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Proc. R. Soc. B (2008) 275, 2723–2732
doi:10.1098/rspb.2008.0882
Published online 12 August 2008
Incipient allochronic speciation due to
non-selective assortative mating by
flowering time, mutation and genetic drift
Céline Devaux1,2,* and Russell Lande2
1
Division of Biological Sciences, University of California, San Diego, La Jolla, CA 92093-0116, USA
Division of Biology, Imperial College London, Silwood Park Campus, Ascot, Berkshire SL5 7PY, UK
2
We model the evolution of flowering time using a multilocus quantitative genetic model with non-selective
assortative mating and mutation to investigate incipient allochronic speciation in a finite population. For
quantitative characters with evolutionary parameters satisfying empirical observations and two
approximate inequalities that we derived, disjunct clusters in the population flowering phenology
originated within a few thousand generations in the absence of disruptive natural or sexual selection. Our
simulations and the conditions we derived showed that cluster formation was promoted by limited
population size, high mutational variance of flowering time, short individual flowering phenology and a
long flowering season. By contrast, cluster formation was hindered by inbreeding depression, stabilizing
selection and pollinator limitation. Our results suggest that incipient allochronic speciation in populations
of limited size (satisfying two inequalities) could be a common phenomenon.
Keywords: sympatric speciation; assortative mating; flowering time; multilocus trait;
quantitative genetic model
1. INTRODUCTION
Speciation through a reduction of gene flow and an
increase in genetic variance between populations can
occur by evolution along a spatial, temporal, behavioural
or morphological axis ( Fisher 1958). Temporal partitioning of activities, especially reproduction, promotes
assortative mating and may reduce competition for
resources among individuals separated in time. Assortative mating and disruptive selection have been regarded as
hallmarks of sympatric speciation (Bolnick & Fitzpatrick
2007). Hence, divergence of breeding times, or allochronic separation, constitutes a plausible mechanism for
sympatric speciation.
Allochronic speciation has been documented for
insects that partition mating activity among days (Santos
et al. 2007) or even among times of day (Miyatake et al.
2002; DeVries et al. 2008), and for other animal taxa
(fishes: Kirkpatrick & Selander 1979; bats: Adams &
Thibault 2006; birds: Friesen et al. 2007; corals:
Tomaiuolo et al. 2007). Examples of allochronic speciation are also expected in plants because assortative mating
by flowering time is inevitable as mating can occur only
among simultaneously open flowers ( Fox 2003; Weis et al.
2005; Savolainen et al. 2006; Gavrilets & Vose 2007).
Although most studies on plant speciation have focused
on prezygotic isolating mechanisms such as changes in
self-fertilization rate, sex expression and pollinator or
habitat specialization (Conner 2006), evidence also
exists for temporal isolation. Seasonally or diurnally
displaced flowering times ( Rathcke & Lacey 1985;
Antonovics 2006; Martin & Willis 2007) and/or patterns
* Author and address for correspondence: Division of Biology,
Imperial College London, Silwood Park Campus, Ascot, Berkshire
SL5 7PY, UK ([email protected]).
Received 27 June 2008
Accepted 21 July 2008
of pollinator visitation (e.g. Grant 1971; DeVries et al.
2008) represent important mechanisms of reproductive
isolation between sympatric plant species. For further
review see Coyne & Orr (2004).
Non-random mating can contribute to the origin of
reproductive isolation by increasing genetic variance,
which facilitates splitting of a population into two noninterbreeding parts. Two major forms of non-random
mating are inbreeding, due to mating between closely
related individuals with similar genotypes, and assortative
mating in which phenotypic resemblance determines the
probability of mating. Inbreeding increases homozygosity
within loci without causing linkage disequilibrium (associations between alleles among loci), while (positive)
assortative mating for a multigenic quantitative character
creates (positive) correlations between allelic effects on the
character both within and among loci. Classical models of
mating systems in an infinite population with no mutation
or selection show that strong inbreeding can at most
double the genetic variance, whereas strong assortative
mating can increase the genetic variance by up to a factor
of twice the (effective) number of loci, in comparison with
the genic variance at equilibrium under random mating
( Wright 1921a,b; Crow & Kimura 1970). Therefore,
assortative mating is potentially much more powerful than
inbreeding for increasing genetic variance and contributing to sympatric speciation. Assortative mating is
consequently an important factor in many models of
sympatric speciation (reviewed by Gavrilets 2004).
In finite populations with strong non-random mating,
random genetic drift and mutation can also contribute to
the formation of reproductively isolated groups. The first
multilocus simulations of flowering phenology (Crosby
1970) focused on reproductive character displacement
2723
This journal is q 2008 The Royal Society
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2724 C. Devaux & R. Lande
Incipient allochronic speciation
Table 1. Baseline and perturbed parameter values for the simulations.
parameter
baseline value
perturbed values
genomic mutation rate, U
mutational variance, s2m
number of loci, n
environmental variance, s2e
0.1
0.04
5
4
0.01
0.004, 0.08, 0.02, 0.01
10, 20, 50, 100
population size, N
individual phenology (days)
inbreeding depression, d
expected reproductive success
1000
5
0
constant
deterministic
2000, 500
10, 1
0.05, 0.20
low (ssZ0.1), high (ssZ0.33) stochasticity
Gaussian (qnZ0, snZ200), bimodal
(qbZ95, sbZ123.5)a
density dependent with low (KZ130), high
(KZ260) threshold
density dependent
symbols
PS1, PS2
PH1, PH2
IB1, IB2
ST1, ST2
PO1, PO2
DD1, DD2
a
The reproductive success on day d is expfKðd K qn Þ2 =ð2s2n Þg for the Gaussian distribution and expfKðd K qb Þ2 =ð2s2b ÞgC expfKðd C qb Þ2 =ð2s2b Þg for
the bimodal distribution; both distributions are truncated to zero outside the flowering season.
between pre-existing subspecies with reduced hybrid
viability, and did not observe splitting within populations,
in part owing to the short time scale of the simulations
(less than 100 generations) and the lack of new mutations.
van Dijk & Bijlsma (1994) and Kondrashov & Shpak
(1998) simulated assortative mating for a quantitative
character and found rapid origin of bimodal or disjunct
distributions; their findings are an unrealistic artefact of
initial conditions involving equal frequencies of two alleles
at each locus, and the lack of new mutations, as explained
further in the discussion of our results. Reproductively
isolated groups originated in models of finite populations
with mutation and either sexual selection ( Wu 1985) or
inbreeding (Higgs & Derrida 1992). Other models of the
origin of reproductive isolation by non-random mating,
primarily based on one or two loci, are reviewed by
Gavrilets (2004). Weis & Kossler (2004) and Weis (2005)
developed methods for estimating the phenotypic correlation between mates for flowering time, accounting for
the distribution of individual and population flowering
phenology, and Weis et al. (2005) suggested the need
for further development of multigenic models for the
evolution of flowering phenology, which is the purpose of
the present paper.
In contrast to previous models of sympatric speciation
based on disruptive sexual selection ( Nei et al. 1983; Wu
1985; Higashi et al. 1999; Kondrashov & Kondrashov
1999; Doebeli & Dieckmann 2003), we evaluate the
potential for sympatric speciation in the absence of
disruptive natural or sexual selection. Our model includes
neither disruptive selection caused by pollinators (e.g.
Gegear & Burns 2007) nor adaptation to ecological spatial
gradients (e.g. Stam 1983; Gavrilets & Vose 2007). We
model flowering time as a multilocus quantitative trait
undergoing non-selective assortative mating and spontaneous mutation in a finite population to investigate the
conditions promoting or inhibiting the emergence of
disjunct clusters in the population flowering phenology.
2. THE MODEL
We model flowering time as a quantitative trait in an
annual self-compatible species with discrete non-overlapping
generations and a constant population size. The model
applies for wind- or animal-pollinated species, assuming no
pollen limitation.
Proc. R. Soc. B (2008)
(a) Phenology
The flowering phenology of an individual plant consists of
10 flowers, each open on a single day, distributed over five
consecutive days with 1, 2, 4, 2 and 1 open flower. The
onset of individual flowering, o (the day its first flower
opens), is the integer part of a continuous flowering trait,
z, which is influenced by n completely additive diploid
loci and subject to environmental variance, s2e , with no
genotype–environment correlation or interaction.
Mutation occurs during meiosis independently for alleles
at each locus with probability mZ U =ð2nÞ per generation,
where U is the genomic mutation rate for flowering time.
The mutational variance is s2m Z U s2m , such that changes in
allelic effects of new mutations are normally distributed
with mean 0 and variance s2m . There is free recombination
between loci (see the algorithm in appendix A). The
baseline parameter values used in our simulations (table 1)
are within the range of empirical observations for
quantitative characters. In particular, the genomic
mutation rate, UZ0.1, was calibrated from mutation
accumulation experiments on quantitative characters of
plants including flowering time as twice the gametic
mutation rate (0.09 in Sprague et al. 1960 and 0.056 in
Russell et al. 1963). These high genomic mutation rates
can be explained by either a low mutation rate per locus
with a very large number of loci (e.g. more than 50 in
Laurie et al. 2004) or a high mutation rate per locus with a
moderate number of loci (of the order of 5 or 10) as found
for the onset of flowering (Lande 1981; see references in
Fox 2003) and other traits in plants (Kearsey & Farquhar
1998), for example due to unequal exchange in tandemly
duplicated gene families ( Frankham et al. 1978). The ratio
of the mutational to the environmental variance was taken
from Lande (1975) and Lynch (1988), while the value of
the environmental variance was chosen to match previous
observations for flowering time in herbaceous species
(Lande 1981; O’Neil 1997; Shaw & Chang 2006).
(b) Mating opportunities
All flowers open on a given day have an equal chance to be
pollinated by or to pollinate flowers open on the same day,
including self-fertilization at an expected rate given by the
inverse of the number of flowers open on that day. Mating
is thus random within days and assortative among
days ( Fox 2003; Weis et al. 2005). Each flower has a
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Incipient allochronic speciation
Proc. R. Soc. B (2008)
flowering season
20
0
–20
– 40
0
2500
7500
5000
generation
10 000
Figure 1. (a) Population flowering phenology as a function of
flowering season and generation, with grey representing days
with open flowers and white representing days without
flowers. (b) Full distributions of the population flowering
phenology are plotted at generations 1, 2500, 5500 and 8000.
The simulation illustrated here contained the longest lived
cluster among the 20 replicate simulations with the baseline
parameter values.
5000
2000
1000
500
200
DD1
DD2
0.05 0.60 0 0.85
PO1
PO2
0
IB1
IB2
ST2
PH2
ST1
0.60 0.25 0.80 0.30 0.95 0.55 0.80 0
PS2
PH1
100
probability
PS1
(d) Sensitivity analysis of the model
We investigated the sensitivity of the results by perturbing
single parameters from their baseline values (table 1). To
explore the role of random genetic drift, we either changed
the size of the population or altered the number of days a
plant could flower keeping the same total number of
flowers (table 1). Environmental stochasticity was incorporated by making the expected reproductive success of
flowers among days follow a lognormal distribution with
low or high standard deviation, ss, and no serial
correlation (table 1). Stabilizing and/or disruptive natural
selection on the expected reproductive success of flowers
was introduced in several ways. We implemented inbreeding depression of self-fertilized seeds by decreasing their
survival probability by a constant fraction (d; table 1).
With inbreeding depression, the total population size was
kept constant by replacing self-fertilized seeds that died
with others generated at random from the population. We
modelled stabilizing or disruptive selection corresponding
to pollinator abundance distributions or negative density
dependence of the expected reproductive success per se
due to pollinator limitation. The relative abundance of
pollinators throughout the flowering season was assumed
to be Gaussian or bimodal, symmetrical around day 0
and zero outside the season; parameters were chosen
to generate a maximum of 6 per cent difference in the
expected reproductive success over the season (table 1).
Alternatively, the expected reproductive success of flowers
was density dependent, as commonly found in natural
(a)
40
baseline
(c) Definition of incipient allochronic speciation
Starting from a monomorphic population with no genetic
variance, we quantified the emergence of incipient species,
defined as disjunct clusters in the population flowering
phenology that persisted for at least 100 generations. For
each of 20 replicate simulations lasting for 10 000
generations, we measured the number and duration of the
clusters, the generation at which they started, and the
duration of the disjunction between them. The edges (first
and last 5 days) of the flowering season of 101 days
were omitted because they sometimes included only a few
isolated individuals that mostly self-fertilized. We also
calculated the self-fertilization rate, the heritability, the
genic, genetic and phenotypic variances, and the phenotypic
correlation between mates in flowering time (appendix B).
(b)
40
20
0
–20
– 40
cluster duration
single ovule but a large number of pollen grains so that it
can contribute once as female and several times as male to
the next generation. The expected reproductive success of
flowers is constant within and zero outside the boundaries
of the flowering season of G50 days around the mid-day
numbered 0. Reproductive success is density independent, not pollen limited, and equal among all flowers in
the population regardless of the date of flowering. The
population size remains constant due to competition for
shared resources, such as space, that do not depend on the
date of flowering. These conditions are likely to apply for
species with no pollen limitation, such as wind-pollinated
species producing numerous pollen grains and animalpollinated species with pollinators providing constant
visitation rate per flower. The population size, N, is kept
constant by choosing N random ovules among all open
flowers in the season and fertilizing these by N random
pollen grains from simultaneously open flowers.
C. Devaux & R. Lande 2725
Figure 2. Duration of the longest lived clusters per replicate
simulation with baseline and perturbed parameter values
(table 1). Clusters are defined by disjunctions in the
population flowering phenology that persisted at least 100
generations, excluding the first and last 5 days of the season.
Probabilities of cluster formation among 20 replicate
simulations of 10 000 generations appear along the abscissa.
populations (Burd 1994; Knight et al. 2005), due to
decreasing pollinator visitation rate per flower with
increasing flower number (Geber & Moeller 2006).
Expected reproductive success remained constant up to
a threshold number of flowers, above which it decreased
with increasing flower number. It was described on day
d by the threshold function f1; if Fd % K and K =Fd ;
if Fd O K g, with K the threshold flower number and Fd
the number of flowers open on day d (table 1).
3. RESULTS
With baseline parameter values (table 1), the population
flowering phenology fluctuated greatly among generations
and displayed more than one distinct mode during most
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2726 C. Devaux & R. Lande
Incipient allochronic speciation
(a)
(b) 1.0
0.8
600
400
0.6
0.4
200
0.2
0
0
(c)
phenotypic variance
heritability
800
1000
160
650
(d ) 1.0
180
145
800
160
600
200
400
200
0
2500
5000
generation
7500
phenotypic correlation
genic and
genetic variances
1000
0.8
0.6
0.4
0.2
0
2500
5000
generation
7500
Figure 3. Evolution of (a) genic (lower line) and genetic variances, (b) heritability, (c) phenotypic variance and (d ) phenotypic
correlation between mates for one simulation with the baseline parameter values. Numbers above lines in (c) indicate the
durations of the six clusters that emerged in the population flowering phenology. The simulation depicted here contained the
longest lived clusters among the 20 replicate simulations (the same as given in figure 1).
of the 10 000 generations of each replicate. Gaps in the
population phenologies (i.e. days with no open flowers)
occurred within the first few hundred generations and a
small fraction of them persisted more than 100 generations
and were thus considered as disjunct clusters or incipient
species. Over half the replicate simulations (12 out of 20)
evolved disjunct clusters, and, on average, the longest
lived of these persisted 300 generations (figures 1 and 2).
During the first few hundred generations, mutations
accumulated and the heritability and the phenotypic
correlation between mates increased rapidly to values
near 1 (figure 3). The genetic and phenotypic variances
increased greatly under mutation accumulation and
assortative mating among individuals, while the genic
variance initially increased but then remained relatively
low. Peaks in the genetic and phenotypic variances
coincided with the formation of clusters in the population
flowering phenology (figure 3).
The role of random genetic drift in cluster formation
is demonstrated by comparison with simulations of
populations containing twice or half as many individuals
as the baseline simulations (NZ2000 or 500). The larger
population showed reduced frequency and duration of
clusters, whereas the smaller population displayed
increased frequency and increased persistence of clusters
(up to 1600 generations; PS1 and PS2, figure 2).
Clusters were hindered when plants flowered for twice
as many days as for the baseline simulations, whereas
long-lived clusters (up to 1600 generations) were
promoted when plants flowered for a single day (PH1
and PH2, figure 2). Under short individual flowering
phenology, gaps started near the tails of the population
phenology where a few self-fertilizing individuals could
initiate sub-populations that could slowly grow in size
Proc. R. Soc. B (2008)
and persist for hundreds of generations. Given that
random genetic drift is essential for cluster formation,
one might expect environmental stochasticity to have
similar effects; its influence was, however, more
complex. When weak, environmental stochasticity did
not change the probability of clusters but diminished
their duration, and when strong, it increased the
probability of clusters but not their duration (ST1 and
ST2, figure 2).
The population self-fertilized at an average rate of 1.7
per cent under baseline parameter values. Selection in the
form of inbreeding depression of self-fertilized seeds
prevented the formation of gaps (IB1 and IB2, figure 2).
Inbreeding depression lowered the reproductive success of
flowers at the tails of the population flowering phenology
selecting against the increase in the phenotypic variance,
thus reducing the probability of splitting (figure 4).
Stabilizing natural selection, beyond that implied by
the finite duration of the flowering season, was implemented by a Gaussian function describing the expected
reproductive success of flowers throughout the season.
This selection limited the genetic variance maintained
by mutation and assortative mating, and almost completely prevented the formation of gaps and clusters in
the population flowering phenology (PO1, figure 2).
As expected, disruptive natural selection implemented
by a bimodal function describing the expected reproductive success of flowers throughout the season produced
the longest lived clusters (up to 4300 generations; PO2,
figure 2) and the largest disjunctions between clusters
(up to 30 days). However, because the population size
was held constant, with equal competition among all
phenotypes, the bimodal distribution of flowering
phenology caused by disruptive selection was not stable,
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1.00
flowers
1600
0.75
1200
800
0.50
400
0.25
0
–5
0
5
flowering season
10
reproductive success
Incipient allochronic speciation
0
Figure 4. Population flowering phenology (solid line, left
ordinate) and expected reproductive success of flowers (right
ordinate) for one simulation at generation 1 under negative
density dependence (dotted line) and inbreeding depression
of self-fertilized seeds (dashed line) with all other parameters
at baseline values (table 1).
and the population phenology would eventually become
unimodal with its mean displaced towards one end of the
flowering season.
Competition for pollinators, involving decreasing
expected reproductive success of flowers with increasing
flower number, had opposite effects on cluster formation
depending on its intensity. Strong competition (with low
threshold above which reproductive success decreased)
prevented cluster formation (DD1, figure 2), whereas
weak competition increased their probability but not
their duration compared with the baseline simulations
(DD2, figure 2). During the first generations, density
dependence of reproductive success increased the
expected relative fitness of flowers at the edge of the
population flowering phenology, selecting for an increase
in the variance of flowering time (figure 4) and producing a
nearly even distribution of flowers throughout the season.
Under strong competition, on almost all days of the
season, there were more flowers than the threshold above
which reproductive success decreased (130 flowers
per day). These flowers thus had a reduced fitness
compared with the flowers open on days containing
fewer flowers than the threshold. This pattern prevented
modes in the population flowering phenology from
growing and also restricted the formation of gaps and
disjunctions. Under weak competition for pollinators, on
every day of the season, all flowers fell below the threshold
number above which reproductive success decreased (260
flowers per day) and all of them had equal expected fitness,
and thus the same evolutionary dynamics as in the baseline
simulations (DD2, figure 2).
4. DISCUSSION
We modelled multigenic mutation and non-selective
assortative mating for flowering time in a finite population,
using individual-based simulations to investigate conditions for incipient allochronic speciation in the absence
of disruptive natural or sexual selection. With baseline
parameter values (table 1), the population flowering
phenology fluctuated greatly, displaying transient multimodality and disjunctions lasting hundreds of generations
(figures 1–3). Formation of disjunct clusters was promoted by high mutational variance and multigenic
Proc. R. Soc. B (2008)
C. Devaux & R. Lande 2727
inheritance of flowering time, strong assortative mating
and populations of limited size.
In the following, we review classical formulae concerning
the (purely additive) genetic variance in a quantitative
character maintained by non-selective assortative mating
in an infinite population without mutation, or by neutral
mutation in a finite population. Consideration of these
formulae, and the interactions of the processes they describe
in relation to our results, allows us to derive approximate
conditions for incipient allochronic speciation by mutation
and non-selective assortative mating in a finite population.
We also briefly discuss selective mechanisms for reinforcement of this incipient reproductive isolation.
Fisher (1918) and Wright (1921b) analysed nonselective assortative mating for a quantitative character
in an infinite population with no natural selection or
mutation. Wright (1921b) showed that assortative mating
increases the genetic variance by creating positive
correlations of allelic effects among loci, and also to
some extent within loci. At equilibrium under assortative
mating, the genetic variance, s2g , exceeds the genic
variance that would be expressed under random mating
(with linkage equilibrium
and Hardy–Weinberg equiliP
brium), s2a Z 2 niZ1 s2i , where s2i is the variance of the
(purely additive) allelic effects at the ith locus, and
their ratio is
s2g
1
Z
;
s2a
1Kh2 rð1 K 1=ð2ne ÞÞ
where h2 is the heritability; r is the phenotypic correlation
between mates; and their product h2r is the genetic
correlation between mates in the assortatively mating
population. The effective number of loci,
n 2
P
si
i Z1
ne Z P
n
s2i
i Z1
is always less than the actual number of loci, n, when the
variance of allelic effects differs between loci ( Wright
1921b; Crow & Felsenstein 1968). Under strong assortative mating with a very high genetic correlation between
mates, h2r/1, the ratio of genetic to genic variance
achieves its maximum value, 2ne.
A popular model of mutation in quantitative characters
assumes that mutational changes in (purely additive)
effects of alleles have the same distribution for all alleles at
a locus, although the mutation rates and variance of
mutational changes in allelic effects may differ among loci.
Mutation creates a constant input of new additive genetic
variance each generation, s2m called the mutational
variance of the character, regardless of the amount already
present (Kimura 1965; Lande 1975). In a finite randomly
mating population, this mutation maintains an expected
genetic variance
s2g Z 2Ne s2m ;
where Ne is the effective population size (Clayton &
Robertson 1955; Lande 1980; Lynch & Hill 1986). Owing
to the assumption of random mating, this quantity
also equals the expected genic variance maintained by
mutation. In our model, if all flowers were open on a
single day, the population would closely approximate
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2728 C. Devaux & R. Lande
Incipient allochronic speciation
the ideal population with reproduction by random union
of gametes, under which NeZN, the actual population
size ( Wright 1931; Crow & Kimura 1970). Under nonselective assortative mating as in our model, with a
wide flowering season guaranteeing the absence of
selection and with other parameters at baseline values,
the genic variance in our simulations averaged only
approximately half the value predicted under random
mating (see below), indicating that, with respect to the
maintenance of quantitative genetic variance by mutation,
NezN/2.
The effective number of alleles per locus maintained by
selectively neutral mutations in a finite randomly mating
population is the inverse of the expected homozygosity or
identity by descent, 1C4Nem, where m is the allelic
mutation rate (Malécot 1948; Kimura & Crow 1964).
Under baseline parameter values, after an initial period
of several hundred generations during which mutations
accumulated in the population, both the heritability and
the phenotypic correlation between mates were very
high, with their product averaging approximately 0.95.
It should be noted that the high heritability achieved
under baseline parameter values exceeds heritabilities of
flowering time typically observed in natural populations
(O’Neil 1997; see Geber & Griffen 2003 for a review)
because simulated populations were not subject to
stabilizing selection or only to very weak selection (see
below). Such relaxed selection could occur in low-density
populations with a long reproductive season or in the
absence of competition, for example at the periphery
of a species’ geographical range. However, some of our
simulations with perturbed parameter values (table 1)
achieved heritabilities overlapping the upper end of
observed heritabilities for flowering time while still
allowing splitting of the flowering phenology.
With baseline parameters, the average genic variance
and effective number of alleles per locus were then
approximately 30 and 13, respectively, considerably
smaller than the predictions of 80 and 41 for a randomly
mating population with no selection. This occurred in part
because our simulations involved (very weak) stabilizing
selection caused by the limited duration of the flowering
season, reducing to zero the fitness of flowers open outside
the season. Greatly expanding the flowering season to
minimize stabilizing selection increased the average genic
variance to approximately 40 but hardly changed the
effective number of alleles per locus. Although mutation
parameters were identical among loci, the effective
number of loci averaged neZ4.84, below the actual
number of loci, nZ5, because random genetic drift
created differences in the variance of allelic effects
maintained at loci with equivalent mutation parameters.
The ratio of the genetic to genic variances averaged
approximately 7.4, whereas using the observed average
values of the genetic correlation between mates and the
effective number of loci predicted a ratio of 6.6.
Discrepancies between simulations and theoretical
predictions for the genic variance and the effective number
of alleles occur primarily because assortative mating
magnifies the impact of genetic drift. Nevertheless, the
observed and predicted values are in rough agreement. We
can therefore employ these classical formulae and our
simulation results to derive approximate conditions for
observing frequent disjunctions in the phenotype
Proc. R. Soc. B (2008)
distribution of a finite population with mutation and
assortative mating in the absence of selection. With a
sufficiently large effective population size and high
mutational variance, the population evolves a high genetic
correlation between mates, so we can roughly substitute
the expected genic variance maintained under random
mating into Wright’s formula for the maximum genetic
variance under strong assortative mating, using NezN/2,
to obtain the order of magnitude of the genetic variance,
2ne Ns2m . To evolve a high heritability and a high genetic
correlation between mates, this quantity must be much
greater than the sum of the environmental variance in the
date of first flowering, s2e , and the variance in the
individual flowering phenology, denoted as s2I . To allow
the phenotypic distribution to fit well within the flowering
season, and to avoid appreciable stabilizing selection
caused by its limited duration, say 2L or 2LC1 days, the
genetic variance should also be much less than L2. Thus,
we deduce sufficient conditions for frequent incipient
allochronic speciation by mutation and non-selective
assortative mating in a finite population in the absence
of disruptive selection,
L2 [ 2ne Ns2m [ s2e C s2I :
The validity of these conditions was confirmed by
additional simulations (table 1, not illustrated) perturbing
the genomic mutation rate, the number of loci (and
thus the mutation rate per locus), the mutational variance
and the duration of the reproductive season. In contrast to
models of sympatric speciation including disruptive
selection ( Waxman & Gavrilets 2005), the time scale
for emergence of clusters in the population flowering
phenology showed little influence of the mutation rate per
locus (ranging from 10K2 to 10K4), provided that
the genomic mutation rate for flowering time was in the
realistic range of UZ0.01–0.1 (Sprague et al. 1960;
Russell et al. 1963) and the above conditions were
satisfied. Nevertheless, if the conditions for splitting are
fulfilled, we expect that the formation of clusters in the
population phenology would be delayed in very large
populations with a small mutational variance.
Our model demonstrates how polygenic mutation and
random genetic drift interact with non-selective assortative
mating. Random genetic drift has previously been shown to
accelerate the evolution of reproductive isolation by
facilitating strong linkage disequilibrium between ecological
and mating strategies (Dieckmann & Doebeli 1999;
van Doorn & Weissing 2001) or fostering local adaptation
to adjacent habitats differing in flowering time (Stam 1983).
Previous models of sympatric speciation involved disruptive
natural or sexual selection (Kondrashov & Kondrashov
1999; Drossel & McKane 2000; Doebeli & Dieckmann
2003; Bürger et al. 2006; Doebeli et al. 2007; Gavrilets &
Vose 2007; Leimar et al. 2008). In those models, disruptive
selection and reproductive isolation are caused by
resource competition among similar phenotypes, displacement into distinct ecological niches or selective assortative
mating. In our model, ecological and mating strategies
are determined by a single trait, the flowering date of
individuals, undergoing non-selective assortative mating.
Kondrashov & Shpak (1998) found that strong nonselective assortative mating in an infinite population can
produce permanent splitting of the population into two
Downloaded from http://rspb.royalsocietypublishing.org/ on June 15, 2017
Incipient allochronic speciation
disjunct phenotypic classes. That result arises from their
assumption of two alleles at each locus with equal allelic
effects and frequencies among loci. Strong assortative
mating producing high correlations of allelic effects among
all loci then leads to the production of two phenotypic
classes: one concentrating alleles increasing the character
and the other concentrating alleles decreasing the
character (Crow & Kimura 1970, pp. 148–149). More
alleles per locus with asymmetries in allelic effects among
loci would produce a more nearly continuous phenotypic
distribution under strong assortative mating. For example,
in the extreme case of a normal distribution of allelic
effects at each locus, the equilibrium phenotypic distribution under any degree of non-selective assortative
mating would remain normal. By contrast, for a finite
population with multiple alleles per locus maintained by
mutation, we show that disjunctions in the phenotype
distribution are possible under strong non-selective
assortative mating, but are transient rather than permanent, similar to previous models of inbreeding (Higgs &
Derrida 1992) and sexual selection ( Wu 1985).
Weak stabilizing selection over the flowering season
prevented but did not eliminate incipient allochronic
speciation, whereas weak disruptive selection promoted
long-lived transient clusters (figure 2). This portrait
indicates that the distribution of reproductive opportunities
(e.g. relative abundance of pollinators or pollen density) also
determines the probability of allochronic speciation. Under
the Gaussian distribution of expected reproductive success
throughout the flowering season, often assumed in models
of plant–pollinator interactions (Elzinga et al. 2007),
allochronic speciation in most plant species would be
infrequent. Nevertheless, we found that even with this
expected stabilizing selection, large environmental stochasticity in pollinator abundance among days within flowering
seasons, as observed in natural communities (Motten 1986;
Kunin 1997; Larson & Barrett 1999), can facilitate
incipient allochronic speciation. Strong selection by densitydependent reproductive success due to pollinator limitation also prevented cluster formation in the population
flowering phenology.
The clusters we observed cannot be considered true
species because they are potentially interbreeding entities
(Dobzhansky 1951) that disappeared or coalesced after
some time. For the following reasons, we believe that a
transient cluster in our model represents incipient
allochronic speciation. Our definitions of cluster origination and duration are conservative because we define
gaps in the population flowering phenology by the absence
of open flowers and not only of reproducing flowers, and
these gaps must persist for at least 100 generations. Thus,
gene exchange between clusters is limited for a longer
period than reported here. With baseline parameter
values, the duration of clusters was increased up to 200
per cent when including days with single open flowers in
the disjunctions between clusters. Furthermore, our
model did not allow the joint evolution of other isolating
mechanisms, although some can occur within the duration
of the clusters we observed ( Turelli et al. 2001; Ramsey
et al. 2003; Primack et al. 2004; Hendry & Day 2005;
Savolainen et al. 2006; Franks et al. 2007). For example,
temporal isolation can arise from pollinator shifts or
preferences for specific floral traits (Coyne & Orr 2004;
Proc. R. Soc. B (2008)
C. Devaux & R. Lande 2729
Johnson 2006) or coevolution of pollinators and plants
(Bhattacharyay & Drossel 2005; Sargent & Otto 2006).
In conclusion, our model demonstrates that complex
processes of disruptive selection are not necessary for the
origin of species in sympatry and indicates that the
following factors may be associated with allochronic
speciation in plants: short individual flowering phenology
and a long flowering season; absence of pollinator
limitation; self-compatibility with low inbreeding
depression; high mutational variance and multigenic
inheritance of flowering time; and limited population size.
We thank J.-N. Jasmin for discussion and comments on this
manuscript. This work was supported by grants from the
National Science Foundation and the Royal Society.
APPENDIX A. ALGORITHM FOR FREE
RECOMBINATION AMONG LOCI
The algorithm, designed for fast calculation using
MATHEMATICA ( Wolfram Research, Inc. 2005), generates
haploid gametes from diploid genotypes, assuming
Mendelian inheritance and free recombination. The
algorithm first creates two pools of gametes from the
diploid genotypes of the individuals chosen as female and
male parents and then forms the diploid genotypes of their
progeny. The parental diploid genotypes are described by
two matrices (male and female) with N rows (individuals)
and 2n columns (alleles), where the first and second n
columns provide, respectively, the first and second N
haplotypes for n loci. Four steps are performed to generate
the female gametes from the matrix of the female diploid
genotypes, and similarly to generate the male gametes.
These four steps are described for the females only.
(i) A matrix of N rows and n columns containing 0s
and 1s is built to determine whether each allele
from the first haplotypes of the female genotypes
will contribute to progeny (1 means its inclusion
in gametes while 0 means its rejection). It is
created by partitioning a random vector of length
Nn into N rows containing 0s and 1s with a
frequency of 0.5. That random vector is built by
generating a random base 10 integer, uniformly
distributed between 0 and 1, with the number of
digits log10 ð2Nn K1Þ, and then converting this to a
binary integer,
vector Z IntegerDigits½IntegerPart½Random½Real;
f0; 2Nn K1g; Ceiling½Log½10; 2Nn K1; 2; Nn:
(ii) The complement of the N!n matrix obtained in
step (i) is then created by partitioning into N rows
a vector of length Nn in which 1s replace 0s from
the random vector. The complement matrix is
then appended to the matrix obtained in step (i) to
form an N!2n matrix.
(iii) An N!2n matrix containing the values of the
alleles to be included in the female gamete pool is
generated by multiplying element by element
(ij by ij ) the matrix containing 0s and 1s (obtained
in step (ii)) and the matrix containing the female
diploid genotypes.
(iv) The first n columns of the matrix from step (iii)
are added to its last n columns to form the N!n
matrix of haploid female gametes.
Downloaded from http://rspb.royalsocietypublishing.org/ on June 15, 2017
2730 C. Devaux & R. Lande
Incipient allochronic speciation
For example, assume two female diploid genotypes
with three loci {{a, b, c, d, e, f }, {g, h, i, j, k, l }}. The first
female is ad for locus 1, be for locus 2 and cf for locus 3
( gj, hk and il, respectively, for the second female). A
random inclusion/rejection vector, e.g. {0, 1, 1, 0, 0, 1},
and its complement {1, 0, 0, 1, 1, 0}, partitioned into the
matrices {{0, 1, 1}, {0, 0, 1}} and {{1, 0, 0}, {1, 1, 0}},
when appended, give the matrix {{0, 1, 1, 1, 0, 0}, {{0,
0, 1, 1, 1, 0}} (step (ii)). This matrix is then multiplied
element by element with the matrix of diploid genotypes
to yield {{0, b, c, d, 0, 0}}, {0, 0, i, j, k, 0}} (step (iii)),
and the addition of its first and last three columns
produces the haplotypes {{d, b, c}, { j, k, i}} for the two
gametes (step (iv)).
The above four steps are also performed to generate
the male gametes from the male diploid genotypes. The
female and male gamete pools are then appended into
an N!2n matrix to create the diploid genotypes of
the offspring population; the alleles inherited from
mothers and fathers are in the first and second n
columns, respectively.
APPENDIX B. FORMULAE FOR THE QUANTITATIVE
GENETIC PARAMETERS
Individual flowering is determined by a continuous trait, z,
influenced by n additive diploid loci and environmental
variance s2e . The onset of flowering of the k th individual,
ok, is defined by the floor (integer part) of its phenotypic
value zk Z gk C ek , where gk and ek are the breeding and
environmental values of individual k, respectively.
The
P
breeding value of the individual k is gk Z i ðaik C bik Þ,
with aik and bik being the values of the two alleles of the
i th locus.
The genetic and phenotypic variances
P of the popu2
Z
ð1=ðN
K1ÞÞ
2 and
lation are, respectively,
s
g
k ðgk K gÞ
P
2 , with N the population size
s2z Z ð1=ðN K 1ÞÞ k ðok K oÞ
and x the mean of x for the population. The genic
variance, which does not include correlations between
alleles within or among
P Ploci caused by assortative mating,
is s2a Z ð1=ðN K1ÞÞ i k ½ðaik K ai Þ2 C ðbik K b i Þ2 , where
ð ai C bi Þ=2 is the mean allelic effect at the ith locus. The
heritability is h2 Z s2g =s2z .
Assortative mating among individuals is measured by the
phenotypic correlation between
mates in flowering time,
P
rZ ð1=ðN K 1ÞÞð1=ðsof som ÞÞ i ðof i K of Þðomi Kom Þ, with ojk
the onset of flowering for the kth individual of parental sex j,
and oj its mean.
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