Introduction
A defining feature of the angiosperms is their remarkable floral diversity. (Armbruster 2014; Hernández-Hernández & Wiens 2020). Darwin first recognized that the diversity of floral shape, size, colour and scent could be attributable to selection by pollinators (Darwin 1877). This realization has resulted in a large modern research program, spawning ecological, evolutionary and genetic studies, investigating how interactions between plants and pollinators drive floral evolution (Harder & Johnson 2009; Van der Niet & Johnson 2012). While we are now heading toward a strong mechanistic understanding of how flowers evolve (Moyroud & Glover 2017; Shan et al. 2019; Fattorini & Glover 2020), the ecological processes involved in flower diversification remain poorly understood (Kay & Sargent 2009; Johnson 2010; Van der Niet & Johnson 2012; Armbruster 2017).
Most of our understanding of how flowers diversify derives from two principles, which were combined into the Grant-Stebbins model (Johnson 2006, 2010). First, flowering plants should adapt to the most effective pollinator in a given environment (Stebbins 1970); that is, the pollinator that visits most frequently (number of visits) and efficiently (per-visit pollen transport efficiency) (Armbruster 2014). Second, since pollinator assemblages are geographically variable, plants should be under divergent selection in different environments, resulting in adaptation to different pollinators (Grant & Grant 1965).
Shifts in floral characteristics and pollination systems, however, frequently occur without associated geographical shifts in pollinator assemblages, suggesting that other factors drive flower diversification (Herrera et al. 2006; Van Der Niet et al. 2014; Ferreiroet al. 2017). In the last few decades, several studies have emphasized the importance of community context in understanding the ecology and evolution of plant pollination (Caruso 2000; Sargent & Ackerly 2008; Mitchell et al. 2009; Muchhala et al. 2010). Competition and facilitation among plant species for access to pollinator visitation, and interspecific pollen transfer play important roles in determining the outcome of pollination (Geber & Moeller 2006; Morales & Traveset 2008; Sargent & Ackerly 2008; Mitchell et al. 2009; Pauw 2013). Furthermore, competition via interspecific pollen transfer offers a potential mechanism promoting divergence in pollinator use by favoring reduced pollinator sharing (Muchhala et al. 2010; Moreira-Hernández & Muchhala 2019). However, despite now having a more comprehensive understanding of pollination ecology and several hypotheses having been proposed to explain either variability in pollinator use or degree of specialization (Waser et al. 1996; Johnson & Steiner 2000; Aigner 2001; Gómez & Regino 2006; Sargent & Otto 2006; Muchhala et al. 2010; Moreira-Hernández & Muchhala 2019), we still lack a theory explaining the broad patterns of floral diversity within and among communities.
A general mechanism promoting floral diversity in both pollinator use and degree of specialization might derive from the consideration that several processes governing pollination success (e.g. intraspecific competition for access to pollinator visitation, interspecific pollen transfer and pollen carryover) are modulated by floral abundance, which intrinsically varies among species within communities. It is therefore possible that different plant species of different floral abundances face divergent selective pressures from a same pollinator assemblage. For example, interspecific pollen transfer is expected to have a stronger impact on populations of low abundance, as the proportion of interspecific pollinator visits increases with the proportion of heterospecific individuals in plant communities (Rathcke 1983; Feinsinger et al. 1991; Caruso 2002; Palmer et al. 2003; Sargent & Otto 2006; Mitchell et al. 2009; Runquist & Stanton 2013). Likewise, opportunities for pollen loss, whether passively or due to pollinator grooming, should be greater for rare plant species because pollinators visiting rare plants spend more time between conspecific visits (Minnaar et al. 2019). Therefore, pollen carryover— the proportion of the removed pollen carried to the next conspecific flower—is expected to increase with floral abundance. Conversely, intraspecific competition for pollinator visitation should be stronger at high floral abundance, as more flowers compete for visitation by the same pollinator community (Rathcke 1983; Geber & Moeller 2006; Duffy & Stout 2008; Pauw 2013; Benadi & Pauw 2018). While studies carried out at limited spatial scales often find increased per flower pollinator visitation with increasing floral density (due to greater attractiveness of larger flowering patches) (see Ghazoul 2005), at the landscape scale—the scale that ultimately matters for floral evolution—the opposite trend is observed: the number of visits received per flower decreases with a species abundance (see Pauw 2013 and references therein; Hegland 2014; Benadi & Pauw 2018; Bergamo et al. 2020). The reason for such scale dependency is simple: large floral patches are more attractive to pollinators, but a population composed of multiple large floral patches will saturate the pollinators available, leading to stronger intraspecific competition.
In this article, I propose that the pollination system offering the optimal evolutionary solution for a plant species is a function of the plant’s relative abundance in a community. In this view, different pollinators and degrees of specialization are favoured at different floral abundances. Floral diversification can result from shifts in relative species abundance associated with the colonization of new habitats or geographical ranges, creating new conditions under which floral diversification can occur. Abundance has been previously identified as a potential driver of floral specialization (Feinsinger 1983; Sargent & Otto 2006). However, the potential for variability in species abundance to drive adaptation to different types of pollinators has never been considered before. To demonstrate the potential of variability in species abundance to foster floral diversity, I develop a simple mathematical model of pollen transfer considering the interaction of several pollination processes—pollen carryover, pollen removal, intra- and interspecific competition for pollinator visits, and interspecific pollen transfer. By tracking pollen fate, the model explicitly measures male reproductive success. Female reproductive success should be affected similarly to male reproductive success as long as increased conspecific pollen deposition to stigmas increases seed production. I use the model to assemble plant-pollinator networks from simulated plant and pollinator communities. In a community context, the interactions between plants and their pollinators are generally investigated in terms of networks of interactions. Using this approach, I assess if and how interspecific variation in floral abundance generates diversity in pollinator use and degree of specialization. In addition to supporting the conceptual model, the mathematical model is consistent with, and suggests explanations for, several patterns governing the evolution, diversification and community assembly of flowers.

How floral abundance drives flower diversification: a conceptual model

Since plants produce a finite number of gametes, optimizing reproductive success requires maximizing the number of ovules fertilized by a finite amount of pollen. When limited by pollinator quantity, plants should benefit from being less restrictive in their flower accessibility. Any visitor, regardless of its quality (pollen carryover efficiency and specialization), is likely to increase the number of pollen grains deposited on conspecific stigmas (Thomson et al. 2000; Thomson 2003; Muchhala et al. 2010). However, pollen grains have a higher probability of reaching conspecific stigmas when carried by more efficient pollinators. When enough pollinators are available to remove most pollen grains, pollinator quality becomes more limiting to plant reproductive success than pollinator quantity, and plants should specialize on the most efficient pollinator (Thomson et al. 2000; Thomson 2003).
Here I propose that the selective importance of pollinator quantity and quality is modulated by floral abundance. At high floral abundance, as more pollinators are required for sufficient pollination, plant reproduction is more strongly limited by pollinator quantity. Conversely, pollen loss to inefficient carryover and interspecific visits is reduced as floral density increases. Therefore, increased floral abundance should increase the selective importance of pollinator quantity while reducing the selective importance of pollinator quality. Under these conditions, plants should benefit from generalized pollination where more pollinators, but perhaps more wasteful ones, have access to flowers. At low abundance, the dynamic is reversed and plants should benefit from specializing on efficient carriers of their pollen.
While I have so far considered a dichotomy between low and high abundance, plant populations can exist in any state from extremely rare to abundant. Likewise, pollinators vary in their abundance (quantity component) and efficiency (quality component). As each pollinator offers a unique combination of quantity and quality components of pollination, changes in a plant species’ relative abundance in a community will shift the identity of the pollinator representing the most effective pollinator, and therefore, the pollination system resulting in a fitness optimum. In this view, floral abundance shapes the plant selective landscape. Variations in plant abundance move the fitness peak of the selective landscape toward different pollination systems, fostering floral diversification.

Model of pollen transfer

Here I develop a mathematical model which determines how the optimal pollinator or set of pollinators for a plant population changes as a function of floral abundance. The model measures the proportion of pollen grains produced by a single flower of the focal species (hereafter ‘focal flower’) that reaches conspecific stigmas. This value is influenced by the interaction of several pollination processes that are linked to floral abundance (described above): pollen removal and carryover, intra- and interspecific competition for pollinator visitation, and interspecific pollen transfer. By tracking pollen fate, the mathematical model explicitly determines the pollination system maximizing male reproductive success. However, selection through either male and female function is expected to reach the same solution in terms of optimal pollination system as long as pollen receipt and export are limiting and are governed by the same variables for each sex (i.e. pollinator identity and abundance).
In the model, I treat pollinators as functional groups of pollinator species with similar attributes (morphology, behaviour) that produce similar selection on flowers (e.g. different species of hummingbirds, large-bodied bees) (Fenster et al. 2004). The model assumes that flowers are distributed homogeneously in space (i.e., flowers of the same species are not more likely to be near one another). Variation in floral abundance in the model is therefore the result of variation in the abundance of the focal plant species rather than variation in the number of flowers produced per individual of the focal species. By assuming homogeneous spatial distribution, the model does not consider the potential for facilitation among species sharing the same pollinators, although facilitation is expected to benefit rarer species (Rathcke 1983; Steven et al. 2003; Essenberg 2012), contributing to the predicted increase in quantity limitation with abundance—i.e. facilitation should reinforce the predictions from the conceptual model.
For a given plant species a , the total number of pollen grains produced per flower is represented by Pt . The proportion of grains removed with each pollinator visit to a focal flower is represented by Ri , and depends on both the attributes of the pollinator i and of the focal plant species. While adaptation toward a pollination system could, theoretically, increase R , I am more interested in the causes of shifts between pollination systems rather the mechanisms leading to a subsequent better fit to the system, so the model does not consider evolution of Ri . With each new visit to a focal flower, the amount of pollen remaining on the anthers diminishes, and the amount of pollen picked up with each new pollinator visit diminishes proportionally to the number of visits already received, resulting in an exponential decline of pollen removed with each new visit (Young & Stanton 1990; Robertson & Lloyd 1993). Considering that pollinator i is the only visitor (floral generalization is considered in equation 5), the total amount of pollen removed, Pr , from the focal flower by a given pollinator i is therefore
\({P_{r}=P}_{t}\left[1-\left(1-R_{i}\right)^{V_{\text{ij}}}\right]\)(1)
where Vij represents the number of visits by pollinator i to flower j , which is a subset ofVi , the total number of visits made by the pollinator in the plant community. Assuming that the visits made by the pollinator i , Vi , are equally distributed among all the flowers visited by the pollinator, the number of visits to the focal flower by pollinator i is
\(V_{\text{ij}}=\frac{V_{i}}{A_{a}+\sum A_{i}}\) (2)
where Aa represents the abundance of the focal species and ΣA i represents the total abundance of flowers for each plant species visited by the pollinator iexcluding the focal species. In this model I treat all plant species as being equally attractive to each pollinator (variable attractiveness could be considered by weighting Aa by the relative attractiveness of the focal species).
Considering the role of floral abundance and competition for a limited number of pollinator visits, the number of pollen grains removed from a focal flower by a given pollinator can be expressed as
\(P_{r}=P_{t}\left[1-\left(1-R_{i}\right)^{\frac{V_{i}}{A_{a}+\sum A_{i}}}\right]\)(3)
The proportion of pollen grains removed by the pollinator i that reach a conspecific stigma depends on the pollen carryover capacity of the pollinator, Ci — the proportion of pollen carried over to each subsequent visit. Pollen carryover only accounts for the proportion of pollen that is still in circulation for potential pollination. Pollen groomed and accumulated on the scopae or corbiculae of bees is therefore not considered in pollen carryover as this pollen very rarely contributes to pollination (Thorp 2000). For a given individual of the pollinator i , as with each new visit the amount of pollen remaining on the pollinator body declines at a rateCi , the proportion of grains remaining on the individual pollinator follows an exponential decay (Lertzman & Gass 1983; Campbell 1985; Robertson 1992) (although longer or shorter than exponential tails have been observed; Morris et al. , 1994; Holmquist et al. , 2012). From the pollen grains deposited on the individual pollinator body, as pollen is lost with each visit, the amount that will reach a conspecific stigma is a function of the number of interspecific visits made by the pollinator before reaching a conspecific flower, which is a function of the reciprocal of the proportional floral abundance of the focal species in the community of flowers visited by pollinator i : (Aa + ΣAi ) / Aa . Therefore, assuming that the pollinator does not exhibit floral constancy (floral constancy is considered in Appendix S1 by weighting ΣAi by the reciprocal of the degree of floral constancy exhibited by the pollinator), the proportion of the removed pollen grains by individuals of pollinator “\(i\)” that reach conspecific stigmas is equivalent to
\begin{equation} {C_{i}}^{\frac{A_{a}+\sum A_{i}}{A_{a}}}\nonumber \\ \end{equation}
Floral constancy can be considered as temporal specialization (Waser 1986; Amaya-Márque 2009), and therefore has a similar impact on pollen transport as fixed specialization (not behaviourally flexible), which is represented in the model by Aa + ΣAi . Considering both the amount of pollen removed by the pollinator i and the proportion of this pollen that is deposited on conspecific stigmas, the total number of pollen grains deposited on conspecific stigmas, Pd , is expressed as
\(P_{d}=P_{t}\left[1-\left(1-R_{i}\right)^{\frac{V_{i}}{A_{a}+\sum A_{i}}}\right]\bullet{C_{i}}^{\frac{A_{a}+\sum A_{i}}{A_{a}}}\)(4)
Equation (4) determines the effect of specialization on a given pollinator on the pollination success of the focal plant species. The effect of specialization on different pollinators can be evaluated by comparing the value of Pd for exclusive pollination by different pollinators.
When multiple pollinators visit the focal plant species (generalization on a subset of the available pollinators), considering that the number of pollen grains removed from a flower diminishes with each new visit, the number of pollen grains removed by a given pollinator idepends on visitations by other pollinators. Pollinator itherefore removes a subset of the total amount of pollen removed by all the pollinators visiting the focal flower. Assuming random visitation order between individuals of the different pollinators, the total proportion of pollen grains removed from the focal flower,Pr, can be expressed as the product of the proportion of pollen removed by each pollinator alone.
\begin{equation} \prod_{i=1}^{n}\left(1-R_{i}\right)^{\frac{V_{i}}{A_{a}+\sum A_{i}}}\nonumber \\ \end{equation}
From the total number of pollen grains removed from the focal flower, the contribution of pollinator i to the number of removed pollen grains is relative to the proportion of visits to the focal flower made by the pollinator i
\begin{equation} \frac{\left(\frac{V_{i}}{A_{a}+\sum A_{i}}\right)}{\left(\sum_{i=1}^{n}\frac{V_{\text{ij}}}{A_{a}+\sum A}\right)}\nonumber \\ \end{equation}
and its pollen removal rate relative to the removal rate of the other pollinators visiting the focal flower.
\begin{equation} \frac{R_{i}}{\left(\frac{\sum_{i=1}^{n}R_{i}}{n}\right)}\nonumber \\ \end{equation}
The number of pollen grains removed by the pollinator i can therefore be expressed as
\begin{equation} \left[\prod_{i=1}^{n}\left(1-R_{i}\right)^{\frac{V_{i}}{A_{a}+\sum A_{i}}}\right]\bullet\frac{\left(\frac{V_{i}}{A_{a}+\sum A_{i}}\right)}{\left(\sum_{i=1}^{n}\frac{V_{\text{ij}}}{A_{a}+\sum A}\right)}\bullet\frac{R_{i}}{\left(\frac{\sum_{i=1}^{n}R_{i}}{n}\right)}\nonumber \\ \end{equation}
The proportion of the removed pollen grains that reaches conspecific stigmas is measured in the same way as for equation (4). The individual contribution of each pollinator to Pd is summed and the total amount of pollen deposited on conspecific stigmas corresponds to
\begin{equation} P_{t}=\sum_{i=1}^{n}P_{t}\left\{\left[1-\prod_{i=1}^{n}\left(1-R_{i}\right)^{\frac{V_{i}}{A_{a}+\sum A_{i}}}\right]\bullet\frac{\left(\frac{V_{i}}{A_{a}+\sum A_{i}}\right)}{\left(\sum_{i=1}^{n}\frac{V_{\text{ij}}}{A_{a}+\sum A}\right)}\bullet\frac{R_{i}}{\left(\frac{\sum_{i=1}^{n}R_{i}}{n}\right)}\right\}\bullet{C_{i}}^{\frac{A_{a}+\sum A_{i}}{A_{a}}}\ \ \ (5)\nonumber \\ \end{equation}
Equation (5) is a generalization of equation (4) and determines the effect of pollination by a combination of pollinators on the pollination success of the focal plant species.