have developed a framework for
describing the evolutionary forces acting on the strength
of assortative mating in response to selection at a single
locus. Our goal was to keep the assumptions as general as
possible: allowing for any form of frequency-dependent
selection and for a range of mechanisms by which
assortative mating can be accomplished, with or without
costs. We assume that the level of assortative mating
depends on the alleles carried at a modifier locus, M,
and that assortative mating is based on the trait
influenced by the selected locus itself, A. This framework
describes a one-allele mechanism (Felsenstein 1981)
because assortative mating can arise from the spread of
a single allele at the M locus throughout the population. Our main results are as follows:
I. Higher levels of assortative mating are favored
when homozygotes are, on average, fitter than
heterozygotes.
II. This evolutionary force is countered by any costs of
assortment, but assortative mating can still evolve as
long as the costs are weak.
III. Assortative mating based on group formation is
particularly favorable to the evolution of assortative
mating; complete assortative mating can evolve in
small or large steps from random mating as long as I
continues to hold.
IV. Assortative mating based on female mating preferences complicates matters by inducing sexual
selection on the selected locus, which typically
hinders but can sometimes facilitate the evolution
of assortative mating, as discussed below.
These results are based on two techniques: a separation of timescales assuming weak selection (a ‘‘QLE’’
analysis) and a local stability analysis, the combination
of which provides a fairly complete picture of the
evolution of assortative mating in response to selection
at a single-trait locus. These results are consistent with
other studies (e.g., Matessi et al. 2001; Pennings et al.
2008), but by describing the general conditions under
which higher levels of assortative mating can evolve—
without recourse to specific assumptions about the nature
2106
S. P. Otto, M. R. Servedio and S. L. Nuismer
of frequency-dependent selection or mating scheme—
this article contributes to the large body of theoretical
work on speciation by clarifying the key requirements
for assortative mating to evolve in response to selection
on a focal gene (see also de Cara et al. 2008 for an
analysis of the multilocus case).
Several comments are in order. In an equilibrium
population, homozygotes are, on average, fitter than
heterozygotes (as required by result I) only if there is
underdominance. Frequency-dependent selection favoring rare alleles is, however, necessary to maintain a
polymorphism that exhibits underdominance. Models
of competition can create such conditions (e.g.,
Dieckmann and Doebeli 1999; Matessi et al. 2001;
Pennings et al. 2008), as can a number of other biologically relevant scenarios, for example, mating preferences for unusual mates, antagonistic species
interactions with predators, herbivores, or parasites that
are less able to recognize rare types, and mutualistic
interactions that reward rare types (see, e.g., Doebeli
and Dieckmann 2000).
Our model clarifies, however, that underdominance
is not necessary for the evolution of assortative mating;
in nonequilibrium populations, all that is required is
that the heterozygote be closer in fitness to the less fit of
the two homozygotes, not that the heterozygote be the
least fit. For example, whenever a partially recessive
beneficial allele arises and spreads within a population,
assortative mating would be favored along the way. This
process is self-limiting, however, because once the
beneficial allele is fixed there is only one genotype at
the trait locus and assortative mating is no different
from random mating. Nevertheless, substantial levels of
assortative mating could accumulate should variation be
regenerated at any locus affecting the trait (through
migration or mutation). Similarly, cases of coevolutionary cycles or arms races can generate persistent directional selection, which could drive the evolution of
nonrandom mating. Indeed, we have recently shown
that assortative mating is often favored in parasites,
leading to host specialization (Nuismer et al. 2008).
If assortative mating is based on female preferences
for similar males, the resulting sexual selection alters
the relative fitnesses of heterozygotes and homozygotes.
In certain cases, sexual selection can cause the fitness
of heterozygotes to rise above the average fitness of
homozygotes, preventing the further evolution of assortment. In particular, if heterozygotes are common
(pA near 12) and/or if heterozygous males are relatively
well liked by homozygous females (K ¼ r1 =r2 small),
then heterozygous males have an easier time finding a
mate than homozygous males, counteracting any fitness
advantage of homozygotes generated by natural selection. In this case, as the level of assortative mating rises,
mating preferences eliminate the fitness advantage
of homozygotes necessary for the evolution of further
increases in assortment. As a consequence, assortment
tends to evolve from random mating only up to a point
(4Hns when pA is near 12; see Figure 3), as found
by Matessi et al. (2001) using a particular form of
frequency-dependent selection.
On the other hand, if heterozygotes are rare (because
assortative mating is already common or pA is far from 12)
and if heterozygous males are not well liked by homozygous females (large K ¼ r1 =r2 ), then sexual selection
will itself contribute to the low fitness of heterozygotes
experienced at the A locus. Essentially, this occurs
whenever heterozygotes have a harder time finding a
willing mate than homozygotes. Under these conditions, natural selection need not act at all (or indeed
could induce some overdominance), and yet assortative
mating by sexual selection can create the conditions
(result I) needed for the evolution of more extreme
levels of assortment, as found by Arnegard and
Kondrashov (2004). This phenomenon is observed
in the upper right-hand corner of Figure 4 for pA near 12
and on the right-hand side of supplemental Figure S2
1
for pA near 10
. An important implication is that if
assortative mating is already commonplace, sexual selection can drive the evolution of complete reproductive isolation because heterozygotes have such
difficulties finding mates. Thus, we predict that mechanisms of assortative mating that induce sexual selection
(i.e., based on female preferences rather than group
membership) should be particularly prevalent in cases
where assortment evolves after previously isolated populations come back into contact and have some degree
of reproductive isolation (reinforcement).
Even if total fitness (accounting for both natural and
sexual selection) continues to exhibit underdominance, a further problem arises when assortative mating
is based on mating preferences: carriers of the more
common allele have less trouble finding a receptive
mate than carriers of the rarer allele. Sexual selection
thus favors the fixation of the common allele. When
assortative mating is rare, this force is weak, but if
assortative mating becomes common, sexual selection
on the A locus becomes strong, unless the two alleles are
nearly equal in frequency (pA 12). Thus, to maintain a
polymorphism in the face of sexual selection, frequencydependent natural selection must strongly favor the
rarer allele (as in supplemental Figure S2).
In short, two obstacles to the evolution of assortative
mating arise when females prefer similar mates: such
preferences can (a) generate heterozygote advantage
and (b) lead to the fixation of the common allele. In
contrast, assortative mating that arises because individuals tend to cluster into groups within which they mate
randomly raises no such obstacles, so that the degree
of assortative mating will continue to rise as long as
homozygotes remain fitter than heterozygotes.
It is worth emphasizing that when fitness is frequency
dependent, the relative fitness of heterozygotes and
homozygotes may very well change as assortative mating
Evolution of Assortative Mating
evolves. Depending on the nature of frequency dependence, it may be that underdominance is exhibited
when the population is near Hardy–Weinberg, but
that overdominance is exhibited when heterozygotes
are rare. In this case, natural selection will favor the
evolution of only partial assortative mating rather than
complete assortative mating. This phenomenon was
very nicely described by Pennings et al. (2008), using
a model of competition. As they point out, if each
genotype has a relatively narrow niche, then as homozygotes become increasingly common, a third niche
opens up in the middle that can be filled by heterozygotes. Thus, as can be seen from their Figure 1,
the evolution of complete assortative mating requires
that the spectrum of resources that can be used by a
genotype is narrow enough that the extremes of the
resource distribution remain underutilized in a population at Hardy–Weinberg but not too narrow, or else
the homozygotes will not exhaust the resource supply
available to heterozygotes. Other forms of frequencydependent selection may or may not lead to a switch
from underdominance to overdominance as assortative
mating evolves (for example, the model of rare-allele
advantage used to generate supplemental Figure S2
does not). Clearly, the exact form of frequencydependent natural selection is important to be able to
predict whether partial or complete assortative mating is
expected to evolve.
Overall, our results suggest that assortative mating that
induces sexual selection (e.g., based on female mating
preferences for similar mates) is not very conducive to
the evolution of complete assortative mating starting
from a random-mating population. Only when natural
selection favors homozygotes over heterozygotes, natural selection is strong, the costs of assortment are
low, and females prefer their own genotype over other
genotypes (K ¼ r1 =r2 sufficiently large; see Figure 5
and supplemental Figure S2) do we expect complete
assortment to evolve from no assortment. In contrast,
assortative mating based on group formation is less
restrictive, and we expect to see the evolution of complete assortative mating as long as homozygotes are
more fit, on average, than heterozygotes and there is a
low cost of assortment.
Interestingly, many purported cases of sympatric
speciation can arguably be described as following a
group-based form of assortative mating. Host races
of phytophagous insects are thought to have evolved
because individuals group together and mate on their
preferred plant host (Berlocher and Feder 2002), as
exemplified by the poster child of sympatric speciation:
the apple maggot fly, Rhagoletis pomonella, in which a new
host race formed on apples in the mid-1800s (Filchak
et al. 2000). Cases of sympatric speciation in plants are
often associated with shifts in flowering time (Gustafsson
and Lonn 2003; Silvertown et al. 2005; Antonovics
2006; Savolainen et al. 2006) or shifts in pollen
2107
placement (Maad and Nilsson 2004), such that mating
tends to occur within groups of similar plants. Similarly,
grouping by location or timing during mating has been
implicated in the evolution of assortative mating in
salmon and birds (Hendry et al. 2007).
It is important to emphasize that focusing on a singletrait gene that is the target of both natural selection and
nonrandom mating preferences, as we have done here,
is particularly favorable to the evolution of assortative
mating (Gavrilets 2003). Most importantly, whether
or not assortment is favored does not depend on the
rate of recombination between the modifier locus and
the selected locus (observe that the recombination rate,
r, does not enter into any of the QLE results and does
not qualitatively affect the stability analyses). If mating
preferences were based on similarity at a different locus
(say ‘‘B’’) than the target of natural selection A, recombination between these loci would weaken the link
between assortative mating (at B) and the production of
excess homozygotes (at A). Thus, our results can be seen
as describing the necessary conditions for assortative
mating to evolve in response to selection at a single
selected gene; for models where assortment is based on
a marker gene, not only must these necessary conditions
be met, but also selection favoring excess homozygosity
at the A locus must be sufficiently strong relative to the
effective amount of recombination between the A and B
loci. Because the effective rate of recombination goes
down as the level of assortative mating goes up (recombination is only relevant in double heterozygotes),
we conjecture that the higher the current level of
assortative mating is, the less of a difference should be
seen between our results and those from a model of
assortative mating based on a separate marker locus.
We are grateful for the helpful and insightful comments provided
by Alistair Blachford, Michael Doebeli, Fred Guillaume, Joachim
Hermisson, Jason Hoeksema, Yannis Michalakis, Patrick Nosil,
Michael Whitlock, and two anonymous reviewers. We also thank Nick
Barton, Mark Kirkpatrick, and Joachim Hermisson for graciously
sharing their unpublished manuscripts. Support was provided by the
Natural Sciences and Engineering Research Council of Canada
(S.P.O.), the National Science Foundation (NSF) [grant nos. DEB0614166 (M.R.S.) and DEB-0343023 and DMS-0540392 (S.L.N.)], and
the National Evolutionary Synthesis Center [NSF no. EF-0423641
(S.P.O., M.R.S.)].
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Communicating editor: M. K. Uyenoyama
APPENDIX A
Two variants of the grouping model discussed in the text (the ‘‘standard’’ grouping model) were explored to
determine the sensitivity of the results to model assumptions.
Variant 1—mated males do not join the random-mating pool: In the text, all males were allowed to join the lek,
regardless of whether or not they had previously mated. Consequently, the standard grouping model does not ensure
that all males have the same reproductive success, even though all females do, so the possibility of sexual selection
remains. As an alternative, we considered a model where only unmated males join the lek, so that all females and all
males have equal reproductive success. Specifically, the probability that a male genotype joins the random-mating pool
is one minus the probability that it mated within a group. The QLE results for this alternative model are identical to
Evolution of Assortative Mating
2109
Equations 6–12. Sexual selection is absent to order e (see Equation 6) both in the grouping model considered above
and in this variant model. The standard grouping model does, however, induce sexual selection to lower order than e.
When natural selection is absent, that is, when the fitnesses (1) are equal to one and the costs of assortment are zero, it
can be shown that disequilibria develop and allele frequencies do change, but very slowly, at the modifier and selected
loci in the standard grouping model but not in this variant. Nevertheless, we focus on the standard grouping model
because the recursions are simpler.
Variant 2—the random-mating pool is a separate group: Finally, we considered a model whereby individuals join a
mixed-group lek with probability 1 r during the grouping phase, rather than after the grouping phase. The
remainder of the population, r, form into three groups according to their A-locus genotype, as described in Figure 1.
Mating then occurs randomly within the lek and within the three groups. For example, during the mating season,
individuals could either choose to swarm ( joining the lek) or return to the host plant that they prefer (influenced by
the A locus). In this variant model, the probability that a female of genotype AA at the trait locus and k at the modifier
locus (k ¼ MM, Mm, or mm) mates with a male of genotype Aa becomes
! P3
P3
3
X
r
g
ðfreq
of
Aa;
l
malesÞ
Aa;j
l
l¼1
l¼1 ð1 rl Þðfreq of Aa; l malesÞ
P
P
P
P
1 ð1 rk Þ 3
;
ðA1Þ
rk gAA;j 3
3
3
l¼1
i¼1 rl gi;j ðfreq of i; l malesÞ
l¼1
i¼1 ð1 rl Þðfreq of i; l malesÞ
j¼1
where j sums over the three groups, i sums over the three male genotypes at the trait locus fAA; Aa; aag, and l sums over
the three male genotypes at the modifier locus fMM ; Mm; mmg. A QLE analysis was performed on this variant model,
assuming an initially low level of assortative mating. To leading order, the efficiency of assortative mating, u, is the same
as in the text, except that assortment has twice the effect on D̃AM ;A (doubling Equations 10 and 12) because males that
are in a group are also likely to carry the modifier, doubling the chance that the modifier becomes associated with
excess homozygosity. An additional difference is that assortative mating now occurs in groups that are smaller in
number, because groups are restricted to those individuals who choose to mate assortatively, not all individuals. Thus,
the relative costs of assortative mating are much larger, with 12pM qM cr DR in (7) becoming pM qM cr Dr, whether
or not individuals in the lek pay the relative cost of finding mates. Accounting for these changes, the QLE results
(6)–(12) continue to hold.
APPENDIX B
Here we consider the preference-based model of sexual selection with the generalized preference matrix:
Male genotype
0
aa
Aa
AA
aa 1 r½1; 1
B
Female genotype Aa @ 1 r½2; 1
1 r½1; 2
1 r½2; 2
1
1 r½1; 3
C
1 r½2; 3 A:
1 r½3; 1
1 r½3; 2
1 r½3; 3
AA
ðB1Þ
The modifier locus alters these preferences to r½i; jMM in MM females, r½i; jMm in Mm females, and r½i; jmm in mm
females. While the preference matrix in the text (Equation 3) focused on assortative mating, matrix (B1) is much
more general and can be used to describe disassortative mating (smaller diagonal elements) or mating preferences
that are independent of the female’s trait (rows are equivalent). A QLE analysis assuming weak selection in a
population with initially weak assortative mating was performed, producing results analogous to those in the main text
(Equations 15–21). Here, we present the specific changes caused by using preference matrix (B1) in place of (3). The
frequency of allele A now changes by an amount given by Equation 6 due to natural selection plus
p A qA 2
qA X ½1 1 2pA qA X ½2 1 pA2 X ½3
2
p A qA 2
qA Y ½1 1 2pA qA Y ½2 1 pA2 Y ½3 1 Oðe2 Þ;
ðB2aÞ
cr
2
where
X ½i ¼ ðr½i; 3 r½i; 2ÞpA 1 ðr½i; 2 r½i; 1ÞqA
ðB2bÞ
Y ½i ¼ ðr½3; i r½2; iÞpA 1 ðr½2; i r½1; iÞqA
ðB2cÞ
2110
S. P. Otto, M. R. Servedio and S. L. Nuismer
2
2
and where r½i; j ¼ pM
r½i; jMM 1 2pM qM r½i; jMm 1 qM
r½i; jmm is the average mating preference that females of
genotype i at the A locus exhibit toward males of genotype j at the A locus. The first row in (B2a) measures the
effect of sexual selection on males that carry the A allele instead of the a allele, while the second row measures selection
due to mating costs acting on females that carry the A allele instead of the a allele. The change in frequency of the
modifier allele remains the same (compare to Equation 17), except that the costs of sex must be adjusted for the new
mating scheme:
1
DpA
DpM ¼ pM qM ðcf Dr 1 cr ðqA2 Z ½1 1 2pA qA Z ½2 1 pA2 Z ½3ÞÞ 1 DAM ;A Htot 1 ðDAM 1 DA;M Þ
1 Oðe3 Þ:
2
p A qA
ðB3aÞ
Here, cf Dr measures the fixed costs of the new modifier allele, and Z ½i measures the effect of the modifier on the
degree of choosiness of females of genotype i at the A locus, given the current composition of the male population,
Z ½i ¼ qA2 Dr½i; 1 1 2pA qA Dr½i; 2 1 pA2 Dr½i; 3;
ðB3bÞ
where Dr½i; j ¼ pM r½i; jMM r½i; jMm 1 qM r½i; jMm r½i; jmm measures the effect of the modifier allele M on
r½i; j. If the costs of assortment are substantial (much larger than e), the first line of (B3a) dominates the second, and
modifiers that reduce female preference, r½i; j, will be favored.
Henceforth, we assume that the costs of assortment are small (of order e) and describe the contribution of the
genetic associations in (B3a). The relative fitness of homozygotes vs. heterozygotes at the A locus is Htot ¼ Hns 1 Hss ,
where Hns is given by (9b) and
Hss ¼
1 2
q H ½1 1 2pA qA H ½2 1 pA2 H ½3 :
2 A
ðB4aÞ
Here, H ½i measures the impact of mate choice by females of genotype i at the A locus on Htot :
H ½i ¼ ð1 r½i; 1Þ 1 ð1 r½i; 3Þ 2ð1 r½i; 2Þ:
ðB4bÞ
Equations 19a and 20a continue to describe the QLE values of the disequilibria. The effect of the modifier allele M on
DpA due to a single round of nonrandom mating becomes
p A qA 2
qA DX ½1 1 2pA qA DX ½2 1 pA2 DX ½3 1 Oðe2 Þ;
ðB5aÞ
Dc ¼ 2
where
DX ½i ¼ ðDr½i; 3 Dr½i; 2ÞpA 1 ðDr½i; 2 Dr½i; 1ÞqA :
ðB5bÞ
Dc is positive if the modifier allele M increases the strength of sexual selection favoring the A allele and the A allele is
currently preferred by females (i.e., the first line of B2a is positive). Plugging (B5) into Equation 19a and then into
(B3) indicates that cis and trans linkage disequilibria favor the spread of modifiers that increase the strength of
assortment as long as the sexually preferred allele is increasing in frequency. The final term that is needed to predict
the change in modifier frequency is the effect of the modifier on the departure from Hardy–Weinberg (generalizing
Equation 20b):
1 2
1
1
2 2
DDA;A ¼ 2pA qA qA Dr½1; 1 pA qA Dr½1; 2 1 pA qA Dr½1; 3
2
2
2
2
1
1
1
Dr½2; 2 1 pA pA Dr½2; 3
pA qA Dr½2; 1 2 pA 2
2
2
1
1
1 2
ðB6Þ
1 pA qA Dr½3; 1 1 pA pA Dr½3; 2 pA Dr½3; 3 1 Oðe2 Þ:
2
2
2
Modifier alleles that increase the production of homozygotes cause DDA;A to be positive. Equations B2–B6 reduce to
Equations 17–20 given the symmetric preference matrix (3).
APPENDIX C
Stability analysis in the grouping model of assortative mating: We performed a local stability analysis of the
genotypic grouping model with gAA;1 ¼ gAa;2 ¼ gaa;3 ¼ 1 (Figure 1) in the absence of costs. With m fixed, a potential
equilibrium must satisfy
Evolution of Assortative Mating
D̂A;A ¼
2111
rmm
ð2p̂A ð1 p̂A Þ 1 2D̂ ns
A;A Þ;
4
ðC1Þ
where we use a caret to denote an equilibrium value and ‘‘ns’’ to denote the value of a term after natural selection (all
other terms are measured after offspring production). With m fixed, it can be shown that the equilibrium allele
frequency is the same before and after natural selection in the grouping model (p̂A ¼ p̂ ns
A ). Because 2DA;A can never be
, 2pA ð1 pA Þ if measured at the same point in the life cycle, it follows that D̂A;A is positive (or zero) in the grouping
model. We assume that the form of frequency-dependent selection is such that there is a stable internal polymorphism
satisfying (C1) with allele frequency given by p̂A (depending on the form of frequency dependence, there may be
multiple equilibria, each of which can be analyzed as follows). Introducing a small frequency of the M allele and
linearizing the recursions around the equilibrium, we obtain a 7 3 7 local stability matrix, three of whose eigenvalues
equal zero, and the remainder are the four roots of a quartic equation.
In the symmetric case with p̂A ¼ 12, the quartic factors into two quadratic equations, which are more readily analyzed,
so we consider this special case first. If we were to consider a new modifier that has no effect (Dr ¼ 0), we would expect
one of the eigenvalues to equal one (as expected for a neutral allele); this is true for one of the quadratic equations, so
we concentrate on this equation, which has the form l2 1 bl 1 c ¼ 0. It can be shown that the minimum of this
quadratic lies between 1 and 11. Furthermore, we are guaranteed that the leading eigenvalue will be real and
nonnegative by the Perron–Frobenius theorem (because the matrix is nonnegative; Gantmacher 1989). Thus, the
condition required for a new modifier to invade (i.e., for a solution of l2 1 bl 1 c ¼ 0 to be greater than one) is
satisfied if and only if the value of the quadratic is negative when l is set to one (because the quadratic equation must
rise and become positive as l goes to positive infinity). The value of the characteristic polynomial when l is set to one
can be calculated and equals
1 Hns
DDA;A ;
2 W
ðC2Þ
where DDA;A measures the effect of the modifier on the departure from Hardy–Weinberg following a single round of
mating, given by DDA;A ¼ Dr DA;A =rmm. According to (C2), a modifier allele that increases assortative mating will
invade if and only if homozygotes are favored by frequency-dependent selection at the polymorphic equilibrium
(Hns . 0).
Unfortunately, it is difficult to interpret the characteristic polynomial when pA 6¼ 12 because we are left with a quartic
equation, which does not factor and can have multiple roots greater than one. Nevertheless, we examined three other
special cases, which could be analyzed using perturbation techniques (summarized in Otto and Day 2007): (i) weak
natural selection (i.e., fitnesses in Equation 1 near one), (ii) a weak modifier allele (Dr near zero), and (iii) a low or a
high frequency of allele A (pA near zero or one). In each case, the results obtained were consistent with the conclusion
that a modifier can spread if and only if Hns DDA;A . 0. Furthermore, it can be shown that when Hns ¼ 0, the leading
eigenvalue is always one and rises above one as Hns increases from zero for a modifier that increases assortative mating
(DDA;A . 0). (Detailed results and proofs are provided in the accompanying online Mathematica package.) Thus, in all
cases of the grouping model considered, modifiers that increase assortment invade if homozygotes are more fit, on
average, than heterozygotes at an equilibrium with frequency-dependent selection.
Stability analysis in the preference-based model of assortative mating: With allele m fixed and p̂A ¼ 12 in a model
of female preferences, the system reaches an equilibrium at which the departure from Hardy–Weinberg depends on
the current degree of assortative mating:
D̂A;A ¼
2
ð1=4 1 D̂ ns
A;A Þ r2;mm
ns
2 4ð1=4 D̂ ns
A;A Þr1;mm 2ð1=4 1 D̂ A;A Þr2;mm
:
ðC3Þ
Introducing a small frequency of the M allele and linearizing the recursions around the equilibrium, we obtain a 7 3 7
local stability matrix, three of whose eigenvalues again equal zero, and the remainder are the four roots of a quartic
equation, which factors into two quadratics when p̂A ¼ 12. Again only one of the quadratic equations gives an eigenvalue
of one when the modifier is neutral (Dr1 ¼ Dr2 ¼ 0), so we concentrate on this equation. It can be shown that the
minimum of this quadratic lies between 1 and 11. Furthermore, the leading eigenvalue will again be real and
nonnegative by the Perron–Frobenius theorem. Thus, a new modifier will invade if and only if the value of the
quadratic is negative when l is set to one. The value of the characteristic polynomial when l is set to one can be
calculated and equals
1 Htot
ðC4Þ
DDA;A ;
2 W
where again DDA;A measures the effect of the modifier on the departure from Hardy–Weinberg following a single
round of mating, given by
2112
S. P. Otto, M. R. Servedio and S. L. Nuismer
DDA;A ¼
ns
Dr2 ð2 4ð1=4 D̂ns
A;A Þr1;mm Þ 1 4Dr1 ð1=4 D̂A;A Þr2;mm D̂A;A
:
ns
r2;mm
2 4ð1=4 D̂ns
A;A Þr1;Mm 2ð1=4 1 D̂A;A Þr2;Mm
ðC5Þ
Thus, a modifier allele that increases assortative mating (causing DDA;A to be positive) will invade if and only if
homozygotes have a higher fitness, on average, at the polymorphic equilibrium, accounting for both natural and
sexual selection (Htot . 0). We determined when Htot would remain at zero by solving for equilibria that also satisfy
(C3); the result is illustrated in Figure 5. According to (C4), no further change in assortative mating is expected along
these curves.
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