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
(high – low ) / 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.