Journal of Theoretical Biology 337 (2013) 101–110
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Journal of Theoretical Biology
journal homepage: www.elsevier.com/locate/yjtbi
The effect of dominance on polymorphism in Müllerian mimic species
V. Llaurens a,n, S. Billiard b, M. Joron a
a
Origine Structure et Evolution de la Biodiversité, CNRS UMR 7205, Muséum National d'Histoire Naturelle, CP50, 45, rue Buffon, 75005 Paris, France
Laboratoire de Génétique et évolution des populations végétales, CNRS UMR 8198, Université des Sciences et Technologies de Lille 1, Bâtiment SN2, 59655
Villeneuve d'Ascq Cedex, France
b
H I G H L I G H T S
We investigate the influence of dominance on the maintenance of polymorphism in a Müllerian mimicry system in a spatially heterogeneous
environment.
Complete dominance was shown to extend the parameter space, and thus the breadth of ecological situations, where Müllerian mimicry
polymorphism was maintained.
Overdominance, which promotes polymorphism, can arise given certain levels of toxicity and predator discrimination accuracy.
Dominant alleles were shown to reach lower frequencies than recessive alleles when selection on both homozygotes was symmetrical.
art ic l e i nf o
a b s t r a c t
Article history:
Received 17 May 2013
Received in revised form
7 August 2013
Accepted 9 August 2013
Available online 21 August 2013
Dominance controls the phenotype of heterozygous individuals, and plays an important role in the
maintenance of polymorphism. Here we focus on the dominance acting on warning-pattern polymorphism in species engaged in Müllerian mimicry. Müllerian mimics are toxic species which display bright
colour patterns used as a warning signal to predators and are subject to local positive density-dependent
selection. Some Müllerian mimics are polymorphic due to a selection/migration balance in spatially
heterogeneous communities of prey. Since heterozygotes at a locus controlling warning pattern might
exhibit intermediate, non-mimetic heterozygous morphs, dominance is likely to influence the polymorphism at this locus. Using a deterministic model describing migration, density-dependent predation
and reproduction, we investigated the influence of dominance on the dynamics of alleles at locus
determining mimetic phenotype. Our results suggest dominance may interact with migration and
selection and plays an important role in shaping the conditions of polymorphism persistence and the
frequency of alleles at this locus. Our results thus highlight the important role of dominance in the
dynamics of polymorphism at loci under balancing selection due to environmental heterogeneity.
& 2013 Elsevier Ltd. All rights reserved.
Keywords:
Mimicry
Balancing selection
Spatial heterogeneity
Aposematism
1. Introduction
Dominance among alleles at the same genetic locus is a widespread phenomenon, described by Mendel in his famous study of
pea crosses in the 19th century (Mendel, 1895). By modifying the
phenotype of heterozygotes, dominance can have an important
impact on the fitness of alleles. For instance, alleles with deleterious properties are generally found recessive or partially recessive
to wild-type alleles (Orr, 1991). Dominance has been shown to
influence the invasion of a new mutation arising in a population,
more dominant mutations tending to get fixed more often
than more recessive ones (i.e. Haldane ‘s sieve (Haldane, 1927)).
n
Corresponding author. Tel.: þ 33 01 40 79 38 61; fax: þ 33 01 40 79 33 42.
E-mail addresses: llaurens@mnhn.fr, llaurens.violaine@laposte.net (V. Llaurens).
0022-5193/$ - see front matter & 2013 Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.jtbi.2013.08.006
Dominance also plays a significant role in the shape of allele
frequency clines maintained by frequency-dependent selection:
Mallet and Barton (Mallet and Barton, 1989) demonstrated that
allelic dominance may result in the formation of asymmetrical
clines and thus favour cline movement. Yeaman and Otto (Yeaman
and Otto, 2011) also confirmed the impact of dominance in the
invasion of new alleles in a classical two-population migration/
selection model.
Since dominance only plays a role in the expression of the
phenotype in heterozygous individuals, it is expected to play an
important role in the evolution of polymorphic loci where heterozygotes are at high frequency. However, the influence of dominance
on the persistence of polymorphism and on allele-frequency
distribution has received little attention. Balanced polymorphism
is maintained in well-documented regimes of selection such as
heterozygote advantage, or negative frequency-dependent selection
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V. Llaurens et al. / Journal of Theoretical Biology 337 (2013) 101–110
(Charlesworth, 2006). However, the persistence of polymorphism
due to spatial heterogeneity might also represent a widespread
situation where balanced polymorphism is maintained (Spichtig
and Kawecki, 2004).
In previous theoretical studies on the effect of dominance on
polymorphisms maintained by a migration/selection balance,
dominance was modelled using a coefficient which directly
modulates the fitness of heterozygotes (Yeaman and Otto, 2011;
Otto and Bourguet, 1999; Orr and Betancourt, 2001), and, in turn,
differences in invasion capacities for alleles with a positive vs. a
negative effect on the fitness of heterozygotes (i.e. overdominance
vs. underdominance). With this specific mathematical formalisation of dominance, the biological mechanism of dominance acting
on the fitness of heterozygotes is not straightforward and this
makes it difficult to confront theoretical predictions with empirical
data. More importantly, predictions obtained by these models
cannot be readily applied to cases (such as Müllerian mimicry)
where fitness is frequency and density-dependent, i.e. when the
fitness of heterozygotes depends both on the phenotype expressed
by heterozygotes and on the overall distribution of the phenotypes
in the population (and in the larger community). To address this
limitation, we here consider a dominance coefficient acting on the
phenotype and not directly on fitness.
In this study, we focus on the role of dominance in shaping
phenotypic variation and the maintenance of polymorphism in
prey species engaged in Müllerian mimicry. Müllerian mimicry is
the adaptive resemblance of multiple noxious prey species whose
shared phenotype functions as a common warning signal to
predators (Ruxton et al., 2004). Müllerian mimicry is a widespread
phenomenon documented in many organisms from arthropods
(Millipedes (Marek and Bond, 2009), Hemiptera, (Zrzavy and
Nedved, 1999)) to vertebrates (amphibians (Symula et al., 2001),
snakes(Sanders et al., 2006), birds (Dumbacher and Fleischer,
2001)…). Predators learn to avoid chemically defended, warningly
coloured prey based on previous experience (Chouteau and
Angers, 2011; Mappes et al., 2005) thus favouring the survival of
prey bearing the commonest (most frequent) patterns in a local
community (Pinheiro, 2003), a well-known example of positive
frequency-dependent selection (Speed and Turner, 1999).
Frequency-dependence operates locally and favours resemblance
among co-occurring species, favouring the locally commonest
phenotypes and selecting away rarer variants. However, selection
on warning patterns may act in opposing directions in distinct
localities, and mimetic communities indeed vary widely in their
warning patterns across geographic areas or across habitats.
The evolution and maintenance of warning-pattern polymorphism would not be predicted in the situation where a single,
best-protected phenotype is expected to reach fixation, but polymorphisms may be maintained in the case of spatially variable
selection, e.g. when the composition of the community of toxic
prey changes spatially, as found for instance in the poison frog
Ranitomeya imitator (Chouteau et al., 2011) or in the butterfly
Heliconius numata (Joron et al., 1999). In Heliconius numata, multiple wing-pattern forms co-occur, each one being a precise mimic
of a distinct species in the distantly related genus Melinaea. Finescale spatial variations in the relative abundances of the Melinaea
species are positively correlated with the frequency of the matching wing colour pattern in Heliconius numata (Joron et al., 1999).
The fine spatial heterogeneity in the mimetic community composition is thought to translate into variations in the direction of
selection for warning pattern resemblance in distinct populations.
This variable selection generates balancing selection on wing
colour patterns at a larger spatial scale.
In the fitness landscape of mimicry, shaped by frequencydependent selection, peaks correspond to common, mimetic phenotypes, and fitness valleys correspond to all rare, non-mimetic,
and intermediate phenotypes which are strongly selected against.
The persistence of polymorphisms will be influenced by the
strength of selection acting on heterozygotes. By controlling the
mimicry of heterozygous genotypes, dominance relationships
among alleles at a locus controlling mimetic phenotypic elements
are of particular relevance. For instance complete dominance of
mutant alleles would allow full resemblance of heterozygote
genotypes with this mutant allele to one of the mimetic homozygotes. Moreover, under strong balancing selection, heterozygote
frequency is expected to be high, and dominance may play an
important role in the selection on alleles, and on polymorphism
maintenance.
The evolutionary dynamics and genetics of mimicry have been
well studied in butterflies. The genetic loci determining mimicry
variation in the genera Papilio and Heliconius have received
sustained attention, and both dominance and co-dominance
relationships are observed among alleles coding for alternative
mimetic phenotypes (see (Clarke and Sheppard, 1960) and (Clarke
et al., 1985)). In Heliconius, alleles at homologous loci can have
differently ordered dominance relationships in distinct species
(e.g. Heliconius melpomene vs. Heliconius erato (Nijhout, 1991);
Heliconius cydno vs. Heliconius melpomene (Naisbit et al., 2003)).
Dominance can also vary within species: in Papilio dardanus,
dominance tends to be stronger in crosses between individuals
drawn from the same geographical area than in crosses involving
individuals from different areas (Nijhout, 2003). Such variation in
dominance level in natural populations underlines the lack of
understanding of the role of dominance in the evolution and
maintenance polymorphisms of these adaptive patterns.
To fill this gap, we built a model based on a previously
described theoretical haploid model (Joron and Iwasa, 2005)
proposed to depict the evolution of warning-signal mimicry in
unpalatable prey. We extended this model to a diploid species, in
order to investigate the influence of dominance on warning-colour
polymorphism.
2. Material and methods
The model presented here is based on the previous model of
Joron and Iwasa (Joron and Iwasa, 2005) which was built to
describe Müllerian mimicry in a spatially distributed community.
We extended this model to describe a diploid species with
dominance relationships among alleles at the locus controlling
mimetic colour pattern.
We considered a system with two patches, 1 and 2, each one
containing a single mimicry community (i.e. a number of species
all mimicking each other and bearing a similar wing pattern
signalling their toxicity to predators). The two patches differed
in the mimetic pattern adopted by local species: the mimicry
community exhibited morph A in patch 1 and morph B in patch 2.
The two communities constituted the mimetic environment and
were assumed to have a fixed abundance and a fixed warning
colour pattern (A or B).
We studied the polymorphism dynamics of a focal species
evolving in this spatially distributed system. The focal species was
a Müllerian mimic, mimicking morph A or B, bearing its own
toxicity, and forming two distinct populations exchanging
migrants between patches 1 and 2. This mimetic species was
diploid and the mimetic morph A or B was assumed to be
determined by a single locus with two segregating alleles a and b.
2.1. Modelling dominance
Individuals of genotypes aa and bb displayed phenotype A and
B respectively. The phenotype of the heterozygote ab depended on
V. Llaurens et al. / Journal of Theoretical Biology 337 (2013) 101–110
103
The survival of the individuals in a population not only
depended on the community environment represented by model
species but also on its own abundance in the community. Predator
avoidance was thus assumed to depend both on the density of
each morph and on the unpalatibility of the focal species, l. The
shape of this function was similar to the function used in haploid
model of Müllerian mimicry used by Joron and Iwasa (Joron and
Iwasa, 2005).
Homozygotes and heterozygotes could display similar colour
patterns, and therefore benefited from their mutual abundance, to
an extent which depends on the visual ability of the predator, i.e.
depending on the generalization function, g(h), described above.
The change in the density of homozygote genotypes aa and bb
in population i due to predation, was given by
Fig. 1. Shape of the generalization function according for different values of γ.
X-axis exhibits the dominance coefficient h and y-axis the generalization function
g. Dashed lines represent g(h) and plain lines g(1-h), black lines : γ ¼0.3, dark grey :
γ ¼0.1, grey : γ ¼0.05, light grey : γ ¼0.01.
the dominance coefficient h. The similarity between the homozygotes aa and bb and heterozygote ab morphs was described by a
generalization function with a Gaussian shape, g(h), following
equation:
gðhÞ ¼ eðh1Þ =2γ
with γ determining the width of the Gaussian function:
2
2
ð1Þ
This shape corresponded to the function generally assumed for
predator generalization, i.e. the range for which patterns are too
similar to be discriminated by predators (Ruxton et al., 2008). The
functions g(h) and g(1 h) represented the resemblance of the
heterozygote ab to the homozygote aa and to the homozygote bb
respectively. When h tended to 1, the heterozygote ab displayed
exactly the same phenotype as the homozygote aa. Fig. 1 shows
the effect of γ on the shape of the function with decreasing γ
leading to steeper shapes, in which co-dominance (h ¼0.5) leaded
to high dissimilarity of the heterozygote ab with respect to both
homozygotes aa and bb.
2.2. Modelling allele dynamics
We assumed a continuous time deterministic model where all
events occurred simultaneously. Three types of events occurred in
the populations 1 and 2: migration, viability selection due to death
by predation, and density-dependent reproduction. The production of genotypes was assumed to follow Mendelian segregation
and was modelled explicitly. The change in the density of each
genotype with time during each event was detailed below.
2.2.1. Migration
The change by time unit of Nuvi (density of genotype uv, i.e. aa,
bb or ab, in population i) due to migration between populations i
and j was given by
dNuvi
¼ mðNuvjNuviÞ
dt
dN aai
daai
¼
N ;
dt
1 þ lðN aai þgðhÞN abi Þ aai
ð3Þ
dN bbi
dbbi
¼
N ;
dt
1 þ lðNbbi þ gð1hÞN abi Þ bbi
ð4Þ
The change of the density of heterozygote genotype ab in populations 1 and 2 after predation was given by
dN ab1
gðhÞdaa1 þ ð1gðhÞdbb1 Þ
¼
N ;
1 þ lðN ab1 þ gðhÞN aa1 þ gð1hÞNbb1 Þ ab1
dt
ð5Þ
dN ab2
gð1hÞdaa2 þ ð1gð1hÞdbb2 Þ
¼
N :
1 þ lðN ab2 þ gðhÞN aa2 þ gð1hÞNbb2 Þ ab2
dt
ð6Þ
For the sake of simplicity, we assumed a symmetrical condition
where daa1 ¼dbb2 ¼ d (1 s) and dbb1 ¼daa2 ¼ d (1 þ s) where d
represented the mean predation risk and s the spatial heterogeneity due to the distribution of the two model species.
2.2.3. Reproduction
The parameter r was set as the intrinsic per female capita
growth rate, K the carrying capacity, assumed equal in both
populations, and N i ¼ N aai þ N abi þN bbi the total density of individuals in population i. We also assume that the sex-ratio was
balanced and that only females gave birth to new individuals. We
assumed that reproduction was density-dependent and that the
per capita growth rate in population i was for all genotype:
ðr=2Þð1ðN i =KÞÞ.
Since we aimed to explicitly model sexual reproduction, the
rate at which each genotype was produced depended on the rate
at which each type of cross occurs, i.e. on the density of each
genotype. We assumed Mendelian segregation at the mimicry
locus. The production rate of individuals with genotype uv in
population i was then described as follows:
r
N
1 i f uvi ;
ð7Þ
2
K
with
with ia j and m as the migration rate:
ð2Þ
f aai ¼
ðN abi þ 2N aai Þ2
;
4Ni
ð8aÞ
2.2.2. Survival within population
We assumed daai and dbbi to be the death coefficient of the
homozygotes aa and bb in population i. These death coefficients
depended on the presence of the model species in the population
considered. Hence, daa1 rdbb1 and daa2 Zdbb2 because the genotype aa (phenotype A) was favoured in population 1 and not in
population 2 due to mimicry with model species at different
abundances.
f abi ¼
2ðN abi þ2N aai ÞðNabi þ 2N bbi Þ
;
4N i
ð8bÞ
f bbi ¼
ðN abi þ 2Nbbi Þ2
:
4N i
ð8cÞ
Although colour patterns are likely to play a role in sexual
selection, its influence on mating success was not included in our
model for simplicity.
104
V. Llaurens et al. / Journal of Theoretical Biology 337 (2013) 101–110
2.2.4. General equations for the population dynamics
Combining the density change of each genotype with time due
to the combination of migration, viability selection by predation
and reproduction gave the total density change in a continuous
time model. The dynamics were described by six ordinary differential equations, three for each population. The equations were
for iaj:
dN aai
daai
¼ mðN aaj N aai Þ
N
dt
1 þlðN aai þ gðhÞN abi Þ aai
r
N ðN abi þ 2Naai Þ2
þ 1 i
;
2
K
4N i
dN bbi
dbbi
¼ m N bbj N bbi N
dt
1 þlðN bbi þg ð1hÞN abi Þ bbi
2
r
N ðN abi þ 2Nbbi Þ
þ 1 i
;
2
K
4N i
ð9aÞ
ð9bÞ
and
dN ab1
g ðhÞdaa1 þ ð1g ðhÞdbb1 Þ
¼ mðN ab2 N ab1 Þ
N
1 þ lðNab1 þ g ðhÞN aa1 þg ð1hÞN bb1 Þ ab1
dt
r
N 1 2ðNab1 þ 2Naa1 ÞðN ab1 þ 2N bb1 Þ
þ 1
;
ð9cÞ
2
4N 1
K
dN ab2
gð1hÞdaa2 þ ð1gð1hÞdbb2 Þ
¼ mðN ab1 N ab2 Þ
N
1 þ lðNab2 þ gðhÞN aa2 þgð1hÞN bb2 Þ ab2
dt
r
N2 2ðN ab2 þ2N aa2 ÞðNab2 þ 2Nbb2 Þ
þ 1
:
ð9dÞ
2
4N 2
K
2.3. Analysis of the model
2.3.1. Stability analyses
We first performed a stability analysis of these equations (Otto
and Day, 2007). For a given equilibrium, eigenvalues were computed, the sign of which determined the conditions of its stability.
Combining the analysis of several equilibria and their stabilities
revealed the conditions for the persistence of polymorphism. We
were able to obtain explicit expressions of the equilibria and their
stability for two simple cases regarding migration: (1) when there
is no migration, and (2) when migration rate is very high. In case
(2), we assumed that the two populations behave as a single large
population, where Naa1 ¼ Naa2 ¼Naa, Nab1 ¼Nab2 ¼Nab and Nbb1 ¼
Nbb2 ¼ Nbb. For all other value of the migration, we investigated the
polymorphism maintenance by a numerical analysis detailed
hereafter.
2.3.2. Numerical analyses of equilibria for intermediate
migration rates
To find the equilibria for any value of the migration rate m, we
performed a numerical analysis using the FindRoot function in
Mathematica. The first investigated equilibrium was for m¼ 0.01,
specifying in the FindRoot function that the starting point was
near the equilibrium values for m ¼0, as it had been computed in
the previous section. We computed the second equilibrium for
m ¼0.02 using the equilibrium found for m ¼0.01 as the starting
value, and so on until m ¼0.4 with an increment of 0.01 between
each computation.
We found that the maintenance of polymorphism can depend
on the initial state of the population. To approximately determine
the size of the attraction basin, we performed additional numerical
analyses to investigate the conditions for the maintenance of
polymorphism for 1000 different initial conditions where the
number of individuals of a given genotype was randomly drawn
in a uniform distribution between 0 and 100. We iterated
equations Eqs. (9a),(9b),(9c) and (9d) until the frequency change
between two iterations was below 10 3 for both alleles. Three
possible outcomes were possible: maintenance of polymorphism
(both alleles a and b were present in the population at the end of
the computation); allele a is lost; or allele b is lost.
In the same way, we also computed the equilibrium for varying
values of the dominance coefficient h for a given value of the
migration rate m using the FindRoot function.
Finally, we also investigated the effect of the toxicity, l, and the
heterogeneity of the environment, s, on the maintenance of
polymorphism for different dominance levels, h. For this, we
performed a numerical analysis, starting with the initial conditions
given by the equations from the equilibrium when m ¼ 0, we
introduced a small quantity of heterozygotes to each patch so that
the initial frequency of the heterozygotes was 0.001. We performed 1000 iterative computations of Eqs. (9a)–(9d) and checked
if the densities of all genotypes were higher than 1 in both
populations, which corresponds to a frequency higher than 10 3
in a 1000 individuals population. If so, we considered that the
polymorphism was maintained.
2.3.3. Parameter values
Unless otherwise stated, we used the following values for the
parameters. The generalization function was chosen to be steep
(γ ¼0.01, see Fig. 1) in the numerical analyses to simulate clear
phenotypic distinction of the co-dominant heterozygote ab from
either homozygote (aa and bb). We assumed a low value, because
the few studies testing the ability of predators to discriminate the
different morphs of Müllerian mimic species show that they were
able to discriminate ‘local' versus ‘exotic' morphs quite accurately
(Chouteau and Angers, 2011; Merrill et al., 2012).
Computations were performed assuming a strict symmetry
between the population 1 and 2, using an intermediate general
predation risk d¼ 0.5 and a high spatial heterogeneity parameter
s¼ 0.9, thus simulating a strong predation difference between
populations. This assumption of a greater advantage provided by
the model species with respect to the within-species density
dependence is biologically relevant: several species can be
involved in a mimicry ring leading the total of individuals to
generally outnumber individuals from the focal species only.
We also assumed toxicity l ¼0.0025, growth rate r ¼1 and
carrying capacity K ¼1000. Note that since model species did not
evolve, we used the Kl product as an estimate of the relative
contribution of the mimetic species to the community toxicity.
3. Results
3.1. Polymorphism without migration
First we analysed the case of two isolated populations exchanging no migrants (m¼ 0). Four possible equilibria were detected
(Table 1). A first equilibrium corresponded to the extinction of
both populations and the second equilibrium to two monomorphic populations, each one being fixed for the allele matching
the local community. The other two equilibria described a monomorphic mimic species with the same allele invading both
populations. However the latter two equilibria were always
unstable when predation risk and spatial heterogeneity were
positive (d 40 and s 40), meaning that these equilibria cannot
occur with predation. In the absence of migration, polymorphism
cannot be maintained within each population here, regardless of
the level of dominance (h).
V. Llaurens et al. / Journal of Theoretical Biology 337 (2013) 101–110
105
Table 1
Description of the four possible equilibria obtained through the analytical analyses, assuming no migration (m¼ 0).
Equilibrium
Frequencies of genotypes
Description of equilibrium
Stability conditions
1
Nab1 ¼ Nab2 ¼ Naa2 ¼ Nbb1 ¼ Naa1 ¼ Nbb2 ¼ 0
Extinct populations
r
Stable when d 4 2ð1sÞ
2
Nab1 ¼ Nab2 ¼ Naa2 ¼ Nbb1 ¼ 0
ffi
pffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
Naa1 ¼ Nbb2 ¼ ðKl1Þr þ r ðKl2lrþ 1Þ r8dKlð1sÞ
p
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
pffi
2
Naa1 ¼ ðKl1Þr þ r ðKl2lrþ 1Þ r8dKlð1sÞ
ffi
pffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
Naa2 ¼ ðKl1Þr þ r ðKl2lrþ 1Þ r8dKlð1 þ sÞ
ffi
pffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
Nbb1 ¼ ðKl1Þr þ r ðKl2lrþ 1Þ r8dKlð1 þ sÞ
ffi
pffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
Nbb2 ¼ ðKl1Þr þ r ðKl2lrþ 1Þ r8dKlð1sÞ
Two monomorphic populations
r
Stable when d o 2ð1sÞ
Allele a fixed in both population
Always unstable for realistic range of predation parameters
(d 4 0 and s 4 0)
Allele b fixed in both populations
Always unstable for realistic range of predation parameters
(d 4 0 and s 4 0)
3
4
3.2. Polymorphism with infinite migration
Secondly, we explored analytically the case where migration
was high. There, the two patches were still occupied by different
mimicry communities, but the focal species showed unlimited
migration between the two populations, approaching the behaviour of a single population. Three equilibria were detected: the
first one was the fixation of allele a, the second one the fixation of
allele b, and the third one the extinction of both populations
(supplementary Table 1). For each equilibrium, the sign of the
leading eigenvalue could be examined to infer equilibrium stability and therefore the conditions under which polymorphism was
maintained. However, these conditions were complex because
many parameters were involved (for more detail see
supplementary Table 2).
Briefly, polymorphism maintenance under high migration
depended mainly on spatial heterogeneity s and the shape of
the generalization function γ: when both parameters were high,
polymorphism was maintained. This means that polymorphism
persisted when there was a sufficient spatial heterogeneity in the
distribution of mimicry community (leading to disruptive selection on the mimetic pattern in the focal species) and when the
generalization function was high enough to allow heterozygotes to
be considered similar to the mimetic patterns. In this case,
spatially heterogeneous selection was sufficient to maintain
polymorphism.
The dominance coefficient h played a significant role on
polymorphism only when parameters s and γ are relatively small.
This was because dominance has an influence on predation only if
predators are able to distinguish heterozygotes from homozygotes
(γ small). In this case, a tight equilibrium between dominance,
demographic parameters (growth rate r and carrying capacity K)
and toxicity could allow the maintenance of polymorphism by
balancing the number of different phenotypes matching each prey
community. In particular, the Kl product (the total contribution of
the focal species to overall toxicity at the community level)
modified the strength of selection exerted by the structure of
predation given by parameters s and γ (i.e. spatial distribution of
model species). When Kl was large, the selection regime switched
from balancing selection due to spatial heterogeneity to directional selection due to within-species positive frequencydependent selection.
Dominance thus plays a complex role in the persistence of
polymorphism in a heterogeneous environment, by interacting
with toxicity, community structure and demography.
3.3. Persistence of polymorphism with intermediate migration
As previously shown in studies on spatially distributed populations, migration is a key parameter in the persistence of polymorphism. In this system, migration is balancing the effect of local
directional selection and, therefore, polymorphism is generally
maintained for low migration rates but becomes unstable above a
critical value of migration (Joron and Iwasa, 2005).
Numerical simulations were used to analyse the influence of
migration on polymorphism in our two-population model. As
shown in Fig. 2, polymorphism was maintained in both populations for all dominance levels when migration is relatively low.
When m 40.3, the two populations started to behave like a single
population, and for any dominance coefficient h, one allele became
fixed while the other went extinct.
However, the interaction between dominance and migration
had an influence on the persistence of polymorphism. Indeed,
dominance determined the strength of migration above which
polymorphism was lost, and which allele becomes fixed (a or b). In
the examples showed on Fig. 2, polymorphism was lost for
m Z0.22 when allele a and b are co-dominant (h ¼0.5) (Fig. 2C),
as compared to m Z0.26 when allele a was either recessive (h ¼0)
or dominant (h¼ 1) (Fig. 2A and D respectively). Similarly, Fig. 3
highlighted that polymorphism started to be lost in some simulations for lower migration rates in intermediate dominance conditions as compared to complete dominance. For instance when
h¼0.05, polymorphism was lost in some simulations when
m 40.08 whereas strictly recessive alleles were lost only when
m 40.25. This suggests that complete dominance allowed polymorphism to remain stable at higher rates of migration compared
to intermediate dominance. The same trend was observed for
spatial heterogeneity (s) with polymorphism observed at lower
level of heterogeneity when alleles exhibited complete dominance
as compared to co-dominance (see supplementary Figure 1).
Altogether, this suggested that complete dominance allowed the
persistence of polymorphism with lower levels of ecological and/
or population structure.
3.4. Influence of initial conditions and dominance on allele fixation
For high rates of migration, the fixation of the same allele in
both populations was observed for all dominance coefficients.
Here, because of a perfectly symmetrical situation (opposing but
identically scaled mimicry selection in the two populations), the
identity of the allele becoming fixed depended mainly on the
direction of dominance.
In the case of co-dominance (Fig. 2C), roughly half of all
simulations showed the fixation of allele a whereas the other half
fixed allele b. There, the identity of the allele dominating at high
migration rates was only determined by the initial frequencies of
the alleles.
In contrast, when dominance was complete, the dominant
allele became fixed in the large majority of simulations where
migration was high (allele b when h ¼0 and allele a when h¼1
respectively, see Fig. 2A and D). For migration rate m Z0.26, the
strictly dominant allele reached a frequency close to 1 in 99%
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V. Llaurens et al. / Journal of Theoretical Biology 337 (2013) 101–110
Fig. 2. Simulations for recessive (h ¼ 0) (Fig. 2A), mildly recessive (h ¼ 0.005) (Fig. 2B), co-dominant (h ¼ 0.5) (Fig. 2C) and dominant (h ¼ 1) allele a (Fig. 2D), for a range of
a realistic migration rates [0.01; 0.4] with an increment of 0.01. Each parameter set was replicated 1000 times. Grey: simulations where polymorphism was maintained, i.e.
both alleles had a frequency Z 0.01. White: simulations where allele a was fixed, i.e. the frequency of a overall both populations was Z 0.99. Black: simulations where allele
b was fixed, i.e. the frequency of allele a overall both populations was r 0.01. In all simulations, the shape of the generalization was chosen to be steep (γ ¼ 0.01), the global
death risk was d ¼ 0.5, the predation risk linked to the different abundance of the model species in the two populations was s ¼ 0.9, toxicity was l ¼ 0.0025, growth rate in
each population was r ¼ 1 and carrying capacity in each population was K ¼ 1000.
Fig. 3. Frequency of allele a in population 1 (a) and 2 (b) most commonly observed out of 1000 simulations for a realistic range of migration rates [0.01; 0.3]. To simplify the
visualisation, frequencies were plotted only in cases of polymorphism. The different lines represent values of the dominance coefficient h: red line: h ¼ 0 (the allele a is
strictly recessive), Orange: h ¼ 0.005, Purple: h ¼ 0.01, Green: h ¼ 0.5 (the allele a is exactly co-dominant), Blue: h ¼ 1 (the allele a is strictly dominant). Note that the lines
are interrupted as soon as polymorphism was lost. The shape of the generalization was chosen to be steep (γ ¼ 0.01), the global death risk was d ¼ 0.5, the predation risk
linked to the different abundance of the model species in the two populations was s ¼ 0.9, toxicity was l ¼ 0.0025, growth rate in each population was r ¼ 1 and carrying
capacity in each population was K ¼ 1000.
of cases. Dominance thus had an important influence on the fate of
the alleles. In case of partially recessive alleles (as for instance
when h¼ 0.005), the most dominant allele became fixed in the
majority of simulations although the fixation of the allele was also
influenced to the initial conditions (Fig. 2B). This effect illustrated
the interaction between an initial allelic density advantage and the
phenotypic density advantage provided by dominance. This interaction would be modified in case of asymmetrical selection,
e.g. when the total densities of the different mimicry communities
are not equal.
3.5. Allele frequencies in polymorphic populations
Fig. 3 showed that the interaction between migration and
dominance was shaping the allele frequency in polymorphic
populations. Allele frequencies were shown to depend closely on
V. Llaurens et al. / Journal of Theoretical Biology 337 (2013) 101–110
Fig. 4. Frequency of heterozygotes ab for a realistic range of migration rates,
m : [0.01; 0.3], averaged across both populations. The different lines represent
shapes of the generalisation function (γ): red line: γ ¼ 0.1, Orange: γ ¼ 0.3, Purple:
γ ¼ 0.5, Green: γ ¼ 0.7, Blue: γ ¼ 0.9. Note that the lines are interrupted as soon as
polymorphism is lost. The alleles were strictly co-dominant (h ¼ 0.5), the global
death risk was d ¼ 0.5, the predation risk linked to the different abundance of the
model species in the two populations was s ¼ 0.9, toxicity was l ¼ 0.0025, growth
rate in each population was r ¼ 1 and carrying capacity in each population was
K ¼ 1000.
the level of dominance, with recessive alleles (0 r ho0.5) exhibiting higher frequencies than dominant alleles, resulting in a total
frequency greater than 50%. This negative relationship between
dominance and frequency, leading to a higher frequency for more
recessive alleles held for wider shapes of the generalization
function (see supplementary Figure 2).
However, for very wide generalisation function (γ Z0.3), the
generalisation curves, g (h) and g (1 h), overlapped (see Fig. 1)
and intermediate heterozygotes ab (hE0.5) were perceived by
predators as similar to both model species (and both homozygotes). Fig. 4 showed that the frequency of the intermediate
heterozygote genotype ab increased with the width of the generalisation function, γ, leading to the persistence of the polymorphism
in both populations, even for high migration rates. This overdominance effect depended on the discrimination capacities of
predators with respect to the variable aposematic signal.
3.6. Migration load
It was notable that, for all dominance levels, the total number
of individuals overall both populations decreased when migration
rate increased (for more details see supplementary Figure 3). This
could be explained by a migration load which limits local adaptation (Lenormand, 2002). When there is no migration, all individuals in each population exhibit the matching phenotype of the
local mimicry community, but as soon as migration increases,
individuals are exchanged between populations, introducing nonmatching phenotypes which suffer higher predation and cause a
decrease in the total number of individuals overall both
populations.
More interestingly, our diploid model showed that this
decrease in population size depended on dominance: when h
was close to 0.5 (co-dominance), the decrease was larger, presumably because heterozygotes did not match the mimetic pattern
in any of the two patches, and suffered more predation in both
populations. In the case of high dominance, heterozygotes
matched the mimetic pattern of one of the two mimetic communities, leading to a higher total population size. Dominance thus
also influenced the demography of both populations.
3.7. Influence of toxicity (l) and spatial heterogeneity (s)
As previously described in the haploid model (Joron and Iwasa,
2005), the level of toxicity (l) and spatial heterogeneity (s) are key
107
parameters for local adaptation to spatially distributed mimicry
communities. Briefly, under a haploid model polymorphism could
be maintained when spatial heterogeneity was high due to the
importance of local selection. Similarly, low values of toxicity were
shown to favour polymorphism, because mildly defended prey
gain high benefits from resemblance to the local mimicry community whereas the dynamics of highly defended prey are less
sensitive to local mimicry. Here, we investigated more precisely
the roles of parameters l and s in the interaction between
dominance and migration (Fig. 5). Overall, our diploid model
confirms the positive effect of spatial heterogeneity on the
persistence of polymorphism whatever the dominance coefficient;
it also highlights that for mildly defended preys, complete dominance tends to promote polymorphism.
A particular case emerged for co-dominant alleles where
polymorphism could persist when toxicity was very high (grey
areas in bottom panels of Fig. 5). When h Z0.1, heterozygotes ab
resembled neither homozygotes (and neither corresponding
mimetic communities), but were produced in higher proportions
than homozygotes due to Mendelian segregation. Since prey were
highly defended, positive frequency-dependent selection was thus
mainly influenced by the phenotypic composition of the focal
species populations (as opposed to the local mimicry community).
Since the commonest phenotype was carried by heterozygotes ab,
a strong positive selection on heterozygotes promoted polymorphism. This overdominance effect explained the level of polymorphism observed at high values of toxicity when heterozygotes were
distinct from either homozygote.
Finally, for very high levels of toxicity, polymorphism was
observed regardless of the spatial heterogeneity or dominance
(see bottom of each graph on Fig. 5). In these cases, toxicity was so
high that all individuals became protected irrespective of their
aposematic pattern.
4. Discussion
4.1. Persistence of polymorphism
As already demonstrated in the haploid model (Joron and
Iwasa, 2005), the persistence of polymorphism was mainly driven
by a balance between migration (here represented by migration
rate, m) and selection (here represented by the overall predation
risk, d, and spatial heterogeneity, s). However, our diploid model
showed that complete dominance favoured the persistence of
polymorphism for a larger range of parameters (m, d, s) than codominance. This is in accordance with the general model of Otto
and Bourguet (Otto and Bourguet, 1999) investigating dominance
in patchy environments, which shows that stable polymorphism is
maintained for small values of h, i.e. in cases of complete
dominance. In the case of Müllerian mimics, the generally
observed positive effect of dominance on polymorphism maintenance was due to the higher levels of predation experienced by
co-dominant alleles in both populations, because of their lack of
resemblance to any mimicry community. In the case of partially
dominant or co-dominant alleles, homozygotes had an important
advantage over heterozygotes in both populations and selection
against non-mimetic heterozygotes might thus favour the fixation
of one of the two alleles, leading to the loss of polymorphism.
The positive effect of dominance on the persistence of polymorphism in Müllerian mimics is supported by empirical data: for
instance the toxic mimic butterfly Heliconius numata exhibits high
local polymorphism and complete dominance among sympatric
alleles controlling the wing-patterns forms (Joron et al., 2006).
However, to demonstrate the crucial role of dominance on polymorphism, empirical estimations of the level of dominance and
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V. Llaurens et al. / Journal of Theoretical Biology 337 (2013) 101–110
Fig. 5. Conditions for Mullerian polymorphism maintenance as a function of the migration rate (m) (columns, from left to right: m ¼ 0.001, 0.005, 0.01, 0.1), dominance
coefficient (h) (rows, from top to bottom lines: h ¼ 0, 0.005, 0.01, 0.1, 0.5), habitat heterogeneity (s) and unpalatibility (l) (note the logarithmic scale on the y-axis). In dark,
polymorphism is maintained in both populations, in grey, polymorphism is also maintained in both populations with a higher frequency of heterozygotes compared to
homozygotes (overdominance), in white polymorphism is lost.
V. Llaurens et al. / Journal of Theoretical Biology 337 (2013) 101–110
the level of colour pattern polymorphism in natural populations of
Müllerian mimetic species would be required.
Co-dominance also leads to the persistence of polymorphism in
two special cases: (1) when the discrimination capacities of
predators were not accurate, providing an advantage to intermediate heterozygotes through their resemblance to both mimetic
communities, providing a multi-model advantage (Edmunds,
2000), and (2) when the focal species was highly toxic and thus
protected independently of the mimicry community, favouring
heterozygotes through simple density-dependent advantage. In
such cases, the advantage of heterozygotes (overdominance)
provides a balancing selective pressure, allowing the persistence
of polymorphism in natural populations.
Dominance thus appears to magnify the effect of heterogeneous selection and thus to modulate the dynamics of the
migration/selection balance. However, the influence of dominance
depends closely on the mechanism of selection acting on the
functional locus and more precisely on the heterozygotes in both
populations. For instance, behavioural studies have demonstrated
that the physiological state of predators can affect their discrimination vs. generalization capacities (Barnett et al., 2012; Kokko
et al., 2003; Halpin et al., 2012), and may thus limit selection on
dominance. Generalization behaviour has also been shown to
depend on the diversity of communities (Ihalainen et al., 2012),
stressing the need to investigate the discrimination capacities of
predators in natural populations of polymorphic Müllerian mimic
species to characterize the selection on dominance more precisely.
109
environment, because it allows locally non-mimetic alleles to
persist. Depending on the ecological context (i.e. the composition
of the butterfly community, migration behaviour and relative
toxicity), recessive alleles can thus reach high frequencies in
mimetic species and persist over long evolutionary timescales.
The capacity for invasion and persistence of migrant or new alleles
with varying dominance levels thus needs to be investigated to
distinguish the effects of population history, ecology and
subdivision.
For high migration rates, when predators had accurate discrimination capacities, selection led to the fixation of the dominant
allele in both populations, assuming symmetrical spatial selection
on homozygous phenotypes. In this situation, since the two model
species provided the same advantage, the advantage provided to
the most dominant allele was given by a positive numberdependent advantage because both homozygote and heterozygote
display the same phenotype. This is equivalent to the situation
where two distinct mimicry communities confer unequal advantages and lead to the fixation of the morph mimicking the more
numerous community (Joron and Iwasa, 2005). This asymmetrical
advantage due to dominance has previously been described by
Mallet and Barton (Mallet and Barton, 1989): they showed that the
movement of clines between different alleles of genes involved in
mimicry is enhanced by dominance because the side of the cline
where dominant alleles are most common is more likely to
expand. Dominance is thus an important factor to consider to
understand the evolution of wing colour patterns in mimic
species.
4.2. Allele dynamics
4.3. Evolution of dominance
In our model without migration, each allele became fixed in the
population where it was locally adaptive, and dominance had no
influence on polymorphism stability. However, as soon as migration was introduced, our model showed the key role of dominance
in equilibrium allele frequencies. For moderate migration rates
(lower than 0.2 in the example presented here), the equilibrium
frequency was higher for the recessive allele than for the dominant allele. Since this corresponds to a two-population situation
with strictly symmetrical selection acting on each homozygote,
this might be similar to the situation found in parapatric mimetic
communities where the relative abundance of mimicry communities could be similar on either sides of a suture zone. Empirical
tests comparing allele frequencies with respect to dominance in
localities where the abundances of the different mimic communities are similar would allow confirming this prediction of
the model.
Our diploid model also highlights that dominance can have
complex interactions with overall prey community structure and
with the toxicity of a species relative to other species in a mimicry
community and may therefore modify the dynamics of polymorphism. In particular, relative toxicity with respect to other
species of the mimetic community can lead to heterozygote
advantage, and thus explain the polymorphism observed in
several Müllerian mimic species.
Finally, our model also predicted that the spatial distribution of
the distinct mimicry communities can drive the direction of
dominance. The traditional view of Haldane's sieve predicts that
new establishing alleles are more likely to be dominant over
ancestral alleles (Clarke et al., 1985), so that dominance could
reflect the chronology of the evolution of the different alleles. In
the case of balancing selection in subdivided populations, genetic
drift is more likely to eliminate unexpressed advantageous alleles
than dominant ones (Schierup et al., 1997), thus favouring the
invasion of dominant alleles in a population with homogeneous
selection. However, our model shows that recessivity can be
advantageous in case of mimicry in a spatially heterogeneous
Our model suggested that larger population size was observed
in cases of complete dominance. This suggests that complete
dominance may limit the mortality of intermediate non-mimetic
heterozygotes. Complete dominance can either be due to (1) positive selection of dominant new or migrant alleles only (i.e.
Haldane's sieve) or (2) evolution of expression levels towards
strict dominance among alleles. This last possibility of evolution of
dominance through natural selection, described by Fisher (Fisher,
1928), is still a debated topic. However, Billiard and Castric (Billiard
and Castric, 2011) recently suggested that dominance is likely to
evolve in many “special cases”, including loci under balancing
selection. Indeed, in cases of balanced polymorphism, the frequency of heterozygotes is high enough to allow the evolution of
dominance through natural selection (Otto and Bourguet, 1999).
The possibility of the evolution of dominance in wing colour
pattern genes has been suggested by Clarke and Sheppard
(Clarke and Sheppard, 1960) based on the study of dominance in
controlled crosses of sympatric versus allopatric morphs in the
mimetic butterfly Papilio dardanus.
In our diploid model, we highlighted the importance of
dominance on adaptation of mimic alleles, suggesting that complete dominance would be a cost-effective way to express only
mimetic phenotypes. Here, we assumed that each allele has a fixed
dominance level, corresponding to a situation where dominance is
an intrinsic property of the allele or is encoded by a gene tightly
linked to the locus encoding colour pattern itself. This situation
can arise in a supergene architecture where recombination is low
as is observed in Heliconius numata (Joron et al., 2011). However, it
has also been shown that the level of expression of major genes
encoding wing colour pattern might be influenced by modifier
genes which are not necessarily linked to these loci of major effect.
For instance, improvement of mimicry has been suggested to rely
on modifier genes with relatively small effect on wing colour
pattern, as demonstrated in Heliconius numata (Jones et al., 2012)
and Heliconius erato (Papa et al., 2013). These data raise the
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V. Llaurens et al. / Journal of Theoretical Biology 337 (2013) 101–110
possibility that dominance at major genes could be modified by
independent loci, allowing the rapid adaptation of heterozygotes
to the local community of toxic butterflies. The influence of
recombination between dominance modifiers and loci controlling
aposematic colour pattern should be investigated in the future and
could shed light on the mechanisms of the evolution of dominance.
5. Conclusions
Our theoretical diploid model suggests that dominance can
have an important impact on the dynamics of alleles responsible
for aposematic pattern in mimetic species and, specifically, on the
persistence of polymorphism. In polymorphic mimetic species,
dominance interacts with the structure of the community of
defended prey, migration capacities and demography, and thus
influences the adaptation of aposematic patterns. Our results
stress the importance of investigating the evolution of dominance
in adaptive traits in heterogeneous environments.
Acknowledgements
The authors would like to thank A. Whibley for useful discussions on the manuscript. This work has been sponsored by the
ATM Formes possibles, formes realisées from the National Museum
of Natural History to VL, by the ERC starting grant MIMEVOL to MJ
and the ANR BRASSIDOM to SB.
Appendix A. Supporting information
Supplementary data associated with this article can be found in
the online version at http://dx.doi.org/10.1016/j.jtbi.2013.08.006.
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