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Larger workers outperform smaller workers across resource environments: An evaluation of demographic data using functional linear models


Abstract and Figures

• Behavior and organization of social groups is thought to be vital to the functioning of societies, yet the contributions of various roles within social groups toward population growth and dynamics have been difficult to quantify. A common approach to quantifying these role‐based contributions is evaluating the number of individuals conducting certain roles, which ignores how behavior might scale up to effects at the population‐level. Manipulative experiments are another common approach to determine population‐level effects, but they often ignore potential feedbacks associated with these various roles. • Here, we evaluate the effects of worker size distribution in bumblebee colonies on worker production in 24 observational colonies across three environments, using functional linear models. Functional linear models are an underused correlative technique that has been used to assess lag effects of environmental drivers on plant performance. We demonstrate potential applications of this technique for exploring high‐dimensional ecological systems, such as the contributions of individuals with different traits to colony dynamics. • We found that more larger workers had mostly positive effects and more smaller workers had negative effects on worker production. Most of these effects were only detected under low or fluctuating resource environments suggesting that the advantage of colonies with larger‐bodied workers becomes more apparent under stressful conditions. • We also demonstrate the wider ecological application of functional linear models. We highlight the advantages and limitations when considering these models, and how they are a valuable complement to many of these performance‐based and manipulative experiments.
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Ecology and Evolution. 2021;11:2814–
Received: 20 Novemb er 2020 
Accepted: 8 January 2021
DOI: 10.1002/ece 3.7239
Larger workers outperform smaller workers across resource
environments: An evaluation of demographic data using
functional linear models
Natalie Z. Kerr1,2 | Rosemary L. Malfi3| Neal M. Williams4|
Elizabeth E. Crone1
This is an op en access arti cle under the ter ms of the Creative Commons Attribution L icense, which pe rmits use, dis tribu tion and reprod uction in any med ium,
provide d the original wor k is properly cited.
© 2021 The Authors . Ecology and Evolution published by John Wiley & S ons Ltd.
1Depar tment of Biolog y, Tufts Unive rsity,
Medford, MA, USA
2Depar tment of Biolog y, Duke University,
Durham , NC, USA
3Depar tment of Biolog y, University of
Massachuset ts- Amherst , Amherst, M A, USA
4Depar tment of Entomology and
Nematol ogy, Unive rsity of California, Davis,
Natalie Z. Kerr, Depa rtme nt of Biology, Duke
University, Durham, NC 27710, USA.
Funding information
Tufts Graduate Student Research Aw ard;
NSF, Grant/Award Number: DEB1354 022
and DEB1411420
1. Behavior and organization of social groups is thought to be vital to the function-
ing of societies, yet the contributions of various roles within social groups toward
population growth and dynamics have been difficult to quantify. A common ap-
proach to quantifying these role- based contributions is evaluating the number of
individuals conducting certain roles, which ignores how behavior might scale up
to effects at the population- level. Manipulative experiments are another common
approach to determine population- level effects, but they often ignore potential
feedbacks associated with these various roles.
2. Here, we evaluate the effects of worker size distribution in bumblebee colonies on
worker production in 24 observational colonies across three environments, using
functional linear models. Functional linear models are an underused correlative
technique that has been used to assess lag effects of environmental drivers on
plant performance. We demonstrate potential applications of this technique for
exploring high- dimensional ecological systems, such as the contributions of indi-
viduals with different traits to colony dynamics.
3. We found that more larger workers had mostly positive effects and more smaller
workers had negative effects on worker production. Most of these effects were
only detected under low or fluctuating resource environments suggesting that the
advantage of colonies with larger- bodied workers becomes more apparent under
stressful conditions.
4. We also demonstrate the wider ecological application of functional linear models.
We highlight the advantages and limitations when considering these models, and
how they are a valuable complement to many of these performance- based and
manipulative experiments.
Bombus vosnesenskii, callow size, colony age, development, egg production, functional linear
models, larval survival
KERR Et al .
In animal societies, individuals are often observed performing dif-
ferent tasks, such as guarding nests and burrows (Clutton- Brock,
Brother ton, et al., 20 01), nursing, and caring for young (Kerth, 2008;
Sparkman et al., 2011; Wilkinson, 1992), or reproducing (Faulkes &
Bennett, 2001; Jarvis, 1981). The roles within these social groups are
commonly assigned based on the age (Brent et al., 2015; Jarvis, 1981;
Seeley & Kolmes, 1991; Zöttl et al., 2016), size (Goulson, 2009;
Porter & Tschinkel, 1985; Schwander et al., 2005; Wenzel, 1992),
and/or status (Frank, 1986; Sparkman et al., 2011) of individuals.
For example, in Meerkats, which are cooperative breeders, younger
nonbreeding individuals often stand on “sentinel duty” during group
foraging bouts and care for offspring of the dominant breeding pair
(Clutton- Brock et al., 2004; Clut ton- Brock, Russell, et al., 2001;
Clutton- Brock et al., 20 02). Without the co- operation of these non-
breeders, the survival of individuals within the colonies is likely to
decrease, particularly for the young (Doolan & Macdonald, 1999;
Russell et al., 2007). This social behavior and organization is often as-
sumed to be vital to the functioning and survival of these societies.
The most common approach to understanding the contribution
of roles within social groups is to obser ve the behavior and per for-
mance of individuals. However, observing certain individuals per-
forming a task does not mean they are better than other individuals
at per forming that task. To attempt to tackle the challenges associ-
ated with quantifying trait- based contributions, a few studies have
manipulated colonies in the laboratory to evaluate the effects of the
social organization of age and size polymorphic species, such as mole
rats (Jarvis, 1981; Zöttl et al., 2016), ants (Billick & Carter, 2007;
Porter & Tschinkel, 1985), and bumblebees (Cnaani & Hefetz, 1994;
Cou villon et al., 2010; Ja ndt & Dornhaus, 2009, 2011 , 2014). In lab o-
ratory colonies of a eusocial ant Pheidole dentata, larvae gained more
mass when reared by older workers, suggesting that older workers
contribute more toward worker production in these ant colonies
than their younger sisters (Muscedere et al., 20 09). However, col-
onies within these laborator y experiments were not faced with the
same external environmental stressors as those in the wild. In the
case of bumblebees, larger workers are more susceptible to pred-
ators and parasites (Cartar & Dill, 1991; Malfi & Roulston, 2014;
Muller et al., 1996), despite being better foragers. Therefore, the
behaviors of social organism under artificial conditions might not
capture all the feedbacks associated with size- or age- based roles.
Functional linear models (FLMs) provide an additional method
of inference about high- dimensional ecological systems using ob-
servational data. For example, FLMs can evaluate the contributions
of age- or size- based roles within societies to population dynamics.
These models assume that the effect of a predictor variable (e.g.,
number of workers) on a response variable (e.g., egg production) is
a smooth function of some feature of the predictor variable (e.g.,
size of workers). Past applications of FLMs in ecology have inves-
tigated environmental drivers of plant population dynamics (Teller
et al., 2016; Tenhumberg et al., 2018). These studies evaluated the
effects of environmental conditions (e.g., precipitation) on plant
performance (e.g., growth) assuming the slope of the effect of en-
vironmental conditions and plant performance varies as a smooth
function of the time lag between conditions and performance (e.g.,
precipitation in the past 1, 2, 3… months). For example, the slope of
precipitation versus plant growth could go from positive in recent
months to zero at longer time lags. This method has potential for
wider ecological application to investigate life- history phenomena.
Here, we explore application of FLMs to quantifying the relation-
ship between aspects of new worker production as a function of the
body size of existing workers in bumblebee colonies.
Bumblebees (Bombus spp.) are primitively eusocial insects that
form relatively small colonies and have a discrete life cycle lasting
only for a single season, which makes them a tractable system for
studying trait- based roles within societies. Bumblebees also exhibit
worker size polymorphism, where workers within colonies vary up
to 10- fold in mass (Goulson, 2009). In bumblebee colonies, larger
workers are often found foraging and guarding, while smaller work-
ers spend more time in the colony conducting in- nest tasks such
as fanning and incubating (Cumber, 1949; Goulson et al., 2002;
Inoue et al., 2010; Jandt & Dornhaus, 2009; Richards, 1946). Many
studies have measured the importance of body size in determin-
ing how well workers perform various tasks, ranging from foraging
and flight dynamics to thermoregulating and undertaking. Most of
these have found that larger workers are better at multiple tasks,
such as foraging and nursing (Cnaani & Hefetz, 1994; Goulson
et al., 2002; Ings, 2007; Kerr et al., 2019; Peat & Goulson, 2005;
Spaethe et al., 2007; Spaethe & Weidenmüller, 2002), with a few
studies concluding either that intermediate size is better (Jandt &
Dornhaus, 2014), or that there is no size- based difference in perfor-
mance (Jandt & Dornhaus, 2014). Although these studies demon-
strate that body size affects worker performance at certain tasks,
they do not demonstrate how their size- based per formance at tasks
may, in turn, affect colony growth and development.
No studies have found smaller bumblebee workers to be bet-
ter at performing tasks essential to colony function. However,
smaller workers are more resilient to starvation (Couvillon &
Dornhaus, 2010). Therefore, their value may become more appar-
ent when food resources are limiting. In addition, smaller workers
have lower production costs, so they may be more cost- effective
(Kerr et al., 2019). Here, we used FLMs to evaluate the contribu-
tion of workers of different sizes to worker production in bumble-
bee colonies under three different environments: a low- resource
environment; an environment with an early season pulse followed
by low resources (“high- low”); and a high- resource environment. We
looked at five vital rates relating to worker production: (a) number of
new eggs laid, (b) development time, (c) larval survival, and (d) mean
and (e) variance in worker emergence size, that is, the size of callow
workers. By evaluating the contribution of different- sized workers
under different resources environments to worker production, we
can assess whether larger workers are more beneficial when re-
source conditions are more favorable and whether the benefit of
small workers to colonies is only seen when resources are low, mak-
ing both production cost and resistance to starvation a premium.
   KERR Et al.
2.1 | Study species and sites
We hand reared Bombus vosnesenskii colonies from wild- caught queens
collected at the University of California McLaughlin Reserve (N38
52 25.74, W122 25 56.25) in early spring 2015 and 2016 while they
searched for nest sites. These colonies were the basis for two separate
studies, both of which are previously published (Kerr et al., 2019; Malfi
et al., 2019). Here, we use previously unpublished data (Brood mapping,
below) from these studies to investigate effec ts of worker size on col-
ony growth, so we briefly describe the rearing process.
In 2015 and 2016, we hand- reared colonies in the laboratory in
a dark room at 26– 28°C for 6– 9 weeks until their second or first
cohort of worker bees eclosed. In 2015, we relocated seven colo-
nies outside (N38 32 12.21, W121 47 16.95) at the Harr y H. Laidlaw
Jr. Honey Bee Research Facility (Davis, CA), where the surrounding
landscape consisted of agricultural crops, floral research plots, and a
0.2- ha pollinator garden (Figure S3a). In 2016, we relocated 14 col-
onies outside in agricultural fields at UC Davis E xperimental Farm
proper ty (N38 31 32.3, W121 46 56.54). Half of the colonies (n = 7)
had access to flight cages that provided a pulse of native California
wildflower species for ~4 weeks early in the season (“pulse” treat-
ment) and the other half had no supplemental forage (“control”
treatment) (Malfi et al., 2019). The surrounding landscapes were
croplands consisting of mainly nonflowering cereals, corn, and a strip
of riparian habitat (Figure S3b).
In this study, we broadly categorized the resource environment s
ex p eri e nce d by ou r exp e rim e nta l colo nie s in ea ch of th ese yea r s bas e d
on observational differences in the quality and abundance of forage.
The 2015 colonies, located next to a pollinator garden at the Honey
Bee Research Facility, had the highest resource availability and quality
(“high”), colonies in the 2016 pulse treatment had the second highest
resource availabilit y and quality (“high- low”), and colonies in the 2016
control treatment had the lowest availability and quality (“low”). These
three environments will now be referred to as high, high- low, and low.
Note that comparisons between the 2015 colonies and 2016 should
be interpreted with the caveat that differences could be due to factors
other than nutrition. Based on our obser vations, the most noticeable
difference s among treatments were the quality and abundance of flo-
ral resources (discussed further in the Discussion).
2.2 | Brood mapping
Each week, we photographed the brood from multiple angles (above,
side, diagonal) to fully capture all brood cells. We individually num-
bered each brood cell in the photographs as it differentiated and
tracked the fate of all marked cells throughout colony development
(Figure 1). We classified each living brood cell into five categories: (a)
clump stage, which represents the egg stage where individual cells
have not yet differentiated; (b) predifferentiated stage, which repre-
sents early larval instars where individual cells have begun differen-
tiating; (c) differentiated stage, which represents later larval instars
FIGURE 1 Example of brood mapping photographs used to track the fate of individual cells. These mapping photographs are aerial
photographs for colony 6 in (a) week 5 and (b) week 6 since the first brood photograph. Aerial, side, and diagonal photographs were taken to
capture all cells. Each cell has been individually numbered to track each cell. The larger stand- alone open wax structures are honey pots
KERR Et al .
where individual brood cells are clearly differentiated; (d) cocoon
stage, where cells had darkened indicating that pupae have spun their
cocoons; an d (e) eclosed st age, whe re the cel l ha s op ened and an adult
worker emerged (Figure 2 for stages). We also had two other cate-
gories: (f) dead, where we had observed a dead cell, and (g) unseen,
where the cell could no longer be seen in the brood photograph.
Some brood clumps did not develop into distinct cells before
the end of brood mapping, while other clumps died before cell parti-
tioning. Rather than exclude these indistinct, dead, or undeveloped
brood clumps in our analyses (Nlow = 24/115; Nh i g h - l o w = 36/15 0;
Nhigh = 36/163), which could result in underestimating egg production
and overes timating larval sur vival, we estimated the numb er of cells for
these clumps. We did this by classif ying these indistinct brood clumps
into five size categories (tiny, small, medium, large, and extra- large)
based on comparisons with similarly sized brood clumps that did divide
into individual cells and assigning the mean value of cells for these size
categories to indistinct clumps. From the 322 distinct clumps with a
total of 3,917 cells with known fates, we estimated 432 cells from 96
indistinct clumps appeared to have died before differentiating, which
comprises of less than 10% of total cells in our lar val survival analyses.
From the br ood mapp ing , we es tim ate d th r ee vi tal rate s: eg g pro -
duction, larval development time, and larval survival. We considered
weekly egg production to be the number of newly visible cells in
either clump or predifferentiated stages. We assumed that the num-
ber of distinct cells formed by a brood clump represented the total
number of eggs laid, that is, no eggs died before lar val cells differen-
tiated. We calculated development time for each cell as the number
of days from when it was first seen as an egg (defined as the “clump”
stage) to when it was first seen as an eclosed cell. Cells that were
not detected in the clump stage or that disappeared from view be-
fore visibly eclosing were excluded from our analyses of larval de-
velopment time. Finally, we classified larval survival as the success
of each cell at surviving to eclosion. We excluded 43 unseen brood
cells from our larval analyses because more than 8 days (50% the
normal bumblebee development time) passed between photographs
of them so their fates could not be unambiguously mapped. These
represent 10% of 437 unseen cells or 1% of all 4,640 cells mapped
across the 21 colonies and three resource environments.
2.3 | Worker surveys
We conducted weekly night- time surveys to estimate the mean and
coefficient of variation (CV) in the size of newly emerged workers
FIGURE 2 Brood mapping photos showing each of the six categories of living or dead stages of cell development. The six stages are:
(a) clump stage, which are egg stages; (b) prepopcorn stages, which represents early larval instars; (c) popcorn stage, which are late instar
larvae; (d) cocoon stage; (e) eclosed stage, and (f) a dead cell (dashed circle). These categories assisted with estimating three vital rates
relating to worker production: eggs laid, development time, and lar val survival
(a) (b) (c)
(d) (e) (f)
   KERR Et al.
(hereafter referred to as “callow size”). We assigned each bee a
unique tag using a combination enamel paint and numbered, color-
tags or Microsensys radio- frequency identification (RFID) tags (Kerr
et al., 2019; Malfi et al., 2019). For each newly emerged (“callow”)
worker, we estimated body size by measuring inter tegular (IT ) span
to the nearest 0.01 mm using digital calipers (Cane, 1987; Hagen
& Dupont, 2013) and wet weight to the nearest 0.01 mg using an
analytical microbalance (Mettler Toledo XS205DU). The size of each
worker at initial capture was used to estimate the mean and CV of
callow size. We used these size measurements in combination with
presence/absence data to determine the number of workers of each
size (now referred to as “worker size composition”) present in each
colony for each week of the survey in order to evaluate the effects of
worker size composition on aspects of worker production.
2.4 | Functional linear models
We used functional linear models (FLMs) to estimate how five vital
rates varied with worker size composition. FLMs are a type of re-
gression spline that allows a covariate to vary smoothly over a con-
tinuous domain (Ramsay et al., 2009; Ramsay & Silverman, 2005).
FIGURE 3 Example of functional linear model results showing the smooth function of the slopes of Y versus the number of workers as a
function of worker size, x. Y covariate could be one of the five metrics of worker production: egg production, larval development time, larval
survival, and mean and variance in callow size. We illustrate the following examples: (a) no size- based per capita effect, but more workers of
any size increases (β0 > 0) or decreases ( β0 < 0) Y; (b) positive size- based per capita ef fects on Y; (c) negative size- based per capita effects on
Y; and (d) mixed size- based per capita effects, that is, more workers of one size have negative ef fects and more workers of another size have
positive ef fects. The dotted line on each panel represents no per capita effect s of workers
KERR Et al .
Therefore, instead of restricting our predictors (X) to unidimensional
space (i.e., simple linear models, such as total worker number pre-
dicts number of eggs), we can evaluate the effect of the number of
workers on some response variable (e.g., number of eggs) as a con-
tinuous function of worker size (i.e., a separate at tribute of the pre-
dictor variable), such that the smooth function of size- specific slopes
versus worker size can be described as:
is the expected value of the response variable Y (e.g., num-
ber of eggs);
is the intercept;
is the number of workers n of
size x; and
(s) is the slope of Y versus the number of workers of each
size category x (c.f. methods in Teller et al., 2016). Here, the continu-
ous attribute (i.e., worker size) of the predictor variable (i.e., number
of workers) is discretized into many size categories (14 size categories
for both low and high- low, and 17 for high- resource colonies) to ap-
proximate a continuous distribution of sizes (i.e., the worker size com-
position). The expected value of the response variable is the sum of
the product of the size- specific slopes
(sx) multiplied by the number
of workers of size x (Figure 3). If the slope of Y versus the number of
workers of size x is po si tive, then mor e wo rkers of size x increase values
of Y and vice versa when the slope is negative (Figure 3).
We parameterized the smooth functions of the size- specific
slopes using general additive models (GAMs). We fit our GAMs using
the cubic spline basis for all smooth covariates, so that the coeffi-
cients will be set to 0 if our covariates have no effects on the re-
sponse (see Zuur, 2012, for an excellent textbook introduction to
GAMs). We used worker size composition in the previous week to
predict both the number of eggs laid and lar val survival in the pres-
ent time step for our size composition FLMs. For the other three
vital rates relating to worker production, we quantified worker size
composition as the average number of workers in each size categor y
across their larval development period.
Models were fit separately to data from each study (i.e., low- ,
high- low- , and high- resource environments), and we included col-
ony ID as a fixed effect (i.e., a different intercept term for each
colony) for each model to account for between- colony effect s.
We used negative binomial GAMs to account for overdispersion
for estimating new eggs laid and development time. We offset the
number of new eggs laid by the number of days bet ween brood
photographs. We used binomial and Gaussian- distributed GAMs
for larval sur vival and callow size, respectively. We parameterized
the binomial GAMs for estimating larval survival using successes
and failures, where the total number of trials was defined as the
number of days between brood photographs, and the number of
successes was defined as the total number of days if the cell sur-
vived (i.e., zero failures) and the total number of days minus 1 if
the cell died (i.e., one failure). We restricted the number of knots
for our smooth terms of the number of workers of size j to a max-
imum of five. We also rejected any model structure that did not
produce unimodal functions for our smooth term of worker size
composition, since GAMs are prone to over fitting, and multimo dal
functions generally did not appear to be biologically meaningful.
We used likelihood ratio tests to assess the fit of the parametric
intercept term and the number of knots for each smooth term in
our models given our data. We used cutoff of p < .05 for para-
metric terms and a cutoff of p < .01 for smooth terms, since p
values for smooth terms are only approximate and are likely too
low (Wood, 2017). We ran these general additive models (using
mgcv::gam; Wood, 2004, 2011) in progra m R (R Core Team, 2017);
se e Appe ndi x S1 for ex amp le co d e fo r our f uncti ona l li near mod els .
To evaluate whether size- specific slopes of worker
size differed among treatments, we ran a model with
all data combined and evaluated the AIC of the com-
bined model with an AIC of models separated by treatment
( AICsep
and by year (Table 1). We repeated all analyses with slopes scaled to
size- based worker production costs (see Appendix S2 for methods;
Kerr et al., 2019 for produc tion costs), rather than numbers of indi-
viduals. Because these results were largely parallel (Appendix S2),
we do not discuss them further.
Colony size (i.e., number of observed workers) increased with
colony age across three resource environment s (Figure S2- 4). To
avoid potentially confounding effects due to collinearity between
colony age and worker number, we ran models separately with
colony age and worker size composition as predictors of various
measures of worker production success. Results for colony age
are described in Appendix S3. Relationships between worker size
TABLE 1 dAIC values for functional linear models using data combined (i.e., no effect of treatment or year) for each daily vital rate
Vital rates
dAIC (models fit to all data) dAIC (Pairwise comparisons)1
Combined By treatment By year Low versus high- low Low versus high
High- low
versus high
Daily egg produc tion 23.1 06.4 6.4 15.6 7.9
Development time (days) 352.7 048.2 5.2 96.8 272 .3
Daily larval survival 24,0 04.1 23.2 0−2 3 .1 12,568.6 17,4 8 8 .6
Mean callow size 10.8 3 .1 0−3.1 7. 7 1.25
CV in callow size 41.4 011.5 11. 5 34.2 40.8
1AIC of models fit to data from both groups together, minus AIC of models fit to data from each treatment group separately. Positive values indicate
significant dif ferences bet ween groups.
   KERR Et al.
composition and larval sur vival and mean callow size were some-
what confounded with colony age effects and should be interpreted
with c aution (Table 2, Appendix S4). We found no evidence for po-
tentially confounding relationships of colony age and worker number
on mean worker size or CV in worker size across the three resource
Average worker size increased with available ambient resources
(likelihood ratio (LR) test for models with and without treatment;
χ2 = 14,701, df = 3, p .001). Worker size was smallest in the low-
resource environment (mean and SE in IT span: 3.16 ± 0.049) and
largest in the high- resource environment (IT span: 3.68 ± 0.048)
(multiple comparison of means between high and low; estimated dif-
ference, E = 0.52, Z = 7. 5 , p .001), with the high- low- resource
environment being intermediate (IT span: 3.31 ± 0.049) (multiple
comparison of means between high- low and low: E = 0 .14, Z = 2.1,
p = .09; high and high- low: E = 0. 37, Z = 5.4, p s. 0 01). Th e se re su l ts
broadly recapitulate results of previous analyses of the separate ex-
periments as reported by Kerr et al. (2019) and Malfi et al. (2019) for
the 2015 and 2016 data, respectively.
3.1 | Daily egg production
Worker size composition did not affect egg production in the
low- resource environment (Figure 4a; χ2 = 6.3E−6, e.df = 4.2E−5,
p = .75). More larger workers increased egg production in both the
high- low- and high- resource environments (Figure 4b,c; χ2 = 83.3,
e.df = 2.8, p < .001, and χ2 = 6.4, e.df = 1.3, p = .01 for high- low
and high (respectively)), but more larger workers had greater im-
pact on egg production in the high- low- resource environment than
in the constantly high- resource environment (Table 1). To illustrate
these differences for each vital rate, we plotted the lines predicted
by FLMs for workers of different sizes (see egg production relation-
ships in Figure 5a- c).
3.2 | Larval development time
Larval development time increased with more smaller workers in all
three resource environments (Figure 4d- f; LR test of smooth term
versus constant: χ2 = 124. 6 , e.df = 2. 7, p < .001; χ2 = 42 2 .8, e.df = 2.4 ,
p < .001; χ2 = 21.4, e.df = 1. 9, p < .001 for low, high- low and high
(respectively)). Worker size composition affected larval develop-
ment time differently in each environment (Table 1). More larger
workers decreased development time in both the high- low- and
TABLE 2 Size- specific relationships of the smooth terms of colony age, the number of workers of each size (i.e., worker size composition,
WSC), and standardized (“std”) WSC for each of the five vital rates relating to worker production
Response variable Resource environment Sample size
Smooth terms
effects2Colony age WSC1Std WSC1
Egg production Low 72 Concave ×, × × , ×
H i g h - l o w 74 Concave ±, ↑ ±, ↑ Possibly
High 65 Concave ±, ↑ ±, ↑ No
Development time Low 541 Multimodal ±, ↓ ±, ↓ Possibly
H i g h - l o w 974 Concave ±, ↓ ±, ↓ Possibly
High 1,108 Convex ±, ↓ ±, ↓ Possibly
Larval survival Low 3,521 Multimodal ±, ↕ ±, ↕ Yes
H i g h - l o w 6,045 Decreases ±, ↑ ±, ↕ Yes
High 5,364 Convex −, ×−, ↑ Ye s
Mean callow size Low 65 Decreases ±, ↑ ±, ↑ Yes
H i g h - l o w 59 Multimodal ±, ↑ ±, ↑ Yes
High 57 Multimodal −, ×−, ↑ Ye s
CV in callow size Low 65 Concave ×, × ×, ×
H i g h - l o w 59 Multimodal ±, ↕ ±, ↕ No
High 57 Constant ×, × ×, ×
Note: Relationship descriptions provided are restricted over the observed range of worker body sizes and colony ages including days spent in the
laboratory. Since colony age and population size are correlated, we were unable to determined which smooth term was driving these ef fect s if both
smooth terms have similar effects. Shaded grey cells had a significant fixed ef fect of colony ID on the parametric intercept in the GAM.
1For WSC and std WSC, the firs t symbol refers to whether the relationship has a positive (+), negative (−), mixed (±), or no (×) per capita effect, and
the second symbol refers to whether the relationship increases (↑), decreases (↓), both (↕), or has no effect (×) with worker size. Sample sizes are also
provided for each of the five vital rates.
2The column “confounding effects” describes whether both colony age and WSC had similar ef fect s on the response variable when both smoot h
terms are significant.
KERR Et al .
   KERR Et al.
high- resource environments (Figure 4e,f ) but not in the low- resource
environment (Figure 4d). However, these effects were negligible in
the high- resource environment compared to the low- and high- low-
resource environments (Figure 5).
3.3 | Larval survival
Larval survival decreased with more smaller workers in the low- and
high- low- resource environments (Figure 4g,h; χ2 = 18 .9, e.df = 2.6,
p < .001; χ2 = 10 3.9, e.df = 2.6, p < .001 for low and high- low (re-
spectively)). The difference between the low and high- low envi-
ronments was not statistically significant (Table 1). Larval survival
slightly decreased with more workers of all sizes in the high- resource
environment (Figure 4i; χ2 = 29.1 , e.df = 1 . 7, p < .001). This ef fect
was negligible (Figure 5i), and this relationship for high- resource
colonies (i.e., colonies in 2015) differed significantly from both lower
resource environments (i.e., treatments in 2016) (Table 1).
3.4 | Callow size
In the low- resource environment, mean callow size decreased with
more smaller workers (Figures 4j and 5j; F = 3.3, e.df = 1.9, p = .007),
but worker size composition was unrelated to CV in callow size
(Figures 4m and 5m; F = 2.5E−6, e.df = 1.7E−5, p = .52). In the high-
low- resource environment, mean callow size decreased with more
smaller workers and increased with more larger workers (Figures 4k
and 5k; F = 6.4, e.df = 2, p < .001), whereas more larger workers slightly
decreased the CV in callow size (Figures 4n and 5n; high- low - F = 3.8,
e.df = 3, p < .001). In the high- resource environment, more workers of
any size decreased the mean callow size (Figures 4l and 5l; F = 16.5,
e.df = 1.7, p < .001), but worker size composition was unrelated to the
CV in callow size (Figures 4o and 5o; high - F = 5.2E−6, e.df = 4.6E−5,
p = .59). The effect of worker size on mean callow size of new workers
did not differ between the lower resource environments (i.e., 2016
treatments), but both differed from the high- resource treatment (i.e.,
2015 colonies) (Table 1). The effects of worker size on the CV in callow
size differed among all three treatments (Table 1).
Size- based contributions of bumblebee workers to worker pro-
duction differed among vital rates and resource environments.
Despite these differences, we never detected cases where smaller
workers outperformed larger workers for vital rates relating to in-
nest tasks. Therefore, the fact that smaller workers remain in the
nest is likely not due to their superior skill at those in- colony tasks
(Jandt & Dornhaus, 2014). Instead, colonies with more larger work-
ers often had greater worker production compared to colonies with
smaller workers. This pattern is similar to many performance- based
(Goulson et al., 2002; Ings, 2007; Kapustjanskij et al., 2007; Peat &
Goulson, 2005; Spaethe et al., 2007; Spaethe & Weidenmüller, 2002)
and manipulative experiments (Cnaani & Hefetz, 1994). However, we
found the opposite effect in two cases: more workers of any size
slightly decreased both lar val survival and mean callow size in the
high- resource environment. We discuss each result in turn below, as
well as some advantages and limitations of functional linear models.
For two vital rates, larval survival and mean callow size, both treat-
ments applied in 2016 differed from 2015, and not from each other.
Therefore, these differences could be due to other features that dif-
fered among the sites where the two experiments were conducted or
conditions in the 2 years. For example, the site of the 2016 experiment
was an agricultural field in an agricultural landscape. The field of the
experiment was used only for growing flowers to create the “high” re-
source pulse in the “high- low” treatment. Nevertheless, pesticides and
other factors (such as nest temperatures) may have differed between
the two landscape contexts. In general, conditions for bumblebees in
the 2016 experiment appeared to be more stressful than conditions in
the 2015 experiment. Although the results are not uniquely attribut-
able to floral resources, our analyses provide a reasonable test of size-
based differences under relatively low- to high- stress levels.
4.1 | Functional implications of worker size
Across social organisms, the number of offspring produced often in-
creases with the number of helpers (Biedermann & Tab orsky, 2011;
Brown et al., 1982; Malcolm & Marten, 1982; Young et al., 2015),
particularly when resources are high (Doolan & Macdonald, 1997;
Harrington et al., 1983). We found a similar per capita effect on col-
ony egg production in both our high- low- and high- resource treat-
ments, yet FLMs also revealed that in these environment s more
larger workers increased colony egg production relative to more
smaller workers. Laboratory studies of bumblebees have shown that
colonies consisting of only larger workers produce more eggs than
colonies consisting of only smaller workers (Cnaani & Hefetz, 1994).
Larger workers are known to return more resources to the colony
(Goulson et al., 2002; Kerr et al., 2019), but they are less resilient
against starvation (Couvillon & Dornhaus, 2010). This trade- off
FIGURE 4 Generalized additive model results depicting the smooth function of the size- specific slopes for all five vital rates relating to
worker production versus the number of workers of size x for the low- resource environment (left), high- low- resource environment (middle),
and high- resource environment (right). Dashed horizontal line at zero represent deviations from mean slope values, that is, slopes above the
line means more workers of size x have positive impact on Y. Grey dashed vertical line represents the mean worker size for colonies in each
of the resource environments. Plots with a significant smooth term of WSC are labeled with p < .01. Note dif ferent scales on the Y- axes in
each row
KERR Et al .
   KERR Et al.
might explain why larger workers increased colony egg production
only in the high- low- and high- resource environment but not in the
low- resource environment. The opposite effect has been found in
a fire ant, Solenopsis invicta, where monomorphic colonies of large
workers produced almost no brood compared to monomorphic colo-
nies of small workers (Porter & Tschinkel, 1985). However, the size-
based roles of workers in these two eusocial insec ts differ. Larger
bumblebees are foragers (Cumber, 1949; Goulson, 2009; Goulson
et al., 2002), but smaller fire ant workers do most of the foraging
and feeding (Cassill & Tschinkel, 1999; Wilson, 1978). Larger fire
ant workers live longer than smaller workers (Calabi & Por ter, 1989;
Por ter & Tschinkel, 1985), which is the opp osi te of bumbleb ee work-
ers (Kerr et al., 2019; da Silva- Matos & Garofalo, 20 00). Therefore,
the general mechanism may be similar, despite contrasting patterns.
The smallest observed workers had negative effects on both de-
velopment time and larval survival in the low- and high- low- resource
environments; note that this worker size was not present in the high-
resource colonies. In bumblebees, there seems to be a resource-
driven trade- off between provisioning for developing lar vae and
production of new eggs when resources are low. For example, in the
low- resource environment, egg- laying did not depend on the num-
ber of large workers. In contrast, in higher resource environments,
the number of eggs laid increased with more larger workers. This
contrast suggests that workers in the low- resource environment
are allocating more resources to maintaining larval survival and de-
velopment time, rather than supporting more workers. Results for
small workers in the lower resource environments are similar to
those for cooperative breeding species, in which the presence of
more helpers often reduces offspring survival when resources are
low (Harrington et al., 1983; Woodroffe & Macdonald, 200 0). These
negative impacts of helpers in cooperative breeding species may be
due to them shifting efforts toward increasing their own survival
(Bruintjes et al., 2010), which seems less likely in bumblebees be-
cause workers are nonreproductive. Indeed, bumblebee workers are
reported to switch from nursing to foraging tasks when resources
are low (Cartar, 1992), indicating that workers overall increase (not
decrease) cooperative efforts. Additionally, bumblebee workers pre-
dominantly feed on nectar and larvae predominantly feed on pollen
(Goulson, 2009; Plowright & Pendrel, 1977), which may reduce com-
petition among siblings and enhance cooperative behaviors. It would
be interesting to monitor foraging behavior of bumblebee workers
during resource dearths, that is, changes in nec tar versus pollen col-
lection rates, to better understand their cooperative ef forts.
Across our three environments, observed average size of all work-
ers decreased in colonies with less available resources. In the high-
resource environment, more workers of any size decreased the size
of callow workers. Worker size is known to decrease with colony age
(Couvillon et al., 2010), which correlated with colony size. In the low
and high- low- resource environments, more smaller workers resulted
in callow workers of smaller sizes and more larger workers resulted in
callow workers of larger sizes. Bumblebee workers have been recorded
to be smaller on average in simple, intensively managed landscapes
(Persson & Smith, 2011). Laboratory experiments also show that colo-
nies produce smaller workers during food shortages (Schmid- Hempel
& Schmid- Hempel, 1998). The correlation between worker size distri-
bution and callow worker size in the low- and high- low- resource en-
vironment suggests that stressful resource conditions may produce
a negative feedback loop, where colonies of smaller workers cannot
pr ope r ly fe ed an d car e for broo d (Ca r t ar & Dill , 1991 ) cau sin g the emer-
gence of smaller callow workers. Therefore, the cost and benefits of
helpers within social groups may often regulate the traits of individuals
(e.g., sex ratios, worker sizes) that are expressed (Griffin et al., 20 05).
Func t io n al line ar mo del s ar e onl y a co rre lat ive tec hni qu e , so an al ter na-
tive shared driver could be shif ting the size distribution toward smaller
workers. For example, lower resources could cause differential mortal-
ity of larger workers due to star vation (Couvillon & Dornhaus, 2010)
and cause lar vae to develop into smaller callow workers because of
fewer resources brought back by the remaining workers. Laboratory
monomorphic colonies consisting of only small or large workers had no
difference in the mean and variance in callow size when supplied with
abundant resources (Cnaani & Hefetz, 1994). If these laborator y colo-
nies had to forage for resources and still produced workers of similar
sizes, then we might be able to determine whether a shared driver is
most likely causing these effects in our study.
4.2 | Functional linear models as a statistical
approach in ecology
Previously, FLMs have been used to evaluate the lagged effect s
of environmental drivers on plant population dynamics (Teller
et al., 2016; Tenhumberg et al., 2018). Here, we extend the use of
FLMs to evaluate the size- based contribution of workers in bum-
blebee colonies. FLMs could be applied to understanding many
high- dimensional social systems. For example, they could be used
to explore the contributions of trait- based sociality, such as the
contributions of age polyethism within social groups of different
taxa and levels of sociality, including eusocial honey bees (Seeley &
Kolmes, 1991), semisocial mole rates (Jarvis, 1981; Zöttl et al., 2016),
and cooperative breeding meerkats (Clutton- Brock, Brotherton,
et al., 2001) or cichlid fish (Bruintjes & Tab orsky, 2011). In the
African mole rat, larger groups had higher rates of offspring recruit-
ment ( Young et al., 2015) and cooperative behaviors were found to
increase with age (Zöttl et al., 2016). Therefore, FLMs might be able
FIGURE 5 The relationship between number of workers of three observed worker sizes and the five vital rates relating to worker
production across the three treatments. Three workers sizes range from the smallest size of 2.5 mm (light grey), intermediate size of 3.5 mm
(dark grey), and largest size of 4.5 mm (black line) that are observed in colonies across all three treatments. Each of these lines represents
the function defined by x = 2. 5, 3.5, and 4.5 on the x- axis of Figure 4. Parametric intercepts were used from the GAMs, and intercept s were
averaged on the link function scale if the model had a significant fixed effect of colony
KERR Et al .
to determine how vital rates (e.g., offspring recruitment) differ with
the number of helpers of different ages for the African mole rate.
FLMs provide an alternative way to study these high- dimensional
ecological systems using field observational data, particularly where
manipulative experiments may not be possible.
Correlative techniques, such FLMs, provide a valuable comple-
ment to many manipulative experiments that aim to test similar
hypotheses. However, these separate approaches have their own
set of advantages and limitations that need to be considered when
making conclusions from these models. For example, FLMs can be
data- heavy (e.g., 2025 independent observations of the signal and
response; Teller et al., 2016); only inform us about correlations and
not causations; and may have collinear predic tors that obscure the
true driver of these responses. Collinearity is not specific to FLMs
but is equally problematic for many simple (e.g., multiple regression)
and complex statistical techniques (e.g., structural equation models).
To date, only two studies have reported applying functional smooth-
ing approaches to high- dimensional ecological systems by explor-
ing how lagged environmental drivers influence plant performance
(Teller et al., 2016; Tenhumberg et al., 2018). Teller et al. (2016) pre-
dicted how lagged effects of past precipitation and local competition
influenced plant growth and survival; however, they would not be
able to parse out the true driver of plant performance if densit y and
precipitation covaried across some gradient. When exploring the
trends and collinearity for these several vital rates (Appendix S4),
two of four vit al rates (Table 2) had confounding effects of colony
age and size composition suggesting that either or both might be
driving these trends (Table 2). When using simple or complex correl-
ative methods, it is important to explicitly evaluate the collinearity of
predictor variables as we have demonstrated here.
4.3 | Summary
Overall, we found that the advantages and disadvantages of work-
ers of different sizes on worker production only became apparent
when exploring these effects across these three different resource
environments. We also found that bumblebee colonies shif ted
their worker size distribution across these resource environments.
Among eusocial insects, caste size polymorphism is hypothesized
to be an adaption to expand accessibility of resources, such as seed
size in ant s (Davidson, 1978; Retana & Cerdá, 1994; Traniello &
Beshers, 1991) and flower size in bumblebees (Peat et al., 2005).
However, the shi ft in wor ker size distrib ution across th ese resource
environments could have emerged from the lower tolerance of
larger workers to starvation (Couvillon & Dornhaus, 2010). Prior
to this study, quantifying the contribution of individuals in social
groups has been challenging. Here, we demonstrate that func tional
linear models have the potential to evaluate observational data for
complex, trait- based life histories of social organisms. As such , they
provide a valuable complement to the constraints of experimental
work and a mechanism to focus hypotheses for further experimen-
tal studies.
Our study was jointly funded by Tufts Graduate Student Research
Award and Collaborative NSF Grant to N.M. Williams (DEB1354022)
and E. E. Crone (DEB1411420). We thank Nick Dorian, Sonja Glasser,
Colin Fagan, Jessica Drost, Andrew Buderi, and John Mola for field
assistance, and the L aidlaw Bee Biology Facility at UC Davis for use
of their facilities.
We have no conflicts of interest to declare.
Natalie Z. Kerr: Conceptualization (equal); data curation (lead);
formal analysis (equal); investigation (equal); methodolog y (equal);
project administration (equal); visualization (lead); writing- original
draft (lead); writing- review & editing (equal). Rosemary L. Malfi:
Data curation (supporting); investigation (supporting); project ad-
ministration (equal); resources (equal); writing- review & editing
(equal). Neal M. Williams: Funding acquisition (lead); investigation
(supporting); project administration (equal); resources (equal); su-
pervision (lead); writing- review & editing (equal). Elizabeth E. Crone:
Conceptualization (equal); formal analysis (equal); funding acquisi-
tion (lead); investigation (equal); methodology (equal); project ad-
ministration (equal); supervision (lead); visualization (supporting);
writing- original draft (supporting); writing- review & editing (equal).
Demographic data: Dryad
Natalie Z. Kerr
Rosemary L. Malfi
Neal M. Williams
Elizabeth E. Crone
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How to cite this article: Kerr NZ, Malfi RL, Williams NM, Crone
EE. Larger workers outperform smaller workers across
resource environments: An evaluation of demographic data
using functional linear models. Ecol Evol. 2021;11:2814– 2827.
... In general, access to abundant and high-quality floral resources supports greater mean size at the colony level (Grass et al. 2021). At the population level, with increasing food availability, bumble bees tend to produce larger workers and males (but not necessarily larger queens, Sutcliffe and Plowright 1988, Schmid-Hempel and Schmid-Hempel 1998, Kerr et al. 2021, Zaragoza-Trello et al. 2021. Larvae fed pollen with higher protein content tend to be larger in size (Tasei and Aupinel 2008, but see Vanderplanck et al. 2014), consistent with work in non-Bombus bee species (Roulston and Cane 2002). ...
... However, direct tests of how floral resource availability affects the resource allocation strategy of bumble bee colonies are needed and it is unclear which strategy (many small workers versus fewer large workers) confers a greater overall fitness benefit under different conditions. For example, under experimental conditions, when given access to a greater abundance of resources, colonies of B. vosnesenskii Radoszkowski (a medium-sized species) appear to favor producing fewer, larger workers, particularly during the early stages of colony development (Malfi et al. 2019, Kerr et al. 2021). ...
... Both hypotheses could hold true under different contexts and for different species. Empirical evidence suggests that as mean worker size increases, bumble bee colonies tend to produce more new queens (Herrmann et al. 2018), and the benefits of larger workers hold across resource conditions (Kerr et al. 2021). Although bumble bee colonies may produce a greater variety of worker sizes in less-favorable conditions, these colonies with more worker size variation do not appear to produce more workers or reproductives (Kelemen et al. 2020). ...
Body size is arguably one of the most important traits influencing the physiology and ecology of animals. Shifts in animal body size have been observed in response to climate change, including in bumble bees (Bombus spp. [Hymenoptera: Apidae]). Bumble bee size shifts have occurred concurrently with the precipitous population declines of several species, which appear to be related, in part, to their size. Body size variation is central to the ecology of bumble bees, from their social organization to the pollination services they provide to plants. If bumble bee size is shifted or constrained, there may be consequences for the pollination services they provide and for our ability to predict their responses to global change. Yet, there are still many aspects of the breadth and role of bumble bee body size variation that require more study. To this end, we review the current evidence of the ecological drivers of size variation in bumble bees and the consequences of that variation on bumble bee fitness, foraging, and species interactions. In total we review: (1) the proximate determinants and physiological consequences of size variation in bumble bees; (2) the environmental drivers and ecological consequences of size variation; and (3) synthesize our understanding of size variation in predicting how bumble bees will respond to future changes in climate and land use. As global change intensifies, a better understanding of the factors influencing the size distributions of bumble bees, and the consequences of those distributions, will allow us to better predict future responses of these pollinators.
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In the context of the definition of Specific Protection Goals for bees, risk managers asked EFSA to provide the scientific background to support the decision‐making process about what needs to be protected and to what extent. In this document, information, data and tools have been investigated to support the risk managers in deciding on the Specific Protection Goals for bumble bees and solitary bees. In particular, EFSA summarised the relevant information on biology and ecology and investigated the possibility of performing an analysis of background variability to support the definition of a threshold of acceptable effects. Accounting for the current level of knowledge, EFSA concluded that this analysis is complex, and it would require information that is not yet available and tools that have not yet been fully evaluated. Nevertheless, based on the limited data available, represented by the control groups of few experimental field studies, an analysis of variability of different relevant endpoints was provided. By taking into account the results of this analysis, the biology and ecology of bumble bees and solitary bees, as well as the impact for risk assessment, EFSA illustrated possible approaches that could be followed by risk managers.
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The temporal distribution of resources is an important aspect of habitat quality that can substantially impact population success. Although it is widely accepted that floral resources directly influence wild bee population sizes, we lack experimental data evaluating how resource availability affects colony growth via demographic mechanisms. To achieve this, we tracked marked individuals in bumble bee (Bombus vosnesenskii) colonies to evaluate whether worker survival and reproduction responded to experimentally elevated forage early in colony development. Specifically, we assessed the effect of early resource environment on worker and sexual offspring production, and the survival and body size of individual workers. We also assessed whether responses of colonies differed when exposed to higher or lower resource environments at a relatively smaller (~ 10 workers) or larger (~ 20 workers) size. Resource supplementation always resulted in greater total offspring and male production; however, the influence of supplementation on worker production and quality depended on colony size at the start of supplementation. Among colonies that were initially smaller, colonies that were supplemented produced fewer but larger bodied and longer lived workers compared to control counterparts. Among colonies that were initially larger, colonies that were supplemented produced more workers than corresponding controls, but without changes to worker quality. Collectively, these results provide clear experimental evidence that greater resource availability early in colony development increases overall productivity, and indicate that colonies may pursue different allocation strategies in response to the resource environment, investing in more or better workers.
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Temperature and precipitation determine the conditions where plant species can occur. Despite their significance, to date, surprisingly few demographic field studies have considered the effects of abiotic drivers. This is problematic because anticipating the effect of global climate change on plant population viability requires understanding how weather variables affect population dynamics. One possible reason for omitting the effect of weather variables in demographic studies is the difficulty in detecting tight associations between vital rates and environmental drivers. In this paper, we applied Functional Linear Models (FLMs) to long-term demographic data of the perennial wildflower, Astragalus scaphoides, and explored sensitivity of the results to reduced amounts of data. We compared models of the effect of average temperature, total precipitation, or an integrated measure of drought intensity (Standardized Precipitation Evapotranspiration Index, SPEI), on plant vital rates. We found that transitions to flowering and recruitment in year t were highest if winter/spring of year t was wet (positive effect of SPEI). Counterintuitively, if the preceding spring of year t-1 was wet, flowering probabilities were decreased (negative effect of SPEI). Survival of vegetative plants from t-1 to t was also negatively affected by wet weather in the spring of year t-1, and for large plants, even wet weather in the spring of t-2 had a negative effect. We assessed the integrated effect of all vital rates on life history performance by fitting FLMs to the asymptotic growth rate, log(λt ) log(λt ) was highest if dry conditions in year t-1 were followed by wet conditions in the year t. Overall, the positive effects of wet years exceed their negative effects, suggesting that increasing frequency of drought conditions would reduce population viability of A. scaphoides. The drought signal weakened when reducing the number of monitoring years. Substituting space for time did not recover the weather signal, probably because the weather variables varied little between sites. We detected the SPEI signal when the analysis included data from two sites monitored over 20 years (2x20 observations), but not when analyzing data from four sites monitored over 10 years (4x10 observations). This article is protected by copyright. All rights reserved.
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Significance Nonreproductive group members of naked and Damaraland mole rats are thought to be organized in permanent, distinct castes that differ in behavior and physiology, suggesting that their social organization resembles that of obligatorily eusocial insects. This study tests predictions about the distribution of cooperative behavior based on the suggestion that individual differences represent a caste system. Our data provide no evidence that helpers show fixed, divergent developmental pathways or specialize in particular tasks. Instead, variation in their behavior appears to represent an age-related polyethism. The results suggest that the behavioral organization of social mole-rat groups is similar to that of other singular cooperatively breeding vertebrates, and that similarities to obligatorily eusocial insects have been overestimated.
Size‐number trade‐offs in reproduction are commonly observed in nature. Bumblebee ( Bombus spp.) colonies produce workers that vary considerably in size. This variation suggests that colonies face potential size‐number trade‐offs when producing workers. Here, we estimated size‐based vital rates of Bombus vosnesenskii workers using colonies reared from wild‐caught queens. We conducted a mark–recapture study to estimate worker survival as a function of body size. We also collected data on pollen and nectar loads as well as foraging trips using a radiofrequency identification system to estimate daily resource return as a function of body size. We integrated survival and daily resource return to estimate lifetime resource collection and offset these estimates by the size‐based worker production costs. We found size‐based trade‐offs among workers of different sizes. Smaller workers had higher survival, but larger workers returned with more resources per day. The largest workers made slightly fewer foraging trips per day. Overall, larger workers made the greatest lifetime contribution to both nectar and pollen collection. However, once the benefits of larger workers are offset by their higher production costs, intermediate‐sized workers were the optimal for net resource contribution according to our models. Many previous studies have found that larger workers outperformed smaller workers with foraging and in‐nest tasks, yet these studies have not integrated multiple fitness components or worker production costs to quantify net resource contribution towards colony growth. Accounting for trade‐offs between costs and performance changed our conclusions about optimal body size from being large to being near the observed average. Similar approaches of integrating multiple vital rates may resolve apparently suboptimal life histories in other taxa. A plain language summary is available for this article.
Summary. Recent work by Reiss and Ogden provides a theoretical basis for sometimes preferring restricted maximum likelihood (REML) to generalized cross-validation (GCV) for smoothing parameter selection in semiparametric regression. However, existing REML or marginal likelihood (ML) based methods for semiparametric generalized linear models (GLMs) use iterative REML or ML estimation of the smoothing parameters of working linear approximations to the GLM. Such indirect schemes need not converge and fail to do so in a non-negligible proportion of practical analyses. By contrast, very reliable prediction error criteria smoothing parameter selection methods are available, based on direct optimization of GCV, or related criteria, for the GLM itself. Since such methods directly optimize properly defined functions of the smoothing parameters, they have much more reliable convergence properties. The paper develops the first such method for REML or ML estimation of smoothing parameters. A Laplace approximation is used to obtain an approximate REML or ML for any GLM, which is suitable for efficient direct optimization. This REML or ML criterion requires that Newton–Raphson iteration, rather than Fisher scoring, be used for GLM fitting, and a computationally stable approach to this is proposed. The REML or ML criterion itself is optimized by a Newton method, with the derivatives required obtained by a mixture of implicit differentiation and direct methods. The method will cope with numerical rank deficiency in the fitted model and in fact provides a slight improvement in numerical robustness on the earlier method of Wood for prediction error criteria based smoothness selection. Simulation results suggest that the new REML and ML methods offer some improvement in mean-square error performance relative to GCV or Akaike's information criterion in most cases, without the small number of severe undersmoothing failures to which Akaike's information criterion and GCV are prone. This is achieved at the same computational cost as GCV or Akaike's information criterion. The new approach also eliminates the convergence failures of previous REML- or ML-based approaches for penalized GLMs and usually has lower computational cost than these alternatives. Example applications are presented in adaptive smoothing, scalar on function regression and generalized additive model selection.
The first edition of this book has established itself as one of the leading references on generalized additive models (GAMs), and the only book on the topic to be introductory in nature with a wealth of practical examples and software implementation. It is self-contained, providing the necessary background in linear models, linear mixed models, and generalized linear models (GLMs), before presenting a balanced treatment of the theory and applications of GAMs and related models. The author bases his approach on a framework of penalized regression splines, and while firmly focused on the practical aspects of GAMs, discussions include fairly full explanations of the theory underlying the methods. Use of R software helps explain the theory and illustrates the practical application of the methodology. Each chapter contains an extensive set of exercises, with solutions in an appendix or in the book’s R data package gamair, to enable use as a course text or for self-study.
In observational demographic data, the number of measured factors that could potentially drive demography (such as daily weather records between two censuses) can easily exceed the number of independent observations. Thus, identifying the important drivers requires alternatives to standard model selection and variable selection methods. Spline methods that estimate smooth functions over continuous domains (such as space or time) have the potential to resolve high‐dimensional problems in ecological systems. We consider two examples that are important for many plant populations: competition with neighbours that vary in size and distance from the focal individual and climate variables during a window of time before a response (growth, survival, etc.) is measured. For competition covariates, we use a simulation study based on empirical data to show that a monotone spline estimate of competition kernels via approximate AIC returns very accurate estimates. We then apply the method to long‐term, mapped quadrat data on the four dominant species in an Idaho (US) sagebrush steppe community. For climate predictors and their temporal lags, we use simulated data sets to compare functional smoothing methods with competing linear (LASSO) or machine learning (random forests) methods. Given sufficient data, functional smoothing methods outperformed the other two methods. Functional smoothing methods can advance data‐driven population modelling by providing alternatives to specifying competition kernels a priori and to arbitrarily aggregating continuous environmental covariates. However, there are important open questions related to modelling of nonlinear climate responses and size × climate interactions.
Functional data analysis (FDA) models data using functions or functional parameters. The complexity of the functions is not assumed to be known in advance, so that methods are used for approximating these with as much flexibility as the data require. Keywords: functional parameters; splines; basis functions; roughness penalty; dynamic model; registration