Céline Devaux
email: celine.devaux@umontpellier.fr

Emmanuelle Porcher

Russell Lande

Mating systems and avoidance of inbreeding depression as evolutionary drivers of pollen limitation in animal pollinated self-compatible plants

Keywords: embryo quality, evolutionarily stable equilibria, inbreeding depression, individual flowering phenologies, geitonogamy, pollen quantity, pollen limitation, pollinator limitation

Mating systems and avoidance of inbreeding depression as evolutionary drivers of pollen limitation in animal-pollinated self-compatible plants

Celine Devaux, Emmanuelle Porcher, Russell

INTRODUCTION

Limitation of seed or fruit production is widespread in animal pollinated plants and has three non-exclusive proximate causes related to pollinators, embryo quality and resource availability [START_REF] Bierzychudek | Pollinator limitation of plant reproductive effort[END_REF][START_REF] Ashman | Pollen limitation of plant reproduction: ecological and evolutionary causes and consequences[END_REF][START_REF] Knight | Pollen limitation of plant reproduction: pattern and process[END_REF]. Limited reproductive success of a plant can be caused by insufficient pollen receipt: some flowers or some of their ovules remain unfertilized because pollinators are scarce, little attracted to the plant, or visit only a fraction of the flowers open [START_REF] Geber | The relationship of plant size to self-pollination in Mertensia ciliata[END_REF][START_REF] Charlesworth | Evolution of low female fertility in plants -Pollen limitation, resource-allocation and genetic load[END_REF][START_REF] Burd | Ovule packaging in stochastic pollination and fertilization environments[END_REF][START_REF] Snow | The ecology of geitonogamous pollination In: Floral Biology. Studies on floral evolution in animal-pollinated plants[END_REF][START_REF] Moeller | Reduced pollinator service and elevated pollen limitation at the geographic range limit of an annual plant[END_REF][START_REF] Schreiber | Evolutionary and ecological consequences of multiscale variation in pollen receipt for seed production[END_REF], or because pollinators carry insufficient pollen loads (e.g. generalist pollinators depositing heterospecific rather than conspecific pollen, [START_REF] Waser | Competition for hummingbird pollination and sequential flowering in two Colorado wildflowers[END_REF]. Even with sufficient conspecific pollen receipt, some seeds on a plant may not mature because embryos are of low quality due to inbreeding [START_REF] Husband | Evolution of the magnitude and timing of inbreeding depression in plants[END_REF][START_REF] Angeloni | Meta-analysis on the association of population size and life history with inbreeding depression in plants[END_REF] or outbreeding depression [START_REF] Whitlock | A systematic review of phenotypic responses to between-population outbreeding[END_REF], or because female resources are limited and induce high rates of abortion in otherwise viable seeds or fruits [START_REF] Willson | Adaptive design of floral display in Asclepias syriaca L[END_REF].

Theoretical work has explored ultimate causes of pollen limitation and showed that it can evolve as a result of trade-offs among three costly reproductive functions for plants: ovule production, seed provisioning, and pollinator attraction. Studies of these functions and their trade-offs revealed that within or among plant variation in pollination rates can generate the frequently observed pollen limitation [START_REF] Bierzychudek | Pollinator limitation of plant reproductive effort[END_REF]; [START_REF] Haig | On limits to seed production[END_REF][START_REF] Burd | Ovule packaging in stochastic pollination and fertilization environments[END_REF][START_REF] Burd | The Haig-Westoby model revisited[END_REF][START_REF] Schreiber | Evolutionary and ecological consequences of multiscale variation in pollen receipt for seed production[END_REF], although this is sometimes debated [START_REF] Rosenheim | Parental optimism versus parental pessimism in plants: how common should we expect pollen limitation to be?[END_REF][START_REF] Rosenheim | Modest pollen limitation of lifetime seed production is in good agreement with modest uncertainty in whole-plant pollen receipt (A reply to Burd)[END_REF][START_REF] Burd | Pollen limitation is common-Should it be?[END_REF]. Much of this theory was devoted to the role of environmental stochasticity, i.e. random spatial variation or random temporal variation in pollen deposition rate affecting all individual plants in a population simultaneously, as a primary determinant of reproductive trade-offs and consequent pollen limitation [START_REF] Burd | Ovule packaging in stochastic pollination and fertilization environments[END_REF][START_REF] Burd | The Haig-Westoby model revisited[END_REF][START_REF] Richards | Variation in pollination: causes and consequences for plant reproduction[END_REF][START_REF] Schreiber | Evolutionary and ecological consequences of multiscale variation in pollen receipt for seed production[END_REF]. In doing so, most models took an ecological rather than a population genetics point of view, emphasizing maternal fitness only (e.g. [START_REF] Haig | On limits to seed production[END_REF][START_REF] Burd | Bateman principle and plant reproduction -The role of pollen limitation[END_REF][START_REF] Burd | Ovule packaging in stochastic pollination and fertilization environments[END_REF] but see [START_REF] Bell | On the function of flowers[END_REF][START_REF] Charlesworth | Evolution of low female fertility in plants -Pollen limitation, resource-allocation and genetic load[END_REF][START_REF] Burd | What does the male function hypothesis claim?[END_REF][START_REF] Harder | Floral adaptation and diversification under pollen limitation[END_REF][START_REF] Thomson | Using pollination deficits to infer pollinator declines: Can theory guide us?[END_REF]. They generally gave little consideration to seed quality and mating system, which are however major evolutionary drivers of plant reproductive strategies [START_REF] Ashman | Pollen limitation of plant reproduction: ecological and evolutionary causes and consequences[END_REF]Devaux et al. 2014a) and have proved to correlate with pollen limitation (e.g. self-compatibility for animal-pollinated plant species, [START_REF] Larson | A comparative analysis of pollen limitation in flowering plants[END_REF][START_REF] Knight | Pollen limitation of plant reproduction: pattern and process[END_REF][START_REF] Alonso | Is reproduction of endemic plant species particularly pollen limited in biodiversity hotspots?[END_REF]).

Here we complement previous evolutionary studies of pollen limitation by examining how selection on the mating system can also cause limitation of seed production and fitness in self-compatible animal-pollinated plants. Our aim is not to question the well-established role of environmental stochasticity in the evolution of pollen limitation (see reviews cited above), but instead to demonstrate that additional mechanisms are likely to be at play. We focus on floral display size, the number of flowers simultaneously open on a plant, a key trait that influences pollen limitation [START_REF] Dudash | Plant size effects on female and male function in hermaphroditic Sabatia angularis (Gentianaceae)[END_REF][START_REF] Dudash | Variation in pollen limitation among individuals of Sabatia angularis (Gentianaceae)[END_REF] via its role in pollinator attraction (e.g. [START_REF] Willson | Pollinator limitation, fruit production, and floral display in pawpaw (Asimina triloba)[END_REF][START_REF] Bauer | Floral traits influencing plant attractiveness to three bee species: Consequences for plant reproductive success[END_REF]) and between-flower self-pollination (geitonogamous selfing as in [START_REF] Lloyd | Self-fertilization and cross-fertilization in plants. 2. The selection of selffertilization[END_REF][START_REF] Lau | Selection through male function favors smaller floral display size in the common morning glory Ipomoea purpurea (Convolvulaceae)[END_REF][START_REF] Karron | Effects of floral display size on male and female reproductive success in Mimulus ringens[END_REF] Unlike previous models, we intentionally do not address the role of seed provisioning or environmental stochasticity but instead focus on the role of mating system on the emergence of pollen limitation. Yet, we incorporate the minimal amount of demographic stochasticity to portray the basic elements of pollination ecology, i.e. random variation in daily floral display, pollinator visitation and pollinator behavior, and include them in a mechanistic model of pollinator foraging behavior. Our model distinguishes two ultimate causes of pollen limitation, i.e. whether plants at evolutionarily stable equilibria (Devaux et al. 2014a) are pollen limited because they produce too few flowers in a day to attract pollinators or because they produce too many flowers to avoid geitonogamous selfing and inbreeding depression.

METHOD

We examine pollen limitation at an evolutionarily stable equilibrium (ESS) predicted by the model of Devaux et al. (2014a). The total number of flowers produced by a given plant throughout the flowering period is Poisson distributed with mean . These flowers are open sequentially according to a normal distribution with standard deviation (see e.g. Fig. 1). correspond respectively to seed production via selfing, seed production via outcrossing and cross-fertilization of ovules on other plants.

Standard deviation in individual

We choose a mechanistic model of pollination, instead of phenomenological functions generally used to describe the relationships between ), ) and

). With this model (described below), ),

) and ) depend on individual flowering phenology, pollinator abundance and pollinator behavior. Trade-offs between fitness components, such as pollen discounting, are emerging properties instead of being hypothesized. We make several simplifying but realistic assumptions to keep the mechanistic model of pollination general (Devaux et al. 2014a) and introduce a minimal amount of demographic stochasticity. To derive equation (1) we assume that:

(i) individual pollinators are generalist, such that their density does not depend on the floral density of the focal plant species but on the density of all plant species in a community;

this assumption ensures that there is no environmental stochasticity, (ii) pollinators are constant, i.e. faithful to the focal plant species within a foraging bout [START_REF] Chittka | Flower constancy, insect psychology, and plant evolution[END_REF] smaller floral displays [START_REF] Snow | The ecology of geitonogamous pollination In: Floral Biology. Studies on floral evolution in animal-pollinated plants[END_REF][START_REF] Chittka | Flower constancy, insect psychology, and plant evolution[END_REF][START_REF] Ohashi | Behavioral responses of pollinators to variation in floral display size and their influences on the evolution of floral traits[END_REF][START_REF] Harder | Beyond floricentrism: The pollination function of inflorescences[END_REF][START_REF] Ishii | The size of individual Delphinium flowers and the opportunity for geitonogamous pollination[END_REF]), (vi) the sequence of flower visitation by pollinators on a plant is either random (hereafter "random-rank") or constant (e.g. always visiting flowers from bottom to top, hereafter "constant rank"). These two extreme behaviors are likely to encompass the variability of visiting patterns across pollinators (with e.g. a higher tendency for constant sequences in bumblebees; [START_REF] Best | Pollinator foraging on foxglove (Digitalis purpurea): a test of a new model[END_REF][START_REF] Harder | Mating cost of large floral displays in hermaphrodite plants[END_REF][START_REF] Harder | The mating consequences of sexual segregation within inflorescences of flowering plants[END_REF] and floral architectures (e.g. more constant sequences in racemes vs. umbels; Jordan and Harder 2006), (vii) a single pollinator visit is sufficient to fertilize all ovules on a flower, which requires few ovules per flower, or large and constant pollen loads of A pollen grains for pollinators that groom little among visited flowers (pollen saturation as in de [START_REF] De Jong | Geitonogamy: the neglected side of selfing[END_REF] with A pollen grains on the pollinator body). This assumption, although not always verified [START_REF] Harder | Evolutionary options for maximizing pollen dispersal of animal-pollinated plants[END_REF], best models nectarivorous pollinators [START_REF] Castellanos | Pollen transfer by hummingbirds and bumblebees, and the divergence of pollination modes in Penstemon[END_REF],

(vii) individual pollinators visit a given flower only once, which is observed frequently [START_REF] Best | Pollinator foraging on foxglove (Digitalis purpurea): a test of a new model[END_REF][START_REF] Goulson | Foraging bumblebees avoid flowers already visited by conspecifics or by other bumblebee species[END_REF][START_REF] Ohashi | How long to stay on, and how often to visit a flowering plant? A model for foraging strategy when floral displays vary in size[END_REF] with (,max = 1 + ) 2 ⁄ when = = 1. This second measure is used to portray situations for which all flowers receive sufficient pollen to fertilize all ovules (no pollinator shortage), yet seed production is still increased by manual supplementation with outcross pollen [START_REF] Eckert | Plant mating systems in a changing world[END_REF]. In other words, this second measure is positive only if inbreeding depression, and not pollinator availability, is responsible for fitness loss. We further assess whether an ESS occurs at the strategy ′ that both maximizes total plant fitness ′ and minimizes pollen limitation -. ) ⁄ , given the constraints generated by pollinators. If ≤ . , we determine whether the evolutionary equilibrium (or equilibria when several exist) corresponds to a larger daily floral display (a shorter plant flowering period < . ) or a smaller display (a longer plant flowering period > . ) than the phenology that minimizes pollen limitation at ′ .

We focus on a reference case chosen to match typical observations (see Devaux et al. 2014a for details and Table S1 [Supplementary Information]) and then vary some parameters that most strongly influence pollinator behavior and mating system evolution. In the reference base case, plants produce = 100 flowers, pollen carry-over of pollinators is 1 -% = 0.75. Limitation occurs in pollinator abundance with $ = 100 pollinators per day, 6 = 100 pollen grains on pollinators body and a probability that a pollinator departs a plant after visiting a flower of & = 0.33. Limitation also occurs in pollinator attraction via a positive relationship between the number of visits 8 9) and daily floral display 9 8 9) = 9 : 9 + 1);1 + < =>? @A ⁄

, with < = 50, B = 0.1 (Fig. S1 [Supplementary Information]).

Inbreeding depression either evolves with the selfing rate (i.e. its purging is possible via lower S1 [Supplementary Information]).

RESULTS

None of the evolutionarily stable equilibria maximizes mean total fitness ( < . [Supplementary Information] for the random-vs constant-rank visitation sequence; Table 1). Differences between pollinator visitation patterns of flowers on a plant are intensified with increased flower production per plant. A random-rank visitation sequence generates multiple stable equilibria: flowering phenologies of a few weeks characterized by pollen limitation mostly due to the low quality of selfed seeds rather than a shortage of pollinator visits, and much longer flowering phenologies with high pollen limitation due mainly to low pollinator attraction, rather than pollen genetic composition. In contrast, a constant-rank visitation sequence generates only extended flowering phenologies favoring outcross pollination; these phenologies are strongly pollen limited because of low pollinator attraction (Figs. 3B andS3B [Supplementary Information] for the random-vs constant-rank visitation sequence; Table 1).

DISCUSSION

This study complements earlier theoretical work on pollen limitation by highlighting a potential additional mechanism driving evolution of limited seed production in natural populations of animal-pollinated plants. We show that both the quantity and genetic composition of pollen receipt of self-compatible animal-pollinated plant species control the evolutionary equilibrium flowering phenologies and consequent pollen limitation. Mean fitness is not maximized, and pollen limitation is never minimized at equilibrium. Nonmaximization of mean fitness is explained by frequency-dependent selection, which violates the assumption of constant genotypic fitnesses required for [START_REF] Wright | Evolution in Mendelian populations[END_REF][START_REF] Wright | Evolution and the genetics of populations[END_REF] principle of evolutionary maximization of mean fitness. In this model, as in earlier ones (e.g. [START_REF] Bierzychudek | Pollinator limitation of plant reproductive effort[END_REF]; [START_REF] Haig | On limits to seed production[END_REF][START_REF] Burd | Ovule packaging in stochastic pollination and fertilization environments[END_REF][START_REF] Burd | The Haig-Westoby model revisited[END_REF][START_REF] Harder | Floral adaptation and diversification under pollen limitation[END_REF][START_REF] Schreiber | Evolutionary and ecological consequences of multiscale variation in pollen receipt for seed production[END_REF] pollen limitation is an evolved emergent property; however, here pollen limitation is constrained by a trade-off between the maternal self and paternal outcross components of fitness that involves both genetic and ecological constraints (Devaux et al. 2014a), whereas most previous models only considered ecological constraints. The ecological constraints include pollinator abundance, pollinator attraction to large floral displays, expected pollinator bout length and pollinator visitation patterns of flowers on a plant; the genetic constraints include inbreeding depression on plant viability and Fisher's automatic advantage of selfing.

Our theoretical approach is mechanistic, which allows an analysis of the causes of pollen limitation among several parameters describing pollinator behavior, floral traits and inbreeding depression. However, as all models, it relies on several necessary simplifying assumptions and omits some potentially important ecological mechanisms that can also influence the evolution of pollen limitation. In the following, we first discuss some implications of our results and identify predictions that could be tested in natural populations.

We then outline the main limitations of our approach and some useful perspectives to broaden our evolutionary understanding of pollen limitation.

Relevance of our model to study pollen limitation in natural populations

The predicted flowering phenologies depend on the pollinator foraging behavior among flowers on a plant. These phenologies are expected to be longer when different pollinators visit flowers of a plant in the same order, and generate higher selfing rates, as experimentally found for bees [START_REF] Jordan | Manipulation of bee behavior by inflorescence architecture and its consequences for plant mating[END_REF], and consequently higher pollen limitation than when pollinators visit flowers on a plant in random order. The differences generated by pollinator movements on a plant are reduced if pollinators with constant visitation sequence among flowers also carry more pollen and/or visit more flowers per plant. Higher pollen limitation under the constant vs. random rank visitation pattern is caused both by a lower number of flowers visited by pollinators and by inbreeding depression in selfed seeds. Pollen limitation is thus predicted to critically depend on pollinator species, inflorescence size and architecture, all of which are known to impact the foraging path among flowers on a plant.

Although our model was not designed to examine the effect of inflorescence architecture on the evolution of pollen limitation, it could be used to test the following prediction: plant species with racemes have been demonstrated to elicit more constant pollinator pathways among flowers than plant species with umbels [START_REF] Jordan | Manipulation of bee behavior by inflorescence architecture and its consequences for plant mating[END_REF], such that plant species with racemes are expected to suffer higher pollen limitation. However, an accurate test of this prediction should be based on a model that incorporates explicitly inflorescence architecture.

Regardless of pollinator behavior, geitonogamous (between-flower within plant) selfing imposed by pollinators and its associated inbreeding depression often cause evolution of long flowering phenologies in which plants produce few flowers per day and thus avoid inbreeding depression at the cost of reduced pollinator attraction. Our results therefore predict that plant species with lower inbreeding depression would evolve shorter flowering phenologies with larger daily floral displays, which would enhance pollinator attraction and thus reduce pollen limitation. This could be tested by examining the relationship between pollen limitation and inbreeding depression in natural populations. Note that there are potential caveats (see below), the main one being that the expected positive relationship between pollen limitation and inbreeding depression may also be caused by environmental stochasticity: the intensity of temporal fluctuations in pollinator abundance within season correlates positively with the duration of flowering phenologies [START_REF] Devaux | Selection on variance in flowering time within and among individuals[END_REF], and is also expected with pollen limitation, as observed in temporal cohorts within a season [START_REF] Thomson | Flowering phenology, fruiting success and progressive deterioration of pollination in an early-flowering geophyte[END_REF]. The predicted extended flowering phenologies under higher inbreeding depression suffer a high risk of daily pollination failure due to temporal fluctuations in pollinator abundance or activity among days within seasons (Devaux and Lande 2010), and depend crucially on floral constancy of generalist pollinators among plant species, which is nonetheless frequently observed in plant communities [START_REF] Chittka | Flower constancy, insect psychology, and plant evolution[END_REF].

Pollen limitation measured over the entire individual flowering phenology, as we do here, may differ from that measured over parts of the phenology [START_REF] Knight | A quantitative synthesis of pollen supplementation experiments highlights the contribution of resource reallocation to estimates of pollen limitation[END_REF] or that measured as the difference in seed production under natural and artificially supplemented pollination [START_REF] Knight | Pollen limitation of plant reproduction: pattern and process[END_REF]. The first type of discrepancy highlights the role of resource allocation in pollen limitation. The second discrepancy can be generated by abortion of inbred embryos; therefore pollen limitation may be frequently overestimated for partially selfing species because it is usually measured by supplementing large amounts of outcross pollen [START_REF] Aizen | Expanding the limits of the pollen-limitation concept: effects of pollen quantity and quality[END_REF]. In other terms, comparing the number of viable seeds under supplemental outcrossed and selfed pollen can help distinguish the cause of pollen limitation, between a shortage of pollinator visits (increased seed production with outcrossed or selfed pollen) and inbreeding depression (smaller increase in seed production with self vs. outcross pollen). Estimating pollen limitation in species whose selfing rates are constrained by pollinators (Devaux et al. 2014b;a) while accounting for inbreeding depression of selfed seeds is a difficult but necessary task if the causes of pollen limitation are to be determined, as already mentioned by [START_REF] Eckert | Plant mating systems in a changing world[END_REF], and several experimental methods that also account for plant resources are available [START_REF] Calvo | Pollinator limitation, cost of reproduction, and fitness in plants -a transition-matrix demographic approach[END_REF][START_REF] Aizen | Expanding the limits of the pollen-limitation concept: effects of pollen quantity and quality[END_REF][START_REF] Wesselingh | Pollen limitation meets resource allocation: towards a comprehensive methodology[END_REF][START_REF] Alonso | A piece of the puzzle: a method for comparing pollination quality and quantity across multiple species and reproductive events[END_REF][START_REF] Arceo-Gomez | Patterns of pollen quantity and quality limitation of pre-zygotic reproduction in Mimulus guttatus vary with co-flowering community context[END_REF].

Limitations and perspectives to model the evolution of pollen limitation

Pollen limitation evolves in this study by mechanisms different from those analyzed in previous theory. First and most importantly, we deliberately excluded environmental stochasticity for the sake of simplicity : it proved to be the main driver of within and among plant variation in pollination and the evolution of pollen limitation in previous models [START_REF] Burd | The Haig-Westoby model revisited[END_REF][START_REF] Richards | Variation in pollination: causes and consequences for plant reproduction[END_REF][START_REF] Rosenheim | Parental optimism versus parental pessimism in plants: how common should we expect pollen limitation to be?[END_REF][START_REF] Schreiber | Evolutionary and ecological consequences of multiscale variation in pollen receipt for seed production[END_REF] and is undoubtedly responsible for some pollen limitation in natural populations. Instead we include demographic Predicting how these mechanisms may interact with environmental stochasticity is not straightforward. In self-compatible insect-pollinated plants, both inbreeding depression with selfing and temporal fluctuations in pollinator availability (i.e. pollinator visits per plant, which could be due to fluctuations in pollinators and/or plant population density, Thomson 2010) should contribute to the evolution of pollen limitation. Obviously, in a highly stochastic environment, highly variable pollinator availability is much more likely important than is inbreeding depression with selfing. In a more constant environment with stable pollinator availability, the contribution of inbreeding depression to pollen limitation depends on both the selfing rate and how much inbreeding depression can be purged.

Evolution of individual flowering phenologies, and consequent pollen limitation, may also be driven by genetic and ecological factors not considered here, acting at both the individual and community levels: our mechanistic model of pollinator behavior is simplified, to address ubiquitous genetic and ecological mechanisms responsible for the emergence of pollen limitation, and cannot portray the immense variation among pollinator species. First, plant resources are limited in our model as all plants display the same expected number of flowers, but we neglect allocation to seed provisioning considered by previous authors [START_REF] Bierzychudek | Pollinator limitation of plant reproductive effort[END_REF][START_REF] Haig | On limits to seed production[END_REF][START_REF] Ashman | Pollen limitation of plant reproduction: ecological and evolutionary causes and consequences[END_REF]. Resource allocation may be particularly crucial to understand pollen limitation in iteroparous species [START_REF] Crone | How do plants know when other plants are flowering? Resource depletion, pollen limitation and mast-seeding in a perennial wildflower[END_REF], which are not considered in our model. Instead we model allocation to flowers among days in the flowering phenology of individual plants and allow plants to mature all seeds without reproductive compensation. Second, autonomous selfing has been proposed many times as a reproductive assurance strategy under pollinator limitation [START_REF] Fishman | Pollen limitation and natural selection on floral characters in the yellow monkeyflower, Mimulus guttatus[END_REF][START_REF] Fenster | Pollen limitation and reproductive assurance in Antillean Gesnerieae: a specialists vs. generalist comparison[END_REF][START_REF] Thomann | Flowering plants under global pollinator decline[END_REF]; its evolution towards increased selfing was found in natural populations [START_REF] Moeller | Geographic structure of pollinator communities, reproductive assurance, and the evolution of self-pollination[END_REF]) and in experimental populations experiencing pollinator abundance limitation (Bodbyl Roels and Kelly 2011).

Allowing autonomous selfing and its evolution can have complex effects on the evolution of flowering phenologies (Devaux et al. 2014a) and pollen limitation [START_REF] Morgan | Self-fertilization and the escape from pollen limitation in variable pollination environments[END_REF]Harder et al. 2008). We also do not account for facilitated selfing (c.f. Lloyd and Schoen 1992) as little empirical information exists on this intra-flower component of selfing except for specific flower morphologies [START_REF] Johnson | Pollen fates and the limits on male reproductive success in an orchid population[END_REF][START_REF] Owen | Facilitated autogamy and costs of selfing in the perennial herb Bulbine bulbosa (Asphodelaceae)[END_REF][START_REF] Vaughton | Does selfing provide reproductive assurance in the perennial herb Bulbine vagans (Asphodelaceae)?[END_REF]. Evolution of sterile flowers can reduce pollen limitation by increasing attraction of pollinators at low energetic and genetic costs [START_REF] Morales | Sterile flowers increase pollinator attraction and promote female success in the Mediterranean herb Leopoldia comosa[END_REF]. We further do not address pollen competition between self and outcross pollen, or among multiple sires [START_REF] Lankinen | Pollen competition reduces inbreeding depression in Collinsia heterophylla (Plantaginaceae)[END_REF][START_REF] Richards | Variation in pollination: causes and consequences for plant reproduction[END_REF]) possibly complicated by pollen precedence [START_REF] Waser | Pollen precedence and stigma closure -A mechanism of competition for pollination between Delphinium nelsonii and Ipomopsis aggregata[END_REF], the evolution of aggregated pollen that occurs in orchids [START_REF] Harder | Function and evolution of aggregated pollen in angiosperms[END_REF], or mechanisms such as dichogamy or herkogamy that can prevent geitonogamous selfing. Our model also omits several factors operating at the community level among plant species that can affect both the amount and genetic composition of pollen receipt: e.g. facilitation and competition among species [START_REF] Moeller | Facilitative interactions among plants via shared pollinators[END_REF][START_REF] Vamosi | Pollination decays in biodiversity hotspots[END_REF][START_REF] Hegland | Is the magnitude of pollen limitation in a plant community affected by pollinator visitation and plant species specialisation levels?[END_REF][START_REF] Devaux | Displacement of flowering phenologies among plant species by competition for generalist pollinators[END_REF][START_REF] Sargent | Effect of local community phylogenetic structure on pollen limitation in an obligately insect-pollinated plant[END_REF][START_REF] Lazaro | Experimental reduction of pollinator visitation modifies plant-plant interactions for pollination[END_REF], which partly depends on the constancy of pollinators to a plant species and the transfer of heterospecific pollen, and can potentially affect the evolution of autonomous selfing. 56.9 1 Note: Base case: $ = 100 pollinators are available, they carry 6 = 100 pollen grains, their probability of leaving a plant after visiting a flower is & = 0.33, their pollen carry-over is 1 -% = 0.75, and their visitation rate is defined by < = 50 and B = 0.1 (Fig. S1 [Supplementary Information]); inbreeding depression is due to deleterious mutations that can or cannot be purged (C = 0.2, ℎ = 0.02, and E = 0.25), plants produce = 100 flowers.

CONCLUSION

Evolutionary equilibria are ranked according to increased (flowering period of plants).

FIGURE LEGENDS

  , hence post-zygotic inbreeding depression. Evolution of floral display size is analyzed by modelling how individual plants allocate a constant total number of flowers through the season (i.e. the individual flowering phenology) under the constraints of pollinator foraging behavior among and within plants. Therefore, we analyze how ecological and genetic mechanisms (insufficient ovule fertilization via pollinator attraction and insufficient embryo quality via inbreeding depression, respectively) jointly constrain evolution of a trait influencing pollen limitation.

  flowering phenology, , is the trait under selection: it describes how individual plants distribute their flowers among days within a season, and relates directly to floral display and duration of flowering time: small values of σ correspond to large floral displays over short periods (e.g. mass blooming), whereas large values of σ produce a small floral display over long periods. The standard deviation in flowering phenology therefore modifies all components of plant fitness through floral display size: male and female outcrossed fitness via pollinator attraction, and geitonogamous (between-flower) self-pollination rates via the foraging behavior of pollinators among flowers within plants. Pollinator behavior and the severity of inbreeding depression of selfed seeds following geitonogamous self-pollination impose trade-offs between maternal and paternal components of plant fitness that govern the ESS. We investigate the ESS by examining the fate of an initially rare modifier of flowering time ( * ) in a resident plant population at equilibrium, assuming infinite population size and a uniform distribution of the average flowering time, i.e. aseasonal reproduction. The ESS can be expressed in terms of standard deviations in individual flowering phenology, , derived from the maximal expected relative fitness of the rare modifier ( * ) in the resident plant population ( * / * = 0 at * = ): with ), ) and ) respectively the fraction of selfed seeds produced by geitonogamous selfing (i.e. geitonogamous pollination rate), number of flowers visited (~ ovules fertilized) and pollen export of genotypes with standard deviation in flowering phenology , and and the mean fitnesses of selfed and outcrossed individuals controlling inbreeding depression ( = 1 -/ !"# ). The three terms of equation (1)

  all plants have the same ovule number per flower, (ix) because of infinite population size, reproduction is never limited by mate availability, regardless of plant phenology.With these assumptions,) and ) can be expressed either in numbers of flowers visited or numbers of seeds produced, as they are proportional. Although all these assumptions may not always be observed in natural populations (see Devaux et al. 2014a for more details), changing them should not affect our main conclusions that pollen limitation evolves as a byproduct of selection for increased pollinator attraction but also for avoidance of inbreeding depression. The few assumptions that can change qualitatively (and not just quantitatively) the results of the model are either explored by changing parameter values (see below), or thoroughly discussed. Devaux et al. (2014a) showed that this model predicts two types of equilibria: (1) evolutionarily stable equilibria (ESS) determined by a trade-off between pollinator attraction to large floral displays and avoidance of inbreeding depression due to selfing, with intermediate geitonogamous selfing rates and (2) ecologically stable equilibria, corresponding to extremely long or short flowering phenologies constrained by pollinator behavior only, which yield minimal or maximal selfing rates. The latter equilibria are rarely observed in natural populations, in which they are constrained by mechanisms not included in the present model. Therefore we ignore them for our study of mechanisms leading to pollen limitation.We explore the causes of pollen limitation by inspecting three fitness components at the evolutionarily stable standard deviation in flowering time: (1) the total number of flowers pollinated , i.e. not including inbreeding depression and thus reflecting only a limitation in the number of pollinator visits, (2) viable seed production (first two terms in equation 1,hereafter "maternal fitness"), including pollinator shortage and inbreeding depression but not pollen export, and, (3) plant total fitness (expressed in number of flowers, as it is proportional to seed production), thus including limitation in the number of seeds produced and the amount of pollen exported accounting for pollinator shortage and inbreeding depression. Note that even if the paternal outcross component of fitness is not included in the first two components, it does constrain the existence and position of the ESS. We quantify pollen limitation at the ESS with two measures, within which most empirical estimates fall. We exclude pre-zygotic effects of pollen quality on pollen limitation (e.g., slow pollen tube growth) by assuming that all conspecific pollen fertilizes equally ovules and that inbreeding depression acts only on post-zygotic components of fitness (i.e. seed viability). We also exclude components of pollen limitation due to costs of producing ovules or maturating seeds (i.e. plants have enough resources to mature all viable seeds), and thus focus on the joint effects of the number of pollinator visits and inbreeding depression on pollen limitation. Pollen limitation is first measured at an ESS as the fraction of unfertilized ovules or equivalently in our model the fraction of unpollinated flowers, -) ⁄ . This measure of pollen limitation describes the potential shortage in pollinator visits. Second, we incorporate embryo quality by measuring pollen limitation as the reduction in plant maternal fitness ( at the ESS (viable seeds) due to inbreeding depression: (,max -( ) (,max ⁄

  survival of individuals carrying more deleterious mutations; genomic rate to nearly recessive lethal alleles C = 0.02, dominance coefficient ℎ = 0.02) or is constant (background inbreeding depression, E = 0.25). This parameterization is chosen to match values observed in natural populations, in which not all flowers are expected to be pollinated because plants receive a finite number of pollinator visits and pollinators leave a plant before visiting all of its open flowers[START_REF] Ohashi | How long to stay on, and how often to visit a flowering plant? A model for foraging strategy when floral displays vary in size[END_REF]. For example with these parameter values ca. 3 flowers among 10 displayed are visited in a single bout for the random-rank model of pollinator foraging behaviour. We investigate the relative contribution of mechanisms driving pollen limitation over a wide range of parameter values that govern pollinator abundance limitation, pollinator attraction limitation, pollinator movements among flowers of the same plant, severity of inbreeding depression and total number of flowers produced per plant (Table

  Fig.2A).

  stochasticity (operating independently among individuals, unlike environmental stochasticity) to produce variation in the number of open flowers a given day, variation in the number of pollinator visits to a plant, and variation in the number of flowers visited per plant per pollinator visit. Because our model involves an infinitely large population, such demographic stochasticity has little impact on evolutionary equilibria in comparison to the temporal environmental stochasticity that is synchronized among all individuals in other models. Pollen limitation in our model evolves because of a genetic trade-off between pollinator attraction (as well as other aspects of pollinator behavior) and inbreeding depression after zygote formation.

  Ecological constraints alone predict that many flowers remain unpollinated because pollen or plant resources for fruit production and seed maturation are limited. Beyond the role of trade-offs among costly reproductive functions, our results show that pollen limitation is an evolved property that depends also on genetic mechanisms and pollinator behaviors that constrain mating systems and the evolution of plant flowering phenologies. Our model suggests that, despite strong pollinator attraction limitation, plants do not evolve short phenologies with an excess of flowers to attract pollinators, but instead evolve long flowering phenologies with relatively few flowers open per day to favor outcross pollination, resulting in pollen limitation due to reduced pollinator attraction to daily floral displays. Future research on pollen limitation should examine how genetic processes interact with more commonly studied ecological processes (resource limitation and environmental stochasticity) to drive evolution of pollen limitation.

Fig. 1 :Fig. 2 :

 12 Fig. 1: Fitness components and floral display under two pollinator visitation sequences of

Fig. 3 :

 3 Fig. 3: Fitness components and floral display under decreased ( = 10, panels A) and increased ( = 1000, panels B) flower production per plant. Random-rank visitation

  . Relaxing this assumption would likely lower the number of pollinator visits and the amount of conspecific pollen deposited on stigmas and exported to conspecifics,

	with the same predicted effects as a variation in pollen abundance M or pollen loads A on
	pollinators (see below),
	(iii) the number of daily pollinator visits per plant is Poisson distributed, with a mean
	determined by both pollinator abundance ($) and the function of pollinator attraction to
	plants with a given floral display (e.g. Klinkhamer et al. 1989; Klinkhamer and de Jong
	1990); this assumption introduces a minimal amount of demographic stochasticity, which,
	unlike environmental stochasticity, operates independently among individual plants in the
	population. It also depicts the observed correlation between pollinator visitation rates and
	inflorescence size (e.g. Willson and Schemske 1980; Bauer et al. 2017),
	(iv) pollinators have the same pollen carryover 1 -%, where ρ is the rate of pollen
	uptake and deposition by pollinator; this assumption ensures the observed decay of pollen
	deposition from a single flower to subsequent flowers (Price and Waser 1982),
	(v) pollinators have a constant probability & to leave a plant after each flower visited,
	such that they visit more flowers, but a smaller proportion of flowers, on plants with larger vs.

Table 1 .

 1 TABLES Minimal vs. realized pollen limitation, expressed either as percent of unpollinated in the base case and for several deviations from the base case. For a given combination of parameter values, the minimal pollen limitation appears on the first line ("Min").

	flowers ; G HI = 100 ×	-) ⁄ @ or percent of deviation from theoretical maximal
	maternal fitness ; G E 8(<K = 100 × (,(<K -( (,(<K ⁄	@, at evolutionarily stable
	equilibria described by the individual flowering phenologies Random-rank visitation	Constant-rank visitation
				sequence				sequence
	Parameter values		σ *	PLunpoll PLdevmax		σ *	PLunpoll PLdevmax
			Min	5.2	30.7		Min	29.8	50.6
	Base case 1		0.6	7.1	32.1		-	-	-
			34	29.6	57.8		53	31.4	58.6
	Lower inbreeding depression		Min	5.2	6.7		Min	29.8	34.3
	(U = 0.02, d = 0)		0.59	7.1	8.7		10	30.9	35.2
			Min	72	79.5		Min	85.8	89.7
	Reduced pollinator abundance (M = 10)	0.59	72	79.5		0.8	86	89.8
			20	88	92.6		22	88.1	93.3
	Reduced pollinator leaving rate	Min	9×10 -6	32		Min	9×10 -6	32.1
	(τ = 0.01)		0.5	0.1	34.2		0.5	0.15	34.5
	Reduced attraction (a = 20, b = 0.01)		-	-	-		-	-	-
	Smaller floral display		Min	23.7	44.0		Min	29.8	50.1
	( = 10)		3.35	29.5	57.3		5.25	31.1	56.4
			Min	4.6	30.2		Min	29.9	50.6
	Larger floral display ( = 1000)		5.65	4.8	30.3		-	-	-
			350	29.7	57.7		550	31.4

*

100 pollinators, τ = 0.33 and 6 = 100, pollen deposition rate ρ = 0.25, = 100 flowers per plant, C = 0.2, h = 0.02 and E = 0.25 for inbreeding depression.

FUNDING

This work was supported by the French CNRS program PICS grant #5273 to E.P., and a grant from the Balzan Foundation and a Royal Society Research Professorship to R.L.

SUPPLEMENTARY DATA

Supplementary data are available online at https://academic.oup.com/aob and consist of the following. Table S1: Ecological and genetic parameters, along with the values investigated and their units.