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Functional
Ecology 2002
16, 492–503
Visit larger displays but probe proportionally fewer
flowers: counterintuitive behaviour of nectar-collecting
bumble bees achieves an ideal free distribution
Blackwell Science, Ltd
K. OHASHI*†‡ and T. YAHARA§
*Biological Institute, Graduate School of Science, Tohoku University, Sendai 980-8578, Japan, and §Department
of Biology, Faculty of Science, Kyushu University, Fukuoka 812-8581, Japan
Summary
1. Patterns of pollinator responses to variation in floral display size have significance for
pollen flow among plants. Here we test a theoretical model for explaining such patterns
by simultaneously assessing bumble bee behaviour and nectar availability in two native
stands of Cirsium purpuratum with different spatial densities.
2. A bumble bee (Bombus diversus) foraging on a plant remembered and avoided only
one or two flower heads that it had probed before, so that the flower-head revisitation
rate increased as it stayed longer on a plant. Moreover, the revisitation rate increased
less rapidly on larger displays.
3. The number of heads probed per plant increased less than proportionally with display
size, and this increase was smaller at higher plant density. This pattern is consistent with
our expectation that a bee leaves a plant when the cost of flower-head revisitation exceeds
that of interplant movement. However, bees left plants slightly earlier than predicted.
4. As predicted, the visitation rate of bees per plant showed a decelerated increase with
floral display size, and this increase was greater at higher plant density.
5. As a result of these complementary changes in the number of heads probed per plant
and visitation rate per plant across plant densities, nectar rewards per head were equalized
among displays (an ideal free distribution was achieved).
Key-words: Bombus, floral display size, foraging behaviour, flower revisitation, ideal free distribution, plant–
pollinator interaction
Functional Ecology (2002) 16, 492 –503
Introduction
Plants within a population often vary considerably
in the numbers of open flowers (Willson & Price 1977;
Pleasants & Zimmerman 1990). Variation in floral
display size causes two types of pollinator response which
can influence pollen dispersal. First, larger floral displays attract more pollinators per unit time (Dreisig 1995;
Klinkhamer & de Jong 1990; Klinkhamer, de Jong &
Bruyn 1989; Ohara & Higashi 1994; Ohashi & Yahara
1998; Robertson & Macnair 1995; Thomson 1988).
This increased attractiveness promotes increased
pollen receipt or removal, or potential mate diversity
(Harder & Barrett 1996). Second, the number of flowers
that individual pollinators probe per plant within
a foraging bout also increases with floral display size
(Geber 1985; Harder & Barrett 1995; Klinkhamer
© 2002 British
Ecological Society
†Author to whom correspondence should be addressed.
E-mail: [email protected]
‡Present address: Department of Zoology, University of
Toronto, 25 Harbord St, Toronto, ON M5S 3G5, Canada.
et al. 1989; Robertson 1992), which increases selfpollination among flowers on the same plant (geitonogamy; de Jong, Waser & Klinkhamer 1993; Richards
1986). Thus the relation of plant fitness to floral display
size depends on the details of the two types of pollinator
responses.
In order to predict how pollinators respond to variation in floral display size, we developed a theoretical
model based on considerations of optimal foraging
(Ohashi & Yahara 1999). First, we considered a situation
in which a pollinator foraging on a plant is increasingly
likely to revisit previously probed flowers until the
cost of flower revisitation exceeds that of moving to
another plant. Such behaviour results in the number of
flowers probed per plant increasing less than proportionally with the size of floral display. Next, based on
previous empirical studies (Dreisig 1995; Robertson &
Macnair 1995), we assumed that pollinators distribute
themselves on displays of different size according to
an ideal free distribution so that they can gain equal
nectar rewards per flower. Given this behaviour, the
rate at which individual plants receive pollinator visits
492
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bumble bees
achieve an ideal
free distribution
© 2002 British
Ecological Society,
Functional Ecology,
16, 492–503
increases in a decelerating manner with the size of their
floral display.
Our model (Ohashi & Yahara 1999; Ohashi &
Yahara 2001) made unique predictions concerning the
effects of plant density on the two types of pollinator
response to variation in floral display size. First, the
number of flowers probed per plant should increase less
strongly with floral display size at higher plant density.
Although some authors have argued that a pollinator
probes a smaller proportion of flowers as plant density
increases (Cibula & Zimmerman 1984; Cresswell 1997;
Heinrich 1979; Klinkhamer & de Jong 1990; Zimmerman
1981), they did not address the effect of plant density on
the functional relation between the number of flowers
probed per plant and floral display size. Second, visitation
rate per plant should increase more strongly with display
size at higher plant density. Although we found that a few
previous observations on changes in pollinator behaviour
across plant densities were consistent with our predictions (Klinkhamer & de Jong 1990; Klinkhamer et al.
1989), these studies did not test their findings statistically, nor did they investigate whether gain per flower
was equalized among different-sized displays.
In this paper we assess the effects of plant density on
the responses of bumble bee behaviour in two native
stands of Cirsium purpuratum (Maxim.) Matsum. First,
we quantified the revisitation rate of bees to flower
heads as a function of the number of previously probed
flower heads on a plant, and the relative cost of interplant movement at each density. Incorporating these
estimates into the model, we predicted the number of
flower heads probed per plant in relation to floral
display size, and compared it with observed behaviour.
Second, we compared the relations between floral display
size and the number of flower heads probed per plant,
the visitation rate per plant, and the visitation rate per
head in high- and low-density areas. Finally, we quantified nectar productivity and standing crops per head
at each density to assess whether bees gain equal average
nectar rewards per head regardless of variation in
floral display size. We ask the following questions: (i) Is
a bee increasingly likely to revisit flower heads as it stays
longer on a plant, and is the increase smaller on larger
displays? (ii) Is the increase in the number of flower
heads probed per plant with floral display size smaller
at higher plant density? (iii) Is the increase in visitation
rate per plant with floral display size greater at higher
plant density? (iv) Is the average visitation rate per head
on each plant balanced by the per-head nectar productivity of the plant, irrespective of its floral display
size? (v) Can the relationship between the number of
flower heads probed by a bumble bee per plant and floral
display size (number of flowering heads per plant) be
quantitatively predicted by the model?

We first outline our model (Ohashi & Yahara 1999;
Ohashi & Yahara 2001). We initially predicted the
number of flowers probed by a pollinator before leaving
a plant when display size is invariable. We considered
a nectar-collecting pollinator making sequential decisions while foraging on a plant, and choosing between
staying on the current plant or moving to another.
This formulation may differ from well-known ratemaximizing models (e.g. ‘marginal value theorem’;
Charnov 1976). We chose this approach because of
its expandability; it holds even when display size is
variable, so long as pollinators’ visitation rate per flower
is proportional to nectar productivity per flower (see
below). In contrast, rate-maximizing models essentially
require a condition that the forager randomly encounters
each patch, which is rarely fulfilled in nature.
Because plants are distributed as discrete patches,
for a pollinator moving to another plant is more costly
than moving within a plant. We incorporated this effect
into the model by defining the discounting rate for
moving to another plant (k) as
k = [(flight time between flowers within a plant) +
(handling time per flower)]/[(flight time between
plants) + (handling time per flower)]
eqn 1
In addition, we assumed that a pollinator remembers
probing a maximum of m flowers on a plant and avoids
revisiting them, but arbitrarily chooses among the
remaining (F − m) flowers where F is the size of floral
display. Hence the flower revisitation rate (r) will increase
linearly with the position of the flower in a visit sequence
(t) (Fig. 1a). That is,
r = (t − m − 1)/(F − m)
r=0
(t > m)
(t ≤ m)
eqn 2
Because revisited flowers yield little or no nectar, this
will cause a gradual decrease in the rate of energy gain
per flower with increasing t. We refer to such a decrease
in the rate of energy gain as ‘patch depression’ (originally
termed ‘depression’ by Charnov, Orians & Hyatt 1976).
Assuming that nectar productivity per flower is
constant, we predicted the number of flowers that a
pollinator will probe before leaving a plant as
tc = (1 − k)F + mk
tc = F
(F > m)
(F ≤ m)
eqn 3
where tc is the number of flowers probed per plant
(for further details see Ohashi & Yahara 1999; Ohashi
& Yahara 2001). The predicted pattern is shown in
Fig. 1(b). The number of flowers probed per plant
(tc) increases with floral display size (F). When k < 1,
the increase in tc is slower than the increase in F. At
higher plant density, the increase in tc with increasing
F is smaller.
Next, we expanded the model for variable display sizes
by considering visitation rate per plant as a combined
response of many pollinators (Ohashi & Yahara 1999;
Ohashi & Yahara 2001). We assumed that pollinators
(b)
Small
display
m+1
t c–s
Large
display
t c–1
Flower sequence (t )
(c)
High density
Vf
Low density
High density
Visitation rate per
plant (Vp )
(a)
No. of flowers probed
per plant (tc )
494
K. Ohashi &
T. Yahara
Flower revisitation rate
FEC644.fm Page 494 Thursday, July 18, 2002 11:32 AM
1 – kh
Vf
1 – kl
Low density
Vf
m
m
Floral display size (F )
Floral display size (F )
Fig. 1. Predictions of Ohashi & Yahara’s model (Ohashi & Yahara 1999; Ohashi & Yahara 2001). (a) Relationship between
flower revisitation rate, r = (t – m –1)/(F – m) and the position of a flower in a sequence (t). Dashed line represents marginal level
of revisitation for leaving a plant. The number of flowers probed on small and large displays is denoted as tc–s and tc–l, respectively.
(b) Relationship between number of flowers probed per plant (tc) and floral display size (F ) at high and low plant densities. (c)
Relationship between pollinator visitation rate per plant (Vp ) and floral display size (F ) when visitation rate per flower (Vf ) is
equal between high and low plant densities. The discounting rate for interplant movement (equation 1) at high and low density
is denoted as kh and kl, respectively.
distribute themselves on plants varying in display size
according to Fretwell & Lucas’s (1970) theorem of the
ideal free distribution (IFD): the average visitation
rate per flower to be directly proportional to its nectar
productivity. This is the simplest case of the IFD (continuous input model; Parker & Sutherland 1986), which
is often consistent with observed pollinator behaviour
(Dreisig 1995; Robertson & Macnair 1995). Then the
relationship between the visitation rate per plant (Vp)
and floral display size (F ) was predicted as
Vp = Vf F/[(1 − k)F + mk]
Vp = Vf
(F > m)
(F ≤ m)
eqn 4
where Vf is constant (for further details see Ohashi &
Yahara 1999; Ohashi & Yahara 2001). The predicted
pattern is shown in Fig. 1(c). Pollinator visitation rate
per plant (Vp) increases in a decelerating manner with
floral display size (F ), which counterbalances the decline
in tc /F on larger displays. At higher plant density, the
increase in Vp with increasing F is greater.
These predictions largely depend on the assumptions
that the pollinator’s ‘memory size’ is a constant that is
smaller than display size, that nectar productivity per
flower is equal among displays, and that pollinators
distribute themselves on plants according to an IFD.
To test our model, therefore, the assumptions and the
predicted patterns should be supported empirically.
Therefore we investigated the foraging behaviour of
bumble bees and nectar distribution among plants in
dense and sparse stands of C. purpuratum.
Materials and methods
   
© 2002 British
Ecological Society,
Functional Ecology,
16, 492–503
Cirsium purpuratum is a herbaceous perennial that
inhabits flood plains or volcanic barrens in the Kanto
and Central Districts of mainland Japan. It produces
large, purple, nodding flower heads (400 –700 florets per
head) on several erect flowering stalks that elongate
from a basal rosette. All florets within a flower head
are hermaphroditic and protandrous. The number of
flowering heads provides a practical measure of floral
display size for a C. purpuratum plant (Ohashi & Yahara
1998).
We studied C. purpuratum on a floodplain along
the Kinu River (≈800 m), Tochigi Prefecture, Japan.
During 1997 and 1998, main blooming occurred from
September to October. Floral display size varied from
one to 35 heads per plant, and 78·5% of plants had fewer
than five flower heads. The most frequent pollinators
were the bumble bee, Bombus diversus Smith. In addition,
Bombus honshuensis Tkalcu, Bombus consobrinus wittenburgi Vogt and Bombus hypocrita Pèrez also visited
C. purpuratum, but only rarely. The other rare visitors
of C. purpuratum have been described elsewhere (Ohashi
& Yahara 1998). Cirsium flowers offer nectar and pollen,
but most bees do not actively collect pollen in the sense
of manipulating the anthers. Thus we considered nectar
to be the primary reward.
For several reasons, the Cirsium–Bombus system seems
particularly appropriate for testing our model. (i) Flower
heads are irregularly distributed in three-dimensional
space within a plant, so that flower-head revisitation
by bees is more likely than if heads had a regular distribution. (ii) Individual plants are dispersed so that
each plant represents a distinct patch. (iii) Plants receive
frequent visits by bees, so we could record numerous
sequences of flower-head visitation on each plant. (iv)
Bumble bees appear to compete extensively for nectar,
and the visitation rate per head is almost equal among
different-sized displays, suggesting an IFD (Ohashi &
Yahara 1998).
-    
 
During early September 1997 and 1998, we selected
two areas with different spatial densities at our site. In
the high-density area, mean distance to the nearest two
neighbours was 1·48 m (n = 18, SE = 0·11) in 1997 and
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achieve an ideal
free distribution
1·31 m (n = 22, SE = 0·13) in 1998; and in the low-density
area, 10·2 m (n = 18, SE = 0·82) in 1997 and 8·15 m
(n = 22, SE = 0·50) in 1998. These areas were separated
by >50 m, so that few bees foraged in both areas.
During each daily observation in 1997, we selected
one or two pairs of one large plant (six to 16 flowering
heads) and one small plant (one to three flowering heads)
in each density area. For each plant we numbered
flowering heads consecutively and observed them for
80 min, either directly or indirectly with 8 mm video
cameras (Handycam CCD-TR250, Sony). For each
plant we recorded the number of bees that visited, the
number of flower heads probed per visit, and the sequence
of flower heads probed during each visit. We repeated
these 80 min observations three times a day for six
focal plants, separated by 30 min intervals (0700–0820,
0850 –1010 and 1040–1200), so that each focal plant
was observed for 4 h. These daily observations were
conducted at three separate times during the flowering
season in 1997 (6–7, 13–14 and 19–20 September). We
also recorded the date of anthesis of each head on 12
plants to describe flower-head age distribution within
plants.
In 1998 we conducted similar observations on 7–12,
14, 17–18, 20 –22, 24, 26–27, 29–30 September, and 2
October. For each daily observation we selected one or
two plants in each density area and monitored their
bee visits continuously for 6 h (0830–1430). Because
we primarily sought to collect data on flower-head
revisitation during 1998, we did not always include
large and small displays in pairs. Display size of focal
plants ranged from eight to 18 heads.
     
  
To estimate the discounting rate for interplant movement (k), we timed inter- and intraplant bee flights,
and the probing time of individual heads (time from
landing until leaving a head) to 0·01 s with a digital
stopwatch (ATC-1100, CASIO) in the high- and lowdensity areas. These measurements were repeated
during the peak of pollinator activity (1100–1400) on
24 –25 September 1997.
     

© 2002 British
Ecological Society,
Functional Ecology,
16, 492–503
In parallel with monitoring bee visitation in 1997
(6 – 7, 9, 13 –14, 16 and 19–21 September), we selected
a pair of displays (large, six to 13; small, two or three
heads per plant) in each density area, and measured
nectar standing crop in all flower heads on them. On
each observation day we sampled nectar at 06.30, 08.20,
10.10 and 12.00 h. We roughly divided each head into
inner, middle and outer florets, then picked five middle
florets from each head with forceps. This influenced
neither nectar productivity of other florets, nor bee
visits to the head (K.O., unpublished results). Sampled
Table 1. Variation in nectar production rate of florets due to
differences in sexual stages (male, intermediate and female) and
relative locations within a head (inner, middle and outer) among
florets. Mean ± SE nectar sugar (sucrose) production rate
(µg floret−1 h−1). Twelve heads were investigated in each case
Location of florets within a head
Sexual stage of florets
Inner
Middle
Outer
Male
Intermediate
Female
23·5 ± 3·7
17·7 ± 2·4
15·0 ± 2·7
22·8 ± 2·6
15·2 ± 2·5
16·6 ± 2·2
21·5 ± 2·6
16·5 ± 3·1
13·2 ± 3·0
Table 2. Variation in nectar production rate of florets due to
differences in sexual stages (male, intermediate and female)
and relative locations within a head (inner, middle and outer)
among florets. A model I 3 × 3 factorial 
Source
df
SS
F
P
Position
Stage
Position × stage
Residual
2
2
4
99
54·41
1179·06
80·64
9290·29
0·29
6·28
0·22
0·75
0·0027
0·93
florets were placed in 1·5 ml microcentrifuge tubes and
taken to the laboratory in a chilled airtight container.
We also sampled five middle florets from each of 20
flower heads just after they had been probed by bees.
In the laboratory, we collected floral nectar by
inserting a 1–10 µl GELoader Tip (Eppendorf, Germany) into a floret and sucking up the nectar with a
5 ml syringe. We then flushed the tip with 3–5 µl distilled water and soaked it up with a rounded wick
(6 mm diameter) of chromatography paper (Whatman
3MM Chr). We thus soaked up nectar from five florets
with a wick and dried it at 37 °C.
Nectar of C. purpuratum primarily contains sucrose
(>90% of solutes), so we measured the sucrose content
of each sample. We dissolved the sugar in each wick in
20 µl distilled water, and quantified sucrose content
with an automated enzyme electrode analyser (Biotec
Analyzer M-110, Sakura Seiki Co. Ltd, Japan). Preliminary analysis showed that sucrose content in five
middle florets correlated strongly with that in 25 randomly picked florets from the same head (n = 19,
r = 0·91, P < 0·0001).
   
On 17 September 1997, we examined nectar sugar
(sucrose) production rate of flower heads on differentsized displays of C. purpuratum. In the high- and lowdensity areas we selected 18 and 20 plants, respectively,
with 1–19 flowering heads per plant. Nectar sugar
production rate is highest when the flower is in the
male phase, which does not vary among the locations
within a head (Tables 1 and 2). To control for this effect
of sexual stage, we selected one flower head on each
plant which had just passed 3–4 days after anthesis,
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K. Ohashi &
T. Yahara
and collected five middle, male-phase florets. After the
first sampling at 1100 h, each focal head was bagged
(2 mm mesh, vinylon 35%, polyester 65%) to allow nectar
to accumulate until we sampled five additional florets
at 1500 h. Sugar production rate was then determined
as the difference between nectar contents before and
after bagging. Nectar production over 4 h was much less
than the maximum nectar capacity of florets (K.O.,
unpublished results).
We also checked whether floret number per head
varied with head number per plant. We randomly selected
one flower head for each of 20 plants with different
sizes of floral display (1–29 flowering heads per plant)
and counted the total number of florets.
We calculated the optimal number of flower heads
probed per plant per bee (t*)
c based on equation 3. Again,
we carried out a Monte Carlo procedure involving
10 000 randomly drawn input variables for each size
of floral display. Values of m were drawn randomly
from a truncated normal distribution (1 ≤ m) with the
parametric mean and standard deviation based on
observed values (see above), and k was calculated as
described above (but 0 ≤ k ≤ 1). For both high- and
low-density areas, the predicted t*c with its 95% confidence limits was compared to the observed tc for
each display size.
 
We previously found that the per-head and per-plant
visitation rate on a given size of display fluctuated
substantially from day to day (Ohashi & Yahara 1998).
Therefore we calculated relative visitation rate per plant
as (number of bee visits to focal plant)/(sum of number
of bee visits to all observed plants on the same day) for
each plant observed during 1997 and 1998. With regard
to the number of flower heads probed per plant, we used
mean data for each plant to avoid pseudoreplication
due to repeated measurements on the same plant. For
similar reasons, we calculated relative mean visitation
rate per head for each plant as (mean number of bee
visits per head on focal plant)/(sum of mean number of
bee visits per head over all observed plants). We then
related these three measures of bee behaviour with
floral display size using linear or non-linear least-squares
regressions with the polytope (simplex) method – a
numerical procedure that can be used to minimize a
function with respect to a set of parameters (Nelder &
Mead 1965; for a  program see Nash & WalkerSmith 1987). Further, we performed s with
either of the three measures of bee behaviour as the
dependent variable, plant density (high or low) as the
independent variable, and display size as the covariate.
Prior to analyses, we logarithmically transformed the
relative visitation rate per plant and the relative mean
visitation rate per head to correct for lack of normality
and/or inequality of variances (Sokal & Rohlf 1995).
When the slopes of regression lines were not significant,
we performed simple comparisons between the highand low-density area using Mann–Whitney U-tests (we
did not use log-transformed variables in such cases).
When variances significantly differed between areas, we
alternatively used tests for equality of medians (Sokal
& Rohlf 1995) to avoid a type I and/or type II error due
to heterogeneity of variances (Kasuya 2001).
Bumble bee behaviour within plants
© 2002 British
Ecological Society,
Functional Ecology,
16, 492–503
From observations of sequences of flower-head visits,
we determined the extent to which bumble bees revisited
flower heads as a function of the number of flower heads
available on a plant and the number already visited by
the bee during one visit. For each plant, we calculated
the flower-head revisitation rate as r = Rt /(Rt + Nt),
where Rt and Nt is the number of visit sequences in
which the tth probing was a revisit and a non-revisit,
respectively. We then estimated ‘memory size’ (m) for
each plant by fitting a linear function, r = b[t − (m + 1)],
to the observed relationship between the revisitation
rate (r) and the position of a head in a sequence (t),
where b and m are constants. Because the number of
heads probed before leaving (the maximum t) varied
among visit sequences, even on the same plant, the
number of sequences including the tth probing (Rt + Nt)
decreased with increasing t. This would increase the
variance of the calculated r with increasing t. Therefore, to estimate b and m as accurately as possible, we
used the weighted least-squares method with 1/(Rt + Nt)
as the weight (Draper & Smith 1998). We also calculated the overall rate of flower-head revisitation for
each plant as a ratio of the sum of the number of revisits
to the sum of the number of visits over all flower heads
on the plant: ro = ΣRt /Σ(Rt + Nt) (t = 3, 4, … ). Data on
plants with fewer than three flower heads were omitted
from these analyses.
We estimated the discounting rate for interplant
movement (k) using the observed values of time per head,
flight time between heads, and flight time between
plants. From the observed distribution of each parameter,
we randomly selected one value and calculated the
discounting rate (k) based on equation 1. We repeated
this procedure 10 000 times for each density area to
estimate the mean k and its 95% confidence limits.
We adopted such a Monte Carlo procedure because
the average of a non-linear function of random variables differs from the function’s value for averages of
these variables (so-called ‘fallacy of the averages’;
Wagner 1969).
Bumble bee visitation rate per plant/head
Nectar
We performed an  on sugar production rate
with plant density (high or low) as the independent variable and display size as the covariate. We
also performed a model I 3 × 3 factorial  with
nectar standing crop as the dependent variable, floral
FEC644.fm Page 497 Thursday, July 18, 2002 11:32 AM
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bumble bees
achieve an ideal
free distribution
display size and plant density as the independent
variables. Prior to these analyses, we logarithmically
transformed the dependent variables to correct for lack
of normality.
Results
  
© 2002 British
Ecological Society,
Functional Ecology,
16, 492–503
The overall rate of flower-head revisitation by bumble
bees on a plant (ro) averaged 0·025 (n = 24, SE = 0·002)
at high density and 0·074 (n = 27, SE = 0·009) at low
density. At both densities, overall revisitation rate (ro)
increased with the size of floral display (F ) (high density: ro = 0·011 + 0·001F, n = 24, R2 = 0·21, P = 0·024;
low density: ro = 0·005 + 0·007F, n = 27, R2 = 0·26,
P = 0·0069), with a stronger increase at low density
than high density (df = 1,47, F = 5·43, P = 0·024). For
each plant, the likelihood of revisitation (r) increased
linearly with the position of a head in a visit sequence
after probing a few non-revisited heads (Fig. 2). The
transition from no revisits to increasing revisits (t = m + 1)
occurred after an average (±SE) of 3·0 ± 0·26 heads or
2·3 ± 0·23 heads in the high- and low-density areas,
respectively. The risk of revisitation increased less
rapidly with successive probings on large displays than
on small displays (Fig. 2). The estimated slope of the
regression line was negatively correlated with display size
(high density: n = 24, Kendall’s tau = −0·36, P = 0·015;
low density: n = 27, Kendall’s tau = −0·52, P = 0·0002).
The estimated memory size (m) was marginally larger
in the high- than in the low-density area (U = 219·0,
P = 0·048, Mann–Whitney U-test). On the other hand,
the estimated memory size did not vary significantly
with floral display size (high density: n = 24, r = 0·13,
P = 0·55; low density: n = 27, r = 0·095, P = 0·64).
Flight time between plants varied significantly between
high and low densities (high density: n = 60, mean ±
SE = 4·21 ± 0·28 s; low density: n = 49, mean ± SE =
7·43 ± 0·41 s; U = 522·5, P < 0·0001, Mann–Whitney
U-test). On the other hand, no significant densitydependent difference was detected in time per head
(U = 1105·0, P = 0·40, Mann–Whitney U-test; n = 99,
mean ± SE = 27·8 ± 2·8 s) and in flight time between
heads within plants (Fisher’s exact probability = 0·098,
test for equality of medians; n = 60, mean ± SE =
1·90 ± 1·2 s). In our Monte Carlo calculation, the
mean k (±95% CL) was estimated as 0·88 ± 0·37 at high
density and 0·76 ± 0·39 at low density.
Bumble bees probed fewer heads per plant than
expected, although the observed means always fell
within the 95% confidence limits of the predictions
(Fig. 3). Bees usually left a plant spontaneously, but
interference by other pollinators (including bumble
bees and other visitors) or vespid wasps caused
about 12% of departures (data on 17 plants in 1998).
The number of flower heads that bees probed per plant
(tc ) increased linearly with, but less rapidly than, floral
display size (Fig. 3). The slope of this relation in the
Fig. 2. Relationships between the likelihood of revisitation (r)
and the position of the head in a visit sequence (t). Bending point
corresponds to (m + 1). (a) High density, data on small displays
(, five plants with five to eight heads, bending point = 3·27,
slope = 0·23, R2 = 0·87, n = 6, P = 0·0064); large displays (, seven
plants with 13–18 heads, bending point = 3·35, slope = 0·068,
R2 = 0·94, n = 11, P < 0·0001). (b) Low density, data on small
displays (, five plants with six to seven heads, bending
point = 2·76, slope = 0·15, R2 = 0·93, n = 8, P < 0·0001); large
displays (, eight plants with 11–16 heads per plant, bending
point = 2·48, slope = 0·076, R2 = 0·93, n = 11, P < 0·0001).
Memory size (m) was also estimated for each plant using the
same regression method (see text).
low-density area was 2·3 (1997) or 3·0 times (1998)
greater than in the high-density area (df = 1,32,
F = 23·0, P < 0·0001 in 1997; df = 1,38, F = 8·76,
P = 0·0053 in 1998).
The relative visitation rate per plant increased in a
decelerating manner with floral display size during
1997 (Fig. 4), but this pattern was not evident during
1998, when we observed few small displays. Visitation
rate per plant increased more rapidly in both years
and reached a higher maximum in the high- than in
the low-density area (intercept: df = 1,33, F = 18·5,
P = 0·0001 in 1997; df = 1,13, F = 34·4, P < 0·0001
in 1998; slope: df = 1,32, F = 0·55, P = 0·47 in 1997;
df = 1,12, F = 0·044, P = 0·84 in 1998,  for
log-transformed dependent variable).
During 1997, the average visitation rate per head
per hour varied among plants from 0·84 to 14·4. The
relative mean visitation rates per head (Vf ) at both
densities did not vary significantly with floral display
size (Fig. 5a; high density: n = 16, F = 2·00, P = 0·18;
FEC644.fm Page 498 Thursday, July 18, 2002 11:32 AM
498
K. Ohashi &
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Fig. 3. Observed and predicted relationships between number
of flower heads probed per plant (tc, t *)
c and floral display size
(number of flowering heads per plant, F ). (a) Data in 1997: high
density (), tc = 1·04 + 0·11F, R2 = 0·91, n = 16, P < 0·0001;
low density (): tc = 0·82 + 0·25F, R2 = 0·84, n = 20, P < 0·0001.
(b) Data in 1998: high density (), tc = 1·25 + 0·078F, R2 = 0·51,
n = 20, P = 0·0003; low density (), tc = 0·65 + 0·24F, R2 = 0·51,
n = 20, P = 0·0003). (c) Predicted mean t *c (high density, ;
low density, ) and 95% CL (high density, dashed lines; low
density, solid lines) for each F.
© 2002 British
Ecological Society,
Functional Ecology,
16, 492–503
low density: n = 20, F = 0·51, P = 0·49). We detected no
significant difference in Vf between the high- and the
low-density area (U = 121·0, P = 0·21, Mann–Whitney
U-test). During 1998 the average visitation rate per head
per hour varied among plants from 2·42 to 16·8. The
relative mean visitation rate per head (Vf) did not vary
with floral display size (Fig. 5b; high density: n = 8,
F = 0·30, P = 0·60; low density: n = 8, F = 0·039, P = 0·85),
nor between the two areas (U = 31·0, P = 0·92, Mann–
Whitney U-test). Thus bumble bees utilized large and
small displays equally in terms of average visitation
rate per head. Within each plant, however, variation
in visitation rate differed greatly among flower heads
(CV = 0 –77·3%, data on 21 plants in 1997).
Fig. 4. Observed relationships between relative visitation rate
per plant (Vp) and floral display size (number of flowering heads
per plant, F ). (a) Data in 1997: high density (), Vp = 0·13F/
(1 + 0·15F ), R2 = 0·61, n = 16, P = 0·00037; low density (),
Vp = 0·19F/(1 + 0·72F ), R2 = 0·28, n = 20, P = 0·018. (b) Data
in 1998: high density (), Vp = 0·067F/(1 + 0·10F ), R2 = 0·30,
n = 8, P = 0·16; low density (), Vp = 0·019F /(1 + 0·038F ),
R2 = 0·15, n = 8, P = 0·35.
 
Mean nectar sugar production rate of a flowering head
(µg sucrose floret−1 h−1) did not vary with floral display
size (Fig. 6). Furthermore, plants in the high- and lowdensity area produced equivalent amounts of nectar
sugar (high density: n = 18, mean ± SE = 16·7 ± 2·1 µg
floret−1 h−1; low density: n = 20, mean ± SE = 16·0 ±
1·5 µg floret−1 h−1; Fisher’s exact probability = 0·63,
test for equality of medians). In addition, the number
of florets per head did not vary with floral display
size (n = 20, r = 0·12, P = 0·63). Thus, irrespective of
floral display size, C. purpuratum provided similar
nectar rewards per head for pollinators. On each
plant, however, there seemed to be some variation
in nectar productivity among heads, derived from
their asynchronous flowering (CV of flower-head age
within a plant per day = 0–47·1%, data on 12 plants
in 1997).
Standing crop of nectar in flowering heads (n = 144
heads, mean ± SE = 16·9 ± 1·8 µg floret−1) varied
neither with display size (df = 1,140, F = 0·60, P = 0·44)
FEC644.fm Page 499 Thursday, July 18, 2002 11:32 AM
(a) 1997
Relative mean visitation rate per head (Vf )
499
Nectar-collecting
bumble bees
achieve an ideal
free distribution
Discussion
1
0·9
0·8
0·7
0·6
0·5
0·4
0·3
0·2
0·1
0
High density
Low density
0
2
4
6
8
10 12 14 16 18 20
6
8
10 12 14 16 18 20
(b) 1998
1
0·9
0·8
0·7
0·6
0·5
0·4
0·3
0·2
0·1
0
0
2
4
No. of flowering heads per plant (F )
Fig. 5. Observed relationships between relative mean visitation
rate per head (Vf ) and floral display size (number of flowering
heads per plant, F ). Data in (a) 1997; (b) 1998. During both
years and in both areas these variables were not significantly
correlated with each other (see text).
Fig. 6. Relationships between nectar productivity of a flowering
head and floral display size (number of flowering heads per
plant, F ). In both areas these variables were not significantly
correlated with each other (see text).
© 2002 British
Ecological Society,
Functional Ecology,
16, 492–503
nor with plant density (df = 1,140, F = 0·14, P = 0·71).
The interaction of the two independent variables was
also non-significant (df = 1,140, F = 3·19, P = 0·076,
model I 3 × 3 factorial ). On average, bumble
bees left a head after removing about 80% of the initial
crop (nectar sugar in a head just after probed: n = 20,
mean ± SE = 3·03 ± 0·56 µg floret−1; nectar sugar in a
randomly sampled head: n = 20, mean ± SE = 15·2
± 2·0 µg floret−1; Fisher’s exact probability = 0·00016,
test for equality of medians).
-   

The estimated memory size of bumble bees was between
one and two heads (Fig. 2). This suggests that bumble
bees store spatial information on previously probed
flower heads in spatial working memory (or short-term
memory) with limited capacity and persistence. This
appears to contradict the fact that bees often exhibit
large long-term memory capacity for spatial information, such as the location of the nest and flower patches
(Menzel et al. 1996). In nature, however, bees often
probe hundreds of flowers successively during one
foraging trip, in which case spatial working memory
may be more efficient than long-term memory in terms
of processing speed and flexibility. Honeybees also use
spatial working memory to avoid flower revisitation
instead of long-term memory (Brown & Demas 1994;
Brown et al. 1997). It is unclear why bumble bees began
revisiting previously probed heads slightly later on a
plant in the high-density than in the low-density area.
This may reflect some flexibility in bees’ foraging strategy
in response to plant density. Otherwise, bees might use
neighbouring plants as temporal signposts or landmarks
in the high-density area. For example, Redmond &
Plowright (1996) have reported that bumble bees avoid
flower revisitation more efficiently when landmarks
are available.
As a consequence of this small memory size, the risk
of flower-head revisitation increased linearly with the
position of a head in a visit sequence after just two or
three heads had been probed (Fig. 2). Because bees
depleted the nectar in heads to 20% of the initial crop,
our result indicates that increasing risk of flower-head
revisitation caused a gradual decrease in the rate of gain
per head (patch depression). This role of flower revisitation in patch depression has been poorly appreciated
because bees seldom revisit flowers: 2·9% (Pyke 1979);
3·5% (Pyke 1982); 0·2 and 3·0% (Galen & Plowright
1985). However, we observed patch depression even
though bees revisited 5·1% of the heads they probed.
Several previous studies have also found that a pollinator is increasingly likely to revisit flowers as it stays
longer on a plant or inflorescence (Pyke 1978; Pyke 1981;
Pyke 1982; Redmond & Plowright 1996).
Flower-head revisitation rate increased more slowly
on larger displays (Fig. 2). This is simply because larger
displays offer more flower heads from which to choose
and relieve the risk of flower-head revisitation, as
predicted in equation 2 and Fig. 1(a). Pyke (1982) also
reported a similar result in bumble bees foraging on
Aconitum columbianum (Table 14 of Pyke 1982). Previous
authors have often considered the spatial gradient of
nectar crops per flower as the cause of patch depression
(Best & Bierzychudek 1982; Hodges 1981; Hodges 1985).
However, as Pyke (1982) pointed out, spatial structuring
of nectar crops alone cannot explain why pollinators
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K. Ohashi &
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tend to be more tenacious on larger displays, because
flowers on large displays often contain the same nectar
distribution as those on small displays (Pyke 1978;
Pyke 1982; Wolf & Hainsworth 1986; this study, but
see Hodges 1985). We suggest therefore that the risk
of flower revisitation, as well as nectar distribution
pattern, is important in understanding pollinator
behaviour on plants.
     
    
© 2002 British
Ecological Society,
Functional Ecology,
16, 492–503
As has been observed repeatedly for other species
(reviewed by Goulson 2000; Ohashi & Yahara 2001),
the number of heads that bumble bees probed per
plant increased less than proportionally with floral display size, so that the proportion of flower heads probed
per bee (tc /F ) declined with display size (Fig. 3). This
increase in the number of heads probed per plant was
significantly slower in the high- than in the low-density
area (Fig. 3). Ohashi & Yahara (1999) (see Fig. 1b)
proposed that this change would result because bees
decrease the marginal level of revisitation for leaving a
plant (dashed line, Fig. 1a) at higher plant density,
which reduces the difference in number of flower heads
probed per plant between large and small displays
(tc–l − tc–s).
We also found that visitation rate per plant increased
at a decelerating rate with floral display size (Fig. 4). A
decelerating increase of visitation rate per plant with
increasing display size has been reported in previous
studies (reviewed by Iwasa, de Jong & Klinkhamer 1995;
but see Andersson 1988; Ohara & Higashi 1994; Sih &
Baltus 1987). As predicted, visitation rate per plant
increased more strongly with increasing displays size at
higher plant density (Fig. 4).
The preference for visiting larger displays counterbalanced the decline in the proportion of flower heads
probed by a bee on larger displays, so that the average
visitation rate received per head did not vary with display size, irrespective of plant density (Figs 3–5). This
strongly suggests that bees preferred to visit large
displays because a decline in the proportion of heads
probed per plant reduced the competition among bees
on these plants. Because neither nectar production rate
per head, nor head size (floret number per head), varied
with floral display size of the plant, an equalization
of the visitation rate per head means that bumble bees
gained equal nectar rewards per head on all sizes of
display – they achieved an IFD. As a result of this,
flower heads on large and small displays contained the
same amounts of nectar crop. A few researchers have
also proposed reliable evidence that nectar foragers
approximate an IFD on plants. Delph & Lively (1992)
found that perfect flowers on hermaphroditic plants
of Hebe stricta had higher nectar productivity than
pistillate flowers on female plants, but frequent pollinator visits to perfect flowers depleted nectar to the
point where the two sexual morphs contained the same
amounts of nectar crop. Dreisig (1995) found that
foraging bumble bees achieved an IFD by visiting
individual Anchusa officinalis and Viscaria vulgaris
according to its nectar productivity and the proportion
of flowers that a bee probed on that plant. Similarly,
with regard to pollen foragers Robertson et al. (1999)
reported that bumble bees responded to genetic variation in pollen quality (proportion of inviable pollen
grains) among Mimulus guttatus plants by visiting the
high-quality patch more often and by visiting more
flowers within the patch.
We do not emphasize that the distribution of bumble
bees conformed strictly to an IFD. Some authors have
noticed that the restrictive assumptions for the continuous input IFD model (see above) are often violated
under natural conditions (Kennedy & Gray 1993; Milinski
1994). In addition, we found that the overall rate of
flower-head revisitation (ro) increased with display
size, especially at low plant density. Ohashi & Yahara
(1999) predicted that this would result because the
ratio of m to tc declines with display size and plant density. If bees could detect these subtle differences in the
rate of energy gain among plants and adjust their distributions to a complete IFD, the average visitation
rate per head might have varied with display size.
However, such effects seemed too small to affect actual
bees’ visitation. Thus an equalization of the visitation
rate per head among plants observed here can be
regarded as a quasi-IFD.
The strategies that individual bumble bees might use
to achieve an IFD are still open to question. As Dreisig
(1995) suggested, bees’ preferences for visiting large
floral displays may partly explain the IFD. It should be
noted, however, that the observed visitation rates per
plant cannot be viewed simply as an amplified pattern
of individual bees’ choice in relation to floral display
size. Individual bumble bees frequently confine their
foraging to small areas within a larger plant population, and they appear to overlap their foraging areas
(Williams & Thomson 1998). Therefore a bee has to
adjust the degree of overlap in its foraging area with
others, while deciding how often it should visit each size
of display. It has been suggested that bumble bees have
some abilities needed for such complicated adjustments.
For example, bees are known to fly longer distances
after encountering lower rewards (area-restricted searching; reviewed by Motro & Shmida 1995). By adopting
this rule while foraging along its own foraging route
(trapline; Thomson 1996 and references therein), bees
may efficiently reduce the spatio-temporal bias in nectar
distribution. To clarify how these foraging patterns
contribute to an IFD, a detailed description is required
of individual bees’ behaviour and their use of space.
    
    
The predicted optimal number of flower heads probed
per plant was generally larger than that observed
FEC644.fm Page 501 Thursday, July 18, 2002 11:32 AM
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Nectar-collecting
bumble bees
achieve an ideal
free distribution
(Fig. 3). The most probable reason for this discrepancy
is the variation in nectar productivity among heads
within a plant. All available flower heads on a plant
were used to quantify the size of floral display. However,
flower age varied greatly among heads within a plant,
so that this might result in a substantial variation in
nectar productivity among heads. If bees probed older
and less rewarding heads infrequently, as has been
reported in other systems (Cruzan, Neal & Willson 1988;
Gori 1989; Oberrath & Böhning-Gaese 1999), the
realized mean number of heads probed per plant would
be fewer than that predicted.
Another possibility is the assumption of our model
that the bees have ‘complete information’ on nectar
distributions within plants – that information obtained
while foraging on a plant is of no value for their decisionmaking. In nature, pollinators rarely have complete
information and their environment is usually stochastic,
so that they appear to use full or partial information
gained at each flower to decide when to leave a plant;
pollinators often leave a plant just after probing one or
two flowers with little or no nectar (Hodges 1985; Pyke
1978; Pyke 1982; Thomson, Maddison & Plowright 1982).
Such simple probabilistic rules may provide a practical
method to approach a mathematical optimum in nature
(see also Iwasa, Higashi & Yamamura 1981; McNamara
& Houston 1980). We assumed complete information
because our aim was simply to find an optimum plant
departure, rather than patch-leaving rules that visitors
actually follow. However, this difference might result
in a quantitative discrepancy between expectation and
observation.
Conclusions
© 2002 British
Ecological Society,
Functional Ecology,
16, 492–503
Recent empirical research on plant–pollinator interactions has identified a puzzling feature of pollinator
behaviour in response to increasing display size –
pollinators visit larger displays more often, but leave a
greater proportion of flowers behind on larger displays
(Ohashi & Yahara 1999). The results obtained here
strongly support our basic idea that this counterintuitive
behaviour can be understood as an optimal strategy to
avoid the risk of flower revisitation, and to utilize nectar
in flowers left behind during one visit. By comparing
different plant densities, we showed that these two
types of pollinator behaviour interact with each other
through the distribution of floral reward. These results
suggest that decisions on patch choice and patch
residence time should be considered inclusively when
resource patches vary in size.
Another novel finding of this study is that the reduction in plant density decreases pollinators’ preference
for visiting larger displays, while it increases their tenacity
on larger displays. Pollinators are said to alter their
foraging behaviour in response to the decrease in plant
density. First, pollinators may be attracted less strongly
to plants growing at lower densities (Kunin 1997). Second,
pollinators are more likely to behave as generalists on
sparsely distributed plants (Kunin 1993). In addition
to these population-level effects, we showed that pollinators respond to decreasing plant density by altering
their behaviour on each size of display within populations.
Such alterations may greatly affect pollen dispersal from
plants with different size of displays (Harder & Barrett
1995; Rademaker & de Jong 1998). Our study suggests
that observations on pollinator behaviour and/or
pollen dispersal in relation to floral display size should
be viewed with caution unless the effects of plant density
are explicitly described.
Acknowledgements
We are grateful to many colleagues for contributions
to the ideas herein; Drs L. F. Delph, L. D. Harder and
A. W. Robertson for helpful comments on the manuscript;
Dr E. Kasuya for writing  programs and for help
in statistical analyses; Dr O. Tadauchi for identification
of bumble bee species; Drs K. Muraoka, M. Watanabe
and S. Nawashiro for useful advice on nectar measurement; Y. Fujita, C. Kikuchi, M. Kinoshita, A. Konuma,
K. Murayama, Y. Osone, S. Shiokawa and H. Taneda
for patient help with field work; and K. Shibata, Dr M.
Tateno and the other staff of the Nikko Botanical
Garden for invaluable assistance throughout our field
work. This work was partly supported by a fellowship
(no. 1597) of The Japan Society for the Promotion of
Science for Young Scientists.
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Received 6 September 2001; revised 24 January 2002; accepted
4 February 2002