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**|**Proc Biol Sci**|**v.275(1652); 2008 December 7**|**PMC2605824

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Proc Biol Sci. 2008 December 7; 275(1652): 2723–2732.

Published online 2008 August 12. doi: 10.1098/rspb.2008.0882

PMCID: PMC2605824

Received 2008 June 27; Revised 2008 July 20; Accepted 2008 July 21.

Copyright © 2008 The Royal Society

This article has been cited by other articles in PMC.

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.

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 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 non-interbreeding 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 1921*a*,*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 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.

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.

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, ${\sigma}_{\text{e}}^{2}$, with no genotype–environment correlation or interaction. Mutation occurs during meiosis independently for alleles at each locus with probability $\mu =U/(2n)$ per generation, where *U* is the genomic mutation rate for flowering time. The mutational variance is ${\sigma}_{\text{m}}^{2}=U{\sigma}_{\mu}^{2}$, such that changes in allelic effects of new mutations are normally distributed with mean 0 and variance ${\sigma}_{\mu}^{2}$. 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, *U*=0.1, 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).

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 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 ±50 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 animal-pollinated 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.

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 10000 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).

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, *σ*_{s}, 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 (*δ*; 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 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 $\{1,\phantom{\rule{0.25em}{0ex}}if\phantom{\rule{0.25em}{0ex}}{F}_{d}\le K\phantom{\rule{0.25em}{0ex}}and\phantom{\rule{0.25em}{0ex}}K/{F}_{d},\phantom{\rule{0.25em}{0ex}}if\phantom{\rule{0.25em}{0ex}}{F}_{d}>K\}$, with *K* the threshold flower number and *F*_{d} the number of flowers open on day *d* (table 1).

With baseline parameter values (table 1), the population flowering phenology fluctuated greatly among generations and displayed more than one distinct mode during most of the 10000 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 and22).

(*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 **...**

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).

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 **...**

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 (*N*=2000 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 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).

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 **...**

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, 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).

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 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 (1921*b*) analysed non-selective assortative mating for a quantitative character in an infinite population with no natural selection or mutation. Wright (1921*b*) 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, ${\sigma}_{\text{g}}^{2}$, exceeds the genic variance that would be expressed under random mating (with linkage equilibrium and Hardy–Weinberg equilibrium), ${\sigma}_{\text{a}}^{2}=2{\sum}_{i=1}^{n}{\sigma}_{i}^{2}$, where ${\sigma}_{i}^{2}$ is the variance of the (purely additive) allelic effects at the *i*th locus, and their ratio is

$$\frac{{\sigma}_{\text{g}}^{2}}{{\sigma}_{\text{a}}^{2}}=\frac{1}{1-{h}^{2}\rho (1-1/(2{n}_{\text{e}}))},$$

where *h*^{2} is the heritability; *ρ* is the phenotypic correlation between mates; and their product *h*^{2}*ρ* is the genetic correlation between mates in the assortatively mating population. The effective number of loci,

$${n}_{\text{e}}=\frac{{\left(\sum _{i=1}^{n}{\sigma}_{i}\right)}^{2}}{\sum _{i=1}^{n}{\sigma}_{i}^{2}}$$

is always less than the actual number of loci, *n*, when the variance of allelic effects differs between loci (Wright 1921*b*; Crow & Felsenstein 1968). Under strong assortative mating with a very high genetic correlation between mates, *h*^{2}*ρ*→1, the ratio of genetic to genic variance achieves its maximum value, 2*n*_{e}.

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, ${\sigma}_{\text{m}}^{2}$ 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

$${\sigma}_{\text{g}}^{2}=2{N}_{\text{e}}{\sigma}_{\text{m}}^{2},$$

where *N*_{e} 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 the ideal population with reproduction by random union of gametes, under which *N*_{e}=*N*, the actual population size (Wright 1931; Crow & Kimura 1970). Under non-selective 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, *N*_{e}≈*N*/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, 1+4*N*_{e}*μ*, where *μ* 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 *n*_{e}=4.84, below the actual number of loci, *n*=5, 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 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 *N*_{e}≈*N*/2, to obtain the order of magnitude of the genetic variance, $2{n}_{\text{e}}N{\sigma}_{\text{m}}^{2}$. 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, ${\sigma}_{\text{e}}^{2}$, and the variance in the individual flowering phenology, denoted as ${\sigma}_{\text{I}}^{2}$. To allow the phenotypic distribution to fit well within the flowering season, and to avoid appreciable stabilizing selection caused by its limited duration, say 2*L* or 2*L*+1 days, the genetic variance should also be much less than *L*^{2}. 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,

$${L}^{2}\gg 2{n}_{\text{e}}N{\sigma}_{\text{m}}^{2}\gg {\sigma}_{\text{e}}^{2}+{\sigma}_{\text{I}}^{2}.$$

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 10^{−2} to 10^{−4}), provided that the genomic mutation rate for flowering time was in the realistic range of *U*=0.01–0.1 (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 non-selective assortative mating in an infinite population can produce permanent splitting of the population into two 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 density-dependent 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; 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.

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 2*n* 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.

- 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 ${\text{log}}_{10}({2}^{Nn}-1)$, and then converting this to a binary integer,$$\text{vector}=\text{IntegerDigits}[\text{IntegerPart}[\text{Random}[\text{Real},\{0,{2}^{Nn}-1\},\text{Ceiling}[\text{Log}[10,{2}^{\text{N}\text{n}}-1]]]],2,\text{N}\text{n}].$$ - 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*×2*n*matrix. - An
*N*×2*n*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. - 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.

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*×2*n* 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.

Individual flowering is determined by a continuous trait, *z*, influenced by *n* additive diploid loci and environmental variance ${\sigma}_{\text{e}}^{2}$. The onset of flowering of the *k*th individual, *o*_{k}, is defined by the floor (integer part) of its phenotypic value ${z}_{k}={g}_{k}+{e}_{k}$, where *g*_{k} and *e*_{k} are the breeding and environmental values of individual *k*, respectively. The breeding value of the individual *k* is ${g}_{k}={\sum}_{i}({a}_{ik}+{b}_{ik})$, with *a*_{ik} and *b*_{ik} being the values of the two alleles of the *i*th locus.

The genetic and phenotypic variances of the population are, respectively, ${\sigma}_{\text{g}}^{2}=(1/(N-1)){\sum}_{k}{({g}_{k}-\overline{g})}^{2}$ and ${\sigma}_{z}^{2}=(1/(N-1)){\sum}_{k}{({o}_{k}-\overline{o})}^{2}$, with *N* the population size and $\overline{x}$ the mean of *x* for the population. The genic variance, which does not include correlations between alleles within or among loci caused by assortative mating, is ${\sigma}_{\text{a}}^{2}=(1/(N-1)){\sum}_{i}{\sum}_{k}[{({a}_{ik}-\overline{{a}_{i}})}^{2}+{({b}_{ik}-\overline{{b}_{i}})}^{2}]$, where $(\overline{{a}_{i}}+\overline{{b}_{i}})/2$ is the mean allelic effect at the *i*th locus. The heritability is ${h}^{2}={\sigma}_{\text{g}}^{2}/{\sigma}_{z}^{2}$.

Assortative mating among individuals is measured by the phenotypic correlation between mates in flowering time, $\rho =(1/(N-1))(1/({\sigma}_{o\text{f}}{\sigma}_{o\text{m}})){\sum}_{i}({o}_{\text{f}i}-\overline{{o}_{\text{f}}})({o}_{\text{m}i}-\overline{{o}_{\text{m}}})$, with *o*_{jk} the onset of flowering for the *k*th individual of parental sex *j*, and $\overline{{o}_{j}}$ its mean.

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