Celine Devaux
email: celine.devaux@univ-montp2.fr

Emmanuelle Russell Lande
email: r.lande@imperial.ac.uk

Céline Devaux

Emmanuelle Porcher
email: porcher@mnhn.fr

Pollination ecology and inbreeding depression control individual flowering phenologies and mixed mating

Keywords: Flowering phenology, pollinators, and mixed mating Pollinator attraction, pollinator limitation, geitonogamous selfing, floral display, purging

Introduction

Understanding the maintenance of mixed mating systems, in which species with perfect flowers produce both selfed and outcrossed seeds, remains a challenge in plant evolutionary biology.

Classical genetic models of inbreeding depression opposing the 50% automatic advantage of selfing [START_REF] Lande | The evolution of self-fertilization and inbreeding depression in plants. 1. Genetic models[END_REF] cannot explain frequently observed intermediate selfing rates [START_REF] Schemske | The evolution of self-fertilization and inbreeding depression in plants. 2. Empirical observations[END_REF][START_REF] Goodwillie | The evolutionary enigma of mixed mating systems in plants: occurrence, theoretical explanations, and empirical evidence[END_REF]. Additional constraints have thus been invoked for the maintenance of intermediate selfing rates [START_REF] Johnston | Evolution of intermediate selfing rates in plants: pollination ecology versus deleterious mutations[END_REF][START_REF] Cheptou | Can varying inbreeding depression select for intermediary selfing rates?[END_REF][START_REF] Charlesworth | Evolution of plant breeding systems[END_REF]; reviewed in [START_REF] Goodwillie | The evolutionary enigma of mixed mating systems in plants: occurrence, theoretical explanations, and empirical evidence[END_REF]. Recent theoretical studies have suggested that mixed mating systems of most species may simply be maintained by functional relationships or trade-offs among fitness components, such as trade-offs among components of female and male fertility [START_REF] Johnston | Correlations among fertility components can maintain mixed mating in plants[END_REF], or correlations between selfing rate and viability [START_REF] Jordan | Functional pleiotropy and mating system evolution in plants: frequency-independent mating[END_REF]. In many cases the genetic and/or ecological mechanisms creating such relationships among fitness components for plant breeding systems have not been elucidated experimentally (but see [START_REF] Mojica | Viability selection prior to trait expression is an essential component of natural selection[END_REF].

Pollination biology mechanisms play a central role in determining mating systems, and should contribute to trade-offs among fitness components [START_REF] Uyenoyama | Ecological and genetic factors directing the evolution of self-fertilization[END_REF][START_REF] Devaux | Constraints imposed by pollinator behaviour on the ecology and evolution of plant mating systems[END_REF]. For example, trade-offs between selfing rate and pollen export caused by competing selfing [START_REF] Lloyd | Some reproductive factors affecting the selection of self-fertilization in plants[END_REF]) can produce mixed mating systems [START_REF] Porcher | The evolution of self-fertilization and inbreeding depression under pollen discounting and pollen limitation[END_REF][START_REF] Harder | Effects of reproductive compensation, gamete discounting and reproductive assurance on mating-system diversity in hermaphrodites[END_REF][START_REF] Johnston | Correlations among fertility components can maintain mixed mating in plants[END_REF]). Here, we examine the role of pollinator behavior as a general mechanism creating trade-offs among fitness components that can maintain mixed mating systems in animal-pollinated hermaphroditic plants.

Pollinators select for multiple traits simultaneously [START_REF] Bell | On the function of flowers[END_REF][START_REF] O'neil | Natural selection on genetically correlated phenological characters in Lythrum salicaria L. (Lythraceae)[END_REF][START_REF] Elzinga | Time after time: flowering phenology and biotic interactions[END_REF], sometimes in conflicting directions, as in the case of floral display. Pollinator sensory capabilities and preferences may select for increased floral display, as indicated by a positive correlation between the number of open flowers (floral display) and pollinator visitation rate [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]. Larger floral display may also increase the seed production of plants by increasing the number of flowers per plant visited by individual pollinators [START_REF] Hessing | Geitonogamous pollination and its consequences in Geranium caespitosum[END_REF][START_REF] Robertson | The relationship between floral display size, pollen carryover and geitonogamy in Myosotis colensoi (Kirk) Macbride (Boraginaceae)[END_REF][START_REF] Robertson | The effects of floral display size on pollinator service to individual flowers of Myosotis and Mimulus[END_REF]Snow et al. 1996 and references therein;[START_REF] Mitchell | The influence of Mimulus ringens floral display size on pollinator visitation patterns[END_REF][START_REF] Grindeland | Effects of floral display size and plant density on pollinator visitation rate in a natural population of Digitalis purpurea[END_REF]). These pollinator behaviors can promote geitonogamous selfing (cross-fertilization among flowers on the same plant, [START_REF] Holsinger | Dispersal and plant mating systems: the evolution of self-fertilization[END_REF][START_REF] De | The consequences of pollination biology for selection of mass or extended blooming[END_REF][START_REF] De | Geitonogamy: the neglected side of selfing[END_REF][START_REF] Harder | Mating cost of large floral displays in hermaphrodite plants[END_REF][START_REF] Snow | The ecology of geitonogamous pollination[END_REF] and expression of inbreeding depression in the many self-compatible species that display numerous flowers [START_REF] Geber | The relationship of plant size to self-pollination in Mertensia ciliata[END_REF][START_REF] Mazer | The quantitative genetics of life history and fitness components in Raphanus raphanistrum L. (Brassicaceae): ecological and evolutionary consequences of seed-weight variation[END_REF][START_REF] Johnston | Natural selection on floral traits in two species of Lobelia with different pollinators[END_REF][START_REF] Mitchell | The influence of Mimulus ringens floral display size on pollinator visitation patterns[END_REF][START_REF] Ashman | Genetic constraints on floral evolution: a review and evaluation of patterns[END_REF][START_REF] Eckert | Plant mating systems in a changing world[END_REF]. In animal-pollinated plants, these conflicting selective forces may often result in the evolution of intermediate floral displays characterized by intermediate selfing rates. Darwin (1876 quoted above) first suggested that flowering phenology and mixed mating systems evolve as a trade-off between the benefits of attracting more pollinators and the detrimental effects of geitonogamous self-fertilization. The role of pollinators in determining plant mating systems was modeled by several authors including [START_REF] Lloyd | Some reproductive factors affecting the selection of self-fertilization in plants[END_REF][START_REF] Lloyd | Self-fertilization and cross-fertilization in plants. 2. The selection of selffertilization[END_REF], [START_REF] Geber | The relationship of plant size to self-pollination in Mertensia ciliata[END_REF], [START_REF] De | The consequences of pollination biology for selection of mass or extended blooming[END_REF], [START_REF] Robertson | The relationship between floral display size, pollen carryover and geitonogamy in Myosotis colensoi (Kirk) Macbride (Boraginaceae)[END_REF], [START_REF] Porcher | The evolution of self-fertilization and inbreeding depression under pollen discounting and pollen limitation[END_REF], and [START_REF] Jordan | Functional pleiotropy and mating system evolution in plants: frequency-independent mating[END_REF]. However, these works were based on simplified models for pollinator behavior or the dynamics of inbreeding depression. The joint evolution of floral display, geitonogamous selfing and inbreeding depression has not previously been modeled, although the dynamics of inbreeding depression can affect mating system evolution. Empirical evidence supports purging of inbreeding depression in selfing populations, particularly its early-acting component due to nearly recessive lethal mutations [START_REF] Husband | Evolution of the magnitude and timing of inbreeding depression in plants[END_REF][START_REF] Charlesworth | The genetics of inbreeding depression[END_REF].

Population genetic theory supports the finding that the late-acting component of inbreeding depression due to slightly recessive mildly deleterious mutations is far less subject to purging, but that substantial reduction in the overall inbreeding depression strongly favors the further evolution of increased selfing [START_REF] Lande | The evolution of self-fertilization and inbreeding depression in plants. 1. Genetic models[END_REF][START_REF] Charlesworth | Inbreeding depression and its evolutionary consequences[END_REF][START_REF] Lande | High inbreeding depression, selective interference among loci, and the threshold selfing rate for purging recessive lethal mutations[END_REF][START_REF] Charlesworth | The genetics of inbreeding depression[END_REF][START_REF] Porcher | The genetic consequences of fluctuating inbreeding depression and the evolution of plant selfing rates[END_REF]).

To clarify the ecological and genetic mechanisms that prevent evolution of complete selfing in partially selfing plant species we model the joint evolution of inbreeding depression and selfing rate determined by individual flowering phenology, floral display, and individual pollinator behavior, generalizing each of these mechanisms in the model of [START_REF] De | The consequences of pollination biology for selection of mass or extended blooming[END_REF].

We derive the expected relative fitness of a rare modifier of flowering phenology in a plant population, and investigate the evolution of flowering phenology and geitonogamous selfing rate (excluding within-flower selfing). By analyzing the occurrence and stability of equilibrium geitonogamous selfing rates we find that a wide range of pollinator behaviors can produce stable intermediate selfing rates, and identify parameters exerting the greatest impact on plant mating system evolution.

Models

We assume a large (effectively infinite) plant population evolving in a non-seasonal environment. All plants produce the same expected total number of flowers , but may differ in their flowering phenology, i.e. how they spread flowers through days. We define the floral display of a plant as the number of simultaneously open flowers. We assume that the expected flowering phenology of plant genotype is normally distributed in time. Stochasticity in the number of flowers open on a particular plant on a given day occurs because of developmental noise and temporal environmental variation. Pollinators are assumed to be generalist [START_REF] Waser | Generalization in pollination systems, and why it matters[END_REF] such that their density is independent of the abundance and flowering phenology of the focal plant species. Pollinators are also assumed to be constant such that they visit flowers on a single species within a day (a widespread pattern across generalist pollinator species; [START_REF] Chittka | Flower constancy, insect psychology, and plant evolution[END_REF]) and they carry exclusively conspecific pollen between plants. Thus, the number of pollinator visits to an individual plant on a given day depends only on the plant floral display that day and the pollinator abundance. Individual flowers in the population differ in their probability of being visited by pollinators, depending on three stochastic factors: the realized floral display of a plant, the number of pollinator visits to a given plant, and the number of flowers visited per plant per pollinator visit (as detailed below).

To obtain the equilibria of flowering phenology and geitonogamous selfing rate that result from the conflicting selective forces imposed by pollinators, we derive the expected relative fitness of a rare modifier genotype caused by a small increase or decrease in standard deviation of individual flowering phenology compared to the resident population. The expected relative fitnesses of the rare modifier ( * ) and common resident ( ) genotypes depend on the expected total number of flowers pollinated , the expected amount of pollen exported by pollinators that fertilizes ovules of other plants, and the geitonogamous selfing rate , as well as the inbreeding depression in fitness of selfed vs. outcrossed ovules. Below, we describe the pollination model, analyzing two mechanisms of reduced seed production due to pollen limitation, and two modes of pollinator foraging behavior among flowers within plants on a given day.

Individual flowering phenology

We assume that the total number of flowers produced by a given plant throughout its .

[1]

The realized number of flowers a plant opens on day follows a Poisson distribution with mean (Table 1). Therefore, the probability that a plant displays a random number of flowers on day is ( ) = ! / !. Each flower is open for a single day (as in Hibiscus moscheutos, [START_REF] Snow | The ecology of geitonogamous pollination[END_REF]), but the model should still be approximately correct if the longevity of flowers is much shorter than individual flowering periods. We assume that the average flowering time of plants with the same is uniformly distributed through time, which models aseasonal reproduction.

Pollination ecology

We assume that self-fertilization occurs only through geitonogamy, and not through autonomous or facilitated selfing (within-flower self-fertilization without or with pollinator involvement).

The widespread occurrence of dichogamy and herkogamy limits the opportunity for withinflower selfing but does not prevent among-flower geitonogamous selfing [START_REF] Snow | The ecology of geitonogamous pollination[END_REF][START_REF] Rademaker | Selfing rates in natural populations of Echium vulgare: a combined empirical and model approach[END_REF][START_REF] Duan | Reproductive ecology of the Qinghai-Tibet Plateau endemic Gentiana straminea (Gentianaceae), a hermaphrodite perennial characterized by herkogamy and dichogamy[END_REF]. Predominance of geitonogamy (with negligible withinflower selfing) has been observed in a number of species, such as Cyclamen creticum [START_REF] Affre | Population genetic structure and levels of inbreeding depression in the Mediterranean island endemic Cyclamen creticum (Primulaceae)[END_REF], Lupinus arboreus [START_REF] Kittelson | Outcrossing rate and inbreeding depression in the perennial yellow bush lupine, Lupinus arboreus (Fabaceae)[END_REF] and Aquilegia coerulea [START_REF] Brunet | Impact of insect pollinator group and floral display size on outcrossing rate[END_REF]. We also assume that a single pollinator visit is sufficient to fertilize all ovules of a flower because pollinators deposit a large quantity of pollen at each visit. This assumption is most appropriate for nectarivorous pollinators, which groom less frequently and tend to transfer more pollen grains than pollen-feeders [START_REF] Castellanos | Pollen transfer by hummingbirds and bumblebees, and the divergence of pollination modes in Penstemon[END_REF], or for plant species with few ovules per flower; it does not preclude pollen limitation at the plant level (see below). Finally, within a single bout at a particular plant an individual pollinator visits a given flower at most once, a pattern commonly observed [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][START_REF] Stout | The use of conspecific and interspecific scent marks by foraging bumblebees and honeybees[END_REF].

Pollinator visitation rate

Pollinator visits influence the three components of fitness of a plant genotype: the number of flowers pollinated , geitonogamous selfing rate , and amount of pollen exported . The with parameters * and . defined in Tables 1 and2, and &(0) = 0 and &(∞) = 1 (Fig. S0 left panel).

Pollen transfer

For simplicity, we assume that pollinators always carry the same (constant) amount of pollen A (pollinator saturation, de [START_REF] De | Geitonogamy: the neglected side of selfing[END_REF]), but allow the fraction of self-pollen in the pollen load to change with the number of flowers visited on a plant during a single bout. When a pollinator visits a flower, it deposits ρA pollen grains on the stigma and collects the same amount from the anthers, so that 1 -2 is the (constant) pollen carryover (identically 2 holds for both the deposition and uptake rates of pollinators; Table 1).

The fraction 3 4 of self-pollen on a pollinator that has just visited the k th flower, which strongly influences the geitonogamous selfing rate, has been observed to follow a damped geometric pattern with increasing k [START_REF] Cresswell | Models of pollinator mediated gene dispersal in plants[END_REF] and references therein):

3 4 = 1 -(1 -2) 4 .

Pollinator behavior among flowers of the same plant

We allow stochastic variation in the number of flowers visited per pollinator visit (or bout) to an individual plant on a given day, even for plants with the same realized floral display F, by assuming that on each bout a pollinator leaves a plant with a constant probability 5 after each flower visited (Table 1). Within a single pollinator bout, the probability of visiting the k th flower is (1 -5) 4 6 . Accordingly, the probability 7 4 (conditional on pollinator visitation) that the pollinator leaves the plant after the k th flower is 012345.

7 4 = 8 0 for < =
[3]

We model two contrasting pollinator visitation patterns among simultaneously open flowers of the same plant that differ by the order in which flowers are visited (random versus constant rank). The preceding equations apply to both visitation models, but the expected number of flowers fertilized and the geitonogamous selfing rate differ between the visitation models when multiple pollinators visit multiple flowers of the same plant on a given day.

RANDOM RANK VISITATION MODEL

Within each pollinator bout the visitation rank of flowers is random, and the visitation ranks of flowers on the same plant on a given day are independent among bouts. For a plant displaying flowers the mean number of flowers pollinated on a single bout is

> = ∑ <7 4 4@ = 6 A '1 -(1 -5) -with > = 0.
The probability of a flower not being pollinated in m (independent) pollinator bouts on a given day is (1 -> / ) B . Averaged over the Poisson distribution of bouts, the probability of a flower not being pollinated on a plant with realized floral display F is

∑ C DEF(G) 'HI( )-J B! K B@ L1 - M G N B = HI( )M G / for ≥ 1.

CONSTANT RANK VISITATION MODEL

Each pollinator visits flowers on a single plant on a given day in the same order, e.g. from bottom to top inflorescences for 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]. Recall that in a given bout the probability that the k th flower on a plant is visited is (1 -5) 4 6 . With m independent bouts the probability that the k th flower is not

pollinated is '1 -(1 -5) 4 6 -B
. Averaging this probability over the Poisson distribution of number of pollinator bouts, the probability that the k th flower is not pollinated is

∑ C DEF(G) 'HI( )-J B! K B@ '1 -(1 -5) 4 6 -B = HI( )(6 A) PDQ .

Plant fitness components

Amount of pollen exported

The quantity of pollen exported by a pollinator leaving a plant after visiting < of its flowers does not depend on the pollinator visitation pattern among flowers and is simply R3 4 ; this facilitates calculating the expected total pollen exported by all visiting pollinators and that is available for outcrossing with other plants. The expected pollen exported during the entire flowering period of plants with a given value of is

( ) = ∑ ∑ ( ) K @ K K @ ∑ C DE F(G) 'H I( )-J B! K B@ $ &( ) ∑ 7 4 4@ R3 4 = ST 6 (6 T)(6 A) ∑ ∑ ( ) K @ K % &( )'1 -(1 -2) (1 -5) - K @ . '4-
With pollinator limitation, P is a non-monotonic function of 2, 5 and (not shown).

Number of flowers fertilized

Under the assumptions of pollen saturation and large pollen loads carried on pollinators, total seed production is proportional to the expected number of flowers fertilized through the entire flowering period ( ). For the random rank visitation model

( ) = ∑ ∑ ( ) K @ K (1 -HI( )M G / ) K @6 , [5a]
and for the constant rank visitation model

( ) = ∑ ∑ ( ) K @ K ∑ L1 -HI( )(6 A) PDQ N 4@6 K @6
.

[5b]

Pollen limitation occurs whenever ( ) < , and can be caused by two different processes.

First, seed set may be limited by pollinators leaving a plant (at rate 5) before visiting all flowers, and by the total abundance of pollinators, M; these two conditions define pollinator abundance limitation. Seed set may further be limited by pollinator attraction if pollinator visitation rates to plants increase with floral display. With either form of limitation, is a non-monotonic function of 2, 5 and (Fig. S0, right panel).

Geitonogamous selfing rate

The geitonogamous selfing rate of plants with a given is the expected total selfed progeny divided by the expected total progeny for all such plants, ( ) = 3( )/ ( ), for any (constant) number of ovules per flower. The expected quantity of self-pollen deposited across flowers 3( ) depends on the pollinator visitation pattern among flowers on the same plant, and on the assumption of a constant amount of pollen 2R deposited per flower per pollinator visit.

For the random rank visitation model, the pollination rank of flowers is random within a single pollinator visit and independent among pollinator visits to a given plant on any day. With these assumptions, the expected proportion of self-pollen on any flower visited (regardless of the number of visits) equals the expected self-pollen deposited across flowers visited by a single pollinator. This quantity does not depend on the degree of pollen precedence among pollinator visits to a given flower. For example, with complete precedence by the first pollinator visit to a flower, the rank order of the flower on the first pollinator visit is still random, with the same expectation. Consequently, the expected quantity of self-pollen deposited on flowers by a single pollinator is given by

3( ) = ∑ ∑ ( ) K @ K (1 -HI( )M G / ) K @6 ∑ V P ∑ W XDQ P XYQ G PYQ M G = ∑ ∑ ( ) K @ K (1 -HI( )M G / ) K @6 Z1 - 6 M G L 6 (6 T) G (6 A) G 6 (6 T)(6 A) N[. [6a]
For the constant rank visitation model, the expected quantity of self-pollen deposited is calculated for all visited flowers as

3( ) = ∑ ∑ ( ) K @ K ∑ 3 4 6 L1 -HI( )(6 A) PDQ N 4@6 K @6
.

[6b]

Again, this quantity does not depend on the degree of pollen precedence among visits to a given flower, and accounts for variation in the composition of pollen deposited per flower among pollinators and among visits to different plants by a given pollinator. Under the constant rank model, the geitonogamous selfing rate of an individual plant on a given day can change only when a pollinator visits more flowers than previous pollinators did on the same day.

Flowering phenologies at equilibrium

We track the fate of an initially rare modifier genotype with standard deviation in flowering time * close to that of the resident population, . We partition the fitness of modified genotypes * into male and female fitness components gained through selfing and outcrossing of ovules from the focal and the resident plants, following [START_REF] Charlesworth | A model for the evolution of dioecy and gynodioecy[END_REF], [START_REF] Lloyd | Some reproductive factors affecting the selection of self-fertilization in plants[END_REF], [START_REF] Lande | The evolution of self-fertilization and inbreeding depression in plants. 1. Genetic models[END_REF], ( ).

[7]

with \]^_ and `ab the mean fitness of selfed and outcrossed progeny, respectively.

Approximate evolutionary equilibria of , and hence the equilibria of geitonogamous selfing rate, occur when the selection gradient vanishes, The derivatives of the total number of flowers pollinated, number of selfed flowers, pollen export and geitonogamous selfing rate with respect to are given in the Appendix.

Inbreeding depression in natural populations is caused by a combination of nearly recessive highly deleterious mutations, which can be purged by selfing and selection, and slightly recessive mildly deleterious mutations, which undergo little purging in response to increased selfing [START_REF] Lande | The evolution of self-fertilization and inbreeding depression in plants. 1. Genetic models[END_REF][START_REF] Charlesworth | Inbreeding depression, genetic load, and the evolution of outcrossing rates in a multilocus system with no linkage[END_REF][START_REF] Husband | Evolution of the magnitude and timing of inbreeding depression in plants[END_REF][START_REF] Charlesworth | The genetics of inbreeding depression[END_REF][START_REF] Porcher | Evaluating a simple approximation to modeling the joint evolution of self-fertilization and inbreeding depression[END_REF]. We include the first component of evolving inbreeding depression using the [START_REF] Kondrashov | Deleterious mutations as an evolutionary factor. 2. Facultative apomixis and selfing[END_REF] model of an infinite number of loci in an infinitely large population mutating to nearly recessive lethal alleles. Each new mutation is assumed to be unique and becomes homozygous only through geitonogamous selfing. This genetic model produces results similar to models with a finite number of loci mutating to recessive lethal alleles in an infinite population, or for a large finite population in which each new recessive lethal mutation is one that is not currently segregating in the population [START_REF] Lande | High inbreeding depression, selective interference among loci, and the threshold selfing rate for purging recessive lethal mutations[END_REF]. We incorporate the second component of inbreeding depression due to slightly recessive mildly deleterious mutations via a constant background inbreeding depression ( , Table 2). Both components of inbreeding depression are assumed to act on plant viability (from seed to flowering).

Internal equilibria in flowering phenology occur at values of where the constraint function crosses (and thus equals) the inbreeding depression, which can then be mapped onto the corresponding geitonogamous selfing rate ( ). The stability of internal equilibria is given by the relative orientation of the constraint function and the inbreeding depression curve at the crossing point. Simple graphical analysis of equilibria of the mating system [START_REF] Yahara | Graphical analysis of mating system evolution in plants[END_REF], without using an explicit genetic model of mating system evolution, is an approximation that gives reasonably accurate results with moderate genomic rate to recessive lethals (U ≤ 0.2;

Porcher and Lande 2013). This approximation contains elements of Evolutionarily Stable

Strategies, as well as inclusive fitness by incorporating the automatic advantage of selfing.

Edge equilibria do not correspond to a crossing point of the constraint function and the inbreeding depression curve and occur at extreme (minimal or maximal) geitonogamous selfing rates as a function of for a given set of ecological parameters. Their existence and stability depend on the internal equilibria and the relative orientation of the constraint function and the inbreeding depression curve at extreme geitonogamous rates.

Scenarios investigated

We examined edge and internal equilibrium geitonogamous selfing rates for many parameter combinations involving (i) pollinator attraction limitation &( ), (ii) pollen transfer, 2, and pollination abundance limitation, 5 and %, (iii) pollinator visitation pattern (foraging behavior within plants), (iv) genomic rate to nearly recessive lethal alleles, v (with dominance coefficient h = 0.02), and constant background inbreeding depression, , and (v) mean total number of flowers per plant, . Parameter values were either assigned according to experimental data, or varied across a wide range including experimental estimates (Table 2). Cases with no pollen limitation were generated by using a flat attraction function making pollinator visits to plants independent of floral display (no pollinator attraction limitation) while greatly increasing pollinator abundance (% = 10 x ) and bout length (5 = 10 y ; no pollinator abundance limitation).

Results

The model predicts equilibrium geitonogamous selfing rates constrained either by pollination ecology (edge equilibria) or by a trade-off between pollinator attraction and inbreeding depression (internal equilibria). We describe the properties of the ecological model (how flowering phenology and pollinator behavior influence geitonogamous selfing rates and plant fitness components) followed by the equilibria obtained from the evolutionary model. We focus on a baseline case with both pollinator abundance limitation and pollinator attraction limitation, a random rank visitation pattern of pollinator visits among flowers on the same plant, and a moderately high genomic rate of nearly recessive lethal mutations (v = 0.2, ℎ = 0.02) associated with a constant background inbreeding depression ( = 0.25). Results obtained under the constant rank visitation model are mainly presented in the Supplementary Online Material. flowers open on a single day, and the selfing rates then depend greatly on pollinator behavior; maximum geitonogamous rates range from near zero, when pollen deposition and bout length are small, to near one under the opposite conditions (Figs. 2 andS2).

Ecological constraints on geitonogamous selfing and fitness components

With no pollinator attraction limitation but a given intensity of pollinator abundance limitation, longer individual flowering periods produce greater pollen export (eq. 4) and seed production, because a higher proportion of total flowers per plant is visited (eq. 5). Under

Evolutionary equilibria of geitonogamous selfing rates

The evolutionary model combining pollination ecology and inbreeding depression predicts three types of equilibrium geitonogamous selfing rates: major and minor internal equilibria, and edge constant rank visitation models, respectively). When floral display influences pollinator attraction, many combinations of bout length, pollen carryover and visitation patterns among flowers generate at least one (and sometimes multiple) major stable equilibria associated with at least two unstable equilibria. Flowering phenologies at major evolutionarily stable equilibria have longer periods when plants produce a larger total number of flowers, because the trade-off between geitonogamous selfing rate and standard deviation in flowering time is weaker with larger total number of flowers. For realistic pollen carryover and leaving rates (between 1/3 and 2/3, Table 2), stable intermediate selfing rates correspond to flowering periods of several days for plants producing few flowers, up to several months or years for plants producing hundreds of flowers (Fig. S6 and S7 for the random vs. constant rank visitation models, respectively). The existence of stable mixed mating systems is conditional on high inbreeding depression:

decreasing the genomic mutation rate to recessive lethal alleles (v) and the background inbreeding depression ( ) condenses the pollination parameter space that allows major stable internal equilibria (Fig. 5 and S5 for the random vs. constant rank visitation models, respectively).

Stable mixed mating can also be observed when pollen limitation is caused by pollinator abundance limitation only (Fig. S0), without pollinator attraction limitation. In this case, with low leaving rate of pollinators, a single stable major internal equilibrium is predicted under both pollinator visitation models (not shown). With no pollen limitation at all (Fig. S0), a single unstable internal equilibrium is predicted under both models of pollinator visitation; this corresponds to short flowering periods ( < 1), producing a geitonogamous selfing rate close to the maximum for a given pollination model.

Finally, the evolutionary model shows that minimal and maximal geitonogamous selfing rates constitute edge equilibria. All pollination models produce a stable edge equilibria at = 0, with maximal geitonogamous selfing rate, although as explained above the position of this edge equilibrium strongly depends on pollinator behavior. These edge equilibria at = 0 correspond to stable mixed mating system for any combination of pollination parameters (Figs. 4-5 and S4-S5 for the random vs. constant rank visitation models, respectively).

Discussion

Our results demonstrate that ecological mechanisms of pollination biology can balance the strong automatic genetic advantage of selfing to produce stable mixed mating systems. This can occur even for selfing rates high enough to purge most of the recessive lethal component of inbreeding Stable internal equilibria result from a balance between ecological and genetic factors, but the selection is frequency-dependent and generally does not maximize mean fitness [START_REF] Wright | Evolution and the genetics of populations[END_REF][START_REF] Lande | Natural selection and random genetic drift in phenotypic evolution[END_REF]). In particular, the automatic genetic advantage of selfing is strongly frequencydependent, diminishing from a 50% advantage in an outcrossing population to 0 in a completely selfing population. Thus directional selection is also likely to prevail at stable internal equilibria.

Pollinator behavior as a general mechanism maintaining mixed mating [START_REF] Johnston | Correlations among fertility components can maintain mixed mating in plants[END_REF] previously analyzed phenomenological trade-offs among plant fitness components to explain mixed mating systems, postulating single-valued functional relationships without specifying the underlying mechanisms. We show here that pollinator foraging behavior generates mechanistic trade-offs among plant fitness components, and that these mechanistic relationships can be multi-valued. Our model also reveals how the constraints among plant fitness components depend on empirically measureable parameters of pollination biology and floral traits (Figs 3,5,S3 and S5).

The existence of evolutionarily stable intermediate selfing rates in our model requires substantial total inbreeding depression and pollen limitation . With no inbreeding depression, we find internal equilibria only for plants producing numerous flowers with short flowering phenologies (mass blooming of de Jong et al. 1992 who assumed no inbreeding depression and τ = 0). Evolution of inbreeding depression by purging its recessive lethal component through partial selfing was found to alter or even eliminate stable equilibrium selfing rates (e.g. [START_REF] Charlesworth | Inbreeding depression and its evolutionary consequences[END_REF][START_REF] Porcher | The genetic consequences of fluctuating inbreeding depression and the evolution of plant selfing rates[END_REF]), but for a wide range of ecological and genetic parameters our model predicts stable mixed mating systems . These intermediate stable geitonogamous selfing rates involve plants producing up to hundreds of flowers in total, and are consistent with estimates between 8 to 70% in natural and experimental populations (Robertson 1992 and references therein;[START_REF] Schoen | Self-fertilization and cross-fertilization in plants. 3. Methods for studying modes and functional aspects of self-fertilization[END_REF][START_REF] Leclerc-Potvin | Modes of self-fertilization in Mimulus guttatus (Scrophulariaceae): a field experiment[END_REF]Snow et al. 1996 and references therein;[START_REF] Eckert | Contributions of autogamy and geitonogamy to self-fertilization in a massflowering, clonal plant[END_REF][START_REF] Karron | The influence of floral display size on selfing rates in Mimulus ringens[END_REF][START_REF] Karron | Pollinator visitation patterns strongly influence among-flower variation in selfing rate[END_REF][START_REF] Brunet | Impact of insect pollinator group and floral display size on outcrossing rate[END_REF].

Our study further shows that pollinator foraging behavior interacts with flowering phenology to constrain geitonogamous selfing rates at the edge equilibria, which comprise stable

Expected mating systems in natural populations

Which among the major internal and edge equilibria is the most likely for a plant species is governed by several parameters in our model. We expect major internal equilibria characterized by intermediate selfing rates and moderate or long flowering periods (from a few days to a few months) in populations with: (i) significant pollen limitation, which is widespread in natural populations [START_REF] Knight | Pollen limitation of plant reproduction: pattern and process[END_REF]), (ii) relatively high genomic rate of lethal mutations v and high inbreeding depression, both of which have been documented (up to U = 0.2, references in [START_REF] Lande | High inbreeding depression, selective interference among loci, and the threshold selfing rate for purging recessive lethal mutations[END_REF]see Johnston and Schoen 1995;[START_REF] Husband | Evolution of the magnitude and timing of inbreeding depression in plants[END_REF] for inbreeding depression), even in populations with intermediate selfing rates (Winn et al. 2011), and (iii) intermediate rates for pollinator leaving and pollen carryover consistent with experimental estimates [START_REF] Geber | The relationship of plant size to self-pollination in Mertensia ciliata[END_REF][START_REF] Robertson | The relationship between floral display size, pollen carryover and geitonogamy in Myosotis colensoi (Kirk) Macbride (Boraginaceae)[END_REF]Snow et al. 1996 and references therein;[START_REF] Duan | Reproductive ecology of the Qinghai-Tibet Plateau endemic Gentiana straminea (Gentianaceae), a hermaphrodite perennial characterized by herkogamy and dichogamy[END_REF][START_REF] Brunet | Impact of insect pollinator group and floral display size on outcrossing rate[END_REF][START_REF] Ishii | The size of individual Delphinium flowers and the opportunity for geitonogamous pollination[END_REF].

In populations with low v and low inbreeding depression, we expect either of the two edge equilibria produced by pollinator behavior, depending on the intensity of pollen limitation. In populations with little pollen limitation (abundant pollinator visits independent of floral display)

we expect lower edge equilibria with a long individual flowering phenology and small daily floral display producing low geitonogamous selfing rates. A long phenology with low pollen limitation maximizes the number of flowers pollinated and pollen export, but is unlikely to evolve for animal-pollinated species because a small flower display usually entails low pollinator attraction and high extinction risk [START_REF] Devaux | Selection on variance in flowering time within and among individuals[END_REF], except with reliable trap-lining pollinators [START_REF] Schemske | Breeding system and habitat effects on fitness components in three neotropical Costus (Zingiberaceae)[END_REF].

In contrast, upper edge equilibria, with a large floral display during a short individual phenology producing the maximal geitonogamous selfing rate, are expected in populations with low inbreeding depression and strong pollen limitation, due to the combined benefits of pollinator attraction and the automatic advantage of selfing. However, mass flowering phenologies are rare, partly because their short duration increases the risk of pollination failure in a stochastic environment [START_REF] Devaux | Selection on variance in flowering time within and among individuals[END_REF]. The evolution of such phenologies, referred to as semelparous, monocarpic, or "big-bang", involves additional constraints on ecology and life-history not included here [START_REF] Young | Ecology and evolution of long-lived semelparous plants[END_REF].

Limitations and perspectives

In our model, the existence of stable mixed mating systems depends on mechanisms that are supported by experimental observations in many natural populations: high inbreeding depression, pollinator attraction limitation, and pollinators visiting more than one (but not all) flowers on a plant. Relaxing various assumptions of the model would change the position but not the existence of stable equilibrium selfing rates.

Two assumptions could significantly alter our predictions and deserve a full analytical treatment. First, we assumed inbreeding depression for plant viability only. Including inbreeding depression for flower number would reduce the number of pollinator visits to inbred plants, and therefore decrease their geitonogamous selfing rate. A lower selfing rate could increase the equilibrium inbreeding depression maintained by deleterious mutations and could promote evolution of long flowering phenologies, if pollinator attraction limitation is weak. Second, we assumed no within-flower selfing, whereas autonomous selfing occurs in many self-compatible species. The outcome of a model allowing evolution of both flowering phenology and autonomous selfing will depend on the mode of self-fertilization (prior, competing or delayed selfing, Lloyd and Schoen 1992), interacting with inbreeding depression, pollen limitation and life history. Including evolving autonomous selfing in the model would likely cause selfing rates to evolve above the maximum geitonogamous limit in our model, extending the upper edge equilibrium to complete selfing. Autonomous selfing would not eliminate the trade-off between pollinator attraction and avoidance of inbreeding depression, so internal equilibria should still exist. Facilitated selfing (within-flower fertilization caused by pollinators) may similarly increase net selfing rates; it has received little theoretical or empirical attention.

We assumed that a single pollinator visit was sufficient to fertilize all ovules of a flower.

Allowing pollen limitation within flowers would not change geitonogamous selfing rates, which are assumed to be independent of the order of pollen deposition within plants. Within-flower pollen limitation would likely strengthen overall pollen limitation, promoting stable intermediate selfing rates.

We also assumed a single flowering period per plant rather than a perennial life history in which resource allocation among years may influence the evolution of floral display and mating system [START_REF] Morgan | The evolution of self-fertilization in perennials[END_REF]. A normally distributed flowering phenology for individual plant genotypes is assumed, with continual flowering of the population in an aseasonal environment.

In a seasonal environment population flowering phenologies are often approximately normal [START_REF] Schemske | Flowering phenology and seed set in Claytonia virginica (Portulacaceae)[END_REF][START_REF] Schmitt | Density-dependent pollinator foraging, flowering phenology, and temporal pollen dispersal patterns in Linanthus bicolor[END_REF]references in Elzinga et al. 2007). Non-normal population phenologies may be caused by a combination of unpollinated flowers remaining open longer than one day [START_REF] Ashman | How long should flowers live?[END_REF] and inclement weather at the end of the temperate-zone flowering season. Extending longevity of individual flowers in our model would likely smooth the oscillations in the geitonogamous selfing rates for low values of (Figs. 1 andS1) and thus eliminate the minor internal equilibria close to the maximal geitonogamous selfing rates.

Although our model incorporates a detailed description of pollinator behavior, further work should include pollinator learning and population dynamics, interspecific ecological interaction and co-evolution. For example, we assume a constant probability of leaving a plant per flower visited, whereas it can depend on floral display [START_REF] Robertson | The relationship between floral display size, pollen carryover and geitonogamy in Myosotis colensoi (Kirk) Macbride (Boraginaceae)[END_REF] in Myosotis; Harder and Barrett 1995 in Eichhornia paniculata; [START_REF] Mitchell | The influence of Mimulus ringens floral display size on pollinator visitation patterns[END_REF] in Mimulus) and on both the quantity and quality of rewards obtained from previously visited flowers [START_REF] Cresswell | How and why do nectar-foraging bumblebees initiate movements between inflorescences of wild bergamot Monarda fistulosa (Lamiaceae)?[END_REF][START_REF] Johnson | Pollen carryover, geitonogamy, and the evolution of deceptive pollination systems in orchids[END_REF]). Pollinator leaving rates that increase with the number of flowers visited on a plant would reduce the geitonogamous selfing rate below that in the present model, but would not eliminate unavoidable geitonogamy.

  flowering period follows a Poisson distribution with mean . The flowering phenology of plants with a given genotype follows a normal distribution with standard deviation . A standard deviation in flowering time of zero describes plants that display all their flowers on one day, whereas large values of standard deviation characterize plants that display few flowers per day over many days. Within the individual flowering period of a plant, the expected number of flowers open on day (floral display at day ) is

  number of daily pollinator visits $ to a plant follows a Poisson distribution with mean %&( ), where % is pollinator abundance (the expected number of individual pollinators encountering a plant) and &( ) is an attraction function arising from sensory detection and preference of pollinators (the probability that an individual pollinator encountering a plant does visit it). The attraction function depends on the realized floral display ,

  and Johnston et al. (2009), * = ( * ) \]^_ (

  Solving this equation we find the constraint function j k that balances the inbreeding depression and produces equilibrium. The constraint function includes both the automatic genetic advantage of selfing and the ecological constraints on geitonogamous selfing rate caused by pollinator behavior (for details on the method see Porcher and Lande 2013),

For a wide range

  of pollinator behaviors and intensities of pollen limitation, geitonogamous selfing rates generally decrease with longer individual flowering phenology and lower total flower production (Fig. 1 and S1, top panels), because fewer flowers are open simultaneously. However, the impact of flowering phenology on geitonogamous selfing rate diminishes with a large total number of flowers per plant. For very short individual flowering phenologies with < 1, the geitonogamous selfing rate ( ) may show steep changes or be a non-monotonic or even oscillatory function of (Figs. 1 and S1, bottom panels). For such short individual flowering periods, most flowers are open on a single day and are highly self-fertilized, while the rare flowers open in the extreme tails of the individual phenology are strictly outcrossed; a small change in the flowering period then alters the expected total number of flowers fertilized and the proportion fertilized by outcrossed pollen, depending on the shape of the pollinator attraction function. In addition to the flowering phenology ( and ), the average number of flowers visited per plant and the pollen deposition rate (via 5 and 2) exert the greatest impact on geitonogamous selfing rates (eqs. 5 and 6 and Figs. 1, 2, S1and S2). Geitonogamous selfing rates are also influenced by pollinator visitation patterns, with higher selfing rates under constant vs. random movements of pollinators among flowers (eqs. 5 and 6; compare Figs. 1 and S1). This difference is mainly explained by pollinators visiting identical (under the constant rank model) or different (under the random rank model) initial flowers, which disproportionately affect the geitonogamous selfing by receiving the largest amount of outcross pollen. In contrast, changing pollinator abundance % affects total and selfed progeny by the same factor, and thus has no impact on geitonogamy. The range of possible geitonogamous selfing rates is constrained by the total number of flowers produced, the plant flowering phenology and the behavior of pollinators. A geitonogamous selfing rate of zero is always produced by an extremely long flowering phenology ( → ∞) resulting in floral displays of at most one flower per day. In contrast, the highest geitonogamous selfing rates correspond to the shortest phenology ( = 0), with all

  pollinator attraction limitation, changes in total fitness and its components become nonmonotonic functions of individual flowering period, with maxima at intermediate values of individual flowering standard deviation (Fig.S0). Total pollen export and number of pollinated flowers are larger under the random than the constant rank visitation model, because multiple pollinators are expected to fertilize more (different) flowers when they move randomly among them. Constraints and trade-offs imposed by pollinators generate non-monotonic and multivalued relationships among plant fitness components (Figs.3 and S3). The shape of these relationships is most strongly influenced by pollinator attraction limitation and total number of flowers produced, especially for small .

  equilibria. Minor internal equilibria are produced by changes in the sign of }3( )/} and loops in the constraint function j k when plotted against , which both correspond to steep or oscillatory changes in geitonogamous selfing rates for short individual flowering phenologies ( < 1; Figs.1 and S1). Their number and stability depend on the interplay between the foraging behavior of pollinators among flowers and the total production of flowers per plant, (Figs.3,4, S6 and S3, S4, S7 for the random vs. constant rank visitation models, respectively). Since minor internal equilibria are practically indistinguishable from neighboring edge equilibria at maximal geitonogamous selfing rates (e.g. Figs. 5 and S5), they have limited biological significance and will not be mentioned further.The existence of major internal equilibria with > 1 depends most strongly on pollinator attraction limitation and inbreeding depression(Figs. 4, 5, S6 and S4, S5, S7 for the random vs.

  depression. Stable mixed mating systems result from (i) intermediate flowering phenologies maintained by a trade-off between floral display for pollinator attraction and inbreeding depression due to geitonogamous selfing as proposed by Darwin (1876) (internal equilibria), and (ii) extreme flowering phenologies constrained by pollinator behavior (edge equilibria). Stable edge equilibria are maintained by directional selection on flowering phenology and selfing rate.

  mixed mating systems maintained without inbreeding depression. Edge equilibria predicted for extremely long individual flowering phenologies with few flowers open per day (producing minimal geitonogamous selfing) have little biological significance as flowering seasons are usually limited[START_REF] O'neil | Natural selection on genetically correlated phenological characters in Lythrum salicaria L. (Lythraceae)[END_REF]. In contrast, edge equilibria at the opposite extreme of mass blooming produce maximal geitonogamous selfing rates that depend greatly on the pollination parameters (pollinator abundance, foraging behavior, and pollen transfer) and the total flower production of individual plants; these selfing rates are less than one because the first flower a pollinator visits on a plant is certainly (at least partially) outcrossed.

Figure 1 :

 1 Figure 1: Geitonogamous selfing rate, , under pollinator attraction limitation (* = 50, . = 0.1) and pollinator abundance limitation (% = 10; τ = 0.01, dashed, or τ = 0.25, solid), for the random rank visitation model (eqs. 5a and 6a), as a function of flowering phenologies for three

Figure 2 :

 2 Figure 2: Geitonogamous selfing rate, , as a function of pollen transfer parameters (τ and ρ), for two values of standard deviation in flowering time ( ), under the random rank visitation model and pollinator attraction limitation (* = 50, . = 0.1) with = 100 flowers per plant, and % = 10.

Figure 3 :

 3 Figure 3: Relationships between maternal selfed and outcrossed components of fitness under the random rank visitation model with pollinator attraction limitation (* = 50, . = 0.1) and pollinator abundance limitation (% = 10 and 5 = 0.33), for 2 = 0.25 , R = 100, and three values of total number of flowers per plant, . Edge (squares) and internal (circles) stable (full)

Figure 4 :

 4 Figure 4: Stable (full) and unstable (open) internal (circles) and edge (squares) equilibrium geitonogamous selfing rates under pollinator attraction limitation (* = 50, . = 0.1) and the random rank visitation model for three values of total flowers per plant, , as a function of leaving probability, τ , for ρ = 0.25 (top panels), and as a function of pollen carryover, 1 -ρ, for τ = 0.33 (bottom panels). Other parameters % = 10, R = 100, v = 0.2 and = 0.25. The edge equilibria at maximal geitonogamous selfing rate (grey symbols) can overlap with minor

Figure 5 :

 5 Figure 5: Stable (full) and unstable (open) internal (circles) and edge (squares) equilibrium geitonogamous selfing rates under pollinator attraction limitation (* = 50, . = 0.1) and pollinator abundance limitation (% = 10 and τ = 0.33) for the random rank visitation model at the intersection of the inbreeding depression curves (black thin solid lines for v = 0.2 with = 0.25 or = 0; v = 0.02 and v = 0 with = 0; and v = 0 with = 0.25) and the constraint function (black thick solid line; eq. 8) for three values of total flowers per plant, . Other parameters ρ = 0.25 and R = 100; dotted grey vertical lines show maximal geitonogamous selfing rates given τ and ρ.

Acknowledgments

We thank three anonymous reviewers who helped to clarify the manuscript. 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.

Appendix

Derivatives used in evaluating the constraint function (eq. 8).

Poisson probability that a plant displays flowers with expectation on day

…( + 0.5)e ( . ) ⁄ -( -0.5)e ( . ) ⁄ ‡ For = 0 this derivative does not exist, but it cancels out in the numerator and denominator of eq. 8 which is then defined in this limit.

Pollen exported per plant, for both the random and constant rank visitation models:

Total number of flowers pollinated and number of flowers selfed for the random rank visitation

Total number of flowers pollinated and number of flowers selfed for the constant rank visitation model

Approximations for numerical computation

We approximated the realized individual flowering phenologies to a finite flowering period of 10σ days (potentially missing < 5.