We model the evolution of allochronic isolation between sympatric animal-pollinated plant species via displacement of their flowering times. The plant species share generalist pollinators and either produce inviable hybrid seeds or do not hybridize at all. Displacement of flowering times between reproductively isolated species reduces competition for pollinators and the formation of inviable hybrid seeds. Under strong pollen limitation, competition for pollinators causes rapid evolution of allochronic isolation both for hybridizing and nonhybridizing species. Under weak pollen limitation, allochronic isolation evolves rapidly for hybridizing species but more slowly for nonhybridizing species. Positive density-dependent pollinator visitation rate at low flower densities facilitates allochronic isolation under weak pollen limitation. Allochronic isolation among sympatric species sharing generalist pollinators could be common under any intensity of pollen limitation if the flowering season is sufficiently long.
Allochronic reproductive isolation between plant species, because of temporal displacement of their flowering phenologies is common in plant communities. It occurs before mating and can evolve to reinforce post-mating isolating barriers. Wind-pollinated populations adapted to different ecological conditions can undergo displacement of flowering times to reduce gene flow between them (McNeilly & Antonovics, 1968; Savolainen et al., 2006). Plant species with specialist pollinators can evolve divergent flowering phenologies if pollinators have distinct distributions of abundances within seasons. Generalist pollinators can promote divergence of flowering times, even among plant species that do not hybridize, because of competition for limited pollinator visitation (Macior, 1973; Heinrich, 1975, 1976; Reader, 1975; Stiles, 1977; Rathcke, 1983; Motten, 1986; van Dijk & Bijlsma, 1994; Moeller, 2004; Martin & Willis, 2007; Pascarella, 2007). This study clarifies the conditions under which competition for generalist pollinators produces allochronic isolation among species that are already reproductively isolated and ecologically distinct.
We model processes causing allochronic reproductive isolation in a simplified community composed of two or more plant species that share generalist pollinators. The plant species either do not hybridize or produce inviable hybrid seeds. We analyse the impact of pollen limitation and density-dependent pollination on the rate of evolution of allochronic isolation. Previous models have shown the potential for reproductive character displacement between reproductively isolated species. However, these models either were not designed for temporal assortative mating in flowering plants (Sved, 1981b; Spencer et al., 1986; Goldberg & Lande, 2006), assumed constant quantitative genetic parameters (Spencer et al., 1986), or included assortative mating in both time and space leading to ambiguity concerning the main factor accounting for displacement (van Dijk & Bijlsma, 1994). We employ a realistic quantitative genetic model of flowering time, parameterized with observations in plants and allowing evolution of positions and shapes of the population flowering phenologies (Devaux & Lande, 2008). In most simulations, we slightly displace the mean flowering times among species, because the initial condition of identical phenologies represents an unstable equilibrium (Sved, 1981a; Spencer et al., 1986). We find that species producing inviable hybrid seeds rapidly evolve allochronic isolation regardless of the intensity of pollen limitation. For species that do not hybridize, allochronic isolation evolves rapidly under strong pollen limitation, but more slowly under weak pollen limitation.
Inheritance of flowering time
We analyse annual plants with perfect flowers reproducing by outcrossing or partial self-fertilization. Each flower contains a single ovule but a large number of pollen grains. Generations are discrete and nonoverlapping. The potential reproductive season for all species lasts 101 days symmetrically distributed around day 0. An individual plant produces 10 flowers, each open for a single day during 10 consecutive days. The individual flowering phenology cannot evolve. The onset of flowering for an individual plant is the integer part of a continuous polygenic trait undergoing non-selective assortative mating, mutation and recombination. The trait follows a quantitative genetic model with parameters based on observations in plants (Devaux & Lande, 2008). Individual flowering onset is determined by n = 5 unlinked loci with purely additive effects, environmental variance , mutational variance , and genomic mutation rate U = 0.1.
Reproductive success of flowers
Evolution of the flowering phenologies depends on whether or not species hybridize, and on the availability and behaviour of generalist pollinators. The first model is designed for two closely related hybridizing species that produce inviable hybrid seeds, whereas the second model concerns two nonhybridizing species. Gene flow between species is absent. Clogging of stigmas with heterospecific pollen (Waser, 1978), pollen precedence (Howard, 1999), differences in pollen tube growth (Carney & Arnold, 1997) or gametic selection/meiotic drive (Haldane, 1932) are not considered. We also assume no cost for heterospecific pollination (Campbell & Motten, 1985) beyond the possible formation of inviable hybrid seeds for hybridizing species.
Pollination occurs by visits of generalist pollinators shared between plant species, with no pollen carry-over past the last flower visited. Expected reproductive success of flowers depends on the number of visits they receive and p, the frequency of conspecific pollen on the day the flowers are open. For both hybridizing and nonhybridizing species, each flower receives k visits, k being drawn at random from a Poisson distribution with mean pollinator visitation rate θ. For hybridizing species, reproductive success of flowers depends on the first visit by a generalist pollinator, which determines whether a zygote is formed by conspecific or heterospecific pollen. The formation of an inviable hybrid seed reduces flower reproduction to zero, with probability 1−p. For nonhybridizing species, reproductive success does not depend on the amount of heterospecific pollen but is determined by the probability that a flower receives at least one conspecific pollen grain among the k realized visits, 1−(1−p)k.
Pollen limitation arises when seed or fruit production is lower than it would be under full pollination, which can be detected empirically by pollen supplementation (Knight et al., 2005). Pollen limitation may be caused by low pollen production per flower or low pollinator abundance, the latter increasing competition among flowers for pollinator visitation. We model different intensities of pollen limitation, or competition for pollinators, among flowers of all species by altering the mean pollinator visitation rate (θ). This alteration is independent of flower density (Campbell et al., 1991) and can be produced by different overall abundances of pollinators. We assume pollinator abundances to be constant throughout the season and not regulated by flower abundances. Long-lived pollinators, such as birds, or long-lived colonies of short-lived pollinators, such as social bees, may be little influenced by seasonal fluctuations of the flower densities in plant communities. Pollinators that are highly mobile could rely on plants growing in other locations; for example, bees and humming birds can feed in several distant patches composed of different plant species (Jordano et al., 2006).
Density-dependent pollinator visitation rate
Although we assume that pollinator abundance is constant throughout the season, density of flowers may influence pollinator foraging behaviour (Rathcke, 1983). Positive density-dependent pollinator visitation rate can be caused by increased attraction of pollinators to large groups of flowers (Levin & Anderson, 1970). In contrast, negative density-dependent pollination is produced at high flower density when the handling time of flowers by pollinators, and pollinator abundances, are constant. Density dependence of pollination differs among pollinator species because of differences in mobility and efficiency (Rathcke, 1983). Accurate information is currently unavailable on the functional shape of the density dependence of pollination, and a diversity of functions has been fitted to experimental data (Bosch & Waser, 2001; Forsyth, 2003) and used in models (Haig & Westoby, 1988; Feldman et al., 2004; Bhattacharyay & Drossell, 2005; Morgan et al., 2005). We model density dependence of pollination using simple functions of the mean pollinator visitation rate θ and Fd, the total number of flowers open in the plant community on day d. Positive density dependence is given by
where a = 2 and b = 0.01. Negative density dependence is given by
where a = 10−3 and b = 5 × 10−4. Positive and negative density-dependent pollination operating together are described by the minimum of θ+ and θ− (Fig. 1).
Regulation of population sizes
We enforce constant population size Ni for plant species i (with exceptions explained below). We randomly choose Ni ovules among open flowers, according to their expected reproductive success, and mate them at random with conspecific flowers open simultaneously. When the expected reproductive success of flowers of species i is very low for a given generation, this procedure may fail to produce Ni ovules and thus lead to a decreased population size for that generation. For hybridizing species, the production of inviable hybrid seeds is not compensated at the plant level but at the population level. When species self-fertilize, and for both hybridizing and nonhybridizing species, inviable seeds produced by self-fertilization (i.e. a few seeds per generation) are not compensated at all. Note that, except for finite population sizes in our model, differences in abundances of plant species are identical to differences between species in flower production per plant or pollen production per flower.
Species with initially identical flowering phenologies represent an unstable equilibrium (Sved, 1981a; Spencer et al., 1986). To avoid waiting for demographic stochasticity and random genetic drift to displace initially identical flowering phenologies, and to focus on the phase during which divergent selection and character displacement happen, in most simulations we separated the initial mean flowering times among species by 2 days.
We simulated the evolution of flowering times for two species with equal population sizes (N = 2000), and the same initial genetic variance () and heritability (h2 = 0.71). For each individual, purely additive allelic effects were randomly sampled from a normal distribution with a mean 0 and variance 1. The evolution of allochronic isolation was investigated for nonhybridizing species, and for species producing inviable hybrid seeds, under different intensities of pollen limitation. The mean pollinator visitation rate (θ) was assigned one of five values between 1 and 100 (Waser, 1978; Campbell et al., 1991; Price et al., 2005; Stehlik et al., 2008), with or without dependence on flower densities. Five replicate simulations were run for 2500 generations when considering a mean of one visit per flower and longer for higher mean visitation rate (e.g. 7000 generations for θ = 100 visits per flower).
We also analysed asymmetries in population sizes between species (N = 1000 for one species and N = 2000 for the other species) and allowed self-fertilization with inbreeding depression of 10%. Finally, we extended the model to a larger number of plant species in the community while keeping the same duration for the reproductive season and the same parameters for the density dependence of pollination.
Displacement of the flowering phenologies between a focal species and one or more competing species was evaluated using an index of allochronic reproductive isolation adapted from Martin & Willis (2007). For the focal species i, with j representing all other species in the plant community, the index is:
where Hid is the proportion of heterospecific pollen received by species i on day d of the reproductive season and Fid is the number of flowers of species i open on day d, assuming all species to have the same pollen production per flower. For two equally abundant species i and j, RIi≈ RIj, which both equal 0.5 when flowering phenologies are identical and tend towards 1 as phenologies diverge through time. Significant differences in RI between species can be observed only when species are not equally abundant.
With a single actual visit per flower, the expected reproductive success for both hybridizing and nonhybridizing species is determined only by the frequency of conspecific pollen. Under strong pollen limitation with a mean of one visit per flower, the displacement of flowering phenologies is therefore similar for both types of species, and results are presented together. In contrast, under weak pollen limitation the expected reproductive success differs between hybridizing and nonhybridizing species, producing differences in the displacement of their flowering phenologies. Effects of weak pollen limitation combined with density-dependent pollination, mating system and relative abundances of species are illustrated for hybridizing and nonhybridizing species separately. For each set of parameters, one of five replicate simulations was chosen at random for the figures.
Starting from identical phenologies with no selection (no pollen limitation and no hybrid formation), the variance of flowering time among species relative to the phenotypic variance within species produced by t generations of random genetic drift equals h2t/(2Ne) (Lande, 1976), where Ne is the effective population size. Accounting for assortative mating by flowering time, Ne ≈ N/2 (Devaux & Lande, 2008). Under the genetic parameters of our model, displacement of flowering phenologies between species by about one day requires on the order of N generations, as we verified by simulations. However, under strong pollen limitation, simulations showed that the mean waiting time until significant displacement was much shorter. Larger or smaller (N = 1000 or N = 4000) population sizes for two equally abundant competing species had little impact on the dynamics of displacement in their flowering phenologies, once an initial displacement on the order of 2 days was achieved. We also found that allochronic isolation between species was not sensitive to the number of loci affecting flowering time (n = 5 or 10), when keeping the genomic mutation rate and total mutational variance constant, thus changing the mutation rate per locus.
Rapid allochronic isolation under strong pollen limitation
With a mean of one visit per flower, 37% (= e−1) of flowers received no pollen grain. Strong pollen limitation caused rapid displacement of flowering phenologies between both hybridizing and nonhybridizing species (Fig. 2a,b). Low pollinator visitation rates generated strong selection in opposite directions between plant species (Fig. 3a). The intensity of directional selection was determined by the overlap of the flowering phenologies between species, measured by RI, and thus decreased through time (Fig. 2). Once allochronic isolation was complete, flowering phenologies of species could drift through the reproductive season until their inner tails overlapped and selection displaced them again, or their outer tails encountered the beginning or the end of the reproductive season (Fig. 2a,b).
Under strong pollen limitation, neither differences in abundances between species nor self-fertilization of species affected the rapid completion of allochronic isolation within a few dozen generations, for hybridizing or nonhybridizing species. With unequal species abundances, the less abundant species had lower initial allochronic isolation. It was displaced faster than the more abundant species; nevertheless, the two species rapidly became completely isolated (Fig. 2c).
Positive density-dependent pollination led to small displacement of flowering phenologies, even under strong pollen limitation (Fig. 2d), but did not prevent complete allochronic isolation between species. Strong stabilizing selection against the tails of the flowering phenologies (Fig. 3b) maintained a short flowering period for each species, which compensated for the smaller displacement of their phenologies. In contrast, negative density-dependent pollination generated disruptive selection within species (Fig. 3c), which greatly increased the flowering periods of both species and their final displacement (Fig. 2e). Positive and negative density-dependent pollination together generated strong disruptive selection within species (Fig. 3d) and rapid displacement of flowering phenologies. This also produced persistent selection against the outer tails of the flowering phenologies, which prevented their complete displacement (Fig. 2f).
Allochronic isolation under weak pollen limitation
Under weak pollen limitation, regardless of density-dependent pollination, all flowers received at least one pollen grain. For hybridizing species, the expected reproductive success of flowers thus equalled the frequency of conspecific pollen on any given day. Allochronic isolation evolved rapidly for these species and was not affected by the intensity of pollen limitation or the density dependence of pollination (Fig. 4a).
In contrast, for nonhybridizing species weak pollen limitation delayed allochronic isolation and decreased its maximal value (Fig. 4a). Under weak pollen limitation, almost all flowers of two nonhybridizing species open on any day had equal reproductive success. Flowering phenologies could thus drift through the season, whereas their variance steadily increased by mutation accumulation. After a few hundred generations, the flowering phenologies of the two species together occupied the entire reproductive season. Phenologies eventually experienced weak stabilizing selection at the edges of the reproductive season that displaced them back towards the middle of the season. The displacement of the flowering phenologies between species occurred in three steps. First, the flowering period of one species was suddenly reduced by selection against one of its tails because of low frequencies of conspecific pollen (Fig. 5a). Directional selection then increased for that species during the next few hundred generations as the frequency of heterospecific pollen increased in its tail overlapping with the other species (Fig. 5b,c). Finally, as flowering phenologies of the two species became substantially displaced, their overlapping tails were strongly selected against (Fig. 5d), until displacement was completed with a gap between flowering phenologies.
Neither unequal abundances nor self-fertilization affected allochronic isolation between nonhybridizing species under weak pollen limitation (Fig. 4b,c). Positive density-dependent pollination promoted reproductive isolation (Fig. 4d) as it increased pollen limitation and thus selection against the tails of the flowering phenologies for both species. In contrast, negative density-dependent pollination had little impact on the evolution of allochronic isolation under weak pollen limitation (Fig. 4d).
Allochronic isolation in a four-species community
In simulations with four species, the parameters of density-dependent pollination were identical to those with two species. With more plant species initially packed into the middle of the reproductive season, divergent selection between overlapping phenologies was stronger, whereas stabilizing selection against the tails of the phenologies was weaker. Negative density-dependent pollination combined with strong pollen limitation caused rapid extinction of the two species in the middle of the reproductive season, for both hybridizing and nonhybridizing species.
In communities of four hybridizing species, complete allochronic isolation evolved rapidly regardless of the intensity of pollen limitation or positive density-dependent pollination. With four nonhybridizing species, allochronic isolation also evolved rapidly under strong pollen limitation (Fig. 6a), regardless of positive density-dependent pollination. Weak pollen limitation delayed allochronic isolation for nonhybridizing species and even prevented it for two of the four species (Fig. 6b,c).
We simulated plant species that share generalist pollinators to investigate causes of diversification of flowering phenologies among species in a community. Flowering time was modelled as a polygenic trait undergoing mutation, recombination and assortative mating. Our study focused on selective factors influencing allochronic isolation between plant species. Nonhybridizing species and species that produce inviable hybrid seeds were analysed. To account for pollen limitation and density dependence of pollination, reproductive success of flowers depended on the quantity and specific composition and amount of pollen deposited. We found that the production of inviable seeds or strong pollen limitation caused rapid evolution of complete allochronic isolation between species. Below, we discuss the mechanisms of reproductive character displacement and consider their potential operation in plant communities.
Classical ecological character displacement evolves because of resource competition among species (McArthur & Levins, 1967; Lawlor & Maynard Smith, 1976;Roughgarden, 1976; Slatkin, 1980; Taper & Case, 1992). Similarly, competition for pollen among plant species sharing generalist pollinators arises because of the limited number of flowers pollinators can visit. This competition depends on the proportions of conspecific and heterospecific pollen on a given day and may also depend on total flower density. The initial condition of two species with identical flowering phenologies constitutes an evolutionary unstable equilibrium (Sved, 1981a; Spencer et al., 1986), so that only random genetic drift and demographic stochasticity can begin to separate flowering phenologies before divergent selection operates. In most of our simulations, we therefore initially displaced mean flowering times among species by 2 days. This allowed us to focus on the most rapid phase of reproductive character displacement caused by divergent selection. Some initial displacement in flowering phenologies seems likely for species that are already completely reproductively isolated.
For nonhybridizing species, reproductive success of flowers depends on the deposition rate of conspecific pollen pθ, where θ is the mean number of pollinator visits and p the frequency of conspecific pollen. Smaller pθ increases pollen competition and generates higher expected rates of displacement. In our model, strong pollen limitation produces complete allochronic isolation between nonhybridizing species in a few dozen generations. Displacement also evolves rapidly in a rare species competing for pollen with a common species. The rapidity of reproductive character displacement and its dependence on the relative abundances of competing species have already been observed in a variety of models (Sved, 1981a; Spencer et al., 1986; van Dijk & Bijlsma, 1994; McPeek & Gavrilets, 2006). The rate of displacement also depends on heritability and assortative mating for flowering time, which are both high but substantially below one in our model, consistent with observations in natural populations subject to stabilizing selection (O’Neil, 1997; Geber & Griffen, 2003). Our results demonstrate that pollen limitation and density-dependent pollination also strongly influence the evolution of allochronic isolation in plants. Under weak pollen limitation, the displacement of flowering phenologies between nonhybridizing species evolves more slowly because a larger separation of phenologies is required to cause substantial divergent selection.
For hybridizing species, the production of unfit hybrid offspring, mediated by assortative mating, can cause evolutionary reinforcement of reproductive isolation (Crosby, 1970; Caisse & Antonovics, 1978; Felsenstein, 1981; Butlin, 1987; Kirkpatrick, 2000; Gavrilets & Vose, 2007). For hybridizing plant species that are already reproductively isolated, displacement of flowering times does not occur to reduce gene flow. Displacement rather prevents the formation of inviable hybrid seeds, and also reduces interspecific competition for pollination. In our model of hybridizing species, flowering phenologies diverge more rapidly than for nonhybridizing species, as previously shown (van Dijk & Bijlsma, 1994; Goldberg & Lande, 2006). We also show that the evolution of allochronic isolation between hybridizing species is nearly independent of the intensity of pollen limitation because the expected reproductive success of a flower is determined by the first visit of a generalist pollinator.
Partial displacement of flowering phenologies commonly occurs among plant species in a community that shares generalist pollinators (Heinrich, 1975, 1976; Reader, 1975; Stiles, 1977; Rathcke, 1983; Motten, 1986; Moeller, 2004). Lack of information on pollen limitation and the potential for hybridization among species in these studies prevents clear conclusions concerning the mechanisms involved. Our theoretical results suggest that partial displacement of flowering phenologies can be explained by observations of intermediate intensity of pollen limitation (Campbell et al., 1991; Knight et al., 2005; Price et al., 2005; Wilson et al., 2006). Complete displacement of phenologies also can be prevented by stabilizing selection because of a limited reproductive season, especially under weak pollen limitation. Strong pollen limitation can circumvent the constraint of a limited reproductive season, allowing complete displacement of flowering phenologies, by reducing the variance in flowering time within species.
With pollen limitation, density dependence of pollination is more likely to be negative than positive. Negative density-dependent pollination results from a simple ubiquitous mechanism, i.e. pollinators cannot visit all flowers when those are numerous. It must be associated with strong pollen limitation to generate disruptive selection within species and rapid displacement of flowering phenologies among species. In our model with strong pollen limitation, positive density-dependent pollination decreases both the duration and the displacement of flowering phenologies. In contrast, with weak pollen limitation, positive density-dependent pollination facilitates displacement of flowering phenologies by reducing reproductive success in the outer tails of phenologies.
The number of ecologically similar species that can coexist in a constant environment is determined by the niche overlap between species along a limiting resource axis (McArthur & Levins, 1967). In our model this axis corresponds to flowering time, with plant species competing for generalist pollinators. In our four-species model with strong negative density-dependent pollination, extinction of two species occurs because the limited reproductive season cannot accommodate the displacement of flowering phenologies among all the species. However, quantitative comparisons cannot be made to classical models of species packing because in our model the regulation of population sizes is largely independent of niche separation between species. Thus, with no pollen limitation or density-dependent pollination, any number of nonhybridizing species, and numerous hybridizing species can coexist.
This study elucidates the conditions causing allochronic isolation between animal-pollinated species that are reproductively isolated. For plant species sharing generalist pollinators, divergence of flowering phenologies is driven by the formation of inviable hybrid seeds, competition for pollen because of pollen limitation, and density-dependent pollination.
We thank J.-N. Jasmin and the reviewers for comments that helped to clarify the manuscript. This work was supported by grants from the US National Science Foundation and the Royal Society of London.