Download How do natural and sexual selection contribute to sympatric

doi:10.1111/j.1420-9101.2004.00776.x
How do natural and sexual selection contribute to sympatric
speciation?
S. GOURBIERE
Galton Laboratory, University College London, London, UK and Laboratoire de Théorie des Systèmes, Université de Perpignan, Cedex, France
Keywords:
Abstract
competitive speciation;
genetic architecture;
hypergeometric model;
individual based model;
natural and sexual selection;
pleiotropy;
reinforcement;
sympatric speciation.
I use explicit genetic models to investigate the importance of natural and
sexual selection during sympatric speciation and to sort out how genetic
architecture influences these processes. Assortative mating alone can lead to
speciation, but rare phenotypes’ disadvantage in finding mates and intermediate phenotypes’ advantage due to stabilizing selection strongly impede
speciation. Any increase in the number of loci also decreases the likelihood of
speciation. Sympatric speciation is then harder to achieve than previously
demonstrated by many theoretical studies which assume no mating disadvantage for rare phenotypes and consider a small number of loci. However,
when a high level of assortative mating evolves, sexual selection might allow
populations to split into dimorphic distributions with peaks corresponding to
nearly extreme phenotypes. Competition then works against speciation by
favouring intermediate phenotypes and preventing further divergence. The
interplay between natural and sexual selection during speciation is then more
complex than previously explained.
Introduction
Sympatric speciation has been a subject of debate for a
long time, motivating many theoretical studies that
specify the conditions necessary for the evolution of
reproductive isolation between incipient species. Longstanding efforts have been made to elucidate the interplay between ecological and genetic factors involved in
disruptive selection, mating processes and potential
mechanisms for post-zygotic isolation (see Special Issue
on Speciation in Trends Ecol. Evol., 2001 and Target
Reviews on Wu’s genetic view of the process of
speciation in J. Evol. Biol., 2001). A few years ago, after
a long period of disfavour, the idea that speciation can
proceed in sympatry became very popular, mostly
because numerous theoretical papers appeared claiming
that natural selection (e.g. Doebeli, 1996; Dieckmann &
Doebeli, 1999; Doebeli & Dieckmann, 2000, 2003;
Drossel & McKane, 2000), sexual selection (Kondrashov
& Shpak, 1998; Higashi et al., 1999; Kondrashov &
Correspondence: Sébastien Gourbiere, Laboratoire de Théorie des Systèmes,
Université de Perpignan, 52 Avenue Paul Alduy, 66860 Perpignan Cedex,
France
Tel.: +33(0)4 68 66 1763; fax: +33(0)4 68 66 1760;
e-mail: [email protected]
Kondrashov, 1999) and sexual conflict (Gavrilets, 2000;
Gavrilets & Waxman, 2002) can all lead to the required
disruptive selection.
Recent developments in adaptive dynamics and quantitative genetics theory allow for the simultaneous
description of ecological interactions based on quantitative traits and of the genetics underlying the traits
involved in these ecological interactions (see Dieckmann
& Doebeli, 1999; Doebeli & Dieckmann, 2000, 2003 for
the development of Adaptive Dynamics, and Doebeli,
1996; Kondrashov & Shpak, 1998 and Drossel &
McKane, 2000 for the development of Quantitative
Genetics). These studies use earlier ecological models of
selection devoted to the study of ecological character
displacement (e.g. Bulmer, 1974, Roughgarden, 1976,
Slatkin, 1980, Taper & Case, 1985) and add a description
of the genetics of reproductive isolation. They lead to
compelling theoretical studies of speciation triggered by
different types of ecological interactions. The majority of
these studies demonstrated that sympatric speciation
can easily occur because of competitive interactions
(Doebeli, 1996; Dieckmann & Doebeli, 1999; Doebeli &
Dieckmann, 2000, 2003; Drossel & McKane, 2000), but
predation and mutualism could also lead to such
ecological speciation processes (Doebeli & Dieckmann,
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2000). Indeed, frequency-dependent selection due to all
these different kinds of ecological interactions provides
the disruptive selection which itself leads to the evolution
of reproductive isolation to avoid production of unfit
hybrids. These studies then strongly support the idea that
sympatric speciation is likely to occur because of
disruptive natural selection and subsequent evolution
of assortative mating.
Surprisingly, little attention has been given to the
relative importance of natural selection and sexual
selection during the speciation process. Obviously, these
models are not meant to be models of sexual selection as
they do not include loci for a male specific trait and loci
underlying female preference for that trait (as has been
done with similar genetic frameworks by Higashi et al.,
1999, and Kondrashov & Kondrashov, 1999), but they do
implicitly account for sexual selection. Indeed, they
consider evolution of quantitative traits in the female
part of the population and males are assumed to have
exactly the same phenotypic and genotypic distribution
for these traits. ‘Female preference’ and ‘male trait’
involved in this choice are thus determined by the same
set of loci. As pointed out by Kirkpatrick & Ravigné
(2002), such assortative mating in and of itself is
equivalent to sexual selection. In this paper, sexual
selection is thus considered in a broad sense to include
more or less any kind of assortative mating. Thus, given
that both natural and sexual selection are included in
these models, we can actually determine how each of
them contributes to speciation.
Drossel & McKane (2000) made the only attempt to
investigate the relative importance of sexual selection
and natural selection in competitive speciation. They
worked out the impact of assortative mating alone on the
phenotypic distribution, before including competitive
interactions. They demonstrated that neither selective
assortative mating (hereafter ‘SAM’), i.e. when rare
phenotypes are less likely to mate, nor nonselective
assortative mating (hereafter ‘NSAM’), i.e. when all
individuals are equally likely to mate, can lead to
speciation. Accordingly, they conclude that formation
of new species, after natural selection was subsequently
included into the model, was because of natural selection. However, using another kind of genetic model,
Kondrashov & Shpak (1998) clearly showed the possibility of speciation by means of NSAM and they argued
that increasing the number of loci makes speciation
easier. These two studies produced conflicting results:
why did Drossel & McKane never obtain assortative
mating speciation, even though they are implicitly
dealing with an infinite number of loci? Drossel &
McKane (2000) have proposed that such discrepancies
may be due to the lack of genetic flexibility in
Kondrashov & Shpak’s model. This could either be due
to the artificial constraint used by Drossel & McKane
(2000) by assuming that the genetic variance of the
offspring distribution does not vary through time and
with respect to the parental genotypes (a common
assumption of the infinitesimal model, Bulmer, 1980)
or it could be due to the assumption of equal allele
frequencies across the loci that Kondrashov & Shpak
(1998) made in order to use the hypergeometric framework. So, in the absence of any clear explanation for
these conflicting results, there is no coherent picture of
how assortative mating can split a population into a
bimodal distribution and (with the notable exception of
Drossel & McKane, but using a nonexplicit genetic
background) no attempt has been made to work out
the influence of the mating process itself on competitive
speciation models.
These results leave us with the question: how do
assortative mating and competition contribute to speciation when the quantitative trait involved in both
natural and sexual selection is determined by a finite
number of loci? Clearly, to sort out the relative importance of assortative mating and natural selection in
explicit genetic models of competitive speciation, where
the number of loci and the allelic effect can vary, would
also address the related question: does the genetic
architecture change the relative importance of natural
and sexual selection in competitive speciation?
Here, I investigate these questions using a hypergeometric model and an individual based model (IBM) of
competitive speciation. First, I look at these models to
sort out the effect of different levels of NSAM and SAM,
before I include natural selection. Interestingly, as the
number of loci (and the allelic effect) can vary,
comparisons can be made between results obtained using
explicit genetic models (Doebeli, 1996; Kondrashov &
Shpak, 1998, and this study) and Drossel and McKane’s
quantitative genetic model. Secondly, I check the sensitivity of the results obtained with the hypergeometric
framework to the assumption of equal allele frequencies
at all the loci using the IBM. The questions are then
ultimately addressed within a framework relaxing
assumptions made to set up the analytical frameworks.
In the first part of this paper I consider no competition
between individuals having different phenotypes. I show
that assortative mating can drive speciation when considering a finite number of loci. However, as expected,
rare (extreme) phenotypes’ disadvantage in finding
mates and intermediate phenotypes’ advantage because
of stabilizing selection strongly impede speciation requiring higher levels of assortativeness and particular initial
conditions to proceed. Interestingly, increasing the number of loci also decreases the likelihood of speciation. This
probably explains why Drossel and McKane did not
obtain speciation by means of assortative mating alone,
since they implicitly assumed an infinite number of loci.
In the second part of this study I include competition
between individuals having different phenotypes.
As suggested by earlier investigations, this generates
frequency dependent competition which provides disruptive selection that contributes to speciation. However,
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Sympatric speciation
the relationship between the strength of competition (i.e.
the range of neighbouring phenotypes an individual
competes with) and the disruptiveness of selection is
highly nonlinear. This is mostly because both local and
global competition (that is, competition between individuals having only very similar phenotypes and competition between all the individuals) lead to weak
frequency dependence. It means that the usual criterion
for speciation (which is that competition occurs only
between individuals whose phenotypic differences are
smaller than an upper limit) is not valid. There must also
be a lower limit for the range of phenotypes an individual
competes with, since competition that is not localised
does not allow for speciation, because the phenotypic
distribution then simply fits the carrying capacity.
From these two sets of results I conclude that sympatric
speciation is harder to achieve than is claimed in recent
papers. This is because they usually (1) do not take into
account a mating disadvantage of rare phenotypes and
(2) consider few loci (to save simulation time). I also
suggest that the interplay between natural and sexual
selection during speciation is slightly more complex than
previously assumed. In the early stage, as is usually
thought, natural selection must allow for the evolution of
assortative mating. But as soon as sufficient assortment
has evolved, the population is expected to split into two
monomorphic sets of individuals with extreme phenotypes. Such a split obviously provides a selective advantage to intermediate phenotypes so that, in a second stage,
natural selection actually acts against speciation, by
preventing further divergence. I also argue that natural
selection and sexual selection eventually lead to a nonlinear evolution of reproductive isolation, a dynamical
behaviour which has been briefly reported by Doebeli
(1996).
Method
The hypergeometric model for competitive speciation I
investigated is closely related to Doebeli’s model (1996).
This framework has already been described elsewhere
(Barton, 1992; Doebeli, 1996; Shpak & Kondrashov,
1999) and its use in modelling competitive speciation has
been very clearly presented by Doebeli (1996).
The model deals with the simplest scenario of competitive speciation where mate choice is based on a quantitative character which also determines the competitive
ability of individuals. Organisms are haploid, have sexual
reproduction and generations are discrete and nonoverlapping. The quantitative trait is assumed to be defined
by a set of diallelic loci (with allele 0 and 1). Loci have
equivalent and additive phenotypic effects and recombine freely. I will refer to nl and a as the number of loci
and the allelic effect, respectively. The phenotypes then
lie between 0 and nla.
The mating process is described by the set of mating
probabilities defined as a function of the difference
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between individual phenotypes. The function used is a
normal distribution with a mean of 0 and a variance
denoted r2m . The probability for an individual of phenotype i to mate with an individual of phenotype j is then
proportional to (eqn 12 in Doebeli, 1996):
mði; jÞ ¼ expðði jÞ2 =2r2m Þ:
ð1Þ
Clearly, the lower rm, the stronger the degree of
assortative mating. More precisely, the actual mating
probabilities are derived from eqn 1 in two different ways
to consider either selective or nonselective mating. To
obtain the nonselective mating probabilities, the m(i,j)
are first multiplied by the genotype frequencies. For any
i, they are then divided by their sum over j to ensure that
(for any i) the sum of mating probabilities equals 1.
Pairing then occurs according to these set of normalized
probabilities so that no genotype gets less mating. To
model selective mating because of rare phenotype disadvantage, the m(i,j) are first multiplied by the genotype
frequencies and then divided by their sum over i and j.
This leads to weaker mating probabilities for rare extreme
phenotypes.
The density-dependent fitness function used by Doebeli corresponds to the model of Bellows (1981) describing the population dynamics of species as a function of
three parameters; the intrinsic growth rate of the population, a competition parameter and a parameter influencing the carrying capacity. Unfortunately, the carrying
capacity is then defined as a function of these three
parameters (see eqn 15 in Doebeli, 1996). This made the
sensitivity analysis with respect to the three parameters
impractical. Therefore I used the Lotka and Volterra
equation (e.g. Kot, 2001, p. 51):
Ntþ1 ¼ rNt ð1 ðNt =KÞÞ:
ð2Þ
This function takes only two independent parameters,
r and K, which I refer to as the intrinsic growth rate of the
population and the carrying capacity, respectively. It is a
discrete-time counterpart of the continuous-time Lotka–
Volterra model which has been used in other investigations of competitive speciation (Dieckmann & Doebeli,
1999; Doebeli & Dieckmann, 2000, 2003).
For the purpose of this study, individuals have to
compete with respect to the quantitative character which
also determines their mating probabilities. First, the
carrying capacities of different phenotypes are given by
a normal distribution with a mean corresponding to the
middle of the phenotypic range (i.e. nla/2) and a variance
denoted r2K . The carrying capacity of an individual with
phenotype i is then:
KðiÞ ¼ K expðði nl a=2Þ2 =2r2K Þ:
ð3Þ
Clearly, the lower rK the lower are the carrying
capacities of extreme phenotypes relative to those of
intermediate phenotypes. Secondly, the competition
between individuals is assumed to decrease with the
difference in their phenotypic values. As for the mating
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probabilities, the competition coefficients follow a normal
distribution with a standard deviation denoted as ra. Thus,
the competition coefficient between an individual having
phenotype i and an individual having phenotype j is:
aði; jÞ ¼ expðði jÞ2 =2r2a Þ:
ð4Þ
Obviously, the lower ra, the narrower is the range of
phenotypes an individual with a given phenotype competes with. If ra tends toward 0, only individuals with the
same phenotype compete with one another. If ra tends
towards infinity, all individuals compete with one
another whatever their phenotypes.
Accordingly, the fitness of an individual with phenotype i is:
xi;t ¼ rð1 ðNi;t =KðiÞÞÞ;
ð5Þ
where Ni,t denotes the effective density that an individual
of phenotype i competes with. This density is given by:
X
Ni;t ¼
aði; jÞNj;t ;
ð6Þ
j
Nj,t being the density of individuals with phenotype j at
time t. Equation 6 can also be set as:
X
aði; jÞpj;t ;
ð7Þ
Ni;t ¼ Nt
j
where Nt denotes the population size and pj,t the
frequency of individual having phenotype j before
viability selection.
The above equations are combined to allow the model
to describe the evolution of the phenotypic distribution
during the viability selection stage of the life cycle. The
frequencies are then changed according to:
qi;t ¼ pi;t xi;t =Wt ;
ð8Þ
where qi,t denotes the frequency of individuals with
phenotype i after selection, and where Wt is the mean
fitness of the population.
Equations determining the changes in the phenotypic
distribution during the reproductive stage are more
complex because they describe both the mating process
and the phenotypic distribution of offspring for any pair
of parental phenotypes. The final distribution is given by
(Doebeli, 1996, eqn 13):
pk;tþ1 ¼
X
i
di
X
mði; jÞqi;t qj;t pij ðkÞ;
ð9Þ
j
where pij(k) describes the offspring phenotype distribution from parents with phenotypes i and j and di are
normalizing constants obtained when modelling selective
or NSAM as explained above. The values of pij(k) are
defined according to some binomial and hypergeometric
probabilities described by eqns 8–10 in Doebeli (1996).
The dynamics of the hypergeometric model including
both natural and sexual selection is then given by
iterating eqns 8, 9 together with the equation for the
density of the population:
Ntþ1 ¼ Wt Nt :
ð10Þ
As in any hypergeometric model, the competitive
speciation model described above (and more specifically,
eqns 8–10 in Doebeli, 1996) is built on assumptions that
allele frequencies are equal across the loci and that all
genotypes in the same phenotypic class are equally
frequent. The conditions under which this assumption
would hold have been investigated by Shpak & Kondrashov (1999) and Barton & Shpak (2000). Such symmetrical models are likely to be useful under disruptive
natural selection, as allele frequencies then stay or tend
to become equal across the loci. However, this assumption becomes more problematic under stabilizing selection. Indeed, allele frequencies tend to become
asymmetric because natural selection leads to the
fixation of optimal haplotype with either allele fixed at
different loci. Furthermore, it is not clear whether allele
frequencies tend to be symmetric when assortative
mating alone is considered, although this is strongly
suggested by simulations run by Kondrashov & Kondrashov (1999). Thus, I set up an IBM encapsulating all the
assumptions of the hypergeometric model described
above, but relaxing the assumption of equal allele
frequencies across the loci. This implies that offsprings’
genotypes are no longer defined according to the distribution given by eqns 8–10 in Doebeli (1996). Instead,
any offspring phenotype is determined by random choice
between the two parental alleles at each locus as
expected under Mendelian inheritance and free recombination.
All the simulations run with the IBM start with
asymmetric initial allele frequencies. For any individual,
the genotype is determined according to a nonuniform
pattern of mean allele frequencies across the loci. More
specifically, every 1-allele is randomly assigned to one of
two equally large sets of loci, one set with a probability of
1/3 and the other with a probability of 2/3. Within a set,
all loci are equivalent and the exact position of each
1-allele random.
I investigated the hypergeometric model and the IBM
by first considering assortative mating alone (i.e. considering only sexual selection and neglecting natural selection), then adding natural selection due to the carrying
capacity distribution and finally including frequencydependent competition due to interactions between
individuals having different phenotypes. The first step is
to look at the possibility of speciation by the mean of
NSAM and SAM. I then assume that there is no viability
selection, i.e. that qi,t ¼ pi,t. The second step is to look at
the possibility that stabilizing selection generated by
the carrying capacities distribution prevents speciation
by assortative mating. I then let ra tend toward 0, so
that any individual competes only with individuals
having exactly the same phenotype. Indeed, under these
conditions, eqn 6 becomes Ni,t ¼ Ni,t. The last step was to
check how frequency-dependent competition contribu-
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ted to speciation. The assumption that ra tends toward 0
is then relaxed to allow for nonzero ra values.
In this study, speciation is considered to occur when
the phenotypic distribution becomes bimodal and when
the mating probability between individuals belonging to
each of the two modes falls off below a threshold value
fixed to 10)5.
Below, I first report the results obtained using the
hypergeometric model. I then check whether the conclusions are still valid when the assumption of equal
allele frequencies across the loci is relaxed using the IBM.
Thus, where unspecified, the results described will be
those corresponding to the hypergeometric model.
Results
Can NSAM alone lead to speciation?
Using a quantitative genetic model, Drossel & McKane
(2000) showed that NSAM increases the phenotypic
variance of the population, although it does not allow for
speciation. On the contrary, using a hypergeometric
model, Kondrashov & Shpak (1998) showed that NSAM
may split the population into two subpopulations.
However, the (threshold- and interval-based) mating
functions used by Kondrashov & Shpak (1998) were
different from the one used in most competitive speciation models including Doebeli (1996) and Drossel &
McKane (2000) (but also Dieckmann & Doebeli, 1999;
Drossel & McKane, 2000; Day, 2000; Doebeli &
Dieckmann, 2000, 2003; Mizera & Meszena, 2003). I
therefore checked whether or not NSAM could lead to
speciation by using a hypergeometric framework with
the classical mating function. This also allows comparing
the results obtained with explicit and nonexplicit genetic
frameworks using the same mating function.
I first considered the genetic architecture used by
Doebeli, i.e. a set of nl ¼ 20 loci with the same allelic
effect a ¼ 1. As shown in Fig. 1a, the phenotypic
distribution always evolves to an equilibrium within
200 generations. There are only two types of equilibrium
distributions. As long as individuals mate randomly or
with a weak assortativeness (i.e. rm > 2.4), the asymptotic distribution is an approximately Gaussian distribution (lower panel in Fig. 1b) with a nearly constant
phenotypic variance (slightly >5 in Fig. 1b). When
assortativeness is strong enough (i.e. rm £ 2.4), the
phenotypic distribution becomes bimodal (upper panel
in Fig. 1b). The two modes correspond to the extreme
phenotypes, whose frequencies increase to 0.5. The
reproductive isolation between modes quickly increases
to 1 and speciation is completed. Both species are then
fully monomorphic. Allele 0 is fixed at all loci in the first
species, whereas allele 1 is fixed at all loci in the other
species. The phenotypic variance reaches its maximal
value; (anl/2)2. These findings do not rely on the use of a
hypergeometric model as a very similar nonlinear rela-
(σm)
Fig. 1 Nonselective assortative mating speciation. (a) Temporal
evolution of the phenotypic variances obtained with the hypergeometric model for two different levels of assortativeness, rm. The
asymptotic distribution is reached within 200 generations. The bold
line is for rm ¼ 2.2, whereas the thin line is for rm ¼ 2.6.
(b) Phenotypic variance as a function of rm. The close symbols
correspond to the results obtained using the hypergeometric model,
whereas the open symbols stand for the results of the individual
based model. Square and circles represent results obtained for
population sizes equal to 2000 and 10000 individuals, respectively.
The genetic architecture is as in Doebeli (1996): nl ¼ 20 and a ¼ 1.
In both cases, as long as assortative mating is weak, i.e. rm is high,
the phenotypic distribution remains a gaussian distribution (lower
panel) with a nearly constant phenotypic variance. When assortativeness becomes strong enough, the phenotypic distribution suddenly becomes bimodal with only two extreme phenotypes (upper
panel).
tionship emerges when using the IBM (Fig. 1b). However, there are two interesting differences between
results obtained with these two frameworks. First, when
speciation occurs (for lower values of rm), assortative
mating hardly leads to fixation when using the IBM,
although it does using a hypergeometric framework.
Secondly, when speciation does not occur, loci become
fixed for either loci allele running the IBM whereas (by
assumption) all the loci stay polymorphic in the hypergeometric model. None of these differences alter the
main result about NSAM speciation, as the threshold
value for speciation to proceed is always around 2.4.
Such splits because of NSAM confirm and expand the
previous results of Kondrashov & Shpak (1998). Speciation by means of NSAM is a basic feature of genetically
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explicit models, depending neither on the type of mating
function used nor on the key assumption of equal allele
frequencies across the loci made to set up the hypergeometric framework. These results all together apparently
contrast with those obtained by Drossel & McKane
(2000) who used the same mating function, but a
nonexplicit genetic framework. This suggests that the
possibility of NSAM speciation depends on the genetic
architecture considered.
How does genetic architecture influence NSAM
speciation?
To look at the importance of the genetic architecture for
NSAM speciation, I examine the sensitivity of the
threshold value to the number of loci nl and the allelic
effect a.
Increasing nl or a both lead to a wider range of degree
of assortative mating allowing speciation by pure assortment (Fig. 2a). When speciation occurs, the phenotypic
distribution always splits into a purely dimorphic population with only the two extreme phenotypes. The range
of degree of assortative mating allowing speciation
increases almost linearly with both nl and a. An increase
in a favours speciation much more than an increase in nl.
For instance, starting with nl ¼ 5 and a ¼ 1, a ten-fold
increase in the a or in nl increases the threshold value by
more than 10- and four-fold, respectively.
These results are in good agreement with those of
Kondrashov & Shpak (1998), although the threshold
value does not seem to increase with the square root of
nl, and contrast with the result obtained by Drossel &
McKane (2000). Indeed, I show that an increase in nl (or
a) increases the likelihood of NSAM speciation, whereas
(implicitly) considering a very large number of loci
Drossel & McKane (2000) reached the conclusion that
such NSAM cannot lead to speciation. This contrast
actually disappears when considering an increase in nl
although scaling the allelic effect with 1/n. Indeed, such a
scaling has to be done to keep the phenotypic range as a
constant. Otherwise, for a given value of rm, an increase
in nl or a (which leads to an increase of the total
phenotypic range; anl) would artificially accentuate the
actual level of assortativeness (by decreasing the ratio
between the range of potential mating partners and the
total phenotypic range). An increase in nl (and the
corresponding decrease in a) then does not make speciation by the mean of NSAM easier (as suggested by
Kondrashov & Shpak and by the results of Fig. 2a), but
on the contrary more difficult (Fig. 2b). Again, the results
obtained using the IBM are consistent with those
obtained with the hypergeometric framework, although
speciation then requires stronger degree of assortative
mating (Fig. 2b). This unifies the results obtained from
explicit genetic frameworks (Kondrashov & Shpak, 1998,
and this study) and quantitative genetic models (Drossel
& McKane, 2000). That the relationships displayed in
Fig. 2 Sensitivity of speciation by nonselective assortative mating to
the genetic architecture. (a) Threshold value (of the relationship
between the phenotypic variance and the assortativeness, as
exemplified in Fig. 1b) as a function of the number of loci nl and the
allelic effect a. The results displayed have been obtained using the
hypergeometric model. Diamonds, circles, triangles and squares are
for a ¼ 1, a ¼ 2, a ¼ 5 and a ¼ 10, respectively. The number of loci
corresponds to the x-axis and it varies from 5 to 50. An increase in
the number of loci or an increase in the allelic both lead to a larger
range of assortativeness allowing speciation to occur. The threshold
value increases linearly with both the number of loci and the allelic
effect, but an increase in the allelic effect favours speciation best. (b)
Change in the level of assortativeness required for speciation, when
the number of loci is increased while keeping the phenotypic range
as a constant (here ¼ 50). The closed diamonds correspond to the
results obtained using the hypergeometric model, whereas the open
circles stand for the results of the individual based model. In both
cases an increase in the number of loci decreases the likelihood of
speciation by the mean of nonselective assortative mating.
Fig. 2b have a nonzero asymptote (Fig. 2b) is also
consistent with the analytical results produced by Drossel
& McKane (2000) to describe how the broadening effect
of NSAM increases when sigma rm is decreased. Indeed,
they established that the phenotypic variance increases
with time but reaches an asymptotic value as long as rm
is larger than a (nonzero) threshold value. Otherwise the
variance of phenotypic distribution increases in an
unlimited way, although this never allows for speciation
(i.e. for the evolution of a bimodal distribution). However, it is clear that, by using an explicit genetic
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framework, speciation is allowed because such an
increase in the phenotypic variance leads to the fixation
of either allele. That speciation is always allowed when nl
is increased in my explicit genetic model (potentially
mimicking the framework used by Drossel & McKane,
2000), is then also consistent with their theory.
Thus, when the quantitative trait involved in mate
choice is determined by a finite number of loci, NSAM
can lead to speciation. However, mating can also select
against individuals having rare phenotypes are less likely
to find a suitable mate. We know from Drossel & McKane
(2000) that when mating is selective, not only does
speciation not proceed, but the phenotypic variance of
the population no longer depends on degree of assortative mating. Unfortunately, as Kondrashov & Shpak
(1998) only considered NSAM, we still do not know if
the conclusion of Drossel & McKane (2000) holds for a
genetic system involving a finite number of loci.
Does selection against rare phenotypes prevent
assortative mating speciation?
To look at the importance of rare phenotypes’ disadvantage, I checked the threshold value in the same genetic
conditions as in Fig. 2a, b. Figure 3a, b are then strictly
analogous to Fig. 2, but for SAM.
As expected, rare phenotypes’ disadvantage often
prevents assortative mating speciation to occur. Indeed,
the threshold value is very much lower than when
NSAM is considered. The threshold value still increases
when the phenotypic range is widened by an increase in
a, but surprisingly it no longer depends on nl. To estimate
the actual mating probabilities under SAM, the mating
probabilities (given by eqn 1) are normalized to ensure
that their sum over all i and j phenotypes add up to one.
But, this sum does not significantly increase when nl
increases, because it reaches its maximum as soon as
nl ¼ 5. By contrast, to estimate the actual mating
probabilities under NSAM, mating probabilities are normalized to ensure that their sum over j phenotypes adds
up to one. The sum over j does significantly increase
when nl is increased from 5 to 50. The actual mating
probabilities then depend on nl. This explains why the
threshold value of degree of assortative mating increases
when the phenotypic range is widened by an increase in
nl under NSAM (Fig. 2a), whereas it does not under SAM
(Fig. 3a). However, scaling the allelic effect to keep the
phenotypic range as a constant, I found that (as for
NSAM) to consider more loci with fewer effect decreases
the likelihood of speciation to proceed. Here, it is
important to note that, using the IBM, speciation still
arises for some level of degree of assortative mating, but
only if the initial genetic variance is large enough. The
threshold value of degree of assortative mating and the
minimal value of the initial genetic variance allowing for
speciation are reported on Fig. 3b. As for NSAM, a
comparison with the results of Drossel & McKane (2000)
Fig. 3 Selective assortative mating (SAM) speciation and genetic
architecture. (a) Threshold value (of the relationship between the
phenotypic variance and the assortativeness, as exemplified in
Fig. 1b) as a function of the number of loci nl and the allelic effect a.
These results were obtained using the hypergeometric model.
Legends and ranges of variation of nl and a are the same as in Fig. 2a.
An increase in the number of loci does not lead to significant
changes in the level of assortativeness required for speciation,
although an increase in the allelic effect still favours speciation. (b)
Change in the level of assortativeness required for speciation when
the number of loci is increased while keeping the phenotypic range
as a constant (here ¼ 50). Legends and ranges of variation of nl and a
are the same as in Fig. 2b. As for nonselective mating, when the
allelic effect is decreased and the number of loci increased to keep
the phenotypic range as a constant (here ¼ 50), the likelihood of
speciation by the mean of SAM decreases.
can be made about the asymptote of the relationship
displayed in Fig. 3b. According to Drossel & McKane
(2000), when nl is very large, the phenotypic variance is
expected to take on finite values not depending on rm.
The phenotypic variance is then expected not to increase
infinitely. That speciation appears to happen in more and
more restrictive conditions with apparently no possibility
of split when nl is very large (i.e. a zero asymptote) is thus
consistent with the Drossel & McKane’s theory.
To sum up, both NSAM and SAM lead to speciation
when the quantitative trait involved in the mating choice
is determined by a finite number of loci. For a fixed
phenotypic range, the likelihood of such a speciation
decreases with nl, but never reaches zero if mating is non
selective. As stated by Kirkpatrick & Ravigné (2002), if
mating is selective, speciation also depends on initial
J. EVOL. BIOL. 17 (2004) 1297–1309 ª 2004 BLACKWELL PUBLISHING LTD
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S. GOURBIERE
conditions: the initial genetic variance needs to be large.
So far, I have not considered that the quantitative trait is
under natural selection. To generalize these previous
results, I assessed whether stabilizing selection (because
of the differences in the carrying capacities of phenotypes) can prevent NSAM or SAM speciation from
happening.
Does stabilizing selection prevent NSAM or SAM
speciation?
The strength of stabilizing selection is determined by the
parameter rk in my model. Low rk values lead to sharp
distributions of the phenotypic carrying capacity corresponding to strong stabilizing selection. On the contrary,
high rk values lead to very weak stabilizing selection.
Both NSAM and SAM can still produce speciation
when stabilizing selection is included in the hypergeometric model (see Fig. 4a for NSAM and the upper series
of Fig. 4b for SAM). As expected, the threshold values
obtained are lower than in the absence of selection
because stabilizing selection is opposite to speciation in
that it favours intermediate phenotypes. The way the
threshold value varies with the level of stabilizing
selection is different under NSAM and SAM. Interestingly, including stabilizing selection does not prevent
speciation by the mean of SAM as long as rk is larger than
10 and there is almost no change in the threshold values
obtained with different rk values. This explains why
there is only one series corresponding to the hypergeometric model in Fig. 4b. However, any further decrease
in rk value strongly impedes evolutionary diversification
so that there is virtually no more speciation whatever the
genetic architecture considered (corresponding threshold
values are not displayed as they are always lower than
0.05). On the contrary, when mating is nonselective,
strong stabilizing selection gradually lowers the range of
degree of assortative mating, allowing speciation
(Fig. 4a). However, for both NSAM and SAM, stabilizing
selection does not change the shape of the relationship
between the threshold value and nl: the likelihood of
speciation by NSAM or SAM still decreases exponentially
with nl.
The figure can be very different when using the IBM. It
is well-known that stabilizing selection leads to the
fixation of 0 and 1 alleles at different loci as it allows
producing only the optimum phenotype. Accordingly,
there is a conflict between the effects of assortative
mating and stabilizing selection. Stabilizing selection
tends to produce a population of individuals all having
the same genome consisting of an optimal mixture of 0
and 1 alleles; assortative mating tends to produce two sets
of individuals whose genomes consist only of either 0 or
1 alleles.
When mating is nonselective, stabilizing selection
always prevents speciation. The population starts splitting because of assortative mating, but after a few
Fig. 4 Stabilizing selection and assortative mating speciation. (a)
Threshold value (of the relationship between the phenotypic
variance and the assortativeness, as exemplified in Fig. 1b) as a
function of the strength of stabilizing selection and the genetic
architecture, when non-selective assortative mating (NSAM) is
considered. All the displayed values have been obtained using an
hypergeometric model since NSAM speciation never happens in the
individual based model (IBM). Black squares, open squares, black
circles and open circles are for rk ¼ 10, rk ¼ 15, rk ¼ 20 and rk ¼
50, respectively. Stars indicate the threshold values obtained in
absence of stabilizing selection (i.e. when rk goes to infinity). The
relationship represented by stars then strictly corresponds to the
result of Fig. 2b. The number of loci corresponds to the x-axis and it
varies from 5 to 50. As the number of loci increases, the allelic effect
is decreased to keep the phenotypic range ¼ 50. The demographic
parameters are r ¼ 1.1 and K ¼ 100. As expected, stabilizing
selection makes speciation more difficult. But there is still a range of
assortativeness allowing for speciation. Importantly, these conditions
for speciation to occur depend upon the use of the hypergeometric
model as no non-selective mating speciation was observed using the
IBM (results not shown). (b) Displays the same relationship but
considering SAM. Using the hypergeometric model, speciation
happens as long as rk is higher than 15. For lower levels of rk, there
is almost no variation of the threshold value of assortativeness
obtained for each genetic architecture considered. Accordingly, the
only relationship drawn is for rk ¼ 15 and corresponds to the upper
curve. Others curves correspond to the results obtained using the
IBM. Squares, circles and triangles are for rk ¼ 10, rk ¼ 20 and
rk ¼ 30, respectively. Stars indicate the threshold values obtained in
absence of stabilizing selection (i.e. when rk goes to infinity). The
relationship represented by stars then strictly corresponds to the
result of Fig. 3b. Interestingly, speciation also happens using the
IBM, although this requires both high level of assortativeness and
large enough initial genetic variance. The required levels of genetic
variance are indicated in the vicinity of each symbol.
J. EVOL. BIOL. 17 (2004) 1297–1309 ª 2004 BLACKWELL PUBLISHING LTD
Sympatric speciation
generations individuals with intermediate phenotype
increase in frequency because of natural stabilizing
selection. Most of the loci get fixed, although an
equilibrium between the two processes allows maintaining some level of polymorphism (data not shown).
Accordingly, speciation by the mean of NSAM when
stabilizing selection is included mostly relies on the
artificial assumption of equal allele frequencies across the
loci imposed by the hypergeometric framework. Such an
assumption does not allow for fixation of 0 and 1 alleles
at different loci (as expected under stabilizing selection).
Lowering the effect of stabilizing selection, hypergeometric models then clearly favour evolutionary diversification.
When mating is selective, speciation can still proceed
although it requires specific conditions (Fig. 4b). It
obviously requires stabilizing selection not to be strong
and a high level of assortment. An additional and
important requirement is a large enough initial genetic
variance (see required values displayed in Fig. 4b).
Indeed, a large genetic variance means that individuals
with extreme phenotypes are initially present, although
they are still less abundant than individuals with intermediate phenotypes. Because of assortative mating, those
individuals with extreme phenotypes can increase in
frequency while intermediate phenotypes eventually
start becoming the less abundant ones. As mating is
selective, intermediate phenotypes then experience difficulty in finding mates and the corresponding selective
disadvantage can overcome the advantage those individual phenotypes have because of stabilizing selection.
Clearly, the required level of degree of assortative mating
for speciation is higher than in the absence of natural
selection so that assortative mating is no longer able to
generate speciation when nl is too high. Such a speciation
never happens when mating is nonselective, because
intermediate phenotypes do not suffer disadvantages in
finding mates. Stabilizing selection then prevents the
split of the population. If the initial genetic variance is
weak, SAM no longer allows for speciation. On the
contrary, SAM backs the effect of stabilizing selection as
extreme phenotypes, being rare, also experience difficulty in finding mating partners. Alleles 0 and 1 then
become fixed and the population is only made up of the
optimal phenotype.
These results add to those of Kondrashov & Shpak
(1998), Kirkpatrick & Ravigné (2002) and the previous
results of this study demonstrating speciation by means
of assortative mating in absence of stabilizing selection.
Here, I show that SAM can still lead to speciation when
stabilizing selection acts on the mating trait, although it
requires the initial genetic variance to be large. Interestingly, selective mating can also oppose speciation if the
genetic variance is initially weak.
Now, as long as only stabilizing selection is included,
competition occurs between individual with the same
phenotype. To understand the interplay between assor-
1305
tative mating and frequency dependent competition, I
included interactions between different phenotypes.
Does frequency-dependent competition act for or
against NSAM speciation?
I investigated the threshold value of nonselective degree
of assortative mating allowing for different strengths of
competition between phenotypes, i.e. different values
of ra.
Similar relationships emerge while using the hypergeometric model (Fig. 5a) and the IBM (Fig. 5b), although
quantitative differences exist between them. The
common figure is a nonlinear relationship with a ra
Fig. 5 Importance of frequency-dependent competition on NSAM
speciation: threshold value (of the relationship between the
phenotypic variance and the assortativeness, as exemplified in
Fig. 1b) as a function of the strength of competition between
individuals having different phenotypes (ra). (a) The results
obtained using the hypergeometric model and (b) the individual
based model. Relationships in Fig. 5a correspond to different
strengths of stabilizing selection (rK): circles, diamonds and squares
are for rk ¼ 20, rk ¼ 25 and rk ¼ 30, respectively. The demographic
parameters are r ¼ 1.1 and K ¼ 100. The genetic architecture is as in
Doebeli (1996): nl ¼ 20 and a ¼ 1. In Fig. 5b relationships denoted
by squares, circles and triangles are for rk ¼ 10, rk ¼ 20 and rk ¼
30, respectively. Demographic parameters are the same as in (a), but
the genetic architecture corresponds to nl ¼ 10 and a ¼ 1.
J. EVOL. BIOL. 17 (2004) 1297–1309 ª 2004 BLACKWELL PUBLISHING LTD
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S. GOURBIERE
value allowing for speciation in the largest set of
conditions.
This finding partially contrasts with a canonical result
of the recent theory on competitive speciation: evolutionary diversification happens if the width of the
competition coefficient distribution is lower than the
width of the carrying capacity distribution. That is,
competitive speciation happens if ra < rk. This criterion
has been demonstrated analytically for asexual organisms
(Dieckmann & Doebeli, 1999; Day, 2000; Doebeli &
Dieckmann, 2000) and numerical investigations have
shown this to be a good approximation for sexual
organisms (Dieckmann & Doebeli, 1999; Doebeli &
Dieckmann, 2000). The rationale is that under global
competition (i.e. large ra values), all phenotypes suffer
the same amount of competition as they all compet one
another. As extreme phenotypes have lower carrying
capacities, they never become the more abundant and
the frequency distribution stays a unimodal one. If the
range of competitors is decreased, extreme phenotypes
compete with fewer individuals than intermediate phenotypes do. In this case, they get a selective advantage
which eventually overcompensates for the disadvantage
because of their lower carrying capacity. To decrease the
range of competitors (lowering ra) then widens the range
of degree of assortative mating that leads to speciation.
However, if ra is strongly decreased, the competition
becomes local, i.e. it occurs only between individuals
having very close phenotypes. The actual level of
competition then strongly decreases and the fitness
differences between intermediate and extreme phenotypes are then mostly due to their carrying capacities.
Accordingly, when ra tends toward 0, the phenotypic
distribution simply fits the carrying capacity distribution.
Thus, explanation for the nonlinear relationships
displayed in Fig. 5a, b is simple. Frequency-dependent
selection (required for competitive speciation) decreases
when ra takes on high values (because the impact of
competition on extreme phenotypes is increased), but
also when ra is too weak (because competition is relaxed
on intermediate phenotypes). Consequently, the usual
criteria for speciation (i.e. ra < rk) cannot be a sufficient
condition. There must be a lower limit for the range of
phenotype an individual competes with (i.e. ra), as too
local competition does not allow for speciation.
Discussion
The two main purposes of this study were to sort out (1)
how sexual selection and frequency-dependent natural
selection contribute to influence sympatric speciation
and (2) how this depends on the genetic architecture of
the quantitative trait under both sexual and natural
selection.
I investigated a hypergeometric model of competitive
speciation closely related to Doebeli’s (1996) model
which initiated the recent Dieckmann & Doebeli (1999)
and Doebeli & Dieckmann (2000, 2003) adaptive
dynamic models. This model is interesting for two
reasons. It uses the same hypergeometric framework
that Kondrashov & Shpak (1998) used to demonstrate
the possibility of speciation by the mean of assortative
mating alone and results of this model can also be
compared with those obtained using the quantitative
genetic theory (Drossel & McKane, 2000). All these links
allow addressing the two above questions. However, as
any hypergeometric model, this model is built on the
assumption that allele frequencies are equal at all the
loci. As this assumption may be misleading in generating
unstable solutions (Shpak & Kondrashov, 1999; Barton &
Shpak, 2000), I also used an IBM to back the conclusions
drawn from the hypergeometric framework. This strategy
conforms to the need to unify existing speciation models
(Kirkpatrick & Ravigné, 2002).
Speciation under sexual selection and stabilizing
selection
Looking at the evolution of the phenotypic distribution
under nonselective mating alone, I confirm that a strong
enough level of assortative mating allows an initially
unimodal distribution to split into two purely monomorphic species (Kondrashov & Shpak, 1998). This kind of
assortative mating speciation (with complete loss
of polymorphism) appears to be a general feature of
hypergeometric models, and does not depend on the kind
of assortative mating function used (threshold and
interval-based mode functions in Kondrashov & Shpak
(1998) or the classical exponential function in this study).
Furthermore, speciation by NSAM is also widely obtained
using the IBM, which means that it is a basic feature of any
NSAM model involving a finite set of loci. I further extend
Kondrashov & Shpak’s (1998) results by considering rare
phenotypes’ disadvantage in finding a mate. Such SAM
still allows for speciation, although it requires higher levels
of assortative mating and large enough initial genetic
variance. These results also extend a study by Kirkpatrick
& Ravigné (2002), who showed that SAM can easily lead
to speciation, but working with the most favourable
genetic architecture for speciation to proceed, i.e. a twolocus model (see results on the effect of the number of loci
on the likelihood of speciation discussed below).
That speciation by assortative mating is more likely
when the quantitative trait is determined by a high
number of loci (Kondrashov & Shpak, 1998) apparently
contrasts with the result obtained using the quantitative
genetic theory (Drossel & McKane, 2000). Indeed,
implicitly assuming a very large number of loci by using
quantitative genetic models, these authors never observed speciation. However, this is only an apparent
paradox, which is solved when the allelic effect is scaled
to keep the phenotypic range as a constant. Speciation by
means of assortative mating is then less likely when
considering more loci with smaller effects.
J. EVOL. BIOL. 17 (2004) 1297–1309 ª 2004 BLACKWELL PUBLISHING LTD
Sympatric speciation
Thus, by considering selective and nonselective mating
under different genetic architectures, this study bridges
the gaps between results previously obtained by Kondrashov & Shpak (1998), Drossel & McKane (2000) and
Kirkpatrick & Ravigné, (2002) and provides us with a
coherent picture of how assortative mating can lead to
speciation when neither stabilizing selection nor competitive interactions are considered.
Speciation can still proceed when stabilizing selection is
included giving intermediate phenotypes a selective
advantage. However, this requires the initial genetic
variance to take on high enough values. If the phenotypic
distribution is already broad, assortative mating allows
individuals with extreme phenotypes to increase in
frequency. As they become rare, intermediate phenotypes
then experience difficulty in finding mates and the
corresponding selective disadvantage can overcome the
advantage of intermediate phenotypes because of stabilizing selection. Such a possibility does not appear if
mating is nonselective, so that stabilizing selection generally prevents nonselective mating speciation. Given the
level of genetic variance required for SAM speciation to
occur, it is very likely that this process actually acts only in
a second stage of speciation, after natural selection has
already broaden the phenotypic distribution. As suggested by the final phenotypic distributions I obtained, it
could then be very efficient in splitting incipient species
apart. On the contrary, as long as the initial genetic
variance is low, a selective disadvantage of rare phenotypes in finding mates reinforces the effect of stabilizing
selection. In this case, alleles 0 and 1 quickly get fixed so
that the population contains only the optimal phenotype.
Thus, if no other process allows an initial increase of the
genetic variance, selective mating acts against speciation
preventing extreme phenotypes to increase in frequency.
An important conclusion from these results is that
recent papers supporting sympatric speciation have
probably overestimated the likelihood of sympatric speciation, because they do not account for mating disadvantage of rare phenotypes which, in the initial stage of
speciation, correspond to extreme phenotypes (Doebeli,
1996; Dieckmann & Doebeli, 1999; Doebeli & Dieckmann, 2000, 2003). Another reason why these models
have probably overestimated the likelihood of sympatric
speciation is that they generally deal with a small
number of loci, which strongly enlarges the range of
parameters allowing speciation. These results provide
two possible answers to the question (Bridle & Jiggins,
2000): why is sympatric speciation theoretically so easy?
Importance of frequency-dependent selection for
speciation by reinforcement and pleiotropy
Following Drossel & McKane (2000), I did not explicitly
account for the evolution of assortative mating but
looked at the relative importance of assortative mating,
stabilizing selection and competition. The results of this
1307
study apply directly when a trait determining the habitat
specialization pleiotropically affects the mate choice. But,
these results also give new insights into sympatric
speciation by reinforcement as explicitly modelled in
recent papers (Dieckmann & Doebeli, 1999; Doebeli &
Dieckmann, 2000, 2003).
Pleiotropy is an easy scenario for speciation as we do
not have to consider the evolution of assortative mating
which appears as a by-product of habitat specialization.
The usual picture of this type of speciation is host shift
exemplified by the apple maggot flies, treehoppers,
phytophageous moths and mimetic butterflies (Jiggins
et al., 2004). This generally implies a few morphs
corresponding to a new host. However, there is no reason
why finer adaptive differentiation could not lead to some
prezygotic isolation, as encapsulated in the multiple
phenotypes model presented in this paper. New phenotype then increases in frequency, exactly in the same way
as when an emerging well-adapted morph increases in
frequency by using a still unexploited (unique) host. If
mating is not selective, it then contributes to strongly
increase the genetic differentiation between individuals.
On the contrary, evolutionary diversification can be
strongly impeded if individuals with a new phenotype
or morph suffer a selective disadvantage in looking for still
rare mates. This is especially important if habitat specialization pleiotropically leads to strong level of assortative
mating, for instance when only individuals with exactly
the same niche mate together. However, if natural
selection has previously allowed for the evolution of
some large enough genetic variance, SAM is also expected
to strongly contribute to the genetic differentiation of the
population into two monomorphic sets of individuals
having extreme phenotypes. Interestingly, it means that
sexual selection can induce maladaptive differentiation
by generating ecologically unfit extreme phenotypes.
Natural selection then acts to prevent the split, which
means to unexpectedly reduce evolutionary diversification.
The trickier scenario for sympatric speciation is reinforcement, as we need to explain why assortative mating
would evolve in the first place. The usual understanding
of competitive speciation is that prezygotic isolation
evolves to reinforce post-zygotic isolation due to density
and frequency-dependent competition. The cost of
hybridization is because of the production of offspring
with intermediate phenotype, who experience the
strongest level of competition. This is thought to generate
a gradual increase of prezygotic isolation until the natural
selection gradient acting against intermediate phenotypes
vanishes. Here, I show that if strong enough assortative
mating evolves, it is able to split the population into two
extreme phenotypic clusters. This happens suddenly
around a threshold value of assortativeness. Interestingly,
the evolution of such a level of assortative mating
then generates the conditions for the invasion of
mutant playing a strategy corresponding to lower level
J. EVOL. BIOL. 17 (2004) 1297–1309 ª 2004 BLACKWELL PUBLISHING LTD
1308
S. GOURBIERE
of assortative mating. Indeed, any mutant with a lower
level of assortative mating would get a selective advantage by producing offspring with intermediate phenotypes,
who can then fit the empty space in the phenotypic range.
Evolution of prezygotic isolation is then expected to go
back to a lower level of assortative mating. However, as a
consequence of this backward evolution, the frequencies
of intermediate phenotypes will strongly increase and
conditions for the evolution of a stronger level of
assortative mating will be recovered. Thus, the highly
nonlinear relationship between the level of assortative
mating and the phenotypic variance raises up the possibility for nonlinear evolution of prezygotic isolation with
the possibility of speciation-despeciation cycles. This
might explain the nonlinear evolution of assortative
mating briefly reported by Doebeli (1996, p. 903).
Another point of this paper is the relationship between
the level of differentiation and the strength of competition between phenotypes. This relationship is also nonlinear with an intermediate level of competition
providing the best condition for species differentiation.
Indeed, both low and high levels of competition between
phenotypes do not generate the frequency dependent
selection required for speciation to happen. Although a
simple finding, that low ra values can make speciation
harder or impossible has not been investigated yet. This
contrasts with the classical result that competition lead to
speciation when ra < rK (e.g. Dieckmann & Doebeli,
1999; Day, 2000; Doebeli & Dieckmann, 2000, 2003). I
show that ra must also constraint by a lower limit as
when ra approaches 0 in value, only individuals with
exactly the same phenotypes compete one another and
natural selection then make the phenotypic distribution
fitting the carrying capacity distribution, which is
assumed to be unimodal. More generally, it means that
conditions for a branching process to happen (as determined in the analytical Adaptive Dynamics context) do
not necessarily correspond to conditions for speciation, as
suggested in a recent review on Adaptive Dynamics
(Waxman & Gavrilets, 2004).
To conclude, competitive speciation is harder to
achieve and slightly more complex than claimed in
recent key papers about sympatric speciation. It is harder
to achieve because recent models did not account for
selective disadvantage of rare phenotypes and dealt with
a small number of loci, two conditions which strongly
favour speciation. It is slightly more complex than
previously thought as natural selection favours divergence in the earliest stage of speciation, but latter acts
against divergence to prevent ecological maladaptation
generated by sexual selection.
Acknowledgments
I am deeply grateful to James Mallet for all his comments
throughout the achievement of this work and for his
extremely helpful suggestions on early versions of this
manuscript. I would also like to thank John Welch and
John Maynard-Smith for stimulating discussions. It is a
pleasure to thank Adam Eyre-Walker who kindly invited
me to the Centre for the Study of Evolution (CSE) at the
University of Falmer, during the years 2002 and 2003.
Financial support was provided by a Marie Curie postdoctoral fellowship (HPMF-CT-2001-01230). This contribution is dedicated to the loved memory of my father
whose honesty, ability to observe and knowledge will
stay forever as invaluable landmarks for my professional
and family life.
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