AMER. Zooi... 18:739-752(1978).
Optimal Foraging in Hummingbirds: Testing the Marginal Value Theorem
GRAHAM H.PYKE
Department of Biology,
University of Utah, Salt Lake City, Utah 84112
SYNOPSIS. To a hummingbird, clusters of flowers on inflorescences represent patches and
provide an ideal situation to test prediction of optimal patch-use. The basic question is what
decision rule should a hummingbird use to decide whether or not to leave an inflorescence?
The hypothesis is that hummingbirds will adopt the decision rule that maximizes their net
rate of energy gain while foraging. This hypothesis leads to an analogue of Charnov's marginal value theorem which determines an optimal decision rule. The optimal decision rule
is then used to predict aspects of the hummingbirds' foraging, and these predictions are
compared with field data.
The optimal decision rule is a function of how much information is used by the
hummingbirds. Data indicate that a decision to leave an inflorescence is a function of the
number of flowers visited, the number of flowers available on the inflorescence, and the
amount of nectar obtained at the last flower. The optimal decision rule was calculated
assuming no additional information is used.
INTRODUCTION
During recent years a number of authors have attempted to predict the
foraging behavior of animals from optimization models. The result is known as
optimal foraging theory (for reviews see
Schoener, 1971; Covich, 1977; Pyke et al.,
1977). Animals which feed on floral nectar
are ideal subjects for studies of this kind.
The foraging behavior of nectar-feeding
animals can be readily observed, quantified, and the relevant currency for the
Darwinian fitness often can be argued
to be net rate of energy intake (Schoener,
1971; Pyke et al., 1977) which is easily
measured {e.g., Carpenter, 1976; Carpenter and MacMillen, 1976; DeBenedictis
etal., 1978; Gill and Wolf, \975a,b; Hainsworth and Wolf, 1972a,b; Heinrich,
I975a,b; Pyke, unpublished; Stiles, 1971;
Author's current mailing address is: Zoology Building, University of Sydney, N'.S.W. 2006, Australia.
This research was supported by N'SF Grant DLB
76-20970. F. R. Hainsworth, K. D. Waddington and
L. L. Wolf provided stimulating discussion during the
study. Excellent assistance in the field was provided
by J. Berger, D. Hackenyos, R. Haptonstall and D.
Stephens. E. L. Charnov and J. P. Finerty gave many
helpful comments on an earlier draft of this manuscript.
Wolf, 1975; Wolf and Hainsworth, 1971;
Wolf et al., 1972; Wolfe/ al., 1975). The
aim of the present study was to apply one
aspect of optimal foraging theory to hummingbirds.
An aspect of foraging behavior which
has been treated theoretically (Krebs et al.,
1974; Charnov, 1976) and for which two
empirical tests have so far been carried out
(Krebs et al., 1974; Cowie, 1977) is allocation by an animal of time to a food patch.
The basic question is "When should an
animal leave one patch and move to
another?" This is the question considered
in this study. The patches are inflorescences of plants; the animals are hummingbirds which obtain energy from nectar of the plants, and the question can be
restated as "How many flowers should a
hummingbird visit on an inflorescence before it flies to another?"
The theoretical treatment of timeallocation to patches has resulted in the
"marginal value theorem" (Krebs et al.,
1974; Charnov, 1976). The theorem states
that an animal will maximize its net rate of
energy intake while foraging if it leaves
each patch when its instantaneous net rate
of energy intake in the patch has dropped
to the overall net rate of energy intake in
the entire habitat (Krebs et al., 1974; Charnov, 1976). The form of the theorem is,
739
740
GRAHAM H. PYKE
the number actually probed was determined and an estimate obtained of the
time taken by a hummingbird to move
from one flower to another within an
inflorescence and to remove nectar from
the second flower.
I measured the total observed foraging
time of each hummingbird that was seen
visiting the flower patch and the total
whenever its expected net energy gain ob- number of flowers visited during this time.
tained by moving to the next resource From these data an estimate was obtained
point and removing the food there divided of the overall rate of flower visitation while
by the expected time required to move to the foraging.
next resource point and to remove the
The times spent flying between infloresfood there is equal to the overall net rate of cences by the hummingbirds were conenergy intake in the habitat and if it leaves verted to an average. When this value was
the patch when this ratio is less than the combined with estimates of the overall rate
overall rate.
of flower visitation and the mean number
of flowers visited per inflorescence, a secMETHODS
ond estimate was obtained of the time
This study was carried out in the vicinity taken by a hummingbird to move from one
of the Rocky Mountain Biological flower to another within an inflorescence
Laboratory near Crested Butte, Colorado. and to remove the nectar from the second
The subjects were non-territorial hum- flower.
mingbirds of two species, the Broad-tailed
I recorded the amount of nectar conHummingbird (Selasphorus platycercus) and tained in, and the spatial position relative
the Rufous Hummingbird (S. rufus), and to other flowers on the inflorescence of
the plant species Ipomopsis aggregata (scarlet each flower on random samples of
gilia) which provides a major portion of inflorescences to determine patterns of
the nectar used by these birds at the time nectar distribution.
of the study. Field and laboratory data
The "stems" of artificial inflorescences
were collected. Field observations were for laboratory observations were 2.5 cm
made in a large area of Ipomopsis (approxi- diameter styrofoam about 45 cm long and
mately 50 m x 30 m) between 08:00 and painted green. The "flowers" were surgical
11:00 hr (when the birds fed most exten- needle caps 17 mm long with openings 4
sively from the flowers). Laboratory obser- mm wide and painted red. A variety of
vations were made on artificial inflores- artificial "inflorescences" could be created
cences (described below) placed outside of by pushing the metal portions of the neea window and visited by wild hum- dles into the styrofoam. Artificial infloresmingbirds. The field data reported here cences were constructed as copies of
were collected from 1 July to 9 July, 1977, inflorescences found in the field. The
at a time when Ipomopsis was at peak bloom. copies were made as accurately as possible
The laboratory data were collected from with respect to both the spatial location of
21 June to 2 July, 1977, a period which each flower and its orientation relative to
began when the Ipomopsis was approaching the vertical. Two inflorescence sizes were
full bloom.
used (8 and 12 flowers) and nectar volumes
I recorded the time spent by a hum- were 10/xl per flower of 1.4 M sucrose.
When artificial inflorescences were not
mingbird at an inflorescence, the number
of flowers probed on that inflorescence used, a simple feeder hung outside the
and the total number of flowers available window provided an ad libitum supply of
on the inflorescence. From these data the the same sucrose solution. The humrelationship between the number of mingbirds showed no hesitation in probing
flowers available on an inflorescence and the "flowers" when an artificial inHoreshowever, for a deterministic, continuous
system (see graphs in Charnov, 1976;
Cowie, 1977). The present case of flowers
on inflorescences requires a stochastic discrete form: an animal, which is visiting
distinct food-resource points within
patches, will maximize its net rate of
energy intake while foraging if it remains
in a patch after visiting a resource point
™
OPTIMAL FORAGING
cence replaced the feeder. Each time a
hummingbird visited an artificial inflorescence the sequence of flowers visited was
recorded and another inflorescence was
hung in its place. Ten inflorescences were
created in each size category during the
following time periods which are assumed
to be equivalent: 8 flowers per inflorescence—21 June to 24 June; 12 flowers per
inflorescence — 28 June to 2 July.
From observations of flower sequences
on artificial inflorescences it was possible to
determine the extent to which hummingbirds revisited flowers as a function of
how many flowers were available on an
inflorescence, and how many had already
been visited by the bird in question during
its visit to the inflorescence. The revisitation pattern was assumed to be an accurate
estimate of the natural pattern.
RESULTS AND DISCUSSION
Number of flowers probed vs. number available
To determine the relationship in the field
between the number of flowers probed
by a hummingbird at an inflorescence
and the number of flowers available on the
inflorescence, the data were divided into
several categories on the basis of the
number of flowers available: 6 to 9 flowers
available, 10 to 14, 15 to 18, 19 to 23, 24 to
38, and > 38. They were chosen so the
midpoints of two of the categories were
close to the two sizes of the artificial
inflorescences and so the sample sizes in
most categories were roughly equal. Sample sizes were 8, 13, 13, 13, 12 and 6,
respectively. The mean number of flowers
probed was then plotted against the mean
number of flowers available for each
category (the crosses in Figure 1). When
only 1 flower is available the number of
flowers probed must also be 1, and so this
point is also shown in Figure 1. The
dashed line in Figure 1 is fitted to the data
by eye. The continuous line is the line for
equal numbers of flowers probed and the
flowers available. Figure 1 shows that the
number of flowers probed increased less
rapidly than the number of flowers available increased.
741
Number of I lower*
FIG. 1. Number of flowers visited by hummingbirds
vs. number flowers available on each inflorescence.
The crosses are observed points, the dots are
theoretical (see text). The dashed line is fitted by eye
to the observed data points. The continuous line is the
line for equal numbers of flowers visited and flowers
available.
Frequency distribution of number of flowers
probed
Since the average number of flowers
probed is a function of the number of
flowers available on an inflorescence, the
frequency distribution of numbers of
flowers probed will also be a function of
the number of flowers available. Several
properties of these frequency distributions
are revealed, however, by calculating two
of them—one for inflorescences with — 18
flowers and one for inflorescences with ^
19 flowers. This divides all the observations into two approximately equal samples
(51 and 49, respectively) (Fig. 2).
Figure 2 indicates:
1) For inflorescences with > 19 flowers
only 1 inflorescence of the 51 (i.e., 2%) was
left by a hummingbird after only 1 or 2
flowers had been probed. Most inflorescences were left after between 3 and 9
flowers had been probed.
2) For inflorescences with < 18 flowers a
significantly larger (P = 0.04) fraction of
the inflorescences (10%) were left after 1
or 2 flowers had been probed. This difference between the two frequency distributions has strong implications with respect
to the rule by which a hummingbird decides to leave an inflorescence (see later
discussion).
742
GRAHAM H. PYKE
Z .
B
9
10
II
>2
13
14
,5
16
15
16
n
i
17
18
Number of flowers probed
FIG. 2. The frequency distributions of numbers of
flowers probed by hummingbirds for inflorescences
with ^ 18 flowers available and for those with'^ 19
flowers available.
3) Both frequency histograms suggest
the possibility that the frequency distributions are multimodal with peaks near 3 or
4, 6 or 7, and 13 or 14. There is also the
suggestion that the locations of these peaks
shift to the right slightly with increases in
the number of flowers available. These
patterns do not appear to be consequences
of any multimodality in the frequency distribution of numbers of Mowers available
on the inflorescences visited by the hummingbirds. This possible multimodality
will be discussed later.
not have affected the hummingbirds since
they rarely (5%) visited inflorescences with
fewer than 7 flowers.
Among the inflorescences with > 6 flowers there was no relationship between the
standard deviation of the nectar volumes
and the number of flowers available (P =
0.22). Also, there was no relationship between nectar volume per flower and the
vertical position of the flower on the
inflorescence (P = 0.47). The correlation
between the amount of nectar in a flower
and the amount in its nearest neighbor (r2
= 0.033) was not significantly different
from the correlation (r2 = 0.055) for its
third nearest neighbor (P = 0.61). Thus,
there was no spatial component to any
correlations between nectar volumes
within inflorescences. These three correlations were all significantly greater than
zero (P = 0.02, 0.0001, 0.002, respectively). Hence, a hummingbird could potentially predict (from a knowledge of
nectar yields from past and/or present
flowers) how much nectar it would obtain
at future flowers on an inflorescence.
The nectar volumes in each flower on
inflorescences with ^ 7 flowers and in the
flower immediately below were used to
obtain a precise description of the relationships in nectar content among flowers on
the same inflorescence. The array of nectar volumes was divided into intervals so
the sample size in each interval was about
120, and then the mean nectar volume in
the immediately lower flower was calculated for each interval of nectar volumes in
the higher flowers. The results are shown
in Figure 3 with a fitted curve drawn by
eye.
Distribution of nectar within and between
inflorescences
Gross vs. net rate of energy intake
The data on nectar obtained from 1462
flowers (128 inflorescences) were analyzed
to determine whether any patterns exist.
There appeared to be no relationship between average nectar volume per flower
and the number of flowers available for
inflorescences with > 7 flowers (P = 0.55).
Inflorescences with < 6 flowers did contain
significantly more nectar than the larger
inflorescences (P = 0.001), but this should
The hypothesis generated from the
marginal value theorem is expressed in
terms of net rate of energy intake. It is
possible, however, to transform the
hypothesis into terms of gross rate of
energy intake. This transformation
changes only the word "net" to "gross" and
leaves the hypothesis otherwise unaltered.
Maximization of net and gross rates of
energ) intake are not strictly equivalent
OPTIMAL FORAGING
743
tion of net rate of energy gain. Furthermore, assuming that nectar concentration
is independent of other variables in the
system, maximization of net rate of energy
intake is equivalent to maximization of rate
of nectar (i.e., volume) intake.
This transforms the hypothesis to the
following: A hummingbird should remain
on an inflorescence when its estimate of
'olume per f l c ' I/ill
the nectar volume obtained at the next
FIG. 3. Average nectar in the immediately lower
flower divided by the average time reflower vs. nectar volume in a flower (grouped into quired to move to the next flower and reintervals). The line is fitted by eye.
move the nectar from it is equal to its overall gross rate of nectar (volume) intake in
because the rate of energy expenditure for the flower patch and should leave when
a foraging hummingbird is not constant this ratio is less than the overall rate.
but varies with the flight mode. Hovering
flight (employed while extracting nectar
TESTING THE HYPOTHESIS
from the flowers and partially while moving between flowers within an inflores- Gross rate of nectar intake in a flower patch
cence) involves expenditure of 215 cal/(g X
hr) (Hainsworth and Wolf, 1972; Wolf et
To test the hypothesis we need to deal., 1975) while forward flight (employed termine the gross rate of nectar intake in
while moving between inflorescences) is the flower patch. Since the rate of flower
estimated to be 183 cal/(g x hr) (i.e., 85% of visitation of the hummingbirds and the
the cost of hovering flight) (Wolf et al., average nectar volume per flower are
1975). This small percentage difference in known, I assume this is the product of the
the costs of the two modes of flight two quantities. However, this assumes that
suggests, however, that the overall rate of the hummingbirds encounter the same
energy expenditure can be assumed to be statistical distribution of nectar volumes
approximately constant.
that are found in the flower patch. There
The hummingbirds probed an average are two reasons why this might not be
of 0.89 flowers per second (about 1.12 valid. First, a hummingbird may occasionsec/flower) while foraging at the Ipomopsis ally revisit a flower on an inflorescence that
and each flower contained an average of it has previously probed only seconds be3.26 /i.1 of nectar for a gross rate of intake fore. These revisits should yield little or no
of 2.90 /Lil/sec. The nectar had an average nectar with the effect of making the averconcentration in sucrose equivalents of age nectar volumed obtained perflowerless
22.2% by weight (Watt, et al., 1974). As- than the existing nectar volume per flower.
suming 1 mg of sucrose is equivalent to 3.7 Second, a hummingbird may employ a
cal, the gross rate of energy gain of the sampling scheme and departure rule
hummingbirds averaged 2.38 cal/sec. If where it tends to probe more flowers on
these hummingbirds weighed 3.5 g inflorescences which contain more nectar
(Kodric-Brown and Brown, unpublished) per flower. The effect of this behavior
and hovered constantly while foraging, would be to make the average nectar voltheir rate of energy expenditure would ume obtained per flower more than the
have been approximately 0.21 cal/sec. The measured average nectar volume per
gross rate of energy gain is about 11.4 flower. Since it is not possible to determine
times larger than the rate of energy ex- the extent of either, and since they act in
penditure, so the rate of energy expendi- opposite directions, I will assume that the
ture can be assumed to be constant and average nectar volume obtained per flower
maximization of net rate of energy gain is equal to the measured average nectar
can be assumed equivalent to maximiza- volume per flower.
744
GRAHAM H. PYKE
Time required to move from one flower to
another on an inflorescence and to remove
nectar from the second flower
The next step to test the hypothesis is to
estimate the time required by a hummingbird to move between flowers in an
inflorescence and to remove nectar from
the second flower. When this has been
achieved the hypothesis can be stated in
terms of the expected nectar yield from
the next flower.
Assuming time spent per flower is independent of nectar obtained (about 0.74
sec/flower, see below), time to move between flowers on an inflorescence could be
estimated from the overall rate of flower
visitation or from linear regression of time
at an inflorescence vs. number of flowers
probed. The two procedures give values of
1.10 and 1.22 sec, respectively. These are
not statistically different, and I will use
1.10 sec.
Hypothesis in terms of expected nectar yield at
next flower
The hypothesis indicates that a hummingbird should remain on an inflorescence
when its estimate of the nectar yield at the
next flower is equal to the average time per
flower (1.10 sec) multiplied by the overall
gross rate of nectar intake in the flower
patch (2.90 Ail/sec) or 3.19 /xl. It should
leave if its expected nectar yield is less than
this "expectation treshold."
Departure rule of hummingbirds
tion that is used should be combined to
produce the departure rule. For example,
a hummingbird might make use of knowledge of the amounts of nectar obtained
from just the present flower or of this and
the amount obtained at the previous
flower. If it uses both it may combine them
as a simple arithmetic mean or it might
employ a weighted mean of some kind. In
the following discussion several possible
levels of information use will be considered.
Before considering different possible
levels of information use, it is informative
to examine information that is potentially
available. Any variable which is correlated
with the expected nectar yield at the next
flower is a candidate. These correlations
should result from two fairly distinct patterns: 1) correlations between the nectar
volumes obtained from past and/or present flowers and the nectar volume contained in the next flower, and 2) patterns
of revisitation of flowers within inflorescences by the hummingbirds. The next
flower is more likely to be revisited if more
flowers have already been visited and if
fewer flowers are available on an inflorescence (see later discussion). Hence, the
expected nectar yield at the next flower
should be a function not only of the
amounts of nectar so far obtained but also
of the number of flowers already visited on
an inflorescence and the number available
on the inflorescence. Hummingbirds are,
therefore, likely to use some subset of the
following information: 1) nectar volumes
obtained from each flower so far visited, 2)
number of flowers so far visited and
number of flowers available. Several subsets of this information will now be considered.
This hypothesis, however, raises the
question of how a hummingbird might
estimate its expected nectar yield at the
next flower. A hummingbird may possess
information, such as the amount of nectar
obtained at the present flower, and it may Departure rule as a function only of the amounts
combine information in several ways to oj nectar obtainedfrom present and past flowers
produce a departure rule that results in deHummingbirds do not employ a deparparture from an inflorescence if the expected nectar yield at the next flower is less ture rule which is a function only of the
than the expectation threshold.
amounts of nectar obtained from present
There are two components to deter- and past flowers. This is shown most
mining what the departure rule of the clearly by the observation that the incihummingbirds should be. One is how much dence of departures from inflorescences
information the hummingbirds should after 1 or 2 flowers have been probed is
and do use. The other is how the informa- significantly lower when there are s 19
OPTIMAL FORAGING
_
™
flowers compared with ^ 18 flowers (Figtire 2), since there is no correlation between the number of flowers available and
the average nectar per flower or the standard deviation of the distribution. The
first two flowers probed on an inflorescence are also never revisits. Hence, the
frequency distribution of nectar volumes
encountered in the first two flowers on an
inflorescence should not be a function of
the number of flowers available. The differences in the behavior of the hummingbirds as a function of the number of
flowers available should, therefore, result
from variation in the departure rule as a
function of the number of flowers available. A detailed examination of the frequency distributions of numbers of flowers
probed also indicates that the departure
rule is nor a function only of the amounts
of nectar obtained in the flowers.
Departure rule as a function of only the number
of flowers probed or of only the number of
flowers available
Hummingbirds do not employ a departure rule which is a function only of the
numbers of flowers probed. The distribution of numbers of flowers probed varies
considerably with the number of flowers
available, which suggests that the departure rule is at least a function of the
number of flowers available.
Examination of the shape of the frequency distribution of number of flowers
probed shows that the departure rule is not
a function only of the number of flowers
available. Were this true there would be a
constant probability of departure for each
inflorescence size for each number of
flowers probed. This would generate a
zero-truncated geometric frequency distribution of numbers of flowers probed
with a single mode at one, and the combined frequency distributions for any
ranges of inflorescence sizes would also
have a single mode at one. The observed
frequency distributions (Fig. 2) are distinctly different from geometric distributions and show from one to several peaks
at more than one flower probed.
The situation that departure from an
745
inflorescence {i.e., patch) is not a function
of only a single piece of information contrasts with the study of Krebs et ai, (1974)
who assumed the birds they studied
employed a departure rule which consisted
of a function only of the amount of time
since the last food capture. It is possible
that more information was used by these
birds (Pyke et ai, 1977).
Departure rule as a function of only the number
of flowers probed and the number of flowers
available
It is possible that the departure rule of
the hummingbirds is only a function of
both the number of flowers probed and
the number of flowers available. By adjusting the probability of departure as a
function of these two numbers, frequency
distributions of the numbers of flowers
probed could be generated that are identical to the observed frequency distributions.
This possibility is, however, extremely unlikely. The optimal departure rate which
combines the two will be a rule where the
probability of departure is either 0 or 1
depending on the two numbers (Blackwell
and Girshick, 1954; Aoki, 1967; Kushner,
1971). This optimal rule would generate
distributions of numbers of flowers probed
with all departures at a single number for
each inflorescence size which is not the
case. Since the probability of departure lies
between 0 and 1 for each combination of
numbers of flowers probed and flowers
available, some other factors are also included in the departure rule. The most
likely additional factors are the amounts of
nectar obtained from the flowers.
Departure rule as a function of only the amounts
of nectar obtained and the number of flowers
available.
The departure rule is not a function only
of the amount of nectar obtained at the
present flower and the number of flowers
available. This is shown by examination of
the frequency distribution of numbers of
flowers probed on inflorescences with ^ 18
flowers (Fig. 2). The probability of a
hummingbird leaving after probing only 1
746
GRAHAM H. PVKE
tion which form the departure rule. It is
not possible to test directly the hypothesis
that the hummingbirds are employing this
departure rule but instead I shall generate
predictions concerning the frequency distributions of numbers of flowers' probed
under the optimal departure rule, and I
shall compare these predictions with observed frequency distributions. The possible use of information about more flowers
will be discussed later.
The optimal departure rule should involve a nectar threshold which is a function
of number of flowers probed and flowers
available. If a hummingbird obtained less
than this threshold amount of nectar at the
present flower it should leave the inflorescence, and if it obtained more it should
stay. A threshold is predicted because the
optimal departure rule will consist of leaving an inflorescence with a probability of
either 0 or 1 depending on the values of
the three variables forming the departure
rule (Blackwell and Girshick, 1954; Aoki,
1967; Kushner, 1971) and the expected
nectar at the next flower is an increasing
function of the amount of nectar obtained
at the present flower.
A hummingbird will always leave an
Departure rule as a function of the amounts of inflorescence after a revisit with a nectar
nectar obtained from the flowers, the number of threshold departure rule (assuming it
flowers probed, and the number of flowers avail- empties flowers). Measurements of the
amounts of nectar remaining in flowers
able.
probed by a hummingbird show that the
Since the departure rule of the hum- amounts remaining are almost always less
mingbirds appears to be a function of the than any of the theoretical thresholds calnumber of flowers available and not to be a culated below. In the following analysis it
function only of this number and the will, therefore, be assumed that a revisit
amounts of nectar obtained from the to a flower always results in departure of a
flowers, it may be a function of all of the hummingbird.
variables mentioned. One question still
The relationship between flower revisiremains, however, "How many flowers tations and the numbers of flowers probed
provide information used by hum- and available must now be considered.
mingbirds in their departure rule?" In- More precisely, since a revisit will always
stead of attempting to answer this difficult result in departure, it is the probability the
question directly, I shall assume that the next flower is a revisit (given that there
hummingbirds use a knowledge of the have been no revisits so far) which must be
amount of nectar obtained at the present determined. This was done using data
flower but do not use information of nec- from observations of foraging at the artifitar volumes obtained from previous cial inflorescences. The probability the
flowers. I shall now switch to the second next flower was a revisit was calculated for
component of testing the original both of the inflorescence sizes and for each
hypothesis, namely determining the opti- number of non-revisited flowers probed.
mal combination of the pieces of informa- The results are shown in Figure 4. The
flower (0.02, n = 51) is less than that of
leaving after probing 2 flowers (0.08, n =
50), although the difference is not
significant (P = 0.08). Since revisitation
does not occur up to the second flower
probed, these probabilities should be the
same under this departure rule. The incidence of departures after 3 flowers for
both ranges of inflorescence sizes is
significantly greater than would be expected under this departure rule after
allowing for a small probability of about
0.05 that the third flower probed is a
revisit.
Similar analysis shows that the hummingbird's departure rule is not a function
only of the amounts of nectar obtained
from the last two flowers and the total
number of flowers available, and that it is
not a function only of the mean amount ot
nectar obtained from past and present
flowers and the number of flowers available. It seems likely, therefore, that the
departure rule is not a function only of the
amounts of nectar obtained from the
flowers and the number of flowers available.
OPTIMAL FORAGING
1
=* cc S£ o.
u
c
— o
O
i r
~ a
Number of flowers probed
:I .
Number of flowers probed
FIG. 4. The probability that the next flower is a revisit
given that none of the previous flowers are revisits vs.
the number of flowers already probed for inflorescences with 8 and 12 flowers available. The dashed
lines .\re fitted by eye.
curves are fitted by eye. Also shown in
Figure 4 are the relationships between the
revisitation probability and the number of
flowers probed (given no previous revisits)
that would have occurred if the hummingbirds always selected the next flower
at random from the other flowers (i.e., any
flower except the present). These relationships are linear.
The following properties of the hummingbird's movements between flowers
within inflorescences are revealed by Figure 4: 1) As expected, the probability that
the next flower is a revisit, given no previous revisits, increases with number of flowers probed and with decreases in the
number of flowers available. 2) The revisitation probabilities are almost always considerably lower than they would be if the
hummingbirds selected the next flower at
random.
Estimates of the revisitation probabilities
obtained from the fitted curves can be used
to help determine the optimal departure
rule as a function of the amount of nectar
obtained at the present flower, the number
of flowers probed, and the number of
flowers available. Recall that the expectation threshold is 3.19 (i\, and a hum-
747
mingbird should remain on an inflorescence if it expects to get this amount of
nectar at the next, but should leave if it
expects less. The amount expected at the
next flower will be determined by the
statistical relationship between the nectar
obtained from the present flower and the
expected nectar at the next flower given
that the next flower is not a revisit (Fig.
3) and by the probability that the next
flower is a revisit given that no revisits have
occurred so far (Fig. 4). If the probability that the next flower is a revisit isP then
the expectation threshold given that the
next flower is not a revisit will be 3.19/1 —
P). Since P is a function of the-number of
flowers probed and the number of flowers
available the expectation threshold given
that the next flower is not a revisit will also
be a function of these two variables. These
adjusted expectation thresholds are calculated using/ 1 values estimated from Figure
4 (Tables 1 and 2).
There is now an adjusted expectation
threshold for each combination of numbers of flowers probed and numbers of
flowers available and these thresholds can
be converted into departure rules which
depend on the amount of nectar obtained
at the present flower. The conversion requires calculation of the relationship between possible departure thresholds and
the expected nectar at the next flower
given that it is not a revisit. In other words,
the expected nectar at the next flower must
be determined, given that the amount of
nectar obtained at the present flower is
above some threshold and the next flower
is not a revisit. This relationship was determined from the data used to generate
Figure 3 and is shown in Figure 5.
The departure thresholds corresponding to the various adjusted expectation
thresholds can be obtained directly from
Figure 5. For example, if the departure
threshold is 0.35 /J.\ then the expected
nectar at the next flower given that the
nectar obtained at the present flower was
above the departure threshold (i.e., the
hummingbird stays on the inflorescence)
and given th: t the next flower is not a
revisit is 3.35 /xl. Thus, the departure
threshold when 2 flowers have been
probed on an inflorescence with 8 available
748
GRAHAM H. PVKE
should be 0.35 /xl (see Table 1). The departure thresholds were calculated in this
manner for all combinations of numbers of
flowers probed and numbers of flowers
available (Tables 1 and 2). These departure thresholds, vary, as expected, with the
l
numbers of flowers probed and flowers
available. The set of departure thresholds III
Ifdefines the optimal departure rule under
the original hypothesis and under the assumption that the information used by the
FIG. 5. The expected nectar in the next flower given
hummingbirds consists of the amount of that
the nectar volume in the present flower is above
nectar obtained 'rom the present flower, the departure threshold (see text) and that the next
the number of •lowers probed, and the flower is not a revisit vs. the departure threshold
employed at the present flower.
number of flowers available.
This optimal departure rule cannot be
tested directly because it is not possible to that a hummingbird will leave an infloresdetermine how much nectar each hum- cence without visiting further flowers after
mingbird obtained at each flower. It is, probing n flowers (1 < n 1 < number availhowever, possible to use the optimal de- ble). This probability has two components,
parture rule to predict the frequency dis- the probability that the present flower is a
tribution of numbers of flowers probed for revisit, and the probability that the present
each inflorescence size.
flower is below the departure threshold
The first step toward generating the given that it is not a revisit. Both must be
frequency distribution of numbers of calculated conditional on n flowers having
flowers probed under the optimal depar- already been probed on the inflorescence.
ture rule (i.e., the optimal frequency dis- The first probability will be the probability
tribution) is to determine the probability that the present flower is a revisit given that
ill
Ocporlure thiethold ul
TABl-h 1. Calculation of optimal departure rule and optimal frequency distribution of numbers of flowers probed for
inflorescences with 8flowersavailable.
Number of flowers probed (n)
1
Probability that next flower is
a revisit given no previous
0.000
revisits
Adjusted expectation
threshold (/i.1)
3.10
Departure threshold (pil)
-0.00
Probability that present flower
is below departure
threshold given that
immediately previous
flower was above its
departure threshold and
given that no revisitation
0.000
has occurred so far
Probability of leaving after n
flowers given that n flowers
0.000
have been probed
Unconditional probability of
leaving after n flowers (i.e.,
optimal frequency distribution of numbers of flowers 0.000
probed per inflorescence)
Mean number of flowers
probed per inflorescence
4.68
0.047
0.055
0.075
0.115
0.180
0.350
3.35
0.35
3.38
0.57
3.45
1.00
3.60
1.80
3.89
2.92
4.91
0.108
0.140
0.194
0.284
0.433
0.779
0.108
0.180
0.254
0.366
0.535
1.000
0.108
0.161
0.186
0.200
0.185
0.161
1.000
0.000
OPTIMAL FORACING
no earlier flowers were revisits. The second
probability will be the probability that the
amount of nectar obtained from the present flower is less than the departure
threshold given that neither it nor previous flowers are revisits and given that the
amounts of nectar obtained at previous
flowers were all above the respective departure thresholds. I shall assume that the
last probability depends on the magnitude
of the departure threshold at the immediately previous flower but does not
depend on more previous flowers. This
probability can then be calculated using
the data from which Figure 3 was drawn
(Tables 1 and 2).
The probability of a hummingbird
leaving after probing n flowers, given that
it has already probed n flowers, can be
calculated as the probability that the present flower is a revisit, given no previous
revisits, plus the probability that the nectar
obtained from the present flower is less
than its departure threshold, given that
neither present nor past flowers are revisits
and that the immediately previous flower
was above its departure threshold. The
results of these calculations for all n and
each inflorescence size are shown in Tables
1 and 2. Also shown are the unconditional
probabilities of a hummingbird leaving an
inflorescence after probing n flowers.
These are calculated simply by multiplying
the probability that the hummingbird
leaves after n flowers, given that it has
probed n flowers, by the conditional probabilities that the hummingbird did not
leave after 1 , 2 , . . . , (n-1) flowers.
The optimal frequency distributions of
numbers of flowers probed are given for
each inflorescence size by the unconditional probabilities of a hummingbird
leaving after probing n flowers. These optimal frequency distributions and the observed distributions can now be compared.
1) The mean numbers of flowers probed
under the optimal departure rule (Tables
1 and 2) are both very close to the observed
relationship between numbers of flowers
probed and numbers of flowers available
(Fig. 1). 2) The optimal frequency distribution for 12-Howered inflorescences is
bimodal with one peak near 3 flowers
749
probed and a second peak near £> flowers
probed (Table 2). The optimal frequency
distribution for 8-flowered inflorescences
has a single peak at 3 flowers probed
(Table 1). Hence the optimal frequency
distribution for inflorescences with less
than 19 flowers available should have
distinct peaks near 3 and 6. This is the
case for the observed frequency distribution for inflorescences with less than 19
flowers available (Fig. 2). 3) The observed
frequency distribution for inflorescences
with more than 18 flowers appears to
have distinct peaks near 3 and 7 flowers
probed as might be expected from the
predictions for 12-flowered inflorescences.
In addition, both observed frequency distributions suggest a third peak near 13
flowers probed. However, since no re visitation data are available for inflorescences
with more than 12 flowers, it is not possible
to calculate the optimal frequency distributions for these larger inflorescences.
No comparisons can, therefore, be made
between observed and predicted in these
cases.
In summary, the present analysis shows
a good agreement between expected and
observed frequency distributions of numbers of flowers probed both in terms of
mean numbers probed and the shapes of
the frequency distributions. It is, however,
possible that the hummingbirds are
employing more information about nectar
than just the amount obtained at the present flower. This possibility will now be
discussed.
It is not presently possible to determine
if hummingbirds employ more information than the amount of nectar obtained at
the last flower or if they could benefit by
utilizing more information. There is, however, some suggestion that neither of these
possibilities is fulfilled. First, the success of
the departure rules in predicting the mean
number of flowers visited and the shapes
of the distributions suggests the hummingbirds may not use departure rules
which employ either more or different
information. Second, even an optimal
combination ;>f more information seems
likely to result in a lower rate of energy
gain than the optimal departure rule based
I Mill 2. Calculation of optimal departure rule and optimal frequency distribution of numbers oj flowers probed for injlore\renres with 12 jloxi'ers available.
Number of flowers probed (n)
1
Probability that next flower
is a revisit given no previous
revisits
0.000
Adjusted expectation
threshold (^1)
3.19
Departure threshold (wl)
-0.00
Probability that present flower
is below departure
threshold given that
immediately previous
flower was above its
departure threshold and
{riven that no revisitation
has occurred so far
0.000
Probability of leaving after n
Mowers given that n Mowers
have been probed
0.000
Unconditional probability
of leaving after n flowers
(i.e., optimal frequency
distribution of numbers of
Mowers probed per
inflorescence)
0.000
Mean number of flowers
probed per inflorescence
5.10
2
3
4
5
6
7
8
9
10
II
0.046
0.050
0.062
0.088
0.126
0.175
0.250
0.375
0.580
0.900
3.34
0.25
3.36
0.43
3.40
0.66
3.50
1.22
3.65
15
3.87
2.83
4.25
4.75
5.10
7.60
31.90
12
1.0(
oo
00
c
JO
>
0.108
0.140
0.161
0.205
2.295
0.410
0.639
1.000
0.108
0.180
0.203
0.254
0.357
0.484
0.702
1.000
0.108
0.161
0.149
0.148
0.155
0.135
0.101
0.043
y
7-.
0.000
OPTIMAL FORAGING
on the above assumptions about information use. If the hummingbirds were to use
knowledge of nectar volumes obtained
from the last two flowers, then their departure rule should be a function of the
arithmetic mean of these two nectar volumes in flowers on the same inflorescence
and, hence, each of the two nectar volumes
should be weighted equally.
If the expected nectar in the next flower
is plotted against the mean nectar obtained
from the last two flowers (cf. Fig. 3) a
relationship is obtained which is similar to
that in Figure 3. When plotted on the same
axes this relationship is identical to the
previous one except at the upper and
lower extremes of the abscissa. The new
relationship starts lower and finishes
higher and is linear throughout. The mean
nectar from the last two flowers, therefore,
provides somewhat more information
about the nectar in the next flower than
does the nectar from only the last flower.
Hence, the hummingbirds could achieve a
slightly better prediction of the expected
nectar at the next flower by using the mean
of the nectar volumes obtained from the
last two flowers. However, such a departure rule would sometimes allow foraging
to continue on an inflorescence after a
revisit had occurred. Further revisitation
would probably occur with high probability and result in decreased rate of nectar
yield. It seems likely that the increase in
revisitation would override any benefit in
terms of information used from nectar
volumes obtained from the last two or
more flowers. More data are required before this can be verified.
751
used and the optimal combination of this
information or to make some realistic assumptions about the amount of information used and then determine the optimal
combination with assumptions as constraints. In the present study the optimal
departure rule generates predictions that
lie close to the observed based on the
assumptions that the hummingbirds employ
only information about the number of
flowers already probed, the number available, and the amount of nectar obtained at
the present flower. It may be likely that this
information is the optimum amount, but
this requires verification.
Any consideration of the use of information by an animal is also a consideration
of the animal's perception and memory
capabilities. The present study suggests that
hummingbirds may at least perceive and
remember the number of flowers available
on an inflorescence, the number of flowers
already probed on the inflorescence and
the amount of nectar obtained at the present flower.
This study provides a basis for considering other aspects of hummingbird
foraging. The hummingbirds were clearly
selective in terms of the numbers of
flowers on the inflorescences they visited
and virtually ignored all inflorescences
with fewer than 7 flowers although these
made up about 25% of those available.
This seems to be reasonable since large
inflorescences provide more energy per
unit time than small inflorescences since
the inter-inflorescence flight time is greater than the flight time between flowers
within an inflorescence. To test any quantitative optimal foraging predictions concerning this selectivity will require calCONCLUSION
culating the caloric yield and the time
The present study supports the stochas- taken for inflorescences of all sizes, intic, discrete form of Charnov's (1976) mar- cluding those ignored by the humginal value theorem. It was necessary, mingbirds. Calculations for the infloreshowever, to transform the theorem into a cence sizes ignored by the hummingbirds
hypothesis about the departure rule. Opti- requires also predicting how many flowers
mal foraging hypotheses may have to be the hummingbirds luould visit and how
expressed in terms of optimal decision rules much nectar they would obtain on the average from each flower on the inflorescences.
before they can be tested.
A complication in determining an opti- The marginal value theorem and the remal decision rule is either to determine sults of the present study can be used to
both the optimal amount of information make these calculations and to generate a
752
GRAHAM H. PYKE
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mingbirds. Amer. Nat. 106:589-596.
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