Methods

Importance of pollinator quality and quantity

To determine the effect of floral abundance on the relative importance of the quantity and quality components of pollination for pollination success, using equation (4), I compared the impact of variations in those components on conspecific pollen receipt at different floral abundances. Equation (4) offers an explicit definition of which parameters constitute the quantity and quality components of pollination. Factors affecting the quantity of pollen removed (the left part of the equation)—pollinator abundance and pollen removal rate—are defined as the quantity component. Factors affecting pollen transport efficiency (the right part of the equation)—pollinator carryover capacity and specialization—are defined as the quality component. Given that removal rate does not affect pollen transport efficiency (e.g. a pollen forager will be of low quality despite having a high removal rate), removal is only considering to affect pollination quantity. I used ΣAi , the total floral abundance of all the plant species visited by the pollinator (see equation 4) as a proxy for pollinator specialization. The number of pollen grains produced by the focal flower deposited on a conspecific stigma (equation 4) was compared at low and high values of the parameters (Table 1) while maintaining the other parameters constant. The proportional change (highlow ) / high produced an estimate of the importance of variation of those parameters on pollination success. The importance of the different parameters for pollination success was compared for a range of floral abundances from 2 to 1500 (at least two flowers are required for cross-pollination).
Low, medium and high values of pollen carryover and pollen removal were parameterized based on data from the literature (Table 1). From a literature survey of 18 studies on plant species and pollen vectors, Robertson (1992) reported a range in pollen carryover from 50.2% to 94.7%. I used values of 0.55, 0.73 and 0.9 as low, medium and high values of pollen carryover in the model respectively. The values of pollen removal were selected following Thomson (2003) who modeled pollen delivery as a function of low and high values of pollen removal of 0.3 and 0.7 respectively. In the model, I used values of 0.3, 0.5 and 0.7 as low, medium and high values of pollen removal respectively. These values encompass the pollen removal values measured in various systems (e.g. Wolfe and Barrett 1989, Young and Stanton 1990, Harder 1990, Thostesen and Olesen 1996). Low and high values of total number of visits by the pollinator in the community and abundance of the flower species were not based on empirical data, but were rather selected such that, (1) for plant species of intermediate abundance, most pollen grains were removed (> 90%) at high values of pollinator visits and low abundance of other flower species, while (2) a minority of grains (> 50%) were removed at low pollinator visits and high abundance of other plant species. These scenarios reflect low and high pollinator limitation respectively and should therefore encompass most real-life situations. Medium values of these parameters were determined as the mean between low and high values (high and low values of total number of visits by the pollinator in the community and abundance of the other flower species represent a two-fold increase and decrease from the medium values respectively).
Because the number of pollen grains deposited on conspecific stigmas might be sensitive to the choice of values of the parameters used for the mathematical model, I compared the impact of variations in pollen carryover, pollen removal, pollinator visitation and specialization on conspecific pollen receipt at each possible set of values of the other parameters (low, medium, and high). I used these alternative parameter values to set upper and lower bounds for the estimated importance of the quantity and quality components of pollination. Intermediate values of the importance of a parameter on pollination success correspond to the values obtained while all other parameters were set to medium values. Upper and lower values correspond to the maximal and minimal values obtained among all alternative values of the other parameters respectively. Essentially, the upper and lower values of the estimated importance of the quality and quantity components of pollination indicate the degree to which the estimate varies as a function of variation in the different parameters of the mathematical model and serve as a confidence interval.

Plant-pollinator network simulations

Using equation (5), I verified how variation in floral abundance affects the structure of plant-pollinator networks. Each simulated network was composed of a community of 10 pollinators and 12 plant species. Pollinator communities were assembled by randomly sampling values of pollen carryover and removal for each pollinator from uniform distributions using the runif function in R (R core team, 2020) with maximal and minimal values of 0.9 and 0.55, and 0.7 and 0.3 for carryover and removal respectively. The number of visits made by the different pollinators (relative to their abundance) was sampled from a Poisson log-normal distribution using the rpoilog function in the R package sads (Prado et al. 2018). Poisson log-normal distributions are often used to characterize community species-abundance distributions (Baldridge et al. 2016).
After randomizing plant species order, each plant species colonized the pollinator community successively until all species had colonized the community. For each colonization event, the plant species could evolve to be pollinated by any possible combination of pollinators in the community. The combination resulting in the highest pollination success was selected as the evolutionary outcome for the plant species (assuming no restriction on the evolution of different pollination systems). Considering that new colonization events affect competition and interspecific pollen transfer, after all plant species colonized the community, each plant species could continue evolving different pollination systems. This was simulated by allowing for five successive times each species in random order to evolve a new pollination system. This assured that the networks had the opportunity to reach a stable evolutionary solution.
Plant pollination success associated with the evolution of pollination by the different possible combinations of pollinators was calculated and compared by inputting the simulated parameters (see Table 2) in equation (5). The model used for the simulations incorporated adaptive foraging by using equation (6) and (8) to characterize reward availability and pollinator visitation. The reward production rate,Fa , was set to 1 such thatVrj = Vj . Equation (6) therefore directly corresponded to
\begin{equation} Q_{a}=\frac{1}{1+V_{j}}\nonumber \\ \end{equation}
such that reward quantity was directly related to the number of pollinator visits received. Adaptive foraging was updated dynamically with every change in interaction and affected pollination system evolution. Competition for visits by the different pollinators was dynamically updated with each new colonization event.
Sets of 100 simulations were run for plant communities of either variable interspecific floral abundance or same floral abundance at low (average of 100 flowers), intermediate low (average of 250 flowers), intermediate high (average of 500 flowers) and high floral abundance (average of 1000 flowers). These floral abundance values were selected to encompass a range of situations from the removal of most pollen grains produced by flowers (> 99%) to the removal of a low portion of pollen grains (< 60%) and was given by
\begin{equation} {\overset{\overline{}}{R}}^{\frac{\sum V}{n}}\nonumber \\ \end{equation}
For the simulations with variable floral abundance, plant communities were assembled by randomly sampling plant abundances for each species from a Poisson log-normal distribution.
To analyse the properties of the simulated plant-pollinator networks, I measured network nestedness, connectance and average number of shared pollinators per plant species (measure of niche overlap) using the ‘networklevel’ function in the R package bipartite (Dormann et al. 2020). ‘Networklevel’ produces values of nestedness in degrees (T). Following Bascompte et al. (2003) nestedness, N, was defined as N = (100 – T)/100 with values ranging from 0 to 1 (where 1 represents maximum nestedness). Pollinators in this model were treated as functional groups of pollinator species with similar attributes which prevented the formation of modules of pollinators sharing similar attributes. Network analysis therefore did not include measures of modularity. Variability in degree of specialization within communities was measured as the standard deviation in the number of pollinators visiting the different plant species constituting the plant community of a simulated network.
I ran sets of simulations in which pollinator abundance was either (1) constant and independent of floral resource abundance, or (2) variable and proportional to the resources available to each pollinator. In plant-pollinator systems exhibiting very tight mutualisms where pollinators depends on their plant host throughout the entirety of their life cycle (e.g. figs and fig-wasps, yucca and yucca-moths), pollinator abundance may be tightly linked to its floral host abundance. In most plant-pollinator systems, however, pollinator abundance is weakly linked to floral host abundance since pollinator populations are limited by other factors such as nest sites, larval host availability or territory (Burd 1995; Pauw 2007, 2013; Benadi & Pauw 2018). For this reason, and because the models with and without variation in pollinator abundance produced qualitatively similar results, I present the results from the simulations without variation in pollinator abundance in this paper (results and details on the methodology for simulations with variation in pollinator abundance can be found in the supplementary material; Appendix S1).
I verified the robustness of the conclusions drawn from the model to variation in parameter values beyond the ones used for the standard model and simulations. Additionally, I verified the robustness of the model to the presence of dynamic reward replenishment (equation 7) and the absence of adaptive foraging. Finally, although the model presented here does not incorporate flower consistency, I ran supplementary sets of simulations with different degrees of flower constancy. Given that the general conclusions of the study were robust to those alternative models and parameter values, detailed results from the alternative sets of simulations are presented as supplementary information (Appendix S1).

Pollination system evolution as a function of floral abundance

Using the simulated plant-pollinator networks of the variable floral abundance plant communities with intermediate high average abundances, I tested how floral abundance affects the degree of floral specialization and whether different floral abundances lead to adaptation to different pollinators. For each simulated plant-pollinator network, after all plant species had colonized the community, a new plant colonist invaded the community. I varied the new colonist’s abundance and recorded the subset of pollinators on which the plant evolved at each abundance value.