#### Study system

We studied a population of *Dalechampia schottii* Greenman (Euphorbiaceae) in a lightly disturbed subperennial tropical forest in the botanical garden ‘Dr Alfredo Barrera Marin’ (20°51′11″N, 86°53′43″W) 1 km south of Puerto Morelos, in north-eastern Quintana Roo, Mexico, in September and October 2007. *Dalechampia schottii* is endemic to the Yucatán Peninsula. As illustrated in Fig. 1, the relatively small blossoms are functionally bisexual with three pistillate flowers situated below four staminate flowers. Above the staminate flowers is a resin gland composed of packed bractlets which secrete blue resin. Above and below the blossom are two 5–15-mm, pale-green to white involucral bracts. The blossoms are partially protogynous. During the first 2–4 d after they open, the stigmas are receptive but the male flowers remain closed (‘female phase’); subsequently one to two male flowers open each day, while the stigmas remain receptive (‘bisexual phase’). Resin starts to be produced in the beginning of the female phase.

The study population of *D. schottii* grows in sympatry with *Dalechampia scandens L.*, which has much larger blossoms with transparent to whitish resin and two large pale-green to white bracts. *Dalechampia* resins are collected by several species of megachilid and apid bees for use in nest construction (Armbruster, 1984, 1985, 1988). At this field site, *D. schottii* was pollinated by large resin-collecting female *Euglossa* cf. *viridissima* Friese and small resin-collecting female *Hypanthidium* cf. *melanopterum* Cockerell*.* In the study period, however, we observed very few *Hypanthidium*, so most of the pollination was probably by *Euglossa.* This contrasts to the year previous to this study when *Hypanthidium* was abundant, and probably the main pollinator (R. Pérez-Barrales, pers. obs.). *Dalechampia schottii* has generally been described as a ‘small-bee-pollinated’*Dalechampia* (Armbruster, 1993; Hansen *et al.*, 2000), and in other populations *Trigona* sp. and *Hypanthidium* cf. *melanopterum* have been reported as pollinators (Armbruster, 1985, 1988).

#### Fitness model

Male reproductive success is difficult to estimate in this system, and we focused our study on the female reproductive success of individual blossoms. Our goal was to study the direct selection on blossoms that results from pollinator behaviour. Selection at the blossom level should not be confused with selection at other levels of organization, for example the plant level. Pollinator-mediated selection on average blossom traits on the plant may not be fully explained by selection at the blossom level if pollinators can distinguish between plants within a patch, and this creates a pollination pattern that cannot be explained by within-patch differences in blossom phenotype (i.e. if pollinators choose among blossoms within a plant differently from among blossoms on different plants within a patch). However, the blossom-level selection probably captures most of the selection at the plant level as bees are unlikely to distinguish between plants in a patch except by differences in blossom phenotype. The measures of reproductive success at the blossom level are only valid measures of individual fitness and individual selection if there are differences in mean floral characteristics (i.e. if individuals differ in the distribution of these characters among flowers on a plant). For this selection to induce an evolutionary response there also needs to be heritable variation in these differences. However, for convenience, we refer to our measurements as ‘fitness’ and ‘selection’ and refer to the estimated relationships between fitness and characters as ‘selection gradients’. Note also that this study does not account for other types of selection such as herbivory, selection on the male function, selection during other phases of development, and indirect selection resulting from correlation among traits, all of which will contribute to the total selection acting on the traits.

To build our model of blossom seed set, we combined our field observations of pollen arrival with an empirically established relationship linking pollen arrival to seed set. This relationship corresponds to an increasing function where *P*, the number of pollen grains, maps into *S*, the number of seeds set with an asymptotic value of nine seeds. The general function is described as:

- (Eqn 1)

where *a* describes the increase in the number of seeds with an increase in pollen load. In *D. scandens*, *a* was estimated as 0.0850 (R. Pérez-Barrales *et al.*, unpublished). This value of *a* maps 1 pollen grain into 0.71 seeds, 10 pollen grains into 4.1 seeds, and 50 pollen grains into 7.3 seeds, and explained 24.2% of the variation in seed set in the study on *D. scandens*. Although this relationship might be different for *D. schottii*, the similarity between the two species (same number and arrangement of pistillate flowers; same type of pollination system) suggests that this relationship should be close enough to be useful for comparing seed production under different pollination scenarios.

To understand the effects of the different floral traits on seed set, we modelled the total number, *P*, of pollen grains that arrived at the three stigmatic surfaces during the first day of the bisexual phase as a function of three variables: the probability, *V*, of a blossom being visited during 1 d, the number of pollen grains, *P*_{F}, arriving in the female phase given at least one visit, and the number of pollen grains, *P*_{B}, arriving on the first day of the bisexual phase. These were combined as

- (Eqn 2)

where 1 − (1 − *V* )^{3} gives the probability of being visited during 3 d (i.e. the median duration of the female phase in this population), and (1 − (1 − *V* )^{3}) *P*_{F} gives the expected amount of pollen arriving during the female phase. In the rest of the bisexual phase, not included in this study, the blossom continues to self-pollinate and the arrival of additional cross-pollen is probably less important, because the male flowers are positioned above the female flowers and will form a platform that may hinder contact between the stigma and the pollinator. Although we did not include the total amount of self-pollen, the pollen arrival on the first day in the bisexual phase gives us a relative difference in the ability of the blossoms to self-pollinate. Fig. 2 provides a graphical representation of the model.

The probability, *V*, of a blossom being visited during 1 d can be influenced by the characteristics of each single blossom, but also by the floral display of the patch to which each blossom belongs. Several studies have showed the importance of floral display in the foraging decision of pollinators (Harder & Barrett, 1995; Harder *et al.*, 2001; Harder & Johnson, 2005). We therefore allowed *V* to be influenced by the size of the bract, the size of the resin gland, and the number of blossoms in a patch. To analyse this relationship we used contextual analysis (Heisler & Damuth, 1987), which allowed us to investigate the effects on the probability of being visited during 1 d both among and within groups of blossoms (patches) in the same model. The data have four levels; each observation (*i* ) was taken on a particular day (*j* ) on a particular blossom (*k*), each blossom being nested within a patch (*l* ). We made only one observation per blossom on each day, and hence *V* is given by the probability of observing a visit at observation *i*. We used a mixed-effects model with logit link and binomially distributed errors to analyse the data. The log odds for a blossom of being visited during 1 d is given by:

- (Eqn 3)

for observations *i* = 1,...,484, days *j* = 1,...,38, blossoms *k* = 1,...,159, and patches *l* = 1,...,25. The subscript *j* or *k* with *i* in brackets denotes the corresponding subscript *j* or *k*, respectively, for observation *i*. The subscript *l* with *k* in brackets denotes the corresponding subscript *l* for blossom *k*. One bar denotes the patch mean (for UBL and GA) and two bars denote the grand mean of the trait. *α*_{1j} (= is a random effect that gives the deviation for a day *j* from the grand mean log odds of being visited during 1 d, *α*_{2k} is the predicted log odds of blossom *k* being visited during 1 d, *β*_{0l} is the predicted log odds of being visited for patch *l*, *β*_{1} is the effect of upper bract length within patch, *β*_{2} is the effect of gland area within patch, is a random effect that represents the deviation of the predicted value of blossom *k* from the rest of the model at the blossom level, *γ*_{0} is the grand mean log odds of being visited during 1 d, *γ*_{1} is the effect of upper bract length among patches after the within-patch effect (*β*_{1}) has been removed (the effect of upper bract length among patches is *γ*_{1} + *β*_{1}), *γ*_{2} is the effect of gland area among patches after the within-patch effect (*β*_{2}) has been removed, *γ*_{3} is the effect of the number of blossoms in a patch, and *ε*_{l}^{β0} is a random effect that represents the deviation of the predicted value for patch *l* from the rest of the model at the patch level. The random effects/error terms (*ε*) at each level are assumed to be normally distributed with a mean of zero. In the fitness model we are only interested in the within-patch effects of the morphological traits, while controlling for the among-patch effects. Hence, the probability of being visited during 1 d, *V*, is given by

- (Eqn 4)

where *γ*_{0}, *β*_{1} and *β*_{2} are estimated by Eqn 3, and GA and UBL are centred on their grand means.

The pollen arrival in the female phase, given that the blossom has been visited, *P*_{F}*,* is potentially influenced by the amount of visitation and the fit between the pollinator and the pollen-receiving structures (i.e. the stigma). This fit is influenced by the distance between the reward and the stigma, estimated by GSD. We investigated these effects using a generalized linear model with a log link and quasipoisson-distributed errors to account for overdispersion. The probability of being visited during 1 d and the gland-stigma distance were entered as fixed effects. The estimates (intercept *α*_{1}, and slopes *β*_{1.1}, *β*_{1.2}, *β*_{1.3} and *β*_{1.4}) from this analysis were used to model pollen arrival in the female phase as

- (Eqn 5)

where GSD is centred on its grand mean.

Pollen arrival in the bisexual phase, *P*_{B}, is potentially influenced by the visit of pollinators and by autogamy. Autogamy may be influenced by visitation by pollinators or other biotic and abiotic factors that provoke the fall of pollen on the stigma, but also by the distance between anther and stigma (Armbruster, 1988; Armbruster *et al.*, 2009). We analysed the pollen arrival during the bisexual phase using a generalized mixed-effects model with a log link and quasipoisson-distributed errors to correct for overdispersion, with the probability of being visited during 1 d, the gland-stigma distance, and the anther-stigma distance as fixed effects, and patches as random effects to control for differences in disturbance between patches. These traits, together with visitation, can interact in a complicated way, and indeed, the best model included linear terms and their two- and three-way interactions. We modelled pollen arrival in the bisexual phase, *P*_{B}, from the estimates from this analysis as:

- (Eqn 6)

where GSD and ASD are centred on their grand means.

To aid interpretation, we obtained linear (*β*) and quadratic selection gradients by multiple regression on the predicted values (transformed to relative fitness). We first included only the linear terms to estimate *β.* We then included linear, quadratic and pairwise interaction terms to estimate quadratic selection gradients (Lande & Arnold, 1983). Note that the quadratic terms in the regression were multiplied by 2 in order to obtain the diagonal of quadratic selection-gradient matrix. We also provide mean-standardized selection gradients, as they can be interpreted as percent of selection on fitness (for which *β *= 1) and therefore used to estimate the strength of selection (Hansen *et al.*, 2003b; Hereford *et al.*, 2004). Mean-standardized selection gradients are given in percentages. Selection gradients are estimated for both the predicted amount of pollen and the predicted seed set. Seed set is most probably a better measurement of reproductive success than the amount of pollen, but it does not take into account the effect of pollen competition on seed quality. Pollen competition has been shown to influence seed quality in *D. scandens* (Armbruster & Rogers, 2004).

We investigated the curvature of the fitness function using canonical analysis (Phillips & Arnold, 1989). In a canonical analysis, the axes of the quadratic selection matrix are rotated so that the cross-product terms are eliminated. The new axes are linear combinations of the traits given by the eigenvectors of the quadratic selection matrix and the new quadratic selection gradients are the eigenvalues of this matrix. Hence, the new quadratic selection gradients are fewer and orthogonal. This method has been recently used to study natural selection on *Silene virginica* (Reynolds *et al.*, 2010).

There are several benefits of using such a fitness model instead of estimating selection gradients directly from an observed fitness component, as is normally done in selection studies. First, the fitness model estimates functional relationships, which provide insight into the selection process. Secondly, by taking the functional relationships into account it filters away many environmental correlations and some of the noise that may obscure the relationship between the traits and fitness, and it has a higher probability of avoiding problems of unmeasured correlated traits affecting fitness. Lastly, it can incorporate known biological constraints such as the relationship between the amount of pollen and the number of seeds. The downside of this approach is that the use of link functions without explicit biological bases may affect the estimation of quadratic selection gradients, because the link functions affect the curvature. It is therefore important to consider quadratic terms and interactions in the model.

#### Statistical analyses

We performed model selection by using AICc scores for the model with binomial errors and QAICc scores for the models with quasipoisson error distribution (Burnham & Anderson, 2002). We evaluated quadratic and interaction terms in addition to the linear terms. The only term that was kept in a model even when it did not improve the AICc or QAICc scores was the linear term for upper bract length in the visitation model (Eqn 3).

The analysis of pollen arrival in the female phase given that the blossom had been visited was performed on 36 blossoms in 11 patches after the exclusion of two outliers that had a strong leverage. The analysis of pollen arrival in the bisexual phase was performed on 149 observations (blossoms) in 24 patches.

We estimated the 95% confidence intervals for the selection gradients using parametric bootstrapping (Davison & Hinkley, 1997), re-sampling 1000 times the probability, *V*, of being visited from binomial distributions, and the pollen arrival in the female phase given visit, *P*_{F}, and the pollen arrival in the bisexual phase, *P*_{B}, from negative binomial distributions. The parameters in Eqns 4, 5 and 6 were re-estimated from each of these runs, by exchanging the old variables for the new (obtained from the parametric distributions), both for the response and for the fixed effects. This gave us 1000 different fitness surface estimates, which we used to calculate confidence intervals.