Vol. 127, No.4 The American Naturalist April 1986
PREFERENCE AND PROFITABILITY: THEORY AND EXPERIMENT
D. W. STEPHENS,* J. F. LYNCH, A. E. SORENSEN,* and C. GORDON*
Smithsonian Environmental Research Center, P.O. Box 28, Edgewater, Maryland 21037
Subnlitted November 8, 1984; Revised May 17, 1985; Accepted August 22, 1985
Foraging theory (Krebs et al. 1983; Stephens and Krebs 1986) assumes a simple
hierarchy of feeding decisions. It supposes that foragers first choose among
habitats, then choose among patches within habitats, and finally choose among
prey items within patches. These three decisions have usually been studied
independently. In this paper we begin to integrate these problems by considering
diet choice within patches both experimentally and theoretically. Our theoretical
arguments follow from combining well-known ideas about diet choice (Pulliam
1974; Charnov 1976a) and patch use (Charnov 1976b; Orians and Pearson 1979).
We consider patches that are nothing more than clumps of discrete prey, and we
ask which prey should be attacked and in what order? Despite the simplicity of
this approach, there is some confusion in the literature about how foragers should
choose prey within patches.
Waddington and Holden (1979; see also Waddington 1982) pointed out that
conventional diet theory (Pulliam 1974; Charnov 1976a) assumes that the forager
encounters prey items one after the other (sequential encounter). Waddington and
Holden argued that many foragers experience simultaneous prey encounters (e.g.,
a visual hunter may see many items at once) and that this fact will change the diet
theory's predictions. Waddington and Holden presented a rule for choosing
among simultaneously encountered items. Their rule characterizes a prey item by
the quotient of its net energy value (e) divided by the amount of time (11) required
to pursue and handle it. We use' 'pursuit" to mean the same thing as Waddington
and Holden's term "travel"; like Waddington and Holden, we mean the time from
encounter until the start of actual handling. This quotient (e/h) is the familiar
"profitability of optimal-diet theory (Pulliam 1974; Charnov 1976a). Throughout
this paper we use the term "profitability" only in this restricted sense. Wadding?
ton and Holden argued that the "best" a rate-maximizing forager can do when
simultaneously encountering items is to attack the item with the highest profit?
ability (see fig. lA). We call this the take-the-most-profitable rule. The take-the?
most-profitable rule claims that the relative profitabilities of the items encountered
* Present addresses: D.W.S., A.E.S., Department of Biology, University of Utah, Salt Lake City,
Utah 84112~ C.G., Department of Fisheries and Wildlife, Colorado State University, Fort Collins,
Colorado 80521.
Am. Nat. 1986. Vol. 127, pp. 533-553.
? 1986 by The University of Chicago. 0003-0147/86/2704-0008$02.00. All rights reserved.
534 THE AMERICAN NATURALIST
A
-Time Between Encounters
B
c
'0
0
~
0>Q)
C
L.U
Item 2 Item 1
1/A2 1/Ac 1/Al
- Time Between Encounters
Handling Time-
Handling Time-
FIG. I.-The figure illustrates two models of prey choice when two types of prey are
encountered simultaneously. Both models characterize prey types by net energy gain (ej for
the ith type) and handling time (hJ. A, The take-the-most-profitable rule asserts that the prey
type with the highest quotient e/h j (profitability) should be chosen. The slope of the line
segment drawn from the origin to the point (h j , ej) equals the profitability (e/hJ. A forager
following the take-the-most-profitable rule would always choose item 1 in the case illustrated
here. B, The discrete-marginal-value theorem provides a technique for choosing the type that
maximizes the long-term rate of energy intake, which specifically includes the searching time
between encounters with group~. The absolute value of searching time is plotted to the left of
the origin in the figure. A line drawn through the points representing both alternatives gives
the critical search time l!~c (see the text): for shorter search times (e.g., lI~d the more
profitable type 1 would be preferred by a long-term rate maximizer~ for longer search times
(e.g., 1/~2) the less profitable, but larger type 2 would be preferred.
are sufficient to predict preference (Waddington and Holden 1979). Specifically,
neither the absolute values of the encountered items nor the values of the for?
ager's options elsewhere influence preference. However, we argue that long-term
rate maximizing does not predict the take-the-most-profitable rule, although it is a
special case of the correct result. We also present experimental evidence that
contradicts the take-the-most-profitable rule.
We present the predictions of the theory of long-term rate-maximizing foraging
PREFERENCE AND PROFITABILITY 535
(1)
when simultaneous prey encounters occur. These predictions are important be?
cause they integrate patch-use and prey-choice decisions and because they apply
to dichotomous preference tests. Dichotomous preference tests are probably the
most common type of experiment in the literature of feeding behavior (e.g., Kear
1962; Rachlin and Green 1972; Sorensen 1983). The problem of simultaneous
encounter is also important as a way to test which rate (long-term or per-decision)
foragers maximize. The predictions of foraging models can differ, if different rates
are maximized. This has recently been noted in both the ecological (Templeton
and Lawlor 1981; Gilliam et al. 1982; Turelli et al. 1982) and the psychological
(Staddon 1983; Kagel et al. 1986) literature.
THEORY
The development of the theory proceeds in three steps. First, we use a simple
model to show how the take-the-most-profitable rule may fail. Second, we gener?
alize this model. Finally, we explain how profitability can partially determine the
long-term rate-maximizing diet even if the most profitable item is not always
preferred. To do this we show how diet theory predicts which patches should be
entered and which should be ignored.
A Flock-Encounter Model
A simple model shows how the take-the-most-profitable rule sometimes fails to
maximize the long-term rate of energy gain. Suppose that a bird of prey encoun?
ters flocks of small birds at rate l\., so that it searches for an average of Ill\. s to find
a flock. Also suppose that the flocks consist of two prey types. Type 1 yields el net
cal and takes hI s to pursue and handle (handling includes the time from encounter
with the flock until searching resumes), and type 2 yields e2 cal in h2 s. Assume
that type 1 is more profitable than type 2, that is, ellh l > e21h2. Because an attack
flushes the entire flock, only one prey item can be attacked at each flock en?
counter; prey choices are thus mutually exclusive. If r is the proportion of type 1's
attacked upon encounter with flocks, then the long-term average rate of energy
intake is
R == re I + (1 - r)e2 .
l\. - I + rh I + (1 - r)h2
It can be shown that the rate-maximizing r must be either 0 or 1 (a result analogous
to the zero-one rule of diet theory; see Charnov 1976a). Thus, the take-the-most?
profitable rule can be preserved only if el/(l\. - I + hI) > e2/(l\. - I + h2) whenever
el/h l > e21h2. A little algebra shows that this is not always true. If e2 > el (even
though el/h l > e2Ih2), then the take-the-most-profitable rule disagrees with long?
term rate maximizing when the search time between encounters with flocks is
long. A larger, but less profitable item should be preferred if
(2)
where l\.c is the critical encounter rate. At average search times longer than I/l\.c'
the larger, less profitable item should be attacked; and at average search times
536 THE AMERICAN NATURALIST
shorter than 1/~c, the most profitable item should be attacked (fig. IB). Two
general rules can be found using expression (2): an item both less energetically
rewarding and less profitable should never be preferred; and if two alternatives are
equally profitable, the larger should always be preferred.
This result makes two further points about models of simultaneous encounter.
First, both simultaneous and sequential encounters occur in this model; the bird of
prey encounters flocks (sets of prey items) sequentially, but it encounters birds
simultaneously. The restrictive assumption in conventional diet theory is that the
encountered sets contain just one item, not sequential encounter. Simultaneous
encounter and sequential encounter are not theoretical alternatives. They can and
probably usually do occur within the same foraging bout. Second, the take-the?
most-profitable rule can be the rate-maximizing rule if encounter rates are high or
if the less profitable choice also provides less energy.
The Marginal- Value Theorem for Discrete Choices
This model treats thefiock, a set of simultaneously encountered prey items, as if
it were a patch. The result is like the marginal-value theorem (see fig. IB; Charnov
1976b), except that the patch can be treated in only a limited number of ways:
attack type 1, or attack type 2. Although in the original marginal-value theorem
the forager can choose any patch-residence time in the continuous range of zero to
infinity, we call our model the "discrete-marginal-value" theorem to distinguish it
from the take-the-most-profitable rule. The requirement that e2 > el and el/h l >
e2/h2 is equivalent to negative acceleration in the continuous-gain function of
Charnov's (1976b) continuous-marginal-value theorem. Similar results in the con?
text of central-place foraging were proposed by Orians and Pearson (1979) and
Lessells and Stephens (1983). Engen and Stenseth (1984) dealt with discontinuities
in within-patch gains in a different but apparently equivalent way, but Real (1983)
presented a strikingly different view of simultaneous encounters.
A version of the marginal-value theorem for discrete choices can be" proved (see
the Appendix for details). Imagine that there are m mutually exclusive prey types
in the flock. Because of the results above, we need consider only the subset of
k prey types such that el/h l > e2/h2 > ... > ek-I/hk- I > ek/hk and ek > ek-I >
... > e2 > el (k :S m). Items are ranked by profitability (as in the conventional
theory of diets), but items both less profitable and smaller than the alternatives are
excluded from the ranking because they can never be the rate-maximizing choice.
If R max is the optimal rate of energy gain in the habitat, then the following
relationship must hold if the ith item is the rate-maximizing choice:
(ei - ei-I)/(hi - hi-I) > R max > (ei+ I - ei)/(hi+I - hi).
The quotients that bracket R max are the slopes of the two line segments above and
below (hi, ei). The Appendix proves this result.
Deciding to Attack Patches
Taking the most profitable item does not necessarily maximize the long-term
average rate of energy intake (even though profitability is the rate of gain mea?
sured from the time of encounter until searching resumes), since it ignores the
PREFERENCE AND PROFITABILITY 537
search time between encounters. This leads to the apparent paradox that a long?
term rate maximizer does not prefer larger profitabilities even though the model
for long-term rate-maximizing diet choice ranks items by profitability. The
paradox is resolved by realizing that conventional diet theory views prey items
unrealistically (Stephens and Krebs 1986), by assuming that all prey items oftypej
(for example) yield ej cal and take ~j s to process and that the forager has no
control over how long to handle the item or how much energy to extract. In
contrast, in patch-use theory patches are items that a forager may use in various
ways to yield various amounts of energy. The diet model tells us whether a rate
maximizer should attack an item, and the patch model (marginal-value theorem
and its successors) tells us how a rate maximizer should use the item given that it
is attacked.
The flock-encounter model presented above can be modified to illustrate the
problem. Suppose that th.ere are two types of flocks, A and B. The forager
encounters type A flocks at rate A.A' and they are composed of two prey types,
Al and A2. These prey types have characteristics eAt, hAl, eA2, hA2. Assume that
eAl/hAl > eA;/hA2 and eA2 > eAl' Type B flocks contain only one prey type, eB/hB >
eAl/hAl. The forager is faced with two problems: which flocks should be attacked,
and if type A flocks are attacked, which prey type should be attacked. Type B
flocks should always be attacked because they contain the unambiguously most
profitable prey type. This follows from the "ranking by profitability" result of
conventional diet models.
However, when should flocks of type A be attacked? Type A flocks should be
attacked if excluding them yields an energy intake rate less than the maximum
profitability of A flocks (eAl/hAl ). Ranking by maximum profitabilities is suggested
because di~t theory (e.g., Charnov I976a) shows that including an item whose
profitability is greater than the present rate of intake will increase the rate.
Therefore, if eAl/hAl (== max[eAl/hAl , eA2/hA2]) is greater than the rate of intake
attained by attacking only B flocks, then type A flocks should be attacked, since
taking species Al will increase the forager's long-term rate of energy intake. It
does not follow from this argument that taking species Al will increase the rate of
intake more than taking species A2 will. We have shown above that there are
conditions under which a long-term rate maximizer should prefer the larger but
,.. less profitable item. The suggested generalization (see the Appendix for the proof)
is that the decision to attack can be made precisely as predicted by conventional
diet theory except that flocks (sets of simultaneously encountered items) are
ranked by their maximum profitabilities. The maximum profitability of a set of
simultaneously encountered prey items partially determines whether the set is
attacked at all; at the same time, it is not necessarily the best tactic (according to
long-term rate maximizing) to attack the most profitable item within a patch.
Three cases roughly based on the habitat rate of intake can be specified (see the
Appendix):
1. At high intake rates ignore type A flocks and attack only type B flocks (R max
> eAl/hAl)'
2. At intermediate intake rates attack both A and B flocks, but take only the
more profitable type Al prey from the A flocks (R max < eAl/hAl)'
3. At low intake rates attack both A and B flocks, but take only the less
538 THE AMERICAN NATURALIST
profitable type A2 from the A flocks (R max < eA2/hA2 < eAI/hAl; this is a necessary
but not sufficient condition for attacking A2's instead of AI's).
SUlnmalY of Theory
Our theoretical analysis makes three main points. First, simultaneous and
sequential encounters are not mutually exclusive alternatives; for example, for?
agers that encounter pairs of prey items will encounter these pairs sequentially.
Second, the take-the-most-profitable rule does not maximize the long-term rate of
energy intake because it ignores the time between encounters with sets of items;
generally, it ignores the value of the forager's alternatives elsewhere. Third,
although maximizing profitability is not the long-term rate-maximizing tactic, the
forager's decision to attack or ignore a given set of items depends on the set's
maximum profitability.
An Empirical Test of Rate
Although the take-the-most-profitable rule is not consistent with long-term rate
maximizing, it is a plausible alternative hypothesis. Foraging theory uses long?
term rate maximizing partly because most foraging decisions are repeated many
times and partly because it is a mathematically convenient assumptio~ (for some
recent controversy on this point, see Templeton and Lawlor 1981; Gilliam et al.
1982; Stephens and Charnov 1982; Turelli et al. 1982). However, psychologists
have proposed a model of preference that is nearly identical to the take-the-most?
profitable rule, which they call "momentary maximizing" (see Staddon 1983). The
momentary-maximizing take-the-most-profitable rule might make economic sense
in at least two ways. First, some authors (Staddon 1983; Kagel et al. 1986)
proposed that an immediate reward is fundamentally more valuable than a delayed
reward. This might make the take-the-most-profitable rule (or policies like it) the
optimal behavior. Second, Staddon suggested that momentary maximization
might be a cheap and effective rule of thumb for approximating long-term rate
maximization, claiming that it is a "general principle of enormous simplicity"
(1983, p. 233). In the remainder of this paper, we ask an experimental question:
are a forager's choices influenced only by immediate, relative payoffs, as the take?
the-most-profitable rule predicts, or do the values of the forager's options else?
where affect preference, as the discrete-marginal-value theorem predicts? We use
dichotomous preference experiments, together with the theory developed above,
to distinguish between momentary rate maximizing (the take-the-most-profitable
rule) and long-term average-rate maximizing (the discrete-marginal-value theorem).
THE EXPERIMENT
The model presented above considers only the case in which the forager
chooses mutually exclusive prey items from a set of simultaneously encountered
items. The alternatives analyzed must be mutually exclusive choices, but the
alternatives do not have to be prey items. Imagine that a foraging honeybee
encounters a pair of flowers (fig. 2). Because flowers do not run away, the close
and the distant flowers can both be taken; flowers are not mutually exclusive
PREFERENCE AND PROFITABILITY
Distant
r;;\ More
\:.J Profitable
Flower
Close
r;;\ Less~ Profitable
Flower
Encounter Point
Direction of Travel
539
FIG. 2.-Every 2.44 m within the experimental apparatus the foraging honeybee encoun?
tered a patch as illustrated here. The patch was entered through a small hole, the encounter
point. Behind this hole were two artificial flowers: a close but unprofitable flower and a
distant but profitable flower.
choices. However, there are only a limited number of ways that this pair of
flowers can be used: (1) attack only the close flower; (2) attack only the distant
flower; (3) attack both flowers, the close one first; and (4) attack both, the distant
one first. If at least one flower is taken and if there is no revisitation, then these
four tactics are exhaustive and mutually exclusive. The results from the "flock"
model presented earlier can be directly applied to these four options.
To distinguish between the take-the-most-profitable (momentary-rate-maxi?
mizing) rule and the discrete-marginal-value theorem (long-term rate maximizing),
we studied the choice problem illustrated in figure 2 using foraging honeybees; in
'our experiments we arranged a pair of artificial flowers such that a close flower
was less profitable and energetically less rewarding than a distant flower. The
behavior predicted by the take-the-most-profitable rule is to pass by the close
flower (since it is always the less profitable) and to attack the distant flower. In
contrast, the discrete-marginal-value theorem predicts that "attacking close then
distant" should occur at low habitat rates of energy intake and that "attacking
only distant" should occur at high habitat rates of energy intake. (According to
the discrete-marginal-value theorem, we may ignore the tactics "take close only"
and "take distant then close" because these options would be less rewarding and
less profitable than the other alternatives.)
The two models formulate this choice problem differently: the take-the-most?
profitable rule considers only which flower should be visited first (Waddington and
540 THE AMERICAN NATURALIST
Holden 1979); the discrete-marginal-value theorem considers whether one or both
flowers should be visited. Despite this difference the two models make contrasting
and unambiguous predictions about which flower should be visited first, although
the discrete-marginal-value theorem predicts this indirectly. We can therefore use
the first flower visited to distinguish between the models.
Materials and Methods
The experiment was conducted outdoors on a large lawn at the Smithsonian
Environmental Research Center, near Edgewater, Maryland, from June 1 to
August 17, 1983. The observations were made in the morning, from about 0700 to
1200 h.
The subjects.-The subjects were 12 worker honeybees (Apis mellifera ligus?
tica). The bees were individually marked with spots of paint (Frisch 1967) and
came from two different hives located about 0.5 m apart. We could not tell which
bees were from which hive, although observations of marked individuals showed
that we used bees from each hive.
The alleyway.-We trained the bees to feed in a long alleyway located 16.5 m
from the hives. The alleyway was a plywood trough 11 m long, about 61 cm wide
and 30.5 cm deep, painted white. Fine nylon mesh covered the alleyway to
exclude unwanted bees. When a trained experimental bee arrived at the entrance
to the alleyway, the experimenter could easily admit the bee to the alleyway by
lifting a section of the mesh. Experimental bees always entered the alleyway at the
same end and flew along it until they reached the opposite end or until they were
ready to return to the hive. When a bee flew up to the mesh (anywhere along the
alleyway) we lifted the mesh and released it.
The patches.-Four partitions blocked the alleyway at intervals of 2.44 m.
Each partition contained a rectangular passage, 2.5 cm wide by 3.2 cm tall. The
bottom lips of the passages were painted red to contrast with the white partition.
These restrictive passages controlled the bee's encounters with pair.s of flowers,
which were located immediately behind each partition. We directly observed over
a thousand similar flower choices by controlling encounters in this way. We refer
to the area containing the pair of flowers immediately behind each partition as a
patch. The distance from the second flower to the next patch entrance represented
a between-patch distance.
Theflowers.-We trained the bees to extract "nectar" (a solution of honey and
water) from artificial flowers. The flowers were wooden disks (6 cm in diameter)
painted in three different colors (blue, yellow, and red) and supported on inverted
plastic cups 12 cm high. Since honeybees cannot distinguish red from gray of
equivalent intensity, we used red flowers only in the training phase of the experi?
ment. We drilled a shallow well (7 mm in diameter and about 1 mm deep) in the
center of each flower to hold the nectar. The flowers were coated with transparent
plastic coating to prev,ent the wood from absorbing the honey solution. We
dispensed nectar using a Drummond dispenser (catalogue no. 105).
Preliminary experiments showed that we could not adjust the profitability
(energy gained divided by time to exploit) of flowers sufficiently by varying the
nectar volume. To overcome this, we placed 9-mm-thick doughnut-shaped disks
PREFERENCE AND PROFITABILITY 541
over some of the flowers. These disks resembled the flowers in every way except
that they had a hole (7 mm in diameter) drilled completely through their centers.
The bees learned to crawl through this hole to reach the nectar. These "dough?
nuts" markedly increased the time taken to extract nectar. We call these two
flower types doughnut flowers and open-face flowers, respectively.
The training.-We studied individual bees in sequence. We chose an individual
haphazardly from the marked bees visiting a plate of honey solution placed near
the alleyway. We then trained this bee for 2-3 days to enter the alleyway, to use
the small passages to enter patches (droplets of honey solution were sometimes
placed on the patch thresholds), and to fly between patches to the next patch. Red
flowers were used as required to shape these behaviors. Since the inside of the
alleyway was painted white, these red flowers contrasted with the background.
We trained each individual until it readily entered the alleyway and patches and
consistently drank from both open-face and doughnut flowers (which contained
equal amounts of nectar during training).
The choice .problem.-Each bee encountered many identical pairs of flowers
(fig. 2). All experiments offered a distant open-face flower, 16 cm from the patch
entrance hole and offset to the left, with 3 J.11 of honey solution containing 66%
sucrose equivalents. All experiments also offered a close doughnut flower, 7.5 cm
from the patch entrance hole and offset to the right, with 1 J.11 of either 66%, 33%,
or 16.5% sucrose equivalents (see "Treatments," below). Both flowers were close
enough to the patch entrance that honeybees could easily see them (see Frisch
1967). The distant flower was always a different color from the close flower. These
choices were contrived so that the close I-J.11 doughnut flower was always less
profitable than the distant open-face 3-fJ-I flower. The minimal expectation of the
take-the-most-profitable rule was that the more profitable (distant) flower should
be chosen over the less profitable (close) flower in most encounters (see Stephens
1985). The expectation of the discrete-marginal-value model is that the less
profitable close flower ought to be taken when the marginal value (~e/~h) of doing
so is greater than the rate of intake achieved by ignoring it.
Treatments.-To test the effects of both habitat richness and the relative values
of the members of the flower pair, we manipulated both habitat richness and
flower quality.
,." 1. Habitat-richness manipulations. In rich-habitat manipulations, we placed
five additional high-quality flowers (the same as the distant flower) in the alleyway
between all patches. In poor-habitat manipulations, there were no additional
flowers between patches.
2. Flower-quality manipulations. The distant flower always contained 3 J.1166%
sucrose equivalents (honey solution). We manipulated the quality of the close
unprofitable flower. The close flower always contained 1 J.11 of liquid, but the
dilution of this varied. There were full (66%), half (33%), and quarter (16.5%)
dilution treatments. Each experimental bee experienced only one dilution treat?
ment (four bees in each treatment), but all bees experienced both rich and poor
habitats.
Randomized variables.-Some researchers have suggested that bees may be
biased in favor of flowers of different colors (for review, see Kevan 1983). To
542 THE AMERICAN NATURALIST
control for this possibility we exposed an equal number of bees to "yellow?
distant/blue-close" and to "blue-distant/yellow-close" flower pairs. We assigned
colors at random. Because a bee's experience with the apparatus might affect its
behavior (e.g., by learning; see Laverty 1980), we also randomized the order of
testing in rich and poor habitats. The resulting experimental design has four
"order-of-habitat-richness-flower-color" permutations. Within each dilution
treatment we observed four bees (one assigned to each permutation). Statistically,
individual bees were nested within dilution treatments, but crossed with respect to
habitat richness.
Measurement of parameters.-Both models require a knowledge of many as?
pects of bee behavior, for example, rate of travel, handling times of flowers;"'
pursuit times from the encounter hole to the distant flower. The flight time to the
distant flower (for example) can only be measured if the bee does not stop at the
close flower. To measure this and similar values we performed forced-choice
trials, in which we offered the bee only one type of flower (close or distant). Ten
trials per flower type were performed both before and after the free-choice
observation trials to check for training effects. We chose the order of forced?
choice trials at random. We observed no training (before vs. after) effects. Some
other parameters could not be estimated from the forced-choice trials (e.g., flight
time between flowers); for these we used the first 10 free-choice trials for esti?
mates. These trials were excluded from our analysis of choice behavior. We
recorded approximately 30 free choices for each bee in each treatment. We
converted time and sucrose-equivalent data to energetic costs and benefits
(calories) where appropriate, using Heinrich's (1983) conversion factors.
Summary of treatment.-All bees were treated in the same way. Each bee (1)
was assigned to dilution, habitat-richness, and flower-color treatments; (2) per?
formed 10 forced-choice trials each for both close and distant flower types; (3)
performed 10 free-choice estimation trials; (4) performed 30 free-choice observa?
tion trials (choice data used in analyses); (5) performed 10 forced-ch'oice trials for
each of both flower types; (6) experienced a change in the habitat-richness treat?
ment and repeated the process starting at step 2.
Observed quantities.-Two observers participated in the experiment. One ex?
perimenter cleaned and filled the flowers just ahead of the bee's position in the
alleyway, and the second experimenter recorded all observations on a portable
tape recorder. We decoded the tapes later using a stopwatch. We recorded the
following information: (1) the return time between bouts in the alleyway (including
instances in which the bee returned to the hive to empty its honey stomach and
those in which the bee left the end of the alleyway and flew immediately to the
beginning to start again); (2) which flower (1 J.11 or 3 J.1l) the bee chose upon arrival
in a patch (the bees sometimes landed on the close I-J.11 flower without drinking
the nectar, so we recorded both the first flower visited and the first flower drunk);
(3) all flight times; (4) the time taken to extract nectar from all flowers.
RESULTS
Table 1 presents the estimated handling-time and energy-gain parameters.
These data confirm that the close I-J.11 flower was always less profitable than the
* No significant (5% level) differences between rich and poor treatments. Estimates of these
parameters are pooled appropriately.
t No significant (5% level) differences across dilution treatments. Estimates of these parameters are
pooled appropriately.
--distant 3-J.11 flower. Because the bees did not always drink from flowers that they
visited, there are four possible permutations of first flower choice: (1) visit close,
drink close; (2) visit close, drink distant; (3) visit distant, drink distant; (4) visit
distant, drink close. We never observed the fourth type of behavior, but we did
observe all three other flower choices. The statistical results are the same regard?
less of whether we count the ambiguous "visit close, drink distant" choices as
distant choices, close choices, or not counted. Therefore in the following we
report only analyses on the proportion of "visit distant, drink distant" choices,
which we call the proportion of first visits to the most profitable flower. Counting
visits in this way favors the take-the-most-profitable rule, because it counts fewer
visits to the less profitable flower. Figure 3 presents the proportions of first visits
to distant, more profitable flowers for each of the six manipulations.
The effect offlower quality.-Both'models predict (and intuition demands) that
preference for the close (unprofitable) flower should decrease or stay the same as
544 THE AMERICAN NATURALIST
1.0
'0
c
.~
o
Cl.
e
0..
+.,
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
Poor Habitat\
\
\
1-------- _
114 112
Nectar Concentration in Less Profitable Flower
FIG. 3.-The figure shows the experimental results using the design of fig. 2. The concen?
tration of the 1 IJJ of nectar in the close flower is shown on the abscissa. The proportion of
patch visits when the close flower was ignored and the distant flower was attacked is shown
on the ordinate. The solid line shows the results for the rich habitat~ the broken line, those for
the poor habitat. The difference between rich and poor habitats is predicted by the discrete?
marginal-value theorem, but not by the take-the-most-profitable rule.
its quality (honey concentration) is reduced. Our results confirm this prediction
(fig. 3). The differences between full and half dilutions are not significant (at the
5% level, which we use throughout), but the differences between these treatments
and the quarter-dilution treatments are significant.
Effect of rate: is relative value the only important variable?-The differences
between rich and poor treatments are significant in every possible comparison.
Moreover, they are in the direction that the discrete-marginal-value theorem
predicts. The close (unprofitable) flower is less preferred in high-rate "rich"
treatments. This is an important strike against the take-the-most-profitable rule as
a complete description of honeybee flower choice.
The quantitative effects of quality.-Both models predict threshold behavior:
preference (i.e., first choice) for the close flower below some threshold and
preference for the distant flower above that threshold. Under either model it is
unreasonable to expect 100% or 0% choice (so-called absolute preferences),
which could only happen if one could observe the threshold with no error or
variance (see Stephens 1985). Obviously, any acceptance-rejection threshold
must have some variance composed of between- and within-bee variation as well
as experimental error. We used probit analysis to estimate the mean and variance
of the observed thresholds.
PREFERENCE AND PROFITABILITY 545
The take-the-most-profitable rule predicts that the profitabilities of distant and
close flowers ought to be equal at the threshold point. Similarly, the discrete?
marginal-value theorem predicts that the habitat rate of energy intake (excluding
the close flower) and the marginal value of taking the close flower ought to be
equal at the threshold point.
Test of the take-the-most-profitabLe ruLe.-The difference in profitabilities at
the observed threshold is 0.43 calls, which lies between 0.33 and 0.53 with 95%
confidence. The predicted zero difference (equal profitabilities) lies well outside
the 95% confidence limits. The distant flower had to be (on the average) 5.5 times
more profitable than the close flower before the bees ignored the close flower 50%
of the time.
Test of the discrete-marginaL-vaLue theorem.-The difference between the mar?
ginal value of the close flower and the habitat rate of intake at the observed
threshold is 0.26 calls, which lies between -0.23 and 0.75 with 95% confidence.
The 95% confidence limits include the difference of zero predicted by the discrete?
marginal-value theorem. The bees took the close (unprofitable) flower in some?
what richer conditions than predicted, but this difference was not statistically
significant.
DISCUSSION
Honeybees do not simply take the most profitable flower available. The relative
values (measured in time and energy) of the flowers offered affect honeybee
flower choice, but flower choice is also affected by how well the bees can do
elsewhere in their habitat. OUf data support long-term average-rate maximization
over momentary maximization. The bees sacrificed immediate gain by taking the
close unprofitable flower when the habitat rate of energy gain was low, but they
passed up the close unprofitable flower when the habitat rate of energy gain was
high. Momentary maximizing (the take-the-most-profitable rule) failed most nota?
bly in underestimating the bees' preference for the close unprofitable flower.
Waddington and Holden (1979) studied flower-choice behavior in honeybees,
and they concluded that the take-the-most-profitable rule was well supported.
However, their results are not necessarily inconsistent with ours because they
gave their honeybees dense arrays of artificial flowers, and the discrete-marginal?
value theorem would probably also predict taking the most profitable flower in
these conditions. A thorough analysis of Waddington and Holden's (1979) data
would be complicated because they did not control the sets of flowers that their
experimental bees encountered. We suspect that some of the discrepancies be?
tween Waddington and Holden's data and the take-the-most-profitable rule occur
because the bees reject all visible flowers; the take-the-most-profitable rule cannot
explain these complete rejections. Waddington and Holden's data only support
the take-the-most-profitable rule indirectly, because the researchers did not di?
rectly observe the choices. Instead, they inferred support for their model by
comparing the observed frequency distribution of visits to various flower types
with frequencies predicted by their model. This is not strong evidence for the
take-the-most-profitable rule: many models of choice would predict similar fre-
546 THE AMERICAN NATURALIST
quency distributions since these predicted distributions necessarily incorporate
much information about the experimental flower densities.
Relative Value and Choice
The most significant finding of our study is that there are both theoretical and
empirical reasons to believe that the value of alternatives other than the choices
immediately available affect preferences. Honeybees act like efficient shoppers in
a market whose choices depend not only on what they see in that market, but also
on what other markets have for sale. This finding is important for ecologists who
use preference tests to draw conclusions about predator-prey relationships,
habitat choice, and community ecology, because the frequency and pattern of ...
presentation may affect preference tests. For example, an animal's preferences
among items of a set presented once every hour may differ from its preferences
among the same items presented once a minute. Preference is not absolute; it
depends on the nature and the frequency of alternatives and on the forager's
expectations of unseen alternatives.
Operant psychologists (e.g., Fantino and Abarca 1985) have concluded that the
immediate consequences of choice are more important than the long-term conse?
quences of choice in determining preference, although they concede that long?
term consequences sometimes have an effect. This conclusion may seem at odds
with our results, but the model we present predicts that changes in the long-term
rate of gain need not always affect preference. For example, if one of the forager's
two alternatives is both larger and more profitable, then it should be preferred
regardless of the net habitat rate of energy gain. This interpretation is different
from the view that immediate consequences are most important because it
identifies conditions under which immediate consequences (Le., profitability) are
enough to determine the best long-term foraging tactics. Viewed in this way ,
results showing the importance of immediate consequences are not necessarily
inconsistent with the general importance of long-term rate.
Patches and Diet
Our results show how the floral diet may be influenced by the patchiness of the
habitat. Our theoretical and empirical results show how a patch-use model can
determine prey (or flower) choice; diet choice is sometimes a patch-use problem
(cf. Engen and Stenseth 1984).
This confusion between patch and diet models has appeared in the literature at
least twice. In part, Waddington and Holden seem to have proposed the take-the?
most-profitable rule because of the prominence of profitability in the conventional
diet model. A more subtle confusion has arisen between Schoener's (1979) and
Orians and Pearson's (1979) central-place-foraging models. Schoener asserted
that both models "can be broadly applied to many of the same data" (1979, p.
912). However, these models are not alternatives; they can apply simultaneously
and in complementary fashion. Schoener's model deals with the effects of dis?
tance on profitability, and hence on the diet or "whether to attack at all" problem.
The Orians and Pearson single-prey-Ioader model deals with the effects of dis?
tance on patch use.
PREFERENCE AND PROFITABILITY 547
Schoener's model might be called an encounter-at-a-distance model. The cen?
tral idea is that a prey item encountered at 4 m away is less profitable than the
same item encountered at 4 cm. Schoener studies the effects of "encounter
distances" on profitability, whether an item that is attacked when it is close ought
to be attacked when it is farther away. In contrast, Orians and Pearson's single?
prey-loader model assumes that a forager is repeatedly exploiting the same item
(patch), and deals with the question of how this item should be exploited. The
flock-attacking bird of prey we discussed earlier faces Schoener's and Orians and
Pearson's problems simultaneously. Schoener's model tells us whether a rate?
maximizing raptor should attack a flock encountered half a kilometer away, but
Orians and Pearson's (and more specifically our model) tells us which members of
the flock should be taken, given that the flock is attacked.
Limitations of the Model
Some have criticized foraging theory for only treating patches that have discrete
and well-defined limits (e.g., Bond 1980). These critics argue that many natural
patches are little more than loose clumps of discrete prey items. Our results show
how basic theory can be applied to clumps of discrete prey, by a hierarchical use
of the prey-choice and patch-use models (for different views of similar problems,
see Heller 1980 and McNair 1979). The main limitation of our approach is that we
must assume that the whole clump is encountered (e .g., seen) simultaneously.
Suppose that a honeybee encountered a group of three flowers and chose to attack
the closest one. What should it do, if when flying toward this flower it encounters
a previously unseen flower? Neither our model, nor any other model, treats this
situation. These considerations suggest that lumping pursuit time with actual
handling time (as is usually done) may be unreasonable because pursuit requires
less commitment (i.e., less loss of opportunity) than does actual handling of prey.
Like other forms of the marginal-value theorem, our patch-use model says little
about the tactics an animal can use. Our technique cannot even be applied until all
the ways that the patch (clump, flock, etc.) can be used have been listed, and
plotted as (hi, eJ points. Our experimental situation was especially simple because
it was easy to make an exhaustive list of possible tactics. The marginal-value
theorem (including our discrete version and others) is not a decision rule, but a
technique for choosing among a set of possible decision rules. The view that the
marginal-value theorem is itself a decision rule is the source of much controversy
(Oaten 1977; Green 1980; McNamara 1982).
Momentary Maximizing, Self-Control, and Rules of Thumb
Animal psychologists have studied a broad class of problems under the general
heading of "self-control" experiments. In these experiments, animals (rats or
pigeons) using an operant device choose between keys. Associated with each key
is a food value (a period of access to food) and a delay. These quantities are
comparable to the e and h of our model, so that a profitability (e/h) can be
associated with each choice. Between each choice is a fixed inter-trial interval.
This experimental set-up is analogous to a two-species mixed-flock version of our
patch-use model. Psychologists have traditionally been interested in the effects of
548 THE AMERICAN NATURALIST
delay on choice (will animals delay gratification for a proportionately larger
reward?), and they have performed many manipulations of absolute and relative
delays (handling times) of the two choices. The basic trend observed in these
experiments is consistent with our empirical and theoretical results. Increasing
absolute delay (for both alternatives) increases the preference for the larger (more
energetically rewarding) alternative (Rachlin and Green 1972; Navarick and Fan?
tino 1976; Staddon 1983).
The results of our experiment, although consistent with the general trends
shown in self-control experiments, differed in many ways. We manipulated rela?
tive value rather than handling time (delay), and we explicitly manipulated the
richness of the habitat. We found that overall habitat richness (an ecological
analogue of inter-trial interval) affects bee choice behavior, as a long-term rate?
maximizing model would predict. Many (but not all) self-control results can be
explained by momentary maximizing because at fixed inter-trial intervals the
qualitative effects of changing delays (Le., handling times) are the same under
both momentary and long-term maximizing models. Our experiment should help
to illustrate the kind of natural situation to which self-control results from the
operant laboratory may be relevant. It is a significant and hopeful sign that
animals as different as pigeons and bees have similar behavioral patterns.
As we noted, many, but not all, self-control data can be interpreted according to
momentary maximizing (the take-the-most-profitable rule). Staddon (1983) con?
cluded that animals are momentary maximizers, but that they sometimes take
account of the long-term rate of gain. However, we show theoretically that taking
the most profitable is sometimes the long-term rate-maximizing behavior. Our
experiments show that when taking the most profitable item does not maximize
the long-term rate of energy gain, then foragers do not take the most profitable
item. We conclude, contrary to Staddon's view, that momentary maximizing
cannot be a complete description of behavior. At best it requires a supplementary
decision rule to decide between long-term rate maximizing and momentary max?
imizing, since animals do both.
Staddon was impressed by the general applicability and apparent simplicity of
momentary maximizing as a so-called "rule of thumb." Many other authors have
suggested that the mechanisms required to achieve long-term rate maximization
are too complicated or expensive for animals to implement (e.g., Janetos and Cole
1981; Myers 1983). In their collective haste to find simple and cheap rules, many
authors have ignored the possibility that the rules required to approximate long?
term rate maximization need not be complicated or expensive. For example, a
decision rule based on continually updated achieved rates associated with various
activities may be all that is required. A running tally might be simple and cheap to
operate and maintain. The chief problem is that the intuition of biologists about
which rules are simple and cheap has been unreliable. Animals can often perform
more-subtle tasks than an intuitive view of their simplicity allows (a prominent
example is the discrimination of prey by planktivorous fish simply on the basis of
apparent size, O'Brien et al. 1976; real fish can do better than hypothetically
limited ones, Gibson 1980).
PREFERENCE AND PROFITABILITY 549
Time Discounting
Although long-term rate maximization is generally well supported, there are
reasons to think that it does not completely account for food-choice patterns.
Consider the following thought experiment: a forager is offered two alternatives,
which may be characterized by (hi, eJ pairs, i == 1, 2 (handling times/delays and
energy gain). Assume that hI and h2are fixed. By making e2 sufficiently large there
will always be some point where it is preferred to the smaller el choice regardless
of the values of hI and h2 ? Now suppose that h2 is increased to an extraordinarily
large value (say, longer than the forager expects to live). According to long-term
rate maximizing we could still make e2 the preferred choice by increasing e2 to
infinity. Should a forager prefer to wait for something it has little hope of living to
obtain? The point is that long-term rate maximization is in danger of ignoring the
pattern of energy gain (see Caraco 1980; Real 1980; Stephens and Charnov 1982).
A more sophisticated (and realistic) view would discount the future food value e2
to reflect not .only the forager's mortality schedule, but also its short-term energy
requirements. It has been suggested that foragers ought to discount the future
(Staddon 1983; Kagel et ale 1986) because rewards now are fundamentally more
valuable than rewards in the future. If time discounting is sufficiently drastic,
momentary maximizing may be a sensible strategy. Our bee's tendency to take the
unprofitable flower in richer than predicted conditions may reflect a form of time
discounting.
SUMMARY
We attempt to integrate the related problems of patch choice and prey choice
within patches. Two models of prey choice within patches are discussed. One
based on momentary rate maximizing (the take-the-most-profitable rule) has been
presented previously. We develop an alternative long-term rate-maximizing rule,
the discrete-marginal-value theorem. We present an experimental test of these
two alternatives using honeybee flower-choice behavior. These empirical results
strongly favor the discrete-marginal-value theorem. The relative values of flowers
affect honeybee flower choice as both models predict, but the overall richness of
the habitat also affects flower choice as the discrete-marginal-value theorem
predicts. Our experimental result is consistent with existing data on the so-called
~~problem of self-control" in animal psychology. We consider the generality of
momentary rate maximizing versus long-term rate maximizing, and attempt to
clarify some of the literature's confusion about when the diet and patch-use
models apply.
ACKNOWLEDGMENTS
This work was completed when the senior author was a visiting scientist at the
Smithsonian Environmental Research Center, Edgewater, Maryland. The Smith?
sonian Institution provided equipment and supplies. A. E. Sorensen was sup-
550 THE AMERICAN NATURALIST
ported by a Smithsonian Postdoctoral Fellowship; C. Gordon was supported as a
Work/Learn Intern by the Smithsonian. We are grateful for the Smithsonian's
support. Much of the analysis was performed while the senior author was a North
Atlantic Treaty Organization postdoctoral fellow at the University of British
Columbia (Department of Zoology and Institute of Animal Resource Ecology).
We thank NATO and U .B.C. for their help. We are particularly indebted to C. L.
Gass of V.B.C. Helpful comments were made by T. Caraco, E. Charnov, L. Dill,
J. Krebs, L. Real, P. Schmid-Hempel, J. Staddon, K. Waddington, and R.
Ydenberg. D.W.S. thanks D. Stephens and B. Stephens for their kindness and
support.
APPENDIX
Although the mathematical assertions made in the text may be obvious to many readers
familiar with the subject, we include a brief mathematical appendix for thoroughness and
clarity.
DEFINITIONS AND REQUIRED RESULTS
Consider a case in which pairs of items are encountered and only one item may be
attacked. Call the items 1 and 2.
'A is the encounter rate with pairs.
h; is the time required to attack, eat, and leave the patch if type i is chosen.
e; is the net energy gain if i is chosen. It will always be possible to rename the two items
such that 11.2 ~ hI, that is, type 2 has a longer handling time than type 1.
de/dh is the slope of the line segment with endpoints (11.2, e2) and (h., el); that is,
(AI)
Suppose that when the pair is encountered the forager attacks type i. The long-term rate
of gain is
(A2a)
The terms in brackets represent the gains and time expenditures from encounters with
other patch types. However, because we are interested in a particular patch type, we
summarize these terms using nonnegative constants. Thus,
R; = (c; + 'Ae;)/(k; + 'Ah;). (A2b)
The subscript in c; and k; shows that the way other patches are treated may depend on
whether item I or item 2 is chosen from the patch type of interest. Specifically, we suppose
that c; and k; are chosen to maximize the long-term rate of gain (R;), given that type i items
are chosen from the patch type of interest. It follows that
(c; + 'Ae;)/(k; + 'Ah;) ~ (Cj + 'Ae;)/(kj + 'Ah;) ,
for any feasible (Cj, k) pair.
From elementary algebra,
(A3)
c,.fk; > e;lh; and then c,.fk; > (c; + 'Ae;)/(k; + 'Ah;). (A4)
In words, no item such that c;lk; > e;lh; can be in the long-term rate-maximizing diet,
because the alternatives, represented by c; and k;, give a higher rate of energy gain without
this item.
PREFERENCE AND PROFITABILITY 551
The following expressions are algebraically equivalent for any C ~ 0, k ~ 0, and h2 > hI.
deldh ~ (c + X.e2)/(k + X.h2)
deldh ~ (c + X.el)/(k + X.h d
(c + X.e2)/(k + X.h2) ~ (c + X.el)/(k + X.h l ).
PROOF OF THE DISCRETE-MARGINAL-VALUE THEOREM
(A5a)
(A5b)
(A5c)
Consider two arbitrarily chosen items 1 and 2. The discrete-marginal-value theorem
consists of the following three statements.
1. If deldh > R max , then item 2 is the long-term rate-maximizing choice, that is, R2 > R I.
Proof.-deldh > Rmax implies that deldh > (C2 + X.e2)/(k2 + X.h2) ~ (CI + X.e2)/(kl + X.h2).
Application of equation (A5) yields
R? = C2 + X.e2 ~ CI + X.e2 > CI + x.el R
I
;
k2 + X.h2 kl + X.h 2 kl + X.h l
that is, R 2 > R I by transitivity, as required.
2. If deldh <: R max then item 1 is the long-term rate-maximizing choice; that is, R I > R2.
Proof.-The proof is by contradiction. Assume that deldh < R max , but R2 > R I ? This
requires that deldh < R2. Applying (A4) and (A5) gives
R2 = C2 + X. e2 < C2 + x.el ::::; CI + x.el = R I ,k2 + X.h2 k2 + X.h l kl + X.h l
which implies that R2 < R I and this contradicts the stated assumptions.
3. If deldh = R max , then the long-term rate-maximizing forager is indifferent between
items 1 and 2; that is, R I = R 2 ?
Proof.-There are two cases.
Case 1. Assume that R2 ~ R I. This implies that deldh = R2, which in turn requires that
R2 = C2 + X. e2 = C2 + x.el ::::; CI + x.el = R I ;k2 + X.h2 k2 + X.h l k l + X.h l
that is, given that deldh = R max , if R2~ R I , then R I ~ R2. Both inequalities can hold only if
R I = R2 ?
Case 2. Similarly, if R I ~ R2 , then this implies that R I = R 2 , as required.
EXTENSION OF THE DISCRETE-MARGINAL-VALUE THEOREM TO THE MANY-PREY CASE
The discrete-marginal-value theorem can be extended to include an arbitrary number of
feasible (within-patch) prey by induction. The crucial assertion is that if prey item x is the
'long-term rate-maximizing choice among a set of In prey items, then an 1'1'1 + 1 item can be
the long-term rate-maximizing choice only if choosing the In + 1 item produces a larger
rate of intake than choosing x does. The addition of the In + 1 item to the feasible set
cannot make a previously suboptimal choice optimal.
Since we have shown that the discrete-marginal-value theorem applies for pairs, it
follows by induction that it is correct for an arbitrary number of prey in the feasible set.
Explicitly, it is obvious that if the discrete-marginal-value theorem holds for m prey, then it
holds for m + 1 prey, because this addition only requires a pairwise comparison.
PATCH CHOICE
In this section we present a symbolic argument to support the claim in the text that the
long-term rate-maximizing patch choice can be made according to the same rules as the
long-term rate-maximizing diet choice in the conventional prey model except that patches
must be ranked by their maximum profitabilities.
552 THE AMERICAN NATURALIST
We use symbols modified from Charnov (1976a).
C i is the vector describing the way the forager exploits patch type i. C i may be com?
plicated, possibly including information about the order of prey attack, methods of hunting,
and all conceivable aspects of within-patch tactics.
gi(Ci) is the gain function, specifying energy gain as a function of within-patch tactics Ci?
fi(C i) is the time spent in patches of type i given that the forager adopts tactics Ci.
We wish to show that the following rule is the long-term rate-maximizing patch-choice
rule.
Step 1. Rank the patches such that
max {gl(Cd} > max {g2(C2)} > ... > max {gk(Ck)}.
CI tl(Cd C2 t2(C2) Ck tk(Ck)
Step 2. Add patch types to the "patch diet" in the order of their rank until satisfying
max {[IA;g;(Ci)]/[1 + IA;tlC;)]} > max {gm+1 (Cm+d}. (A6)
CI,C2, ... ,CIIl i=1 i=1 CIIl+1 tm+1 (Cm+d
We call the right side of equation (A6) the maximum profitability of patch type m + 1.
The maximization of the left side must satisfy the discrete-marginal-value theorem.
Proof-It is a claim of conventional diet theory (see Pulliam 1974; Charnov 1976a) that
an In + 1 type should be ignored when encountered if and only if
(A7)
Our only problem is the interpretation of the right side of this expression. In the conven?
tional model this "profitability" is unique, since the forager is supposed to have no control
over e and h. If inequality (A7) is false for any choice ofCm+I, say C~n+ I, then the long-term
rate of energy intake can be increased by taking m + 1 type patches and using tactic C ~n + I.
Clearly, (A7) can be false for some C~l + 1 only if it is also false for the C m + J that
maximizes profitability.
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