w

OPTIMALITY, CONSTRAINTS, AND KIERARCKIES IN THE
ANALYSIS OF FORAGING STRATEGIES

'^' ;-■

V t '

BY
JEFFREY ROBERT LUCAS

A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL
OF THE UNIVERSITY OF FLORIDA IK
PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA
1985

^-

ACKNOWLEDGMENTS

'• This dissertation is the end result of the input and help from many

X people. Dr. James Dufty derived the models I used in the first chapter,

and Dr. Karl Taylor verified the math. Linda Griffin helped a great
deal with Chapter III, and weighed lots of fruit flies and fed lots of
antlions. Steven Frank, Alan Grafen, and Dr. Eric Charnov were
excellent sounding boards for some of my ideas on the theory of foraging
behavior; Steve and Alan were also instrumental in teaching me all I
know about statistical models. Dr. Franl: Mature provided a fellowship
that gave me time to finish the work on Chapter I and also provided easy
access to Sea Horse Key. Dr. Mike LaBarbera was a god-send in showing
me what biomechanics is, and also in explaining the ins and outs of
Stoke 's Law. Thanks to my committee, Drs. Jane Brockmann, Carmine
Lanciani, Brian McKab, Frank Nordlie, and Howard T. Odum, for their aid
and comments on a slightly non-traditional dissertation. The following
people read and commented on one or more chapters: Dr. Robert Jaeger,
Dr. Michael LaBarbera, Dr. A. Richard Palmer, Lynda Peterson, Dr. Nancy
Stamp, Dr. Lionel Stange, Dr. Nat Wheelwright, Steven Frank, Alan
Grafen, Dr. Lincoln Brower, Dr. Mark Denny, and several anonym.ous
reviewers. Most of my ideas about systems are derived from the work of
Dr. Odum, to wnom I am indebted. Special thanks go to my wife, Lynda
Peterson, for many different things. Finally, this work reflects an

il

pose

incredible effort on the part of my chairperson, Jane Brockmann, who
worked nearly as hard on this dissertation as I did, and whose ideas are
integrated in every chapter.

Ill

TABLE OF CONTENTS

PAGE

ACKNOWLEDGEMENTS ii

ABSTRACT vi

INTRODUCTION 1

CHAPTER I - THE BIOPHYSICS OF PIT CONSTRUCTION 5

Introduction 5

General Methods 6

Pit-Construction Behavior 7

Pit Morphology and Prey Behavior 9

Physical Components of Pit Construction 9

Behavioral Components of Pit Construction 23

General Discussion 35

CHAPTER II - MODELS OF PARTIAL PREY CONSUMPTION 38

Introduction 38

Proximate Models 39

Optimal Foraging Models 42

Capture Probability and Ambush Predation 60

Discussion 62

CHAPTER III - PARTIAL PREY CONSUMPTION BY ANTLION LARVAE. 68

Introduction 68

Digestion Rate Limitation (DRL) Model 69

Deterministic Optimality Model 72

Stochastic Optimality Model 95

General Discussion 107

CHAPTER IV - THE ROLE OF FORAGING TIME CONSTRAINTS

AND VARIABLE PREY ENCOUNTER IN OPTIMAL DIET CHOICE 110

Introduction 110

The Cost Model 112

Discussion lAO

Summary 145

IV

CHAPTER V - OPTIMALITY, HIERARCHIES AND FORAGING 148

Introduction 148

Optimality 149

Hierarchy 153

Maximum Power and Foraging Hierarchies 156

Non-Hierarchical Foraging Models 159

Hierarchy and Optimality Models 163

CONCLUSIONS 170

LITERATURE CITED 173

BIOGRAPHICAL SKETCH 182

Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

OPTIMALITY, CONSTRAINTS, AND HIERARCHIES IN THE
ANALYSIS OF FORAGING STRATEGIES

By

Jeffrey Robert Lucas
April, 1985

Chairperson: H. Jane Brockmann
Major Department: Zoology

Three phases of foraging behavior were analyzed: (l) the
preparation for prey (specifically trap construction by antlions), (2)
diet choice, and (3) consumption of prey. Optimal foraging models were
formulated for each phase. Results suggest that behavioral modeling
should be constructed in a hierarchical framework. (1) Antlion larvae
were shown to line their pitfall traps with fine sand, which
significantly increases capture ef ficiencj''. This fine-sand layer is
caused by physical properties of sand (angle of repose and sand
trajectory), and by three components of pit-construction behavior:
regulation of trajectory angle and initial velocity, and pre-sorting of
thrown sand. (2) Two general models were derived that predict diet
choice when foraging time is unconstrained and when external factors
constrain f oraging-bout length. For a two-prey system, the forager
should specialize when K >E^h ./(E . h .-E .h. ) , where i,j=high and

vi

.s^.^

low quality prey (respectively), E=energy, h=handling time. M_. . is

the number of i missed while handling j and is shown to correlate with

relative cost of eating i- When foraging- bout length is constrained

such that one prey is taken per bout, K. . no longer measures the cost

of eating i. Here the predator should specialize when

P(Y.<Z.<t)>E.P(Y.)/E. where P(Y . )=probability that i arrives in a

foraging bout and P(Y .<Z^ <t) = probability that j comes before i in time

t. This model predicts that as foraging time decreases, the diet should

expand. Published data on three predators (salamanders, intertidal

gastropods, and house wrens) support the prediction. (3) Partial prey

consumption by ambush predators was modeled. Assuming a

Michaelis-Menten uptake curve, at high densities optimal handling time

was found to be t ^=(-D t C-(g t C(D C+g -D t ))°'^,
oph pe p max p pe max pe p

where g and C are coefficients from the uptake curve, t is
max -^ p

pursuit time, and D is pursuit minus eating costs. At low prey

pe

0 5
densities, t is t =(CD -(D g C) )/(-D ), where D is
op opl we we max we we

waiting minus eating costs. At medium densities, t will always be

less than t ^ , and varies with prey encounter rate. Each model
opl -^ "^

generates different predictions. The low-density model was tested on

antlion larvae. I calculated t from empirical estimates. Antlions

op

extracted the optimal proportion of each prey, but handling time was
shorter than expected. Results suggest that antlion prey-handling
behavior is adapted to stochastic prey-arrival conditions.

i:^*

Vll

INTRODUCTION

The number of papers dealing with the use of optimality models has
grown rapidly since the study of optimal foraging was first introduced
in 1966 (MacArthur and Pianka 1966; Emlen 1966). Many studies in this
field have been strongly influenced by a few, elegant models. Two of
these models are the marginal value theorem (Charnov 1976a) and
variations of MacArthur and Pianka ' s (I966) optimal diet choice model
(Werner and Hall 1974; Schoener 1971; Charnov 1976b; Pulliam 1974;
Hughes 1979)- In some cases, the models have been modified by various
authors to fit specific situations, but the underlying assumptions were
not challenged.

At the onset of my research for this dissertation, I used these
elegant models as a tool in the study of the foraging behavior of
antlion larvae (a description of this behavior is found in Chapter l).
These models provided specific predictions about the foraging behavior
of antlions. In addition, like any type of model, these also provided a
specific intellectual framework within which questions could be
addressed. During the initial phase of my study, it became clear that
the predictions generated from these models were inappropriate for the
system I was examining. This meant that a number of other studies had
also used these models inappropriately. As a result, the focus of the
work changed from the application of existing models to an evaluation of
the models themselves.

The first four chapters of this dissertation are results of this
evaluation. The first chapter, which discusses the biophysics of pit
construction by antlion larvae, is an illustration of a system in which
the original predictions were upheld. The next three chapters discuss
systems in which the original predictions were rejected. Normally in
studies of optimality, the specific details of the model would be
altered to generate a new model that more closely fit the observed
responses of the organism (Maynard Smith 1978). However, I show that
the restrictive nature of the framework defined by the models may be the
primary cause for the lack of their predictive value in these and other
studies. For example, optimal diet choice models have assumed that
decisions about diet choice concern the addition or deletion of prey
types from the diet. The researcher's perception of how the animal
makes diet-choice decisions will be profoundly influenced by this
assumption. In chapter IV, I show that by focusing on a different
decision, and thereby changing our perception of the system, optimality
models become more realistic and also better predictors of foraging
behavior.

In Chapter V I discuss general features of optimality models, both
my own and those derived by other authors. I come to the conclusion
that generalities about foraging decisions should be derived from
studying the framework of the systems under which foraging behavior
evolves, and not from models which address single aspects of an animal's
behavior. Hierarchical design may be the simplest and best method to
use in studying this framework. The organization of the dissertation
follows the hierarchical framework.

.^

An optimality model, like other models, can be thought of as a
story, or a conceptualization of a particular system. The system can be
anything from diet choice to an ecosystem. The story has two major
parts: a list of factors that are important in the expression of the
trait that is being modeled, and the specific structure or relationship
among those factors. The variables in the model include some of the
factors, but other factors may be implicit in the structure of the
model, such as the underlying distribution of prey in a foraging model.
The mathematical model provides a structure within which specific
predictions can be made.

In testing the model, the condition of optimality is assumed. In
fact, optimality is more of a paradigm than an assumption. The test of
the model is simply a test of the list of factors and the structure of
those factors in the model. As Maynard Smith (1978) pointed out, much
of the confusion about optimality theory has arisen from a
misunderstanding of the exact features that are being examined. The
conclusion that the animal is not foraging optimally if it does not fit
the model is not appropriate for this type of research. If the model
fails, we conclude that the model is either incomplete or incorrect, not
that the animal is non-optimal (Maynard Smith 1978). Thus the model
helps us understand the nature of foraging adaptations and is not used
to determine if the animal is foraging optimally.

Optimality modeling is a technique that can be used to understand a
system. In this dissertation I use optimality theorj' as a tool for
examining the salient features of foraging behavior. If we were to
conclude that an animal was foraging non-optimally because it did not
fit an optimality model, we would be implying that we were om.niscient in

our perception of the system. Clearly this is not the case. If the
model failed to predict the response of an animal, we are always left
with two alternatives: (l) the animal is behaving non-optimally, or (2)
our perception was incorrect. The first alternative is counter
productive, since it yields no further predictions, nor does it provide
any further insight into the behavior of the animal. Since it is very
unlikely that our perception of animal behavior is perfect, then by
taking the second alternative we may learn more about the system than we
already know. This approach is no different than the standard
scientific method. Chapters I-IV all use this approach. Where the
models did not work, I re-evaluated the list of factors I thought might
be important. In each case, either the list proved to be incomplete or
the structure incorrect, and a new testable hypothesis was generated by
expanding the list or changing the model. Thus, the technique is useful
in the study of behavior, but only if optimality is implicity assumed.

CHAPTER I
THE BIOPHYSICS OF PIT CONSTRUCTION

Introduction

Several groups of animals build devices to capture prey. Among
these, spiders are perhaps the best known, but various other arthropods,
such as the larvae of antlions (Neuroptera) , wormlions (Diptera), some
caddisflies fTrichoptera) (Wheeler 1930; von Frisch 1974) and fungus
gnats (Diptera) (Eberhard 1980), also use some form of trap to capture
prey. The trap often represents a major investment, both in terms of
time and energy (Ford 1977; Prestwich 1977; Griffiths 1980), which is
expended before any return is realized. Thus, the predator incurs an
initial cost weighed against some later expected gain. Based on the
predictions of optimal foraging theory (Schoener 1971; Pyke et al.
1977), trap construction should reflect a maximum return rate from prey
per unit cost. Natural selection should act to minimize the cost of the
trap per prey item by optimizing trap design, by using available
materials to their greatest effectiveness, and by minimizing trap
construction costs.

Antlions construct conical- pitfalls in sand or loose soil. When
prey organisms fall into these pits, their escape is impeded by loose
sand on the high-sloped walls. The pit serves several functions: it
funnels prey to the antlion, increasing the "striking distance" over

which the predator can capture prey, and it increases prey escape time
and, therefore, the probability of capture. Pit efficiency depends
strongly on the physical properties of the materials with which it is
built. This study is an analysis of some of the physical principles
that govern the construction and use of the antlion pitfall trap as well
as the behavior of the antlion during pit construction.

All the experiments reported here used one species of antlion,
Myrmeleon crudelis, but I have observed three other species (M.
carolinus, K. mobilis, and M^ immaculatus) in which pit construction
appears to be virtually identical.

General Methods

All observations and experiments reported in this paper were
conducted in the laboratory (Gainesville, Florida) during spring and
summer, 1980. I collected antlion larvae at the University of Florida
Marine Laboratory', Sea Horse Key (Lev;^' County), Florida, and transported
them individually in sand to the laboratory. Here they were kept in
10x10 cm containers filled to about 6 cm depth with sand of mixed grain
sizes (O.l to 1 mm diameter). Each antlion was kept at 24° C for about
two weeks before the experiments were run and fed two dumpy-winged
Drosophila melanogaster per day. Live antlion larvae were weighed with
a Mettler analytical balance. Ants were collected on the University of
Florida campus and held no longer than one day before they were used in
the experiment. The light-colored sand used throughout the study was
obtained commercially as children's sterilized play sand. Lark-colored
sand was mixed with light-colored sand to illustrate the distribution of

sand grain size in the pit. This sand was also purchased commercially
and was sold as "terrarium sand." Sand grains of different sizes were
obtained by sifting through a series of U.S.A. Standard Testing Sieves.

Pit-Construction Behavior

Pit construction consists of several stages. (l ) The antlion moves
just under the surface of the sand, crawling backwards in what appear to
be random directions (Fig. 1.1a). It uses two means of propulsion:
the hind legs are used to pull the antlion through the sand, and
contractions of the wedge-shaped abdomen are also used to plough
backwards through the sand. (2) The antlion then moves in a circular
path during which it flicks sand to the outside of the circle (Fig.
1.1b). (3) By spiralling inward, the antlion deepens and expands the
furrow it creates (Fig. 1.1c), until a conical pit is formed (Fig.
1.1d) (Turner 1915; Youthed and Moran 1969; Tusculescu et al. 1975;
Topoff 1977).

Sand particle size affects the sand throwing behavior. For small
particles, the head and mandibles are loaded with sand which is then
tossed in a clump. This is accomplished when the antlion walks
backwards into the furrow wall causing sand to fall onto its head from
the sloped wall. Larger particles are isolated by sifting away all
smaller particles and then individuallj^ tossing them out, either to the
side or directly behind the antlion. Particles too large to toss are
carried out of the pit on the antlion 's back (Topoff 1977; personal
observation). My paper deals only with the behavior of the antlion
while constructing pits with small particles. This behavior is most

jM

STEP

SURFACE VIEW

B

D

PATH OF AMTllOW

CROSS-SECTIONAL
VIEW

.' '

PIT EDGE

F-i -Vivo .■■-Ti--,.--' : '■.t:\"'^0; ■■■■■'■ '- V, . -

p|?^l||f|0\^:^:>^

Figure 1.1. Steps of pit construction by antlion larvae. (A)
Random movement. (B) Beginning of circular movement; sand thrown
to outside of circle. (C) Antlion continues to circle inward in a
spiralling path, creating a furrow. (D) Finished pit.

common in sandy areas, but is utilized to some degree in any habitat
type.

Pit Morphology and Prey Behavior

If an antlion constructs a pit in sand consisting of several grain
sizes (which is the usual case in nature), the completed pit is
generally lined with the finest sand available (Fig. 1.2). To test
whether this feature functions to increase the efficiency of the pit, I
constructed artificial pits by pressing conical molds into sand of
different grain sizes. Escape time from these pits was measured for two
species of ants, the carpenter ant (Camponotus floridanus) and the
smaller fire ant (Solenopsis invicta).

Ant escape time increased significantly with decreasing sand grain
size for both ant species (Table 1.1, Figs 1.3 and 1.4). Therefore an
antlion pit lined with the finest sand available should serve as a more
efficient trap than a pit lined with the unsorted spectrum of available
sand. Two other variables also significantly affected ant escape time:
pit diameter and slope (Table 1.1).

Two components may affect the distribution of sand in the pit: (l)
the physical properties of sand as they apply to pit morphology, and (2)
the behavioral aspects of pit construction. These components are
examined separately in the following sections.

Physical Components of Pit Construction
Pit morphology is affected by two different physical processes.
The first determines the "behavior" of sand on the furrow walls during

Figure 1.2. Photograph of a completed antlion pit showing the
distribution of fine (white) and coarse (black) sand grains. (a) Pit
wall lined with white sand. (B) Position of black sand 'ring'. The
white line marks the pit edge.

11

12

i»-

u

lij
in

2

UJ

o

t/5

CAMPONQTUS

PtT DIAMETER |

• • 35 min

O- O 50 mm

Ct -A 65 mm

<X

T

A.

:p

i

i

123 - 2S0

250 - 500 500- 1000

SAND GRAIN SIZE (m)

1000- 2000

Figure 1.3. Escape time of the carpenter ant (Camponotus floridanus)
from artificially constructed pits. Slope of pit walls was 40 degrees.

.s^

13

120

lOO-

/

o

80-

UJ

LU 60

a.

<
o

V)

UJ 40

z
<

20-

SOLENOPSIS

PIT DIAMETER
» » 35 mm

O O 50 mm

A A 65 mm

n»l20

/

'•----ggg

125-250

250-500

500^000

\

1000-2000

SAND GRAIN SIZE (//)

Figure 1.4. Escape time of the fire ant (Solenopsis invicta) from
artificially constructed pits. Slope of pit walls was 40 degrees.

i^alX

14

Table 1.1. Multiple Regression Analysis for Ant Escape Times from
Artificially Constructed Pits Differing in Pit Diameter, Slope and Sand
Size*

Solenopsis invicta
Independent variables ** degrees of freedom F value Prob.>F

1.

Pit diameter

1

2.

Slope

1

3.

Sand size ***

3

4.

Regression model

including all variables

5

5.

Error

475

34.35

0.0001

102.60

0.0001

51.04

0.0001

58.06 0.0001

Camponotus f loridanus
Independent variables degrees of freedom F value Prob.>F

1.

Pit diameter

1

2.

Slope

1

3.

Sand size ***

3

4.

Regression model

including all variables

5

5.

Error

235

127.88

0.0001

20.72

0.0001

25.61

0.0001

45.08 0.0001

All regression analyses were run on SAS computer program GLM (Barr
et al. 1979). Each data point represents a different individual.
The following values of the independent variables were used:
Solenopsis and Camponotus: pit diameter: 35, 50 and 65 mm; sand

**

grain size: 125-250, 250-500, 500-1000 and 1000-2000 um;
Solenopsis slope: 30, 35, 40 and 45 ; Camponotus slope: 35 and
40®";
*** Sand size was entered as a class variable and therefore is treated
as a non-continuous variable with four levels.

-^

15

construction. This process will directly affect the morphology of the
pit and is closely related to the angle of repose (as discussed below).
The second process governs the trajectory of thrown particles. Particle
trajectory indirectly influences pit morphology in that it will
determine what types of sand particles leave the pit after being thrown.

Slope: Angle of Repose

The antlion pit is lined with fine sand even before it is completed
(Fig. 1.5). During construction the larger particles tend to fall to
the bottom of the furrow where the animal is digging, leaving the furrow
walls lined with finer sand. The differential response of particles of
different sizes on the furrow walls suggests that particle size may in
itself affect the distribution of sand on the slope. To test this, I
constructed artificial pits of different grain sizes. This was done by
drawing sand through a hole in the bottom of a tray filled with sorted
sand. The slope of the pit walls constructed in this way reflects the
angle of repose of the sand. The angle of repose (0') is the maximum
slope that sand will attain without collapsing.

A significant negative correlation was obtained between sand grain
size and slope (r =-0.771, N=34, P<0.01). Thus, for the sand in which
the antlion was making a pit, larger particles had a lower 0' and
therefore were more likely to fall off a slope than smaller ones.

Although a significant correlation was demonstrated, this
correlation may reflect differences in particle angularity, roughness,
or water content which can covary with particle size. All these factors
will affect the angle of internal friction (0, the minimal angle of

?i^-:

Figure 1.5. Photograph of an antlion pit in construction showing the
distribution of fine (white) and coarse (black) sand grains. (A)
Position of antlion in trough of pit. The white line marks the pit
edge.

17

18

stress where a mass is in equilibrium) (Singh 1976), and therefore will
affect the angle of repose. Marachi et al. (1972) have shown that 0
decreases with increasing particle size, but they note that other
studies suggest either no relationship or an opposite one. However, the
actual physical factors that create the negative correlation between
slope and particle size are unimportant. I have observed antlions
constructing pits in several types of soil and this correlation held in
each case. Thus, a pit will tend to be lined with fine sand through the
differential response of particle size and 0' alone.

Particle Trajectory: Stoke 's Law

If an antlion constructs a pit in sand consisting of a variety of
particle sizes, rings are formed around the pit in order of increasing
particle diameter (Fig. 1.2). This indicates that larger particles are
thrown farther during pit construction than smaller particles. Thus, in
addition to a differential sand sorting on the furrow walls due to the
angle of repose of sand (0'), there appears to be sorting according to
the size of the thrown sand. To understand the basis of the latter
sorting, the physical processes affecting sand particle trajectory must
be understood.

The trajectory of a particle with a given initial velocity is
affected by the drag force imparted on it by friction due to air. As
derived below, the drag force on a particle will vary with particle
radius. The smaller the particle, the higher the drag force due to air
relative to its momentum and the shorter the distance it vill travel.
At Reynolds numbers below 0.1, Stoke's Law defines this force (F )

19

(Bird et al. I96O):

F = u R^(0.5pV^)(24/Re)=6 ^uRV, (I)

where

Re=Reynolds number=(2RVp)/u,

E=particle radius,

p=fluid density=0.00123 g/cm^ for air at about 25°C and 56°/o

relative humidity,
V=particle velocity,
u=fluid viscosity=0.000184 g/cm s.

The relationship between particle trajectory and the characteristics of
particles can be more easily analyzed if Stoke 's Law is expressed in
terms of its effect on the distance a particle travels. Here distance
(D(x)) is defined as the total horizontal distance a particle travels
(see Lucas 1982 for the derivation):

V ^sin29 K'V sin9
D(x)= -2 [i_(4/3)-.2____] (2)

g gR

where

K'=(7.16x10"^)c,

0=initial trajectory angle,

g=acceleration due to gravity=980 cm/s ,

V =initial particle velocity,

c=dimensionless coefficient=4.5 for a sphere (Bird et al. 1960).

20

Equation (2) is a standard Newtonian ballistics equation which
incorporates momentum loss due to drag. According to Stoke 's Law, the
variables that affect the distance a particle travels are initial
velocity, trajectory angle, and particle radius. This equation predicts
the following relationships: (l) Distance (D(x)) will increase
monotonically with increasing initial velocity (V ). (2) The effect of
trajectory angle (9) on D(x) will vary with sand particle size (r). The
trajectory' angle at which distance is maximized will decrease from 45°
as particles become smaller. As sand particle size increases, D(x) is
maximal at ©=45 • (3) As sand particle size increases, distance should
increase monotonically.

At intermediate Reynold's numbers (2<Re<50C), Stoke 's drag force is
underestimated and must be modified as follows (Bird et al. I960):

Fj^= R^(0.5pV^)(l8.5/Re°-^), (3)

which changes the derived Stoke 's equation (equation 2) to (see Lucas
1982 for the derivation)

V ^sin2Q K"V ^ '^c{Q)
D(x)= -2-— [1 - ----° ], (4)

g gR

where

K" = (7.79x10"^)c.

The relationships between particle size, trajectory angle, and distance
listed above for equation (2) are the same for equation (4).

A sand particle with a diameter of 200um would have a Reynolds

21

number of 2 if it travelled only 15 cm/s. Although I did not directly
measure particle speed, a rough estimate suggested that the antlion
threw sand at a much greater velocity than this. Therefore, the
Reynolds number was thought to be greater than 2.

I measured the distances over which antlions threw sand of
different sizes (different R) during pit construction. With empirical
measurements of trajectory angle (0)(see Behavioral Components Section),
thrown distance (D(x)), and particle radius (R), the observed data were
best fit to the modified Stoke's equation (4) using the least squares
method. To measure the distances that particles were thrown, I placed a
card covered with double-sided tape along the edge of the pit and behind
the antlion. I measured the distance thrown for each sand grain on the
tape. The sand grain diameter was measured under a microscope.

With equation (4), the c value that produced the best fit to the
data was 16. 4 (Fig. 1.6). This value is a higher c value than the c of
a sphere (4.5), which suggests that the irregularities of the sand
particle surface increase the drag on the particle. With a c value of
16.4, equation (4) generally produced predicted distances close to the
mean distances that antlions threw particles. When equation (2) was
best fit to the data, a c value of approximately 70 was obtained. This
corroborates the fact that the unmodified Stoke's Law (equation I)
underestimates the friction force in this system. Thus, it appears that
the presence of sorted sand rings around the antlion's pit is due to the
effect of sand particle radius on the drag (F ) to momentum ratio.
This ratio is lower for larger particles. This means that when an
antlion throws mixed sizes of sand, the finest particles fall out first
and the largest particles travel farthest.

79

1000 -I /TN

1©I —

CLEANING PRCOICTCO
UAKING PIT PREDICTED
MAKING PIT OBSERVED
CLEANING OBSERVED

J®

Vm. ioa|
Vc. l«ol

Vm • 76
Vc ■ 119

*°H i° " I \^\. r T T .I..-T--r'-" - 50 ',• T T .1

^ -l.iJ'-rr-*' T T„^u-u^io| ■ 5' k*...4-4-i *23^

1 ' ' r

®

Vm • 85

Vc " 112

Y i ' ■♦•■■■■1.

1®

Vm • 87
ve • 115

io Kt T..I-

I *;

X,7-1S

25 «.5 S.S 8.5 10 5 iZ.5

DISTANCE SAND THROWN FROM ANTLION (cm)

Figure 1.6, Observed and predicted values of diameters of sand grains
thrown during pit construction and cleaning. Bars represent mean
observed values +/- 1 SD. Curves represent values predicted by equation
(4) (see text) . V =best fit value for particle velocity during initial
cleaning phase. V =best fit value for particle velocitv during pit

Tn ' o r

construction. Velocities listed are cm/s.

':^ - 23

Behavioral Components of Pit Construction

Clearly, the structure of a pit is influenced by the physics of
sand, but does the antlion exhibit any behavioral patterns that tend to
increase its efficiency at constructing the pit in terms of these , '
physical laws?

Trajectory Angle During Pit Construction

Based on equation (4), two variables that the antlion can
potentially regulate are trajectory angle (0) and particle velocity
(V^). At 9=45 , large particles will travel a maximum distance.
Small particles, on the other hand, will travel increasingly longer
distances as 9 is reduced from 45 to 0°. This suggests that there will
be a particle size crossover point. At particle sizes below this point,
particles will travel farther as 9 is decreased from 45°, and above
this size particles will travel farther when the trajectory angle is
45 • At an initial velocity of 100 cm/s (approximately the velocity
used during construction; see Fig. 1.6), this crossover occurs within
the range of sand grain sizes used in this study (300-400 um; see Fig.
1.7). Therefore, the predictions derived from equation (4) will affect
the importance of regulating 9 for the antlion. Thus 9 will affect not only
the distribution of sand particle sizes in the final pit but also the
cost of removing these particles. We expect natural selection to act on
antlion behavior in such a way that antlions utilize 9 to enhance pit
structure while keeping construction costs at a minimum.

To measure 9, a cardboard covered with two-sided sticky tape was
placed over a pit during construction. Sand thrown by the antlion stuck

24

TRAJECTORY
ANGLE

30» o o

40*

«0'

DISTANCE (cm)

Figure 1.7. Distances sand grains of different radii travel under
varying trajectory angles as predicted by equation (A) (see text).

V was 100 cm/s.
o

25

to the tape, leaving a record of the dispersion of sand as it was thrown

out of the pit. Thus 0 was estimated by measuring (1 ) the pit angle, (2) the

distance from the antlion's mandible (visible at the bottom of the pit)

to the pit edge, and (3) the distance from the pit edge to the sand on

the tape (Fig. 1.8). Trajectory angle and dispersion of sand during

pit cleaning (described below) were also measured in this manner. This

method tends to underestimate 0 slightly, if sand particle trajectory is

not linear between the release point and the point where it attaches to

the tape. However, since this distance was never more than 2 cm, the

underestimation, if any, would be slight.

The mean 8 was 47 with about a 12 scatter (Table 1.2). Judging
from these data, the antlion appears to be tossing particles so as to
maximize the distance larger particles travel and to maximize particle
dispersion. Particles above 400um will travel farthest at 9=45°.
However, to show that the antlion can truly regulate 9 behaviorally, it
is important to show that an antlion is capable of altering 9. To
demonstrate this, another behavior was analyzed, namely cleaning the pit
after prey handling.

Trajectory Angle During "Pit Cleaning"

When an antlion captures an arthropod prey, it punctures the prey
with its sharp mandibles and injects enzymes that externally digest the
prey. Then it ingests the soft tissues and discards the exoskeleton
(Wheeler 1930). During prey capture and handling, the pit walls are
usually disturbed, causing the bottom of the pit to be partially filled
with sand.

':■ - ',

26

SCOTCH TAPE

A • POSITION OF ANTUION
B ■ EDGE OF PIT

C • POSITION OF SAND
ON TAPE

(AS IN FIGURE l-C)

THE FOLLOWING DISTANCES ARE REQUIRED TO DETERMINE 9:
AB • DISTANCE FROM ANTLION TO PIT EDGE
BC • DISTANCE FROM PIT EDGE TO SAND ON TAPE
AC" DISTANCE FROM ANTLION TO SAND ON TAPE

C B

APPLYING THE LAW OF COSINES:

COS. 9 »

COS. e •

2 a7 '£^

e • e + a

e,' a

THEREFORE TRAJECTORY ANGLE e« 9 + 9

I 2

Figure 1.8. Diagram of method used to estimate trajectory angle (9).
A-position of antlion; B-edge of pit; C-position of sand on tape. The
following distances are required to determine 9: AB-from antlion to pit
edge; BC-from pit edge to sand on tape; AC-from antlion to sand on tape.

27

Table 1.2. Particle Trajectory Angle (9) for Pit Construction and Both
Phases of Pit Cleaning; Numbers in Parentheses are Standard Deviation
(SD).

Pit Pit cleaning, Pit cleaning,

construction initial phase final phase

Mean 0 47.0° (0.7) 46.4° (2.3) 60.0° (4-5)

Range of 9 for

each antlion 12.2° (2.3) 16.0° (2.8) 43.7° (8.6)

N* 10 10 10

* Each data point represents one throw from a separate individual.

^•■-'SilM

28 ■ • . ■

There are two different phases in pit cleaning used in removing the
prey and the excess sand. Initially the antlion throws the prey carcass

and some sand particles from the pit at an angle of approximately 4b .

o /

Finally the antlion cleans the pit by increasing 9 to about 60 (Table

1.2). At this time, the antlion also alternately throws sand to either
side, creating a heart-shaped distribution on the tape (Fig. 1.9). The
sand thrown during the first phase has a significantly greater particle
size than during the final phase (Table 1.3). Thus, the antlion
initially clears the pit of the prey carcass and the larger particles
that have fallen into the pit during prey capture. The trajectory angle
used in this phase (46 ) tends to maximize the distance thrown for
these larger particles. The antlion then increases the trajectory angle
and throws finer sand onto the walls to re-establish the slope of the
walls at the bottom. By increasing 0, the antlion keeps the smaller
particles inside the pit, thus increasing the number of fine particles
lining the pit. As was mentioned at the outset, this directly relates
to the ability of these pits to catch prey (Figs. 1.2 and 1.3). This
final phase of pit cleaning is similar to the final phase of pit
construction.

Particle Velocity

The function of the initial cleaning sequence is to clear the pit
of the prey carcass and any large particles of debris that have fallen
into it. The antlion should use a high V during this phase to
decrease the probability of debris or the carcass blowing back into the
pit. Continually removing these objects would increase the cost of pit

29

POSITION OF ANTLION IN PIT

V— MANDIBLES
— ABDOMEN

INITIAL DISTRIBUTION

OF SAND THROWN WHEN

CLEANING PIT

FINAL DISTRIBUTION

OF SAND THROWN WHEN

CLEANING PIT

Figure 1.9. Distribution of sand tossed by antlion during the initial
and final phases of pit cleaning.

30

Table 1.3. Sand Grain Sizes (in )im) for Initial and Final Phases of Pit
Cleaning.

Antlion mean grain size (standard deviation)

number ,

Initial phase N Pinal phase H Z

Y Y

1 303 (83) 50 206 (71 ) 50 5-40

2 334 (102) 50 236 (64) 50 4.80

3 355 (97) 50 251 (73) 50 5-19

4 323 (107) 50 226 (76) 50 5.05

5 324 (80) 50 238 (41 ) 50 5.65

* Mann-Vhitney U-test for large sample size.
** Significant at the P<0.001 level.

**

31

maintenance. Also, by clearing debris from the pit periphery, there
will be a low chance of the debris falling back into the pit when the
antlion reshapes or enlarges the pit (which they sometimes do once or
more per day; personal observations). Conversely, the function of
tossing sand during pit construction is to empty the pit in order to
construct a funnel. During pit construction, then, the antlion need
only throw sand at a velocity high enough for sand to land outside the
pit. A high V^ during pit construction would increase the cost of the
pit, with no concurrent benefit. A reduction in V during construction

0

would both decrease pit construction costs and tend to retain small

particles within the pit, enhancing the pit efficiency. Thus, if the

antlion could regulate V , it would be advantageous to keep V low

during pit construction and increase V during cleaning. Do antlions

regulate particle velocity?

The V can be estimated using equation (4) and the following

parameters: trajectory angle (0), particle radius (R), particle density

(p ), and the particle coefficient. Methods for estimating Oy R and c

have been described previously. Particle density was obtained by

determining volume displacement of a known weight of sand. With these

parameters, an estimate of V could be obtained by best fitting

particle distance and particle diameter distributions for V . I

0

estimated the velocity that eight antlions used for pit construction and
for cleaning (Fig. 1.6).

The particle velocity during pit construction was approximately 72
percent (+/-5) of that used during cleaning. Thus, it is clear that the
antlions are able to regulate particle velocity. The consequence of
this regulation is that small particle dispersion is reduced during

32

construction, enhancing the differential sorting on the walls due to the
angle of repose. By using a 46 trajectorj' angle the antlion is
maximizing the dispersion of sand grain sizes. By reducing particle
velocity, the antlion enhances pit efficiency by reducing the number of
finer particles that leave the pit and also reduces pit construction
costs. Are there any other behavioral components that may be important?

Foreleg Vibration

An antlion moves backwards under the sand during pit construction.
Sand that falls onto the antlion' s head from the furrow walls is tossed
out of the pit. Individuals of all antlion species hold their forelegs
along the sides of their heads. The legs are vibrated while the pit is
constructed and this appears to aid in the movement of sand onto the
head. The antlion can scoop sand up with its mandibles, but it does
this only in the last stage of pit construction (which resembles the
final phase of cleaning). Turner (1915) showed that the loss of
forelegs did not eliminate the antlion 's ability to construct a pit and
he therefore suggested that the forelegs did not function at all in pit
construction. Although forelegs may not be required to construct a pit,
they appear to provide some finer behavioral regulation during pit
construction. The foreleg movement is clearly a vibration and not a
shovelling movement. The function of these vibrations may be to sort
sand, although I have no direct evidence for this. Preliminary
observations of sand mixtures show that smaller particles tend to sink
when vibrated. The antlion may be sorting out the larger particles,
preferentially moving these onto its head by sifting out the finest

33

particles with its legs. Since the retention of fine particles in the
pit has been shown to be advantageous, the function of this behavior
would be to selectively remove the larger particles. The finest
particles would then tend to stay in the pit and would be used during
the last stage of construction when fine particles are thrown onto the
walls.

Spiralling

Tusculescu et al. (1975) have published the only study to date on
the physics of pit construction. They suggested that the inward
spiralling of the antlion (Fig. 1.1) could be explained solely by
physical factors. They assumed that the slope of the furrow walls was
equal to the angle of repose, and that the area of sand removed by the
shovelling action of the antlion 's head and mandibles was equal on both
the interior and exterior furrow walls. Under these assumptions, the
volume of sand that falls from the walls will be unequal, with more sand
falling off the exterior wall than the interior wall. As a result of
this inequality, the bottom of the furrow would tend to shift inward.
Thus, the antlion need only follow the furrow bottom to spiral inward
without behaviorally modifying the path it takes. Unfortunately, the
assumptions on which Tusculescu et al. (1975) rest their model are
untested. It is extremely difficult to assess the amount of sand
removed from either wall of the furrow. Also, the antlion could easily
regulate the differential flow of sand through the use of the front
legs, mandibles, or head angle. The ability to spiral inward i3
certainly exhibited in the earliest pit construction stages before there

34

is a furrow. Even if these assumptions are correct, the difference in
the shape for the interior and exterior wall would be negligible during
these initial stages. During this phase, it seems very unlikely that
the furrow bottom would shift inward. Thus during this stage, the
antlion must be choosing the path that it takes.

The assumption that all sand above that removed at the bottom will
collapse is also unrealistic. Removing a portion of a body of sand at
the angle of repose will cause that body to shift as Tusculescu et al.
(1975) suggest only if the sand is cohesionless (Krynine 194l).
Particle friction would tend to reduce the impact of sand removal on
sand located above the removal site. I examined Tusculescu and other's
assumptions as follows.

I allowed last instar antlions to construct pits in different sand
grain sizes. The assumption that the pit is always constructed at the
angle of repose is not supported by my data. Multiple regression
analysis showed that a significant proportion of the variation in pit
slope is accounted for by sand grain size with no significant variation
attributable to pit diameter. In addition to sand grain size, the
effect of antlion weight on slope was highly significant (for grain size
F=4.27 and P=0.020, K=24; for weight F=24.01 and P<0.001 , N=24). These
data indicate that the slope of the antlion pit is not at the angle of
repose, although sand size does affect the slope. This suggests that
the antlion may possess some as yet unknown mechanism for regulating pit
slope. For a given volume of sand removed, a reduction in slope will
increase pit diameter. Thus, the optimal slope will not necessarily be
the angle of repose, but may be at some angle that is less than this.

35

where the benefits of increased diameter equal the costs of a reduction
in slope.

General Discussion

Studies on trap construction and use by animals have, with very few
exceptions, dealt with ecological questions such as energetics (Ford
1977; Prestwich 1977), trap-building behavior (Witt et al. 1968;
Youthed and Moran 1969; Witt et al. 1972; Topoff 1977), orientation
(Wilson 1974; McClure 1976; Uetz et al. 1978; Hieber 1979), capture
rates and trapping success (Turnbull 1973; Griffiths 1980; Hildrew and
Townsend 1980). With the notable exception of Denny's (l976) work, the
biomechanical aspects of trap construction and the interface between
biomechanics and trap-constructing behavior have been neglected. Trap
efficiency is strongly affected by trap design and by the materials from
which it is built. Thus, the study of trap biomechanics will increase
our understanding of the advantages and constraints of traps on the
foraging capabilities of the predator. It will also enhance our ability
to evaluate the evolutionary adaptations that the organism exhibits in
the mechanics of trap-constructing behavior.

Trap biomechanics have been studied in only two groups of
organisms, the orb-weaving spiders (Denny 1976) and antlions (this
study). Denny (1976) showed that the spider web conforms to the
principle of least-weight structures, known as Maxwell's lemma. A trap
built with this design will minimize the amount of material needed to
function as a prey-catching device. Thus the spider maximizes the
efficiency of the web per unit silk, which reduces both material and

T<1

36
energy expenditure. Denny (1976) also showed that orb-weaving spiders
construct their webs so that very large and potentially harmful prey fly
through the web. Griffiths (1980) suggested that antlion pits may
function similarly through changes in pit diameter with antlion size:
prey that are too large or dangerous to handle can escape easily. The
potential for damage to antlions due to larger or dangerous prey has
been demonstrated by Lucas and Brockmann (1981 ). The present study
shows that in addition to pit diameter regulation, antlions are able to
regulate the mechanics of pit construction. This is done through the
manipulation of sand particle velocity and trajectory angle in addition
to an initial sorting through foreleg vibration. In so doing, they
enhance the efficiency of the pit in capturing prey by maximizing the
proportion of fine sand on the pit walls.

One prediction of optimal foraging theorj' states that an animal
should forage so as to maximize its net rate of return from prey
(Schoener 1971; Pyke et al. 1977). Both this study on antlions and
Denny's (1976) study on orb-weaving spiders suggest that
trap-constructing behavior supports this prediction. However, the
methods with which they follow this prediction differ due to differences
in the nature of the traps. Spiders exhibit much more control than do
antlions over trap material composition and, in fact, utilize at least
four different silk types in the construction of the orb web (Work
1981). At least two of these silks differ significantly in their
physical properties (Denny 1976). Therefore spiders are able to
manipulate the properties of the trap by using different silk types and
by varying the number of fibers in each web element. They can also
regulate overall trap design to maximize net capture rate (Denny 1976).

37 :^

Antlions, on the other hand, are constrained by the properties of
the material with which they construct their traps. The physical
properties of the sand in which they build their pit will strongly
affect the properties and design of the trap. The angle of repose and
Stoke 's drag force complement each other to produce a pit with enhanced
capturing efficiency regardless of antlion behavior. \ Thus, adaptations
in trap-constructing behavior in antlions will consist primarily of
minimizing energetic requirements of construction, although some control
over the trap design is exhibited. Energy is saved through a reduction
of particle velocity and in the utilization of a trajectory angle that
maximizes the distance that larger particles are thrown. Pit design is
modified by the identical final phases of construction and pit cleaning,
which result in a additional fine-sand layer on the pit walls. Antlions
may also regulate pit diameter and the slope of the pit walls. I

Several distinct taxonomic groups utilize similar trap types.
These include spiders and fungus gnat larvae (Eberhard 1980) (silk webs
and silk thread traps), and antlions and rhagionid wormlions (Wheeler
1930) (sand pitfall traps). Since these groups are morphologically
distinct, it would be interesting to compare their behaviors to see if
the similarity in trap physics has caused a convergence in
trap-constructing behavior. For example, wormlions use the anterior
portion of the thorax to throw sand (Wheeler 1930), while antlions use
only their heads. Yet both groups should use similar trajectory angles
and velocities in order for them to forage optimally.

' ■♦.

CHAPTER II
MODELS OF PARTIAL PREY CONSUMPTION

Introduction

Predators are generally thought to consume their prey whole. This
conception is reflected in the optimal diet choice models developed by
Charnov (l976a), Pulliam (1974, 1975), Rapport (1971, 1980), Werner and
Hall (1974), and others. Each of these authors treats the energy
derived from a single prey item as a constant. However, many predators
either occasionally or predominantly consume only a portion of their
prey. In this case, the energy derived from a single prey will be a
function of the handling time invested in that prey. A number of models
have been proposed that directly address partial prey consumption (Cook
and Cockrell 1978; Griffiths 1982; Holling 1966; Johnson, Akre and
Crowley 1975; Sih 1980a). These models generally fall into two
categories: (l) models of proximate mechanisms that refer to
physiological constraints on foraging, and (2) optimal foraging models.
In this paper I review these models and their predictions. I then
derive a new optimal foraging model that predicts partial prey
consumption by ambush predators. The model illustrates that no global
foraging model can generate predictions for a wide variety of predators.
The models also show that even small changes in predator behavior or
prey conditions may change the expected predatory response to prey.

38

39

Proximate Models

Models concerning the physiological constraints on foraging have
focused on two aspects of the predator, (l) gut size (The Gut-Limitation
Model) and (2) the maximal rate of ingestion (Digestion-Rate-Limitation
Model).

Gut-Limitetion Model

The Gut-Limitation Model (GLM) was first proposed by Rolling (1965,
1966). He suggested that predatory behavior could be thought of as a
number of separate phases, each of which is driven by its own
controlling mechanism. Thus, the behaviors of search, pursuit, strike
and eat are each independently controlled, and each is triggered by a
given threshold of hunger. Holling suggested that hunger thresholds are
determined by the amount of food in the gut and are therefore gut
limited. If the threshold for eating is greater than that for capture,
then the predator may kill prey without eating anything. Eating is
terminated when the predator is satiated, or when the hunger threshold
drops to zero. If the threshold for eating is low, then the predator
may only eat a small portion of the prey before it is satiated and will
only partially consume its prey.

Johnson et al. (1975) proposed modifications of Holling's model for
insect systems. They suggested that there are essentially two levels of
satiation, one involving the filling of the foregut (which affects
eating and striking) and the second involving the filling of the midgut
(which affects striking). Thus, this model differs from that of Rolling
(1966) in that two separate compartments (foregut and midgut) each

^^31

40

affect some portion of predatory behavior, whereas the original model
only assumed one compartment.

Both of these models assume that the degree of satiation directly
controls predatory behavior. In terms of partial prey consumption, the
models predict that partial consumption occurs only when the predator
does not have enough room in its gut to eat an entire prey. As prey
density increases, there is an increasing probability that a predator is
nearly full at the time of prey capture. Thus, there should be a
negative correlation between prey density and percent consumption of
prey. There should also be a concomitant decrease in handling time per
prey item with increasing prey density, since it should take less time
to eat a smaller proportion of a prey item.

The assumption that predators are often satiated appears to be
valid in laboratory studies conducted by Rolling (1966), Johnson et al.
(1975), and Nakamura (1977) on mantids, damselfly larvae, and wolf
spiders (respectively). However, a number of predators that partially
consume prey are clearly not constrained by satiation (ex. Cook and
Cockrell 1978; DeBenedictis et al. 1978; Sih 1980a). In fact, the
predatory mite (Amblyseius largoensis) has been shown to feed to
satiation then return to the same prey after a "digestive pause"
(Sandness and McMurtry 1970). In this case, satiation occurs regularly,
but does not entirely affect prey-consumption behavior. A second
prediction from this model is that there should be a correlation between
inter-capture interval and percent consumption. The importance of this
prediction is discussed in a later section.

41

Digestion Rate Limitation Model

Many predators externally digest their prey and then ingest the
pre-digested material. Spiders, antlions, and hemipterans include many
species that utilize prey in this manner. When feeding on insects, this
type of predator is generally unable to ingest the exoskeleton and
therefore always consumes only a portion of each prey item. Griffiths
(1982) suggested that the rate of digestive-enzyme production increased
with an increase in the rate at which prey were captured and eaten.
This means that the rate of ingestion should increase with increasing
feeding rate (or increased prey density) and the amount of time spent
per prey item should decrease as a result. Thus, handling time is
constrained by the rate of extra-intestinal digestion. Griffiths (1982)
provided data on one species of antlion that supported the model and
suggested that the data from Giller (l980) on backswimmer (Notonectidae)
feeding behavior also could be interpreted in this way. Mayzaud and
Poulet (1978) demonstrated empirically that enzyme production increases
with feeding rate in a number of species of copepods. Thus, this model
may be appropriate for particle feeders as well as the fluid feeders
referred to by Griffiths (1982). Griffiths showed that antlions (which
normally feed at low prey densities) extract the same proportion of each
prey. Thus antlions exhibit no change in partial consumption with
increasing prey density even though there is a decrease in handling
time. So in contrast to the Gut-Limitation Model, the
Digestion-Rate-Limitation Model predicts no change in percent
consumption (for predators that feed at low rates), but a decrease in
handling time with increasing feeding rate.

42 '-^

Optimal Foraging Models

Charnov ( 1976a; also see Parker and Stuart 1976) proposed a model
to predict the behavior of predators foraging at a series of patches.
This model, the Marginal Value Theorem, predicted the amount of time the
predator should stay in a patch, given three sets of information: (l )
the search time required to find the next patch, (2) the average net
rate of benefit accumulated for all patches, and (3) the instantaneous
rate of benefit accumulate for the present patch. Cook and Cockrell
(1978) and Sih (l980a) independently proposed that in many cases, a
single prey item could be treated as a patch with a given search time
required to find the next patch (prey item). They used this analogy to
adapt the Marginal Value Theorem to the study of partial prey
consumption.

In this section, I will review the Marginal Value Theorem as it
applies to partial prey consumption. I then show that this model is
inappropriate for ambush predators (for which it has been used in the
past) and develop an analogous model for this type of predator.

Searching Predators

All optimal foraging models assume that the predator chooses (or is
programmed to select) the sequence of behaviors that maximizes the net
rate of benefit per unit foraging time (Pyke et al. 1977; Schoener
1971). In many cases, benefit has been expressed in terms of energy
(Jaeger and Barnard 1981; DeBenedicts et al. 1978; Hughes 1979; Pyke
1978; Krebs et al. 1977). In terms of the predator that consumes only
a proportion of its prey, optimal foraging theory would predict that the

43

predator should eat only the proportion of each prey that would yield a
maximal net rate of benefit.

The solution for the optimal percent consumption can be expressed
as follows: Let the behavior of the predator consist of three phases,
search, pursuit and capture, and ingestion. The times associated with
these phases are

t =pursuit and capture,

t =search time,
s

t =ingestion or feeding time,

T =total foraging time=t +t +t ,
i p s e

where t and t are constants and T^ is a function of t . Let the
P s i e

gross benefit extracted from the prey (g(t )) be a Michaelis-Menten
function (e.g. Shoemaker 1977; as in Sih 1980a):

g t
/ , \ max e - ^

g(t ) = , (1)

® C + t
e

where g^^^ is the maximal benefit that can be extracted from the prey
and C is a constant that affects the rate of extraction. The
Michaelis-Menten function will be used in all further derivations. This
function, which was originally used by Sih (l9S0a) in his analysis of
partial prey consumption, gives an asymptotic curve and should
generally resemble the extraction rate curves exhibited by foragers.
There should be little qualitative difference between similar asymptotic
functions (e.g. Kichaelis-Menten, exponential, etc.) in the predictions
generated from these models. I will assume that each prey is similar in
that g(t ) is equivalent between prey. 1 also assume that capture

44

probability is unity, although this assumption will be relaxed later in
this paper. I define the costs of foraging as

C =G0st of pursuit and capture per unit time,

C =cost of search per unit time.
Therefore,

C t =total cost of pursuit and capture,
P P

C t = total cost of search,
s s

The total benefit (E) per unit foraging time (T ) is (Sih 1980a)

E g(t )-C t -C t g t C t C t
e_ p p s s ^ max e p p s s

T_ t +t +t (C+t )T T T

T e p s ^ e T T T

The optimal solution is obtained when
d (E/TJ

(2)

= 0. (3)

8 t

From equation

t =[(g C(t +t )+C t +C t )/g ]°*^ (4)
eop ^'^max ' p s' p p s s^^^max"' ^^

(see Fig. 2.1 for the graphical solution to t ).

eop

The predictions from equation 4 are as follows (Sih 1980a);
1 ) there should be a positive correlation between feeding
time (and percent consumption) and the following
parameters

a) pursuit time,

b) search time,

i^-:f*tj

45

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46

c) pursuit cost,

d) search cost;

2) since prey density should be negatively correlated
with search time, feeding time and percent consumption
should decrease with increasing prey density;

3) there should be a negative correlation between feeding
time (and percent consumption) and the extraction rate,
since the extraction rate is inversely proportional to
C.

Cook and Cockrell (1978; also see Parker and Stuart 1976) also
suggested that the predator should respond to the mean encounter rate
and not to each individual inter-arrival interval. This implies that
there should be no correlation between inter-prey interval and feeding
time if the overall rate of prey encounter remains constant.

These predictions are based on the fact that the predator should
weigh any benefit derived from ingestion against the benefit associated
with dropping the prey and searching for another. As the rate of
extraction decreases, there should be a point where searching for the
next prey will be more beneficial than continuing to feed on the present
prey. The optimal solution to this tradeoff between ingestion and
search is expressed in equation 4.

Ambush Predators

For a searching predator, the inter-arrival interval is dependent
on search time, and therefore, the forager has some control over prey

47

encounter rate. For an ambush predator, prey arrive independently of
the behavior of the ambusher. This means that prey inter-arrival
intervals for ambush predators are not the same as those for searching
predators. As a result, there is no tradeoff between search time and
feeding time for ambush predators, and therefore, the Marginal Value
Theorem (MVT) is not an appropriate model for this mode of predation.
Inter-prey interval may affect prey consumption, but for reasons
unrelated to the KVT. This is demonstrated by the models listed below.

For the first set of models, I assume that each prey encountered is
captured, and that the gross benefit derived from each prey per unit
feeding time can be expressed as in equation 1 . The model for ambush
predators generates different predictions at different prey densities.
I will address each of these prey density regions (high, medium and low
prey density) separately.

Ambush model — high density

At high prey densities I assume that prey are continuously
available to the predator, such that as soon as the predator drops one
prey, it can immediately begin pursuit of a second prey. At these
densities, the predator will always be in either the pursuit or feeding
phase of predation. At satiation, the predator may also exhibit some
digestive pause (see Johnson et al. 1975; Sandness and McKurtry 1970).
If the predator is not yet satiated, then the optimal feeding time will
be similar to that predicted by the Marginal Value Theorem, except there
is no search. Thus the net rate of benefit accumulation is

-VOH!;

48

E g(t ) - C t - C t
_ e p p e e

T e P

_ _ P p e e

max e

(c - t )(t + t ) (t - t ) (t + t )
Sep e p ® P

, (5)

where C is the cost of feeding per unit feeding time (t ) and the

e

total cost of eating is a linear function of eating time. The optimal
feeding time is (see Fig. 2.2)

-Dt C-[g t C(DC+g -Dt )]°*^

Dt -g
p max

where

D=C -C
P e

The predictions from equation 6 are as follows:

1 ) there should be a positive correlation between feeding
time (and percent consumption) and the following
parameters:

a) pursuit time,

b) pursuit cost;

2) prey density should have no effect on either handling
time or percent consumption, unless the time or cost

of pursuit is affected by prey density (for example see
Treherne and Foster 1981);

3) there should be a negative correlation between feeding

49

AMBUvSH/HIGHEST DENSITY MODEL

Ll.

LU

Z
LU

LAG
TIME

HANDLING TIME

Figure 2.2. Graphic method for solving the Ambush-Predator Model

with high prey density. C =0 and C =0 for this qraph.

e p ^

50

time (and percent consumption) and the following
parameters:

a) extraction rate (see predictions from eq. 4),

b; cost of eating.

The digestive pause may have a varietj^ of effects on foraging,

depending on how the pause constrains foraging. For example, the

predator may not return to the prey after satiation, in which case the

gut clearance rate and gut size will set an upper bound on feeding time

and percent consumption (as shown by Holling 1966 and Johnson et al.

1975). I will model the simplest case here, where the predator can

return to the prey (as shown in mites by Sandness and McMurtry 1970).

In this case, equation 5 is expanded to include the cost of the

digestive pause (C ) and the time required for the pause (t )

° d

^ s^^J - ^J - c^t, - C t

^ e P_2_ ^ ^ ® s

■^rn t + t + t,

T e p d

(7)

The optimal feeding time is

\op=^^^max^V'd)^^d^d^Vp^/^max^°*' ' ^^^

The predictions from eq. 8 are the same as those from eq. 6. In

addition, increases in t, and C^ should increase te .

da op

51

Ambush model — low density

I define low prey density as densities at which the probability of
encountering a prey during either the lag or ingestive phases is
essentially zero. Here the inter-prey interval is long, but this
interval cannot be treated as it was with the MVT. Prey arrive at given
intervals of time, regardless of how the predator uses that time. With
the MVT, prey arrive at given intervals of search time only. Thus, the
inter-prey interval is influenced by the amount of time the predator
invests in each phase of predation. For ambush predators, prey arrive
at given intervals of total time.

At low densities, the sum of the pursuit time (t ), feeding time

(t^) and waiting time (t =time from the end of feeding until the next

prey encounter) is constant (T) and not a function of t . Here

e

T=t +t +t .
pew

I will also treat pursuit time as a constant.

If waiting costs (C ) and feeding costs (C ) are negligable, then
the predator should hold on to its prey until it is entirely consumed.

Unfortunately, the Kichaelis-Menten function asymctotes to g at

max

infinity, thus assuming that the predator can always extract more from
the prey. If feeding costs are non-negligible, then the predator should
retain the prey approximately until the net rate of benefit accumulation
drops to zero (Fig. 2.3). If feeding cost is a linear function of
handling time, then the total benefit accumulated per unit foraging time
is

k^-

"^^

>,

.

Q)

QJ

U

JJ

&.

tw

3

0

o

^

e

c

^

•H

4J

U

•H

0

3

c

D

iH

U^

QJ

T3

)-l

O

TO

5:

(1)

C

Vj

•H

0

t— i

J-i

«

C3

-o

CD

tfl

^J

•H

CU

1

iJ

x:

E

CO

0

3

a

J3

e

c

<

0

•r-l

0)

J->

,c

cn

J-l

0)

6C

<4-i

C

0

•H

c

1 — 1

0

Cfl

3 -u

0

0

tc

X

4-1

(-1

0

-3

M-i

C

CO

-o

0

J.)

J=

c

4-1

cfl

(1)

j-i

6

CO

c

u

0

•H

0

^

0.

CO

CO

S-i

CO

c

•H

u

m

CN •

3 CO

OC C

•H 0)

53

UJ

Q
O

>-

H

CO

z

LU
D

I-

co

LU

o

CO

D

lld3N3a

54

E g(t )-C t -C t -C (T-t -t )

e eeppw ep

T T

Thus,

E gt Ct etc (T-t -t )

= _?5f_£ _f_£_ _ _P_P_ _ ■** ^ V (q)

T^ T (C + t^) T T T

Here

c(c -c )-[(c -c )g c]°-5

t = __-5_-l^_-L_£_.^_!!?52^___ . (10)

^°P - (C -C )

e w

Equation 10 predicts that the predator should handle the prey until its
rate of net benefit accumulation drops to C . At handling times
greater than this, it will be more costly to feed on the prey than it
would be to drop it and wait for the next prey to come along. Further
predictions from equation 10 are as follows:

1 ) feeding time and percent consumption should be

positively correlated with g and C :

max w

2) feeding time and percent consumption should be
negatively correlated with C and C ;

5) prey density should have no effect on either feeding

time or percent consumption;
4) neither pursuit time nor pursuit costs should have any

effect on feeding time or percent consumption.

55
Ambush model — medium density

At densities intermediate to the low and high density cases, prey
arrive at intervals short enough to overlap with the pursuit or feeding
phases. At medium densities, when a prey arrives, the predator can
either drop the prey item it is currently eating and pursue the second
prey, or ignore the second prey and continue eating the first (Fig.
2.4). We should expect the decision made by the predator to reflect the
maximal net rate of benefit accumulation.

At these densities, the inter-prey interval sets the feeding times.

In fact, the pursuit time (t ) plus the feeding time (t ) are equal to

P e

the inter-capture interval. However, the predator should never hold on

to a prey longer than the time predicted by the ambush/low-density model

(once inter-prey interval drops below this threshold, the

ambush/low-density model predicts predatory behavior). A few new terms

must be defined:

T =inter-prey interval=t +t ,
ir P e

X=the number of intervals before the xth prey is

encountered,
Y=the number of intervals before the yth prey is

encountered,
G(X)=benefit per unit foraging time derived from eating

every xth prey,
G(Y)=benefit per unit foraging time derived from eating

every yth prey.

The net benefit of foraging (E ^) is

net

:,;!

56

>-
UJ

a.
ql

o

z
o
o

UJ
CO

LU
CO

>

111

DC

Q.

K

o

W

<

cc

-J

UJ

oc

Q.

CO

UJ

>

cc
cc

<

liJ

0)

-o
>-

0)
Q.

o;
1)

■D
(U

E
i-

0)

4-1

c

C
(1)

1_

O

*->

ID

in

3

e

ID
(U

J3

ro

«D

>

V)
V
U

O

u

(U

JC

lld3N3S

-a-

CM

L.
3

57

g t
„ max e ^ , , ,

E^^. = - C t - C t . (11)

net c + t ^ P ® ^

e

The net benefit per unit foraging time will be

g (XT^^-t ) C t C t
G(X) = -H?5^-JL_F.__ . __P_P__ . __£_£__ . (12)

(C+XT-tp)(XT^p) XTjp XTjp
If X<Y, then the predator should choose the xth prey when

(XT^p-t )g C t C t

IP p max _P E ^ s

(C.XT^p-t^)(XT^p) XT^p XTjp

(YT^p-t )g C t C t

IP p max p p e e

(C+YTjp-tp)(YTjp) YTjp YTjp

>

(12)

Equation 6 generates the optimal feeding time for an ambush
predator at high densities. This optimal feeding time will also
correspond to the "optimal" inter-prey interval. Since G(X) decreases
monotonically as the inter-prey interval increases above this optimum
(see Fig. 2.5), the following predictions can be made:

1 ) at intervals larger than the "optimal" inter-prey
interval, each prey encountered should be pursued;

2) at intervals less than the optimal, the best interval
will depend on the characteristics of the curve from
eq. 11.

Prediction 1 generates two other predictions that are relevant:
5) as prey density increases, handling time and percent
consumption will decrease until the "optimal" prej'

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59

aau

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v.'"^t'

60

interval is reached;
4) since each prey item is pursued in this region,

variation in prey encounter should be correlated with
variation in handling time and percent consumption.

Prediction 3 is identical to the predictions from the MVT; thus at
medium densities the ambush predators should treat prey similar to
searching predators. However, prediction 4 is different than the
analogous prediction for searching predators and is identical to one of
the predictions from the GLM.

Capture Probability and Ambush Predation

The models listed above illustrate that many factors will affect
the predictions from optimal foraging. I have focused on two factors,
the mode of predation (searching vs. ambush) and the effect of density
on the predicted predatory behavior.

In this section I develop a model for ambush predators which
incorporates capture probability. Griffiths (1982) suggested that there
should be selection for reduced handling time if the capture probability
is lower when the predator is handling a prey than when it is "empty
handed". The model presented below explicitly demonstrates this
relationship. Another important factor is whether or not the predator
returns to a prey item once it is dropped. For example, damselfly
larvae apparently do not return to prey (Johnson et al. 1975), whereas
antlion larvae may cache partially utilized pr'^y on the pit wall,
capture the second prey, then return to the first (pers. observation).
I will assume that the predator can return to the first prey, so that if

61

a prey comes before the predator is finished with a previously captured
prey, it will cache the first prey until it can go back to it and finish
it. Let

A =encounter rate of prey,

P^'^Pi'obability of capture while handling a prey,

P =probability of capture while empty handed,

P,=P -P ,
a 0 w

N=total number of prey handled in a foraging bout

lasting a given length of time, T ,

T,=total time predator devotes to a given prey=t +t .
" p e

To simplify the model, I will also assume that the cost of eating and

pursuit are negligible. This assumption will not affect the qualitative

predictions of the model.

First the number of prey eaten while handling a prey (N ) and the

w

number eaten while empty handed (N ) must be calculated:

N^= A (T^-NT)P^ . (14)

The total number of prey eaten (N) is therefore

N=K^+K^= ANTP^+ A (T^-NT)P^ . (15)
Solving for K,

ATP ATP

,. .....1.2..... . ...iifs.. . (,6)

The benefit associated from each prey.

Kt
g(t ) = — 2___ , (17)

C + t
e

62

times the number of prey yields the gross benefit for the foraging bout;

(Kt )( XT P )

B = g(t )N = ? L2____ . (18)

(CnJ(uXT^P^)

The benefit gained per unit foraging time is

(Kt )( XP )
e 0

^T ^^^^^^^*^Vd^

The optimal solution to equation 19 is

t = (Ct + f'^ . (20)

eop ' p ^p ^ ^ ^

'd

(19)

From equation 20, a predator should decrease handling time as the

difference in capture probability (P -P ) increases. This is because

0 w

there is an added cost to holding onto a prey that must be weighed
against the diminishing return from that prey. Equation 20 is unique
among the ambush models presented here because it is the only model that
requires that the predator "anticipate" the next prey, or at least
modify its behavior before the next prey arrives. Thus, differences in
capture probability should affect how the predator treats variation in
the inter-prey interval.

Discussion

Different predators appear to exhibit a wide diversity in their
responses to prey. Also, as conditions change, the behavior of a single
predator may be predicted to change considerably. Many predators (for
example Plethodon; Jaeger and Barnard 1 981 ) may switch from ambush to

63

searching predators as prey density fluctuates. Some predators may
continuously reach satiation (ex. mantids in Rolling 1966, damselfly
larvae in Johnson et al. 1975), while others may rarely if ever be
satiated (ex. hummingbirds in DeBenedictis et al. 1978; antlions in
Griffiths 1982). This diversity is an important consideration in using
an optimal foraging approach to partial prey consumption, since
predictions change both quantitatively and qualitatively with changes in
predator or prey conditions (see Table 2.1).

One primary focus of a number of papers to date has been the
evaluation of proximate vs. optimal foraging models. The Marginal Value
Theorem (Cook and Cockrell 1978; see also Sih 1980a) was originally
used to show that the Gut Limitation Model was inadequate. Cook and
Cockrell (1978) showed that for a cocinellid and a notonectid, percent
consumption and handling time both decreased with increasing prey
density (predicted by the MVT and GLM) and that individual feeding times
were independent of the previous inter-catch interval (predicted by the
MVT, but not by the GLM). Giller (1980) repeated the experiment on
notonectids and found that individual feeding times were not independent
of the previous inter-catch interval (predicted by the GLM). Giller
(1980) also found that handling time per item decreased through the
foraging bout independent of prey density, suggesting that the predator
may be forming a search image through some optimal feeding mechanism.
Griffiths (1982) proposed the Digestion-Rate-Limitation (DRL) model to
explain this decrease in handling time in notonectids and showed that
the DRL Model applied to antlion larvae as well. He also showed that
ant lion larvae fed at low feeding rates do not change the percent
consumption with changing feeding rates, as predicted by the GLM.

64

Table 2.1. The effect of predator and prey characteristics on
predictions from partial prey consumption models. '+' = positive
correlation, '-' = negative correlation, '0' = no correlation expected,
'N/A' = not applicable.

handling time/percent consumption

variable

GLM

DRL

MVT

A/H

A/L

A/M*

A/CP

prey density

a) near satiation

-/-

-/-

-/-

0/0

0/0

-/-

_/_

b) no satiation

0/0

-/o

-/-

0/0

0/0

-/-

-/-

intercapture interval

a) near satiation

+/+

0/0

N/A

0/0

V+

0/0

b) no satiation

0/0

0/0

N/A

0/0

V-

0/0

cost of pursuit (C )

0/0

0/0

v+

+/+

0/0

0/0

cost of search (C )

s

0/0

0/0

-/-

N/A

N/A

N/A

N/A

cost of eating (C )

0/0

0/0

-/-

-/-

-/-

0/0

cost of waiting (C )

w

N/A

N/A

N/A

N/A

-/-

N/A

pursuit time (t )

a) near satiation

-h

+ /+

*/+

^h

0/0

0/0

V+

b) no satiation

0/0

0/0

V+

^h

0/0

0/0

+/+

extraction coefficient

(c)

a) near satiation

-/-

-/-

-/-

-/-

-/-

0/0

-/-

b) no satiation

0/0

0/0

-/-

-/-

-/-

0/0

-/-

* predictions for interprey intervals greater than the 'optimal'
inter-prey interval only. The predictions for no correlations are due
to the fact that all prey should be pursued (see text).

65

From the models presented in this paper, it appears that the
arguments over proximate and optimal foraging mechanisms in notonectids
addressed the wrong optimal foraging models, since Rotonecta is an
ambush predator (Gittelman 1974). Giller's (1980) results are predicted
hy both the optimal foraging model for ambush predators and the GLM.
The lack of change in percent consumption for antlions (Griffiths 1982)
is also predicted by the ambush optimality model. Thus, the differences
between proximate models and the correct optimal foraging models are
non-existent for the parameters addressed in the literature cited above.
Griffiths (1982) also suggested that in many cases proximate and optimal
models will generate similar predictions, though he incorrectly equated
the predictions from the KVT (which was the incorrect model anyway) and
the proximate models (DRL and GLM) at low prey densities. However, it
seems counterproductive to compare the two sets of models in the first
place, since the goals of the different approaches are dissimilar.
Rolling's (1956) goal in modeling proximate mechanisms of predation was
to generate a realistic model that could be used in a number of
theoretical studies. These studies include an analysis of functional
and numerical responses, and the relative advantages of digestion rate,
prey size or predator size. He also suggested that his model could be
used to test whether the mode of predation exhibited by a predator
maximized energy input or minimized energy output. Thus, his proximate
models required a complete knowledge of predatory behavior, but could
then be used to test other aspects of predation. On the other hand,
optimal foraging models attempt to predict the behavior that should be
expected from an organism based on our knowledge of the factors (or
currencies, Pyke et al. 1977) that may be important in the life of that

k-''

66

organism. The output of these models says nothing about the proximate
mechanisms that drive these behaviors. It is implied that the evolution
of proximate mechanisms should proceed in such a manner as to
approximate the optimal behavior patterns. The models are used to test
how well our understanding of the important factors account for the
evolution of the behavior (Maynard Smith 1978), irrespective of the
exact evolutionary pathway that culminated in the behavior.

All optimal foraging models rely on a set of assumptions. For
example, optimal foraging models have all assumed that the extraction
rate curve is constant. However, the DRL model proposes that the curve
may change with feeding rate. This change does not refute the
optimality approach, it simply requires a change in the assumptions
about the rate curves. In fact, an increase in extraction rate with
increasing prey densities undoubtedly will increase the net rate of
ingestion over the entire foraging bout. Thus predators that can
increase extraction rates will probably do better than predators whose
rates remain constant.

In a review of optimization theory, Maynard Smith (1978) said that
biologists need simple biological models that hold qualitatively in a
number of cases, even if they are contradicted in detail in all cases.
He implied that a qualitative fit to predictions will generally bring
the researcher closer to an understanding of the problem in question.
Unfortunately, generalizations can lead us to accepting models
prematurely. This problem is aptly demonstrated by this review of
models about partial prey consumption. In a sense, part of the question
concerns the definition of detail. For example, one could argue that
the expected correlation between intercapture interval (given a constant

67

density) and handling time is irrelevant detail, in which case the
difference between some of the models presented here is unimportant.
However, I would argue that one of the strengths of optimization theory
is that a quantitative prediction can be explicitly generated and
tested. A number of factors can contribute to the lack of quantitative
fit to a model. Three of the most important of these are constraints on
foraging behavior (including both physiological constraints and
ecological constraints), the failure to include important parameters
into the optimization model and the divergence from an optimal solution
using a satisficing criterion (see Simon 1955). The lack of fit to an
optimization model is bound to yield a greater understanding of the
system when these alternative factors are pursued. But this is a
reasonable pursuit only if models specifically suited to the system are
tested.

CHAPTER III
PARTIAL PREY CONSUMPTION BY ANTLION LARVAE

Introduction

In chapter 2 I addressed existing models of partial prey
consumption and compared two different types, mechanistic and optimality
models. The models were found to generate different predictions under
different conditions. Thus, although some generalizations may be made
concerning partial prey consumption, even qualitative predictions cannot
be formulated without restricting them to a specific system. This
chapter is a test of Griffith's (1982) "digestion rate limitation" model
and the optimality models from chapter 2, using antlion larvae as
predators. Antlions are particularly appropriate for testing the models
since Griffith's mechanistic model was derived with antlions in mind.

I will first present the predictions and tests of Griffith's model.
I then derive and test predictions of an optimality model appropriate
for the antlion system. Antlions construct conical pitfall traps in
sand that aid in the capture of arthropod prey. Once a prey item is
captured, the antlion injects digestive enzymes into the prey and
ingests the predigested material (Wheeler 1930). The exoskeleton is
never eaten, and therefore the antlion never consumes the entire prey.
As I show below, an antlion may also discard a prey before all of the
extractable prey biomass is ingested.

68

69

Digestion Rate Limitation (DP.L) Model; Predictions

Predictions

Griffiths (1982) showed that the rate at which an antlion ingests
prey increases as prey-capture rate increases. This is presumably due
to the fact that antlions produce digestive enzymes at a higher rate
when prey capture rate increases. Handling time was shown to decrease
with increasing capture rate (Griffiths 1982), which is consistent with
this model. Griffiths also suggested that the proportion of each prey
extracted should not change if prey are not simultaneously encountered.
He predicts that at relatively low feeding rates, antlions should simply
extract all they can from their prey irrespective of encounter rate.
The prediction, which originated from the work of Holling (1966), is
that partial prey consumption is caused by the filling of the gut. At
low prey densities, the gut of the antlion will never be full (if the
prey is small enough, as will be true in this experiment). Thus partial
prey consumption should be independent of prey density at low feeding
rates.

Methods

To see whether antlions followed the two simple predictions
generated by Griffith's model, we fed fruit flies (Drosophila
melanogaster, vestigial winged) to antlions (third instar Myrmeleon
mobilis; identified according to Lucas and Stange 1981) at four

Q

different feeding rates. Antlions were kept in the lab at 24 C for at

■y

70

least seven days prior to feeding and fed one fruit fly per day during
this acclimation period. The larvae were then fed one pre-weighed (to
+/- 0.00001 gm) fly per day (FS-1 ) for 3 to 5 days. For each fly, total
handling time was m.easured and the carcass was weighed immediately after
it was discarded by the antlion. The difference between the initial
weight and final weight was calculated as the extracted wet weight.
Percent wet weight extracted (predicted to be constant) was calculated
by dividing the extracted wet weight by the intial wet weight. Antlions
were then divided into one of three groups corresponding to the
remaining three feeding categories: FS-8 (1 fly per 3 hr), FS-24 (l fly
per hr), or YS-A8 (1 fly per 0.5 hr). For FS-8, antlions were fed from
4 to 7 fruit flies in a row; for FS-24 they were fed from 6 to 17, and
for FS-48 they were fed from 5 to 12 in a row. Each run (FS-1 then
FS-8, FS-24, or FS-48) was made with a different antlion. A pilot study
suggested that antlions take, on average, less than 30 min to handle a
fruit fly. Thus, the maximal feeding rate (FS-48) was set at a rate
just low enough to ensure that an antlion never encountered a fly before
it had finished the fly it was handling. Thus, by definition (see
Chapter 2), the antlion was feeding under low density conditions.

Multiple linear regression (GLM procedure in Barr et al. 1979) was
used to test the predictions of the DRL-model. Since handling time
decreases with increasing encounter rate (Griffiths 1982), this should
result in a negative regression coefficient on handling time when
regressed against feeding rate. This v.-as tested to determine whether
our species forages in the same manner as Macroleon quiquemaculatus, the
species studied by Griffiths (1982). We were unable to control for
variance in two factors: initial fly weight and individual variation

71

among antlions. The antlion effect is also compounded by the unequal
number of flies given to each antlion. In an attempt to account for
variance associated with initial fly weight and inter- individual
effects, these two factors were added to the regression model. Thus the
regression model used to test the hypotheses was

T^ = b^ + b^FS + b^I + b AN + e ,

where T = handling time,

FS= feeding schedule,

I = initial fly weight,

AN = antlion "name",

e = error term,

b -b^ = regression coefficients.
0 :)

This model allowed us to test for the effects of feeding schedule on
handling time, independent of the initial fly weight and antlion
differences. This same model was used to test for the effects of FS on
percent of each fly consumed (predicted to be constant) and total
extraction rate (predicted to increase with increasing feeding rate).
Percent consumption was transformed using the arcsin square root
transform (Sokal and Rohlf 1969). Antlion name was treated as a class
variable (see Barr et al. 1979), because it was nominal scale data.
Antlion weight was substituted for antlion number in the above
regression equation to determine if larger antlions could handle flies

72

more efficiently. This information was used to tuild the models listed
below under Optimality Model.

Results

As predicted, handling time decreased with an increase in feeding
rate (Table 3-1). Contrary to predictions, the percent of each fly
consumed dropped significantly with an increase in feeding rate (Table
3.1). Thus, antlions discard prey before they are totally empty even
under low density conditions. As prey capture rate increases, the
antlion extracts less from the fly, even though it never encounters two
at the same time. Thus, the predictions of Griffith's model are not
supported. Antlions appear to be regulating prey handling behavior at a
finer level than that predicted by the mechanistic models listed in
Chapter II. Below I test an optimality model that I derived to test
whether antlion feeding behavior was consistent with the prediction that
they were maximizing net energy during handling time. The optimality
model should not be treated as an alternative to a mechanistic model,
since they address different aspects of the same behavior. They appear
as alternatives here because they have (unfortunately) been treated as
such in the literature.

Deterministic Optimality Model

In chapter 2, I derived an optimality mod il of partial prey
consumption under low prey density conditions. The model predicted that

73

Table 3-1. Linear Regression statistics for handling time and extraction
of biomass from fruit flies. Table A includes antlion name as an
independent variable. In Table E, antlion weight was substituted for
name (see text). Each number is the probability that the regression
coefficient associated with the independent variable is zero. The sign
on the probability is the sign of the regression coefficient. AS % ext =
arcsin square root transform of percent wet weight extracted from fruit
fly. FS = feeding rate, initwt = initial fruit fly weight, lionnm =
antlion 'name', lionwt = antlion wet weight, extrate = extraction rate.

A)

INDEPENDENT VARIABLES

dependent

2
r

0.26

model

total

P>F for

variable

FS

initwt

lionnm*
.0001

df

79

df
552

F
2.1

m.odel

AS % ext

-.0051

+.0254

.0001

\

-.0001

+.0001

.0001

0.74

79

552

17.2

.0001

extrate

+.0001

+.0001

.0001

0.64

79

552

10.7

.0001

* Class variable, no slope estimatable

B) INDEPENDENT VARIABLES

dependant ^ model total P>F for

variable FS initwt lionwt r^ df df F model

AS % ext -.0234 +.0020 -.2898 .03 3 552 6.0 .0006

^h

-.0001 +.0001 -.0015 .46 3 552 156.3 .0001

extrate +.0001 +.0001 +.0020 .30 3 552 80.1 .0001

74 ■ >•

percent consumption should not vary with prey density. However, the
model assumed that the extraction rate curve (ie. extraction rate as a
function of handling time) remained constant with changes in prey
capture rate; this assumption is clearly violated here. Thus,
predictions concerning optimal prey utilization can only be generated
after the prey extraction rate curves are constructed.

All optimality models are couched in terms of a currency or
currencies (Pyke et al. 1977). Energy is by far the most common
currency, although others have been used (see Pulliam 1975; Rapport
1980; Greenstone 1979; Belovsky 1981; Westoby 1978; Owen-Smith and
Novellie 1982). I use energy as a currency here for two reasons: (l )
It is the most likely currency with which to estimate foraging costs to
the predator. Of course, if there is no cost to the animal in terms of
any currency, the forager should simply eat the whole prey or extract as
much biomass as it is capable of extracting. (2) At eclosion, antlion
adults weigh 50 percent of their pre-pupal weight (Lucas, unpubl.
data). Thus the weight of the larva at pupation will determine the
weight of the adult. If adult weight correlates with fitness in
antlions (as it does in many other insects, see Schoener 1971), an
increase in the net rate of energy intake as a larvae should increase
the fitness of the adult.

There are a number of variables that must be incorporated into a
model used to predict optimal prey utilization. These include: (l) an
extraction rate curve (here expressed as wet weight per min handling
time), (2) the conversion of extracted weight to calories, and (3) an

expression of the energetic cost of foraging. The extraction rate curve

, j-

must predict the biomass extracted at any given time during the handling {^.^

75

of a prey. From these three variables, the net energy intake (gross
energy minus the cost of extraction) can be calculated as a function of
handling time. From this function, the handling time that maxim.izes net
energy can be calculated. This optimal handling time can be compared to
the observed handling time to determine whether the criteria on which
the model is based, are good predictors of the foraging behavior. Thus,
the three variables above are descriptive models of foraging. These
models are then combined into a predictive model of optimal behavior.

Methods

(1 ) Extraction rate curves

Wet weight extraction curves were constructed for each feeding
schedule. Antlions were fed flies that were then taken from the antlion
after 2, 5, 10 or 15 minutes. These data were combined with extraction
data for uninterrupted feeding times (Figs. 3.1-3.4) to generate the
extraction rate curves.

From Table 3.1 we know that antlion weight and initial fly weight
affect the extraction rate. Initial fly weight additionally will
influence the percent of each fly consumed. The construction of the
extraction rate model was based on these relationships. I fit the data
to three types of curves: (l ) a Michaelis-Kenten function, (2) an
exponential function, and (3) the power function listed below. The
third model proved to be the best predictor of the data, and was
therefore chosen as the extraction rate curve for the optimality model:

-I

G.GCIG-

^

i

i

1

0. u002-
]

4

O.GCOl-^

i
1

J

o.ooooJ

76

:czn<^^'r.= i

3 „ □ ^

2 ^ :=-^ c

n ~ c

E □ C

n

a

O.OOOS-^ ff ^

0.0005 J 0° i n r- ,

o.aoo7-i ^ A ^ ^ X '^ ^ ^

0.0005 J

□

J n

3 a

•^ A -^ AA - "^

A q ^-.AAA't^A'^A A

> _^A i A

A -.- -1-4- -1-

+ ++■ +

A + -t-* J-

.^ ^ ^ -

S 0

0.0004-3 Q S * ■

j +

O.OOOoJ

n

0 3 6 9 12 15 le 21 2u 27 30 33 35 35 ^g yg yg

TIME

Figure 3.1. Biomass extracted from fruit flies (extrwt) at various
handling times (time) when the feeding rate was one fly per day.
Extracted weight is expressed in grams wet weight, time in minutes.
The symbols represent different initial fly weights: cross: initial
weight -rO.OOO? gm; triangle: 0.0007 gm (initial weight < 0. 0009 gm;
square: initial weight > 0. 0009 gm.

77

EXTRWT
0. 0013-!

"^EnsrH=3

0.0012-

0.001 H

J
0.0010 J

3

o.ooosJ
■i

4
I

1
o.oooeJ

0.0007

0.0006-1

0.0005-i

J

-i
o.oooyJ

0.0003-1

0.0002-1

0.0001-i

D G

n

CD

A

D D

.r A A .A

^'^ A 4-
A a"'

+ +

* + -I-

a.

0.000G-:

I ' ' ' '

0

10

15 20
TIME

25

30

35

Figure 3.2. Biomass extracted from fruit flies (extrwt) at various
handling times (time) when the feeding rate was eight flies per day.
Symbols as in Fig. 3.1.

1

J
J

0, 001 1-

FFEDSCM=24

c

D

3

0.0010-3

oos-3

0.0

0.00C8J

-1

0.0007-q

0.0006^

-i

0.0005-

° D, °

CC

^ _ =

A □ A i

□ ° ^ "^A K. A A '

A A ^ A^ A-^

A ° -IS ^AA ^ £

A ^A A

A A "^AA A *''+

^ ^ A ; \

4- 4-

0.0004-

0.0003J

0.0002-;

B

A

0.0001-

0.0000-

LO

15 20

TIME

25

30 3=

Figure 3.3. Biomass extracted from fruit flies (extrvt) at various
handling times (time) when the feeding rate was 24 flies per dav.
Symbols as in Fig. 3.1.

79

E^''°tv*

'^EEDSCH = 48

O.OOll-

°r.

i
-I

O.GOIO-:

H

0.0009-

0.0008^

0.0007J

3

0.0006-

0,0005-^

o.ooom-

□

^;

□ n

AA
^

A A A

A . A AA
A A
+ A A

0.0003J

3

0.0002-

0.0001-

O.OOOG-

10

20

TIME

25

Figure 3.4. Biomass extracted from fruit flies (extrwt) at various
handling times (time) when the feeding rate was 48 flies per day.
Symbols as in Fig. 3.1.

80

k k k

Gg(T) = XKk^-k^I ')(l-(k^-k^I ^k^W 8)- = ^T-L)) ^ ^^)

where G (t) = the gross gain, in terms of wet weight, extracted from

the fly after time T,
T = time starting from introduction of fly and ending at the

release of the fly,
I = fly initial wet weight,
W = antlion wet weight,
L = lag time from time of introduction of fly to time when the

antlion first begins to extract biomass from the fly,
c, k, -ko = constants,

1 o

X = conversion from wet weight to calories, see below.
The coefficients were estimated using a non-linear least squares method
(program NLIN in Barr et al. 1979).

(2) Wet weight to calorie conversion

Dry weight was measured on 25 fruit flies that had been weighed wet
then dried for 5 days at 60 C. The regression equation fit to these
data was used to convert wet weight to dry weight in the extraction rate
curves. Drj' weight was converted to calories using the following
conversion for fruit flies (Cummins 1967; Jaeger and Barnard 1931 ):

1 gm dry weight = 5797 cal.

v.^

81
(3) Metabolic cost

Both eating costs and waiting costs were measured as rate of oxygen
consumption using a Gilson respirometer. All measurements were made at
a temperature of 24 C. The respirometer was allowed to equilibrate
for one hr before readings were taken. Two cm of sterile sand was
placed at the bottom of 15 ml Gilson flasks into which antlions were
introduced. Waiting costs were measured after antlions constructed pits
in the flasks. Readings were taken every 30 min for 2 hr for each
antlion. The mean oxygen consumption rate from these data was used as
the waiting cost for the antlion. Measurements were used only if the
antlion did not move during the 2 hr measurement period. Eating costs
were measured by placing a live fly in a small open-topped vial glued to
the inside of the Gilson flask. The vial was sealed on top with a steel
ball then the sand and antlion were placed in the flask. When the
respirometer had equilibrated, the steel ball was lifted up with a
magnet. The fly would then jump out of the vial and fall into the pit.
Readings were begun after 2 min, which was enough time for the fly to be
killed by the antlion. I was unable to measure pursuit+pre-ingestion
costs. This cost was assumed to approximate C , since the level of
activity was similar for the two behaviors. Errors in making this
assumption will not be very major since the calculation of the optimal
handling time is independent of C (see below). All antlions were kept
in the lab for two weeks and fed two flies per day before calculating
metabolic rates. Each value represented a single individual.

To calculate an analytical expression of metabolic rate for the two
behaviors, the classic metabolic rate equation (cf. Kemmingsen I960):

82

MR = aW^ ,

where W = weight, and
a,b = constants,

was expanded to include a covariance term between waiting and eating.
By adding both terms to a single model, T am assuming that there is a
linear relationship between the log transformed metabolic rates for
activity (eating) and waiting. This is true of two other species of
antlions (M^ crudelis and M^ carolinus) for which I have a large data
set of metabolic rates (unpubl. data). The addition of both eating and
waiting costs increased the degrees of freedom of the model, which
should decrease the error in the estimates of each coefficient. The
expression is given here in the linear form used to fit the regression
coefficients:

In(MR) = ln(a) + b ln(w) + cE + dE ln(w) , (3)

where E = 1 for an antlion eating and 0 if the antlion was "waiting"
and
a,b,c,d = constants.

The constants were fitted using the least squares technique (program GLM
in Barr et al. 1979). Metabolic rate- was converted to calories by
assuming an R.Q. of 0.8 (which is the most reasonable R.Q. for an
insect of this type; K. Prestwich, pers. comm. 1982). Thus one ul
0 is equivalent to 0.0048 cal (DeJours 1975).

i

4<- "-

83

(4) Net extraction rate curve

Net energetic gain was derived by subtracting the energetic cost of
eating, pursuit + pre-ingestion, and 'waiting' (the non-foraging time
spent between prey handling and pursuit times). Thus net gain, G (T),
is

G^(T) = G^(T)-CpL-C^(T-L)-C^(T^-T) , (2)

where C = energetic cost of pursuit + pre-ingestion behavior,

C = energetic cost of eating,

C = cost of waiting, and
w

T = total time = T + waiting time.

(5) Derivation of Optimal Handling Time

Here I derive the optimal handling time predicted from equation 2.
The derivation assumes that prey arrive at fixed intervals and that the
extraction rate curves can be approximated by the deterministic model
given bj' equation 1 .

The constants from equation 1 can be combined as a shorthand:

k k k \

G (T) = I(k^+k21 ^)(l-(k^+k^I ^+k„W S)-c(T-L)^

= k (1-k, -^^-->). (4)

a D

Thus

G (T) = k (1-k -"^^-^^) - n (T-L) - C (T^-L) - C L ,

where C , = C -C .
a e w

W T

The optimal handling time is defined as the point where the partial rate
of change of G^(T)/T^ as a function of handling time is zero (cf.
Charnov 1976). Thus the optimal handling time is where

d G(T)/T^ -k^k^-"^^-^)(ln(k^))(-c) C^

9 (T-L) T^ " " 't^ " ° '

Replacing the left hand side of equation 5 with zero and rearranging we
get

C, -c(T-L)

— = k ln(k, )

Ck ^

a

Rearranging and taking the natural log of both sides

C

ln( ) = -c(T-L) ln(k ).

ck ln(k, )
a b

From the above equation, the optimal handling time (pursuit + eating

time = T ) is
op

C,

.ln(— J )

ck ln(k, )

T 5...J , . (6)

^ ln(k^)c

'^4

85

The optimal extraction weight and optimal percent extraction can be

estimated by replacing T with T in the equation for gross extraction

op

weight, G (t) (equation 1). Thus both the optimal (predicted) handling
time and the optimal (predicted) percent prey consumption can be found
using equation 1 and 6. Once the variables listed in equation 2 are
measured, the foraging response of tre antlion in terms of handling time
and percent utilisation can be compared to the predictions from the
models.

Results

1 ) Extraction Rate Equations

The gross gain model, G (t), produced a good fit to the data when
initial fly weight and antlion weight were included as variables in the
model. For all feeding schedules, over 99 percent of the variance was
accounted for by the model (Table '5-2). As Griffiths (1982) had found,
extraction rate generally increased with increasing feeding rate (Fig.
3.5).

There was a simple linear relationship between fly wet weight and
fly dry weight. Therefore wet weight can be multiplied by a constant to
convert to calories (Table 3.5). Estimates for eating and waiting costs
are also listed in Table 3.3.

. ;)

a e

•H tot

•T3

0) a>

S-j o

D. O

O

m .

0) o

>

u en

3 TO

u 3

ij

/-, 60

<D -H

4-1 0)

3 S

C

•I-i AJ

a 0)

3

i-i

(U >i

O. --H

14-1

Cfl

0) ..

•H e

1^ OC

O

.-H <r

CO o

u •

^ o

en en

Qi n

> 3

u

3 -U

O J=

oc

(U -H

4-1 (U

n S

V4

c

c o

O -rt

•

•H i-H

4J

4J 4J

X

CJ C

0/

CO <

4J

^

4-1

c

K! •

•H

Qj ^-v

4-1

-a

T2 X

0)

0) (1)

4-1

4J J-1

cn

U

•H

•H 0)

rH

T3 0)

(1) tn

en

^ ^--

u

&H

0)

1—1

4-1

<D

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87

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89

Table 3- 3- Fruit fly wet-weight/dry-weight regression equation and
metabolic cost equations for eating and waiting behavior in antlion
larvae.

Vet weight-dry weight regression
ln(dry weight) = -0.63 + 1.00 ln(wet weight)
dry weight = 0.23 (wet weight).
N=25, F=35, P<.0001, r^ = 0.602.

Metabolic Rate Equations

In(ME) = 1.983 + .910 ln(W) - .255 E - .499 (E ln(V)).

metabolism measured as ;al 0 /hr

N=28 (16 eating, 12 waiting), F=144, P<0.0001, r^=.947

Caloric Cost of Eating* (assuming R.Q. =.8)

eating C = 0.0043 W*'^''''

waiting C = 0.0077 W^^*^
w

* cost calculated as cal/min

90

2) Verification of the extraction rate curves

Before using the optimality model to predict optimal handling time,
I first tested to see whether the extraction rate curves would predict
observed values. If the descriptive rate curves failed to reproduce the
data, the predictions from the optimality model would obviously be
irrelevant. I tested this by comparing predicted values from the
extraction rate curves against observed values from antlions that were
allowed to feed until they discarded the carcass. This was done by
comparing the percent wet weight extracted by the antlion to the percent
wet weight predicted by the model for the time required to handle the
fly. This difference was tested statistically by arcsin-square-root
transforming the percents then subtracting the transformed values. A
one-sample t-test was used to test whether this difference was
significantly different from zero. Data from each feeding category were
divided into four fly weight categories, since percent weight extracted
is influenced by fly initial weight. In 16 comparisons (4 FS x 4 weight
categories), only 2 differences were significantly different at P=0.05
level (Table 3 -A). In fact, with 16 simultaneous tests, this
probability level is an over-estimation of the true alpha level. This
is because, by chance, one in 20 comparisons may be expected to be
significantly different using this test even if the two variables are
drawn from populations that are statistically indistinguishable. Thus,
the model can be treated as a fair estimator of the data.

91

Table 3«4. Optimal and observed values for handling time and percent
prey consumption by antlion larvae. T^p = optimal handling time. T-u ~
observed handling time. OAS %ext = observed mean arcsin square root of
percent extraction. PAS ^ext = predicted optimal arcsin square root of
percent extraction, ^extop = optimal percent extraction (from PAS
^ext). ^extob = observed percent extraction (from OAS text). M-0
AS^ext = difference between the observed percent extraction
(transformed) and the percent extraction predicted from the model for
the observed T^. Mean fly initial weights (initwt) and antlion wet
weights (lionwt) are also given. Fruit fly weight categories (WC) are
as follows: 5= less than 0.0006 gm; 7=. 0006-.0008 gm; 9= .0008-. 0010
gm; 11= greater than .0010 gm. Std=standard deviation.

FS/WC

?AS?ezt 0AS2ext 'extop 2eitc3b M-OAS'ext N initwt lionwt

1/5 29.7 ie.9*** 1.279 ;.22i*** .917 .88-1 -.C15ns V .OOC52 .0278

std 1.5 4.1 .001 .057

.072

.00005 .0094

U" 34.4 21.9*** ■■•230 1.23'*** .913 .889 -.CiOns 93 .00070 .0303

std 1.7 3.4 .001 .052

.054

.00005 .0118

1/9 39.3 25. 6*** 1.23' 1.233*** .916 .890 -.007ns 35 .00088 .034;
std 2.2 5.1 .0004 .064 .0^5 .00006 .0139

. ,' 1 ' -ic . i
std 3.2 5.3

3*. 4*** •.2S1 1.251*** .91s .901
.0002 .043

-.013'
.049

43 .00110 .C35~
.00006 .0108

8/5 23.; 1S.4*» 1.205 1.156ns .872 .838 .026n3

std 0.3 3.5 .001 .070 .076

8/" 24.- 21."*** '.209 1.216ns .875 .879 -.015n3

std Co 2.7 .002 .058 .060

8/9 26.7 23.''** 1.216 1.217ns .879 .880 -.009ns

std 0.7 4.3 .003 .050 .050

3 .00056 .0303
.00002 .0096

33 .0OC7O
.00007

24 .00088
.00005

.0064

.0292
.0064

.066

. 8°6

.381

305 ns
352

25 .00113
.00010

.0301

.0066

24/5 24.* '5.3*** 1.192 I. 139ns .363 .861 -.052*

std 1.3 2.3 .CO' .064 .079

24/" 25.3 19.2*** 1.195 1.153** .365 .846 .0C3ns

std 2.0 3.4 .001 .072 .068

24/9 27.0 21.5*** '.202 •.'0"ns .270 .867 -.Ol6n3

std 2.'' 4.5 .003 .076 .072

24/' • 29.4 24. 2*** '.220 '.133* .382 .360 .012ns

std 5.- 4. a .011 .085 .066

12 .00055 .0350
.00003 .0143

61

00069

00005

.0143

00033

.03=0

00005

.0151

001 1 1

.0339

OOC09

.0133

IT

92

Table 3-4 Continued.

^S/; 21.0 -6. --3 1.2-" • .Zii
3tc: :.l 3.3 .0CC3 .055

^^oTns 1 .0CO5a .0486

061

80

•0001 .0093

% ff^f**'--^'^""' -83 .537 -.01-3 32 .OOO-O .0.05

-° -^ .X,. .06o .063 .00005 .0139

-S/9 23.? 21.1*" 1.231 1.226C3 .389 .886 -.0^3c3 '0 CCCH-^ --a<;

3- 0.0 ..o ..04 .055 .060 .00005 .0132

1(d' f^' 7^°' ^;?" \;?f °^ -^CS .298 ■ .001n3 9.00112 .0368

•056 .00010 .0090

3td 8.5 1.6 .019 .062

ns - difference v.a- significant at ?>0.05
- difference significant at 0.05<?<0.01
•* - difference significant at O.O1<?<.0O1
***- difference significant at p<.001.

W'

93
3) Tests of the optimality model

Using equations 1 and 6, the optimal handling time, and optimal
percent extracted were calculated for each fruit fly fed to the
antlions. These values were then tested against the observed values for
both parameters. As shown above, the percent utilization of each fly
decreased significantly with increasing feeding rate. Handling time
also decreased significantly. How do these changes in foraging behavior
compare to the calculated optimal behavior?

With the exception of FS-1 , the optimal percent extraction was
generally statistically indistinguishable from the observed percent
extraction. I should point out again that the true alpha level of 0.05
is at a P value less than 0.05 due to the use of simultaneous tests.
Only for FS-24 weight category 7 was there a highly significant
difference between predicted and observed values.

Observed values for FS-1 deviated significantly from predicted.
Antlions appeared to extract less than they should have at this feeding
rate. However, we should expect antlions fed at the lowest rate to
extract more from their prey, not less. This underscores one drawback
of using the exponential model. The model assumes that the antlion can
never fully drain its prey.

If the model was modified to reach an upper limit, how would this
affect the predictions? The most parsimonious threshold to set is the
percent consumption exhibited by antlions fed at FS-1 . Antlions fed one
fly a day should be closer to this threshold than any antlion on the
other feeding schedules. This is because the antlion should extract
more when fed at the lowest feeding rate than when feed at higher

i

..a'

94

feeding rates. Thus if there is an upper limit to the amount of
extractable bioraass taken from a prey, this limit should be reflected in
percent utilization at the lowest feeding rate. If the percent
extraction shown at FS-1 is used as the threshold value, the predictions
from the other feeding rates are unchanged. Percent extraction was
lower in the weight categories of the three remaining FS than it was in
?S-1 . The only exception was weight category 5 for FS-48, for which
there are only four data points. More importantly, the predicted
percent extractions were all either similar to or less than the observed
data from FS-1. Thus even if there is a threshold percent utilization,
which is set at the FS-1 levels, this threshold will not affect the
predictions from the other feeding schedules. The reduction in percent
consumption noted in Table 3.1 meets a predicted reduction assuming
antlions forage so as to optimize net rate of energy intake. At the
lowest feeding rate, antlions are either under-utilizing their prey, or
they are simply extracting as much biomass as they can from the prey.

A comparison of optimal versus observed handling time shows a
different relationship (Table 3.4). Antlions generally threw their prey
away a few minutes earlier than predicted. This seems to contradict the
percent consumption comparisons discussed above, since discarding a prey
item early should result in a lower percent extraction from that prey.
Obviously, there was enough variance in percent consumption to offset
this potential reduction in optimal prey utilization. Antlions appear
to discard their prey earlier than might be expected from the
deterministic conditions to which they were subjected. We might expect
handling time to affect the energetics of foraging in a number of ways.
Handling time obviously influences the energetics of foraging on a

95

single prey, as shown above. However, there may be other effects of
handling time on foraging energetics that may select for a reduction
in the time spent eating a prey.

Stochastic Optimality Model

In chapter 2, I analyzed a situation where prey show a Poisson

distribution. In this case, there is always a chance that prey arrival

overlaps with handling time, even at low prey densities. If the forager

can return to a prey after dropping it to capture another prey (as is

true of antlions), then the forager may do better, in terms of net

energetic yield from the diet, if it discards its prey earlier than

predicted by the low density model. This statement is true even if the

mean encounter rate falls in the low prey density category (as defined

in Chapter 2). This tactic is essentially bet-hedging, since the

forager should discard its prey in anticipation of the arrival of the

next prey. The reduction in handling time is generally adaptive only if

there is a difference in the capture probability while handling a prey

(p ) compared to the capture probability while empty handed (P ).
w o

Given this fact, it is possible that the reduction in handling time
exhibited by antlions may be attributed to a bet-hedging tactic due to
the stochastic distribution of inter-arrival times. Here I show that
capture probability does change depending on whether or not the antlion
is handling a prey. I then derive a model that incorporates a
stochastic prey distribution. This model illustrates that a reduction
in handling time can be attributed to the bet-hedging tactic.

96

Prey Capture Probability

(1 ) Methods

To test whether antlions are more efficient at prey capture when
empty handed than when eating a prey, I fed carpenter ants (Camponotus
floridanus) to third instar M^ mobilis. Antlions were given either one
soldier or one worker ant, and then given the same type of ant 10 min
later. Time to capture or time to escape were recorded for most
feedings. Insufficient dexterity on my part accounts for feedings in
which no times were recorded. All ants were caught at Gold Head Branch
State Park and fed to antlions less than 30 hr after capture.

(2) Results

Antlions are more efficient in prey capture if they are empty
handed than when they are eating prey (Table 3.5). This is undoubtedly
due to an increase in the time required to catch a prey when the antlion
is eating. It may also be due to a decrease in ant escape time due to a
difference in pit structure. The pits of antlions that are eating prey
have less fine sand on the walls and will also have a lower slope than
pits of antlions awaiting prey. Both these features affect prey escape
time (Lucas 1982).

Thus, the antlion will gain on average less energy from a prey
organism that arrives when the forager is eating than from a prey that
arrives when the antlion is empty handed. This can be explained as
follows: The benefit associated with increasing handling time is due to

97

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98

an increase in bioraass extracted. However, the costs of increasing
handling time are twofold. There is an energetic cost of eating. There
is also the cost associated with the potential loss of an encountered
prey which would otherwise be available if the antlion were empty
handed. This latter cost may influence antlion behavior even if the
energetic costs associated with increased handling time are negligible.
This is demonstrated in the model derived below.

I should point out that different prey types exhibit different
efficiencies in terms of escape capabilities. This is demonstrated in
the differences in capture probability between worker and soldier
carpenter ants (Table 3-5). Vestigial winged fruit flies never escape.
Thus, the type of prey that antlions forage on influences the relative
cost of increased handling time.

Derivation of Stochastic Optimality Model

For the following derivation, I assume: (l) that prey show a
Poisson distribution, (2) handling time is constant, (3) the forager
always pursues prey that come within striking distance (antlions pursue
all prey dropping into the pit), (4) when a prey arrives while the
forager is eating, the forager drops the prey it is eating and pursues
the live prey, (5) the forager returns to all prey that are dropped
before it has extracted the optimal amount, and (6) the forager has
enough time to extract the optimal amount from all captured prey. 1
also assume that the cost of returning to any stored prey is negligible.
The model is set up this way because antlions will discard prey they are

!f TT^'

99

eating, to capture another prey. After the second prey is discarded,
the antlion will return to the first prey (personal observation).

The model will be developed in several stages. The purpose of the
model is to calculate the relative effect of the overall capture
probability and the difference in this probability between prey-handling
and empty-handedness. This will tell us how much of an effect prey
capture probability should have on the optimal handling time. First, I
calculate for a given prey encounter rate how many prey are captured
while the forager is eating and how many are captured while it is empty
handed. These values will vary with the capture probabilities while
empty handed and while prey-handling. The difference between the
values, when converted to energy, will be used to predict the effect of
capture probability on handling time. From these equations, the energy
derived from the prey and time spent pursuing and eating can be
determined. The net energetic return from the diet is calculated as the
difference between the energy extracted from the prey minus the energy
spent during the entire foraging bout. From these equations, the
optimal handling time can be derived. These stages are similar to the
stages used to derive the deterministic model.
Let

N = total number of prey handled in time T ,
\ = number of prey eaten while handling prey,
N^ = number of prey eaten while empty handed,

T = time spent with a single prey = t + t ,

P e

P^ = probability of capture while eating,

Pq = probability of capture while empty handed, and

X = encounter rate.

100

21^ and N^ can be estimated by calculating the time spent v^ith each
prey and the time spent empty handed (waiting). The expected number of
prey arriving in these time periods is the arrival rate times the length
of the time intervals. For prey arriving while the forager is eating,
this yields

TN = time spent with all captured prey,
A TN = number of prey arriving while handling prey,

N = P X TN.
w w

The time spent empty handed will be the total foraging time (T ) minus
the time spent eating minus the time spent pursuing prey that are
ultimately missed:

(1-P^)ATN = number of prey missed while handling prey,
(l-P )^TNt = time spent pursuing missed prey,

T^-NT-(1-P^)X TNt = time spent empty handed,

N is calculated in the same way:

X (T^-NT-(l-P^) A TNt ) = number of prey arriving while empty
handed,

N^ = P X(T^-NT-(1-P ) X TNt ).
0 0 T w p

So ,

N = N TN = P XTN + P X(t^ - TN - (1-P ) X TNt ) .
0 w w 0 T w p

BSS<»rr'.~

101

Solving for N

N = NP A T + P X T - NP A T - NP X (1 -P ) X Tt ,

W 0 0 0 w p

P :^ T P > T"
0 '^ T 0 ^ T
J = =

1-P XT+P XT+P X Tt (1-P ) 1+P^XT+P X ^Tt ( 1 -P )

woopw dopw

where

P^ = P - P .
d 0 w

The gross benefit from the diet (B (T)) is the benefit derived from

g

each prey (see eq. 4) times the number of prey eaten:

G (T)P XT^

B (T) = G (T)*N = i— -2 1 .

^ ^ 1+P^ Xt+P X'^Tt (1-P )

Foraging costs are the energetic cost spent on each captured prey plus

the cost spent in pursuit of missed prey. I will assume that the

metabolic rate during eating (C ) equals the metabolic rate during

pursuit (C ) and that both are linear functions of time:
P

NT = time spent on captured prey,

C NT = energy spent on captured prey.

M = (l-P ) XTN = number prey missed while handling,

M = (1-P ) X(T„-NT-(1-P ) XTNt ) = number of prey missed while
0 0 T w p - ''

empty handed ,

I :'-■ '

102

T -tM +tM = time spent on missed prey,
C T = energy spent on misser prey.

The net foraging benefit will be the gross benefit minus the energy
spent on prey minus the energy spent waiting for prey. This waiting
cost is

T - T -TN = T^ - NT - (1-P ) X TNt = waiting time,
i m i w p w I

C = metabolic rate while waiting,
C^(T^ - T^ - NT) = waiting cost.

Total energy spent in foraging (E ) is therefore

E- = C T + C TN + C (T^-T -TN) = (C -C )(T +TN) + C T„
I em e wTm e w m w T

The net benefit per unit foraging time (B (T)) is

n

B (T) B^(T) - E-

----- = — ^ -'- . (7)

The optimal time spent per prey item (T ) is generated by taking the

op

partial differential of equation 7 with respect to handling time (T-L)
and setting this partial differential to zero. After some manipulation,
the partial differential of equation 7 becomes

103

" = V^'"'^^Vi*Wi"^ - Vi - h '

Z. = P , X + P X^t (1-P ) ,
1 d 0 p w '

Z^ = ck ln(k, ) ,
^ a b

Z^ = C.(l+t P, X) - C.t ^X^(1-P )(1-P )
3 a p d dp wo

Rearranging

k^Z, * Zj * Z^^jT

The reader should instantly recognize the fact that this is a
transcendental function, for vfhich no algebraic solution for T can be
found. However, T can be estimated using Newton's Method of
approximation (cf. Kaplan and Lewis 1971):

f(x )

V1 = ^n - ------ • ^5)

f (x^)

where

X = nth estimate of the variable (here variable is handling
t ime ) ,

X ^, = improved estimate of variable, ■ '

n+1

f(x ) = value of function at x (here this is the right side of

eq . 6 ) ,

f'(x ) = derivative of f(x ).

n n ,1

i

M

104

I iterated equation 9 ten times for each combination of ? , P and FS.
^ o w

This number of iterations gave an estimate of the optimal handling time

correct to within at least .001 min. The optimal handling time was
derived with the coefficients used in equation 6. The encounter rate

was set to the rate at which fruit flies were fed to the antlions ( X =
1/24, 1/8, 1 and 2 flies/hr).

Results

As shown in chapter 2, the optimal handling time is influenced by
capture probability. However, when P equals P , the optimal handling
time is relatively independent of the value of the capture probability.

The model predicts that as P diverges from P, , the optimal

'^ 0 "

handling time should decrease (Fig. 3.6). This means that the relative
cost of holding on to the prey increases as the probability of missing a
prey while eating increases. This effect is highest when prey encounter

rate is large.

A reduction in handling time below the level predicted by the
deterministic model is predicted if antlions normally forage on prey
that are able to escape occasionally. Unfortunately, I have no
information on the overall probability of escape for the prey of M^
mobilis in the wild. Based on a comparison of the mean handling time
exhibited by antlions and the predicted values from equation 8, the
difference in mean capture probability (Pq-P„) should be small if the
antlions are foraging optimally.

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107
General Discussion

Antlions exhibit two responses to changes in prey encounter rate:
(1) they reduce handling time, and (2) they decrease the percent
extracted i-rom each prey. The second response contradicts the
predictions of the DHL model (Griffiths 1982) and the GLM model (Rolling
1965; Johnson et al. 1975; also see chapter 2). This reduction in
percent extraction was predicted by an optimality model derived for
antlion prey utilization. Thus, antlions appear to be making decisions
that are intrinsically more complicated than the simple
"eat-until-you-are-full" rule posited by Griffiths (1982) and Johnson et
al. (1975). A result of these decision rules is that antlions increase ^-
the net benefit derived from their diet compared to the mechanistic
rules.

The reduction in handling time with an increase in feeding rate is
caused by three factors. Two of the factors are predicted by the
deterministic model that describes the conditions of the experiment.
These are: (l) the increase in extraction rate, and (2) a decrease in
the percent extracted as encounter rate increases. However, the
decrease in handling time was larger than expected based on just these
two factors that were included in the model. One additional factor that "'
may explain this discrepancy is the variance in percent extracted.
There may be no energetic difference between handling the optimal amount
of time (predicted by equation 6) and handling the amount of time
exhibited by antlions. If this is true, then we might expect antlions

- i

to lower handling time if factors other than the energetics of eating - j

.;'>
are influenced by a change in handling time. (3) For antlions, an 1

108

increase in handling time may decrease the probability of catching the
next prey. This potential loss of prey should lower handling. Thus,
giyen a range of handling times over which there is no real difference
in net energy derived from eating, the antlion should choose the lowest
handling time. Of course, if there is a large difference between
capture probability while prey handling and when empty handed, then the
antlion may discard the prey much earlier than predicted by the
deterministic model (Fig. 3.6). In this case, antlions should reduce
the energy derived from any single prey to increase an expected total
net energy derived from all prey.

In two similar papers, Sih (1980), and Cook and Gockrell (1978)
suggested that partial prey consumption could be studied by the use of
optimal foraging theory. More specifically, they suggested that the
marginal value theorem (Charnov 1976) could be used as a tool to help
understand the specific behaviors exhibited by animals that partially
consume prey. The insight in these studies was not that animals may be
thought of as optimizers, since by now the idea is a relatively old one
(cf. Schoener 1971), but that the techniques developed in other areas
of foraging behavior may be useful in understanding partial prey
consumption in particular. Unfortunately, the marginal value theorem is
not as generally applicable as has been suggested, but this problem is
easily, although somewhat tediously, corrected (see chapter 2). On the
other hand, the mechanistic models proposed by several authors (chapter
2), failed in my study on antlions, even though the antlion system was
the primary focus of one of these models (Griffiths 1982). I do not
mean to imply that animals never use simple mechanisms, such as gut
filling, as a cue to cease feeding. Predators such as damselfly larvae

109
(Johnson, et al. 1975) and Rhodnius bugs (van der Kloot I960) probably
do. However, the specific mechanisms that direct foraging behavior in
animals should be evolutionarily labile, since different mechanisms will
work well under certain conditions and poorly under others. For
example, gut filling rules for partial prey consumption may be poorly
adapted under conditions where prey density is low, but prey body size
is large (Cook and Cockrell 1978). It will also be poorly adapted under
conditions where prey encounter rate is relatively high, but prey body
size is small. Thus, animals living under different conditions should
evolve 'rules-of- thumb' that will optimize prey utilization under those
conditions. In other words, we should expect the behavior of an
organism to evolve such that the mechanisms underlying foraging maximize
the fitness of the organism. In a sense, the mechanisms should to some
degree conform to the output of the behavior, since it is the behavioral
phenotype on which selection acts. Animals have been consistently shown
to exhibit foraging behaviors appropriate to the conditions under which
they are tested. This fact suggests that natural selection is a factor
in refining the prey utilization of foragers. The technique of
developing optimal foraging models to study foraging behavior has been a
major contribution in our ability to address this field of study.
Animals have also been consistently shown not to conform precisely to
the predictions from these models. Partial preference is a perfect
example (Krebs et al. 1977). Handling time in antlions is a more
specific example. But the optimization technique allows us to set up
testable hypotheses to study further refinements of our perception of
these systems.

^

CHAPTER IV
THE ROLE OF FORAGING TIME CONSTRAINTS AND VARIABLE
PREY ENCOUNTER IN OPTIMAL DIET CHOICE

Introduction

A variety of optimal foraging models has been proposed within the
last decade (see Schoener 1971; Pyke et al. 1977). Many of these
models have dealt specifically with diet choice and generate the same
general predictions (Charnov 1976b; Pulliam 1974; Werner and Hall
1974): (1) prey types should be ranked according to the ratio E./h.
(the energy derived from a prey divided by its handling time), (2) a
prey type should be added to the diet solely on the basis of the
absolute frequency of encounter with higher ranking prey types, and (3)
prey types should either be eaten on every occasion or not at all.
However, there are conditions under which these predictions change. For
example, prediction 3 may be violated if the diet choice of the forager
is limited by nutrient constraints (Pulliam 1975; Rapport 1971, 1980;
Westoby 1974). Prediction number 2 is violated when prey recognition
time is relatively long or when handling time varies through learning
(Elner and Hughes 1978; Estabrook and Dunham 1976; Hughes 1979; Xrebs
1978).

The most widely used foraging model is a variation of Holling's
disk equation derived by Charnov ( 1976b). This model has been used to

•.*j

fJ

study diet choice in a variety of organisms, including insects, ■>."^«

••J

'■i^

110

Ill

gastropods, birds and mammals (Charnov 1976b; Dunstone and O'Connor
1979; Hughes 1979; Xrebs et al. 1977; Palmer 1981). The model is
based on the premise that foragers choose prey types that maximise the
net rate of return of the limiting currency (or currencies) (see Pyke et
al. 1977). For this discussion, I will assume that there is selection
on the predator to choose a diet which maximizes the net rate of energy
intake. This expected rate of intake is the total net energy gained
from foraging (E) divided by the total time spent foraging (T ):

E X.E.P.
E i ^ ^ ^

T^ 1 + Z A.h.P.
T .111

1

(1)

where E. = energy derived from prey i,

X. = encounter rate of prey i during the search time,
h. = handling time of prey i,
P. = probability of attacking i when i is
encountered.

From this model, Charnov (l976b) proved that prey i should be added to
the diet in order of rank until

E. E *

-i < -^, (2)

h. T*

1

where E^*/t* is the maximum net rate of energy intake. Thus, decisions
concerning the addition of prey items to the diet should be based on th«
effect of these decisions on the energetic return from the entire diet.

B-.:-

112

There is another way of phrasing the decision rule derived from
(1). Instead of asking what prey items would increase the rate of
energetic return from the diet (as in eq. 2), the effect of each
foraging decision can be calculated more directly (see Cost Model
helow). Although the two methods are mathematically similar, I will
show that an emphasis on the costs associated with the foraging decision
can lead to two further predictions about diet choice. The first is
that the time available for foraging may affect diet choice. The second
is that if some of the assumptions of equation (1) are relaxed, then
several characteristics of the prey distribution may be exploitable by
the forager to increase net benefit from the diet. I show that under
these conditions, a forager's perception will affect the predictions
generated from an optimization model.

The Cost Model

Assume that a forager feeds on two prey types, one of low quality
and the other of high quality. Assume also that the forager encounters
prey individually, handles them one at a time and recognizes prey
instantly. When both prey types are present at high densities, there
should be a high probability of missing the opportunity to capture high
quality prey if the predator were to pursue and eat low quality prey.
In fact, the "cost" of taking poor quality prey can be evaluated as the
potential loss of high quality prey items missed while handling the poor
prey. Conversely, the predator should take a poor prey item, regardless
of prey density, if it could be assured that no better prey would be
missed while handling this poorer prey. Any decision involves the

113

coramitrnent of a given period of time to a particular course of action
(Brockmann, et al. 1979). In foraging, a decision is a costly one only
if the forager misses a more profitable course of action otherwise
available during that period of time.

I will first derive this "cost" for a one-prey situation, then
expand it to include a second prey type. Let X. be the expected
harvest rate of prey i over foraging time T (when every prey
encountered is eaten). I will assume that prey show a Poisson
distribution with a mean of X.T (where X. is the encounter rate of
prey i over T^). For a diet composed of one prey, the expected number
of prey taken in time T is

X.T = X .T^ - M. .X.T^, (3)
1 T 1 T 11 1 T

where M = the expected number of other prey i missed while handling

the captured prey i during h. (the total amount of time spent
capturing and eating i, then resuming search).

The expected harvest rate will be the encounter rate discounted by the
rate at which prey are missed while handling captured prey. This will
hold for both sit-and-wait predators and active foragers as long as prey
show a Poisson distribution throughout the foraging bout. Solving for
X.

X = i — - . (4)

1 + M. .
11

Since M

ii '^i*'^i ^°^ randomly distributed prey, then

114

E, \ , E

1 i

X.S. = -i-i-

1 + A .h. T„

1 1 1

for one prey type. Thus, equation 1 and equation 3 are equivalent even
though they address the foraging decision in different ways.

If more prey are added to the diet, then equation 3 expands to

X. = X. - X.M. . - Z X.M. .,
11 1 11 J ij'

where M^ = the number of prey i "missed" while feeding on prey j. If

prey i is the highest quality prey, then in a two-prey system, M is

ij

the relative "cost" of adding the low quality prey (j) to the diet, in
terms of the alternative decision of specializing on prey i. More
specifically, "cost" is a function of the energetic return from the
poorer prey item minus the expected return from the higher quality prey
that was missed while handling the low quality prey. To simplify the
formula, let the handling times and encounter rates of the two prey be

equivalent (so that h. = h. = h and A . = A . = X ). Therefore

1 J 1 J

M = M = M. Then for a two-prey diet, the expected energetic
return from the diet is

E A (E. + E.)

— = X.E. + X.E. = ^-

T ^ ^ '^ "^ 1 + 2M

Under these simple conditions, the predator sliouli specialize (eat only

•Va

prey i) if the energetic return from a single-prey diet is higher than ' '*'^

w^r ^

115
the return from a diet of both prey types;

A,i, X(E. + E.)

^

or

1 + M . . 1 + 2M

11

IT _ V

If we assume that M. . = A .h., Charnov's model can be similarly

ij 1 J

expanded to

E.h.

M. . > — -^-i! . (5)

^^ E.h. - E.h.
1 J J 1

As before, the "cost" of eating prey j is a function of the number of
high quality prey missed while handling lower quality prey (M. .).
Equation 5 shows that M. . is a function of the relative energetic
content and handling times of the two prey types. So if the forager
monitors only mean encounter rates and prey are distributed randomly,
then the rule on which this cost function is based generates the same
three general predictions as those listed above. But under what
conditions might this cost function vary? I will focus on two such
conditions. (1) If the foraging bout becomes exceedingly short due to
constraints on foraging time, the number of higher quality prey
otherwise available to the forager while handling a poor prey will
decrease (here foraging bout is defined as the amount of uninterrupted
time a forager spends searching, pursuing and eating prey). The basis
of this statement will be discussed below. (2) If the forager had more

116

information about the distribution of its prey, then it may be able to
judge variance in 14. .. In this case, partial prey preference is
predicted.

(1) Foraging Time Constraints

Total foraging time for a variety of foragers is often broken into
relatively short bouts. For example, a bird in an area of high
predation risk may often stop foraging to scan for predators (Caraco
1979; Powell 1974). Other foragers, such as some aquatic insects, may
often interrupt foraging to flee from predators (Sih 1980b). Also, the
foraging of intertidal animals, such as predatory gastropods, will be
disrupted by tides (Menge 1974). If these foraging bouts are confined
to an interval of time t, the optimal diet may be affected by the
magnitude of bout length (contrary to the predictions from equations 1
and 5), as demonstrated below. I am assuming here that the factor(s)
that constrains t is extrinsic to decisions about diet choice.

For the following model, I assume that the forager captures only
one prey per bout. This may occur either because t is sufficiently
short to allow the forager to capture and eat only one prey, or because
the forager captures only one prey regardless of the length of t. For
example, house wrens and many other birds foraging to feed nestlings
might capture only one prey per foraging bout before returning to the
nest (Bent, 1964). Here the "cost" associated with foraging on lower
ranking prey is not the number of prey missed while handling these prey
(as in equation 5), but rather a function of the probability of missing

117

any higher ranking prey within the time left for foraging in the bout,
given that a lower ranking prey has been taken.

The length of t can affect foraging in two different ways. If the
foraging time available per bout must include handling time, then the

time available for searching will include only the interval [0,t J,

*

where t is the end of the foraging bout (t) minus the handling time

*
for prey i (h.). No prey item should be caught in the interval [t ,tj

since the predator would not have enough time to handle it.

Alternatively, if the predator can extend handling time past t, such

that it can attend to other requirements while handling the prey, then

the time available for searching includes the entire interval [0,tj.

However, no prey are available after t and the cost of foraging, in

terms of handling time, will terminate at t. In either case, the time

available for foraging may affect diet choice.

?or a two prey system, let g be a higher (good) quality prey and

let b be a lower (bad) quality prey. Also let P(Z <t ) be the

probability that a g arrives before the end of the searching period t .

Here Z is the first arrival time of a good prey. The expected energy
g

(E(g)) that the forager gains in N number of bouts while foraging only

*

on g is the total number of bouts that yield a prey (p(Z <t ) * N)

times the energy derived from one individual prey g (E ), or

g

E(g) = P(Z <t ) ' N • E

If handling cannot extend beyond t, the total time (T^) spent foraging
is

\^/^

118

T = (t* + h ) • N

Thus,

E(g) P(Z <t ) ' N* 3 P(Z <t )' E

_ g g _ g e
^ ^

(t + h ) ' N t + h

m

'T ^' g' -' g

If prey show a Poisson distribution, then

*

# t -X z

P(Z <t ) = / X e ^ dz ,
g 0 ^

or

*

P(Z <t ) = 1 - e ^ .
g

If a second prey is added to the diet, then the probability of
encountering prey g first must be discounted by the probability of first

encountering a lower ranking prey (b) within the time available for

»

foraging. This is represented as P(Y,<Z <t ). Here Y, is the first

b g b

arrival time of a bad prey. This probability will be the "cost" of
foraging on prey b. Thus, the probability of eating prey g (P(Z )) in
a two-prey diet is

P(Z ) = P(Z <t ) - P(Y, <Z <t ;
g g b g

The "cost" of foraging on prey b is

■J
.^1

119

*

* t y -A z -\y
P(Y,<Z <t ) =/ /• X e ^ X, e dzdy
^ S 0 6 ^ b

-X t* > -t*(X^+X )

= (1 _ e ^ ) - (--i— ( 1- e ^ ^ )) .

The first part of this expression is the probability that a g will

* *

arrive in t (P(Z <t )). The second part is the probability that a g

arrives first given that an arrival of any prey type has taken place.

The difference is the probability of missing g as a result of capturing

*

b. As t increases, this probability increases. Thus, as the length

of the foraging bout increases, there is an increase in the probability
that the forager will miss a high ranking prey if it takes a low ranking
prey.

The probability of eating a high ranking prey (P(Z ) ) in a
two-prey system can now be expressed in terms of t :

, -t*(X +X, )

P(Z ) = — -^§— - (1 - e ^ ^ )

^ X + X^
a b

*

If t is the amount of time available for foraging on b, then the

probability of eating b in a two-prey system can be derived similarly:

so a diet of two prey types would yield

E(g,b) P(Z^) • E P(Y, ) • E,
g g , b b

+

T^ t t i

120

The predator should specialize if the net energy derived from eating
only g is higher than the energy derived from eating both g and b:

=g ' ^^V'*^ > \ ' ^^h^ ' h ' ^^\) •

This is also trae if handling time can extend beyond t, in which case t

*

is substituted for t since the time available for searching is t.

Rearranging the above inequality

Eg • (P(Zg<t*) - P(Zg)) > E^ • P(Y^) ,

where P(Z <t )-P(Z )=P(Y. <Z <t ) is the probability of missing a
high ranking prey given that a low ranking prey has been caught. This
is analogous to M. . from equation 5- So the predator should
specialize when

P(Y^<Z <t ) > -— . (6)

b g g

g

Equation 6 generates two predictions that are different than those

of equation 5. (1 ) Diet choice depends on the amount of time available

*

for foraging (t ). As available foraging time decreases, diet breadth

should increase, because the "cost" of catching low-ranking prey (i.e.

the number of high-quality prey missed) decreases as t decreases, ''J

independent of the encounter rate of either prey type. The "cost" is

■- c"

121

(in part) weighed against P(Y, ), which also decreases with decreasing ,.■ •'

* *

t . However, ?(Y <Z <t ) decreases more rapidly than P(Y, ) with

#

changes in t . This can be illustrated by analyzing the ratio

P(Y^<Z <t )/P(Y^) as a function of t . Since the ratio decreases

* »

with decreasing t (Fig. 4.1), diet breadth should expand as t

decreases. (2) In this model, diet choice depends on the encounter »"''

rates of both low- and high-quality prey (X and X , respectively).

In other words, the addition of prey to the diet is affected not only by

the encounter rate of higher ranking prey, but of all prey. As the

density of low-ranking prey increases, the diet should shift from -.C":

generalist to specialist. This may seem somewhat counter-intuitive, -^

since the total number of prey available to the predator increases with

an increase in the density of low ranking prey. However, the

probability of missing a high ranking prey also increases with an '-.

increase in the density of low ranking prey. A numerical example

illustrates the fact that the "cost" of eating low ranking prey \

increases more quickly than the benefit associated with eating more low :,

ranking prey (Fig. 4.2). However, changes in X will have a larger

g

effect on diet choice than changes in X (Fig. 4.2).

I have implied here that the beginning of the foraging bout is set.
The analysis has addressed changes in the time to the end of the bout.
The same predictions hold if the end of the bout is fixed and the
forager continually updates the criterion expressed in equation 6 based '
on the amount of time remaining in the bout. However, bout length in
this case would be the interval [t ,t J where t is present time. *'

0 '.'i|

The diet breadth should increase as the length of the interval [t ,t ] 1

0 "

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Squation 6 explicitly models foragers that eat one prey during the
foraging period. However, even if the forager eats a number of prey
during a foraging bout and if the length uf the bout is set at some time
t, then there will be a point at which the time remaining in a foraging
bout allows the forager to eat only one prey. Thus, the bout consists
of (1 ) a period in which a raulti-prey per bout model expresses the net
energetic return from foraging, and (2) an additional period when a
single-prey per bout model predicts diet choice. Since the models
generate different quantitative predictions, the net rate of energy
intake from any one diet will reflect some combination of the two models
if the foraging time is long enough for the predator to take more than
one prey. If the number of prey types included in the diet does not

change within the foraging bout, we can predict that the diet should

*

become broader as t becomes smaller. This can be demonstrated by

*

simulating the effect of t on the difference between a specialist and

generalist diet (Fig. 4.3). If the length of t is known by the
forager, then we might expect it to exhibit two different choice
patterns under the two different conditions of equations 5 and 6.

(2) Prey Distributions

Most foraging models presented to date (Charnov 1976b; Pulliam
1974; Werner and Hall 1974) assume that prey are distributed randomly
and are encountered one at a time. More importantly, these models
implicitly assume that the forager cannot monitor variance in the number
of prey i missed while handling prey j (M^ ) . in this section, I
demonstrate that if these two assumptions are relaxed, and if as a
result, a forager could monitor this variance, then we should expect

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some degree of partial preference to be exhibited. In other words, a
prey type that is added to the diet will be attacked less than 100
percent of the tims when it is encountered. In fact, partial preference
is predicted from the models derived above. For foraging bouts during
which only one prey is taken, equation 6 predicts that diet choice
should change as the time remaining in the foraging bout approaches
zero. Over many bouts, a forager will exhibit partial preference due to
this change in the optimal diet. In the following discussion, I
consider bouts in which more than one prey is taken.

Two important levels of variance may exist in M. .. The first is
the variance in expected encounter rates of different prey types. The
second is the variance in the actual arrival times of individual prey,
in essence, the variation within the expected encounter rates. I will
illustrate the importance of these sources of variation to diet choice
by using two simple examples.

First let two prey types be distributed in runs of three each where
the time between encounters of each prey is equal. In this case, a
forager might see the following prey in order of occurrence:

g^g^g^b^b^b^g^g^ggb^b^bg ,

at times

rnrnmrnrriryifyimmm m m

^r2^5 4'5'6^7 8^9-10^11^12 '

where T - T = 1. Let the benefit from prey g (S ) be 3 benefit

XIX g

units and E = 1. The handling times for both prey are equal: h =

130

h^ = 1 . So if the predator captures one prey it will miss the next
one. If the forager eats only g, it can start with g^ , then eat g ,
then eat g^, etc. In the first 6 time units (T -T,) it would collect
6 benefit units. If it started with g^, it would collect 3 benefit
units in the first 5 time units, but thereafter it would collect 6
benefit units for each set of 6 time units (starting with g ). So as
the foraging bout gets infinitely long, the forager will get
approximately 6 benefit units in 6 time units. If the forager eats both
g and b, it can start with g^ , then eat g , b^, g,, and so on. With
this sequence it gathers 7 benefit units in 6 time units. It can start
with g^, then eat b^ , b , g , etc. This sequence produces 5 benefit
units in 6 time units. So on average, the predator will take in
(7/6 + 5/6)/2=1 benefit unit per unit time. Now suppose the forager
knows the temporal placement of several prey simultaneously. It should
eat any g that it encounters since it can do no better, but it should
capture b only if no better alternative is available. If the b is
followed by another b, it should eat the first prey, because the time
spent waiting for the second prey (assuming the predator decides not to
take the first prey) should be incorporated into the handling time of
the second prey. Here S/h for the first prey is 1 and E/(h + 1 ) for the
second prey is 0.5. On the other hand, if the second prey is a g, then
E/(h + 1 ) for the second prey is 3/2 so the forager should skip the
first prey (b) and eat the second prey (g). This means that the
relative values of b , b , and b^ are different (actually b = b /
b_). Thus, if the predator could monitor the relative cost of taking
each prey, it would take either of the first two of a prey b sequence
but never the last. This diet would yield 7 benefit units in 6 time

!•>

131

units. Thus, if the animal's foraging decisions were consistent with
the Charnor equation then it should on average take in 1 benefit unit
per time unit. Under the simple conditions of the example above, if the
predator can monitor several prey at a time, it should always do at
least as well as Charnov's rule (eq. 2) and may do better when using a
decision rule that takes into account the relative temporal position of
prey.

If there is any variation in the encounter rate of prey, the effect
of averaging on diet choice (as illustrated above) will be important
even if the predator can not simultaneously detect a number of prey.
This can be demonstrated by simulating the relative effect of the time
the animal estimates M. . on the net energetic intake when the mean
encounter rate of prey varies. I simulated three different decisions
for a two-prey system: (1) take both prey, (2) take only the higher
quality prey, and (3) take lower quality prey when

E,h,
<

M. . < ^-i.

E.h . - E .h.
1 J J 1

I varied the encounter rate of high ranking prey (X ) as a sine

cr
o

function with a range of zero to x (where x varied from 0 to 3.0) (see
Fig. 4.4; note that the abscissa is the mean value of the range of
X ). The simulations illustrate two important properties about
decision rules: (l) the forager always does better if it
instantaneously monitors M. . than if it follows a rule in which it
bases diet choice on the average encounter rate in the foraging bout,
and (2) if the encounter rate of high ranking prey varies, we may expect

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134

to see the diet change through a foraging bout. In other words, the

predator may exhibit "runs" of differing diet choice.

These simulations also illustrate that the amount of time the

forager takes to estimate X will affect the energetic intake from the

diet. For example, if the forager simulated in Fig. 4.4 was to

estimate A based on one cycle (360 time units), then it would base

diet choice decisions on the mean encounter rate from the cycle. For a

range of 0 to 2, the mean would be 1.0. Using the mean encounter rate,

Charnov's equation (2) suggests that the simulated animal should switch

from generalist to specialist at a mean X of 0.5. Yet the

g

simulations show that this decision underestimates the best switching
point. In fact, the switch from generalist to specialist should be
closer to a mean X of 1.50 (see Fig. 4.4). The basis of this
underestimation can be viewed in terms of M. .. With X constant,
most b prey will be taken when the arrival rate of g is low and when
'''^ij ^^ "'"^"" ''^^^ ^'^^ arrival rate of g is high, the probability of
taking b will be small. Thus, summing over all prey b taken, the number
of higher ranking prey missed while handling the lower ranking prey will
be smaller than expected when the arrival rate of g is constant. In
other words, M is reduced when variance in prey encounter rate
increases. This reduction shifts the optimal switching

point to higher levels of X .

O

A second factor is important when the mean encounter rate varies;

the calculation of E/T from equation 2 is overestimated for both the

generalist and specialist diets. If X is varied as a sine function

g

(as above), then the amplitude of the sine wave can be used to simulate
variance in prey encounter rate. As the amplitude increases, the net

135

energy derived from any of the simulated diets decreases (Fig. 4-5).

This is because the energetic intake rate (E/T^) as a function of X

g

changes more rapidly at low values of X than at high values. When

the mean value of a is used to calculate E/T , this nonlinearlity

will cause E/T to be overestimated. This can be shown as follows:

let E , E , h , and h, be constants. Also let A be a constant
g b g b b

(since the level of X will not affect diet choice anyway under the
assumptions of the model). The energetic intake for a two-prey system
is

E EX + E^ X ^
^ g__g b b

T 1 + h X + h X
T ""g g \ b

The change in E/T with changing levels of X is

9 f El E (1 + h^ X ^) - E^h -X ^
___ — = -i -— - Li__^ . (7)

9A I'T^ (1 + h^A ^ + h A )^
g I T J b b g g'

This function decreases monotonically with increasing A (Fig. 4.6).

If Charnov's rule (eq. 2) is considered an estimate of E/T^, then

i

under most circumstances the forager will overestimate the net energetic
return from both the generalist and specialist diets if it requires a
relatively long time to estimate A . Thus, the time the forager takes

o

to respond to changes in prey densities should vary inversely with the
rate of change in mean prey density. Since a number of predators have
been shown to respond to changes in prey density (Elner and Hughes 1978;
Giller 198Q; Goss-Custard 1981; Jaeger and Barnard 1581; Krebs et al.
1977; Sih 1980a; Werner and Hall 1974), it is not unreasonable to

Figure 4.5. Simulations of net energetic intake from three foraging
decision rules when the variance in encounter rate of high ranking prey-
changes. Prey encounter for both high quality and low quality prey was
random with a mean encounter rate of Xg and a^^ (respectively) . X^
varied every 10 time units following the equation: (r- sin (max. time
in interval)+l) • (mean g over 360 time units). Thus, r is a function
of the variance in Ag. Mean X =1.0 for all simulations. The threshold
Mj^^ (see equation 5) for these parameters is 0.5. All other parameters
as in Fig. 4.

(i«r'-T« '

137

UJ

SYMBOL

X X

0—0

FORAGING RULE

GENERALIZE (EAT g AND b)
SPECIALIZE (EAT ONLY g)

FOLLOW M

ij

EQUATION 5

RULE (SEE
IN TEXT)

0.2

0.4

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0.8

1.0

VARIATION OF ^g (r)

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suspect that at least some foragers may be able to monitor M . This
should hold true especially for predators that hunt by sight for prey
which arrive simultaneously (such as planktivorous fish or insects,
salamanders, birds foraging in swarms of insects, etc.)-

Discussion

The models presented here generate the following predictions: (l)
if foraging is confined to a given period of time, t, then as t
decreases, diet choice should become less specialized. (2) Diet choice
should reflect variance in prey distributions if this variance affects
the variation in the relative "cost" of capturing and eating low ranking
prey. This cost in a two-prey system, M. ., is the number of higher
quality prey missed while handling lower quality prey. Short term
changes in prey encounter rates should be incorporated in the decisions
the animal makes about diet choice. This is because decisions based on
long term averages will yield a lower return than decisions based on
short term averages. Thus, variance in the number of prey added to the
diet should increase as variance in prey distribution rises. As a
result of the variance in diet composition, partial preference should
also increase with an increase in the variance in prey distribution.
(3) Diet choice will also be strongly affected by the number of prey the
predator can monitor at any one time. If several prey can be perceived
at once, then the forager should be able to estimate M. . nearly
instantaneously. In this case, the optimal diet changes rapidly and as
a result, partial preference should be exhibited.

Predictions (2) and (3) above illustrate that how the forager

141

estimates prey distributions will strongly affect the optimal diet.
Clumping of prey should select for foraging strategies that restrict the
time required to estimate local prey density (e.g. through short term
rules of thumb as suggested for granivorous birds by Barnard 1980).
Common prey should allow predators to make foraging decisions based on
accurate estimates of M. . if more than one prey can be seen during the
time of the prey choice decision. The importance of these predictions
is illustrated especially in studies on switching in which a predator
tends to forage disproportionately on prey that are most abundant
(Murdoch 1969; Murdoch et al. 1975). Several authors have shown that
switching should occur when some learned foraging response is exhibited
by a forager (Hughes 1979; Murdoch 1969; Oaten and Murdoch 1975). But
in some cases, switching may be predicted if the forager can quickly
monitor changes in prey density, irrespective of any learned behavior.
Visual predators such as fish (Bernstein and Jung 1979; Olmstead et al.
1979) and salamanders (Jaeger and Barnard 1981) have been shown to
respond quickly to changes in prey density and may therefore be able to
monitor M closely (equation 5). One prediction that stems from
these models is that predators should either use visual estimates of
prey density (as shown for salamanders by Jaeger and Barnard 1981), or
should restrict the amount of time over which they estimate prey
encounter rate if visual estimates are impossible. Thus, the length of
the forager's "memory window" (sensu Cowie 1977) should be set small
enough to minimize the effect of averaging unless the costs of
misjudging encounter rates outweighs the costs of averaging. If
foragers are estimating M. . based on short term rules, then as
variance in prey encounter rate increases, the number of prey included

4

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142

in the diet will vary. As a result, foragers should exhibit a higher
degree of partial preference (see Pulliam 1974). Clearly the foraging
decisions exhibited by animals may be much more subtle than foraging
theory has allowed. To understand exactly the nature of diet choice
decisions, we need much more detailed data on foraging behavior.

Three studies support the first prediction. Jaeger et al. (1981)
analyzed feeding selectivity of salamanders foraging on large and small
flies. They showed that a salamander on its own territory
preferentially fed on large flies, but on unmarked territories or on a
conspecific's territory, the salamander exhibited no preference for
either fly type. They also showed that a territory owner spends most of
its time foraging. A salamander on no territory or one on a
conspecific's territory spends a large amount of time either marking or
displaying submissive postures, and feeding is interspersed between
these two behaviors (R.G. Jaeger, pers. comm.). The conclusion Jaeger
et al. (198I; pg. 1100) reached was that salamanders sacrificed
"initial caloric yield until they had established marked territories and
then (switched) to a higher sustained caloric yield". Jaeger et al.
(I98I) viewed the foraging bout as the entire time spent in an area.
However, the decision of diet choice should not be based on time spent
in other activities but solely on time spent foraging. When analyzed in
this way, on average only one fly was taken between bouts of marking
(when the predator was foraging equally on large and small flies), not
including the time spent in a submissive posture. When the predator was
foraging selectively, the foraging bout was uninterrupted. The

salamander data can be reinterpreted as follows: the length of the ,jj

foraging bout was long when the salamander foraged on its own territory '

^

143

but extremely short when on another territory or on an unmarked

territory. In the latter two cases, foraging bouts were broken up by

marking or displaying activities. Thus, if the salamander were to

maximize its net rate of energy return while foraging, we would predict -' ;

that it should increase its degree of specialization with an increase in

foraging bout length. "*»

Freed (1981 ) analyzed diet choice by house wrens foraging for
nestlings. When in the presence of a nestling predator (the fox snake), . ' '
the wrens decreased the amount of time they spent foraging and fed their
nestlings smaller prey than when no predator was present in the area. '\^-

Thus, as foraging bout time decreased, diet selectivity was also . ' ;''

reduced. Freed (1981 ) suggested that, based on these data, the decision
rules under which the wrens were choosing their prey changed when a : r

predator entered the area. However, if the rule governing diet choice
is to minimize the "cost" of foraging on low ranking prey and thus ' ■'■

maximize the net rate at which energy is taken during the foraging bout,
then the models presented here suggest that the decision rules with or '•

without predators are identical. The change in the wren's foraging
behavior is consistent with the predictions from optimal foraging theory
generated from the models presented in this paper.

An exceptionally tractable system for evaluating the effects of
foraging time on diet choice is found in the intertidal predators whose .'.
foraging bouts are constrained by the tidal period. For example, the
predatory intertidal snail, Acanthina, searches only during low tides
(Menge 1974). The models I presented above are particularly suited for
analyzing this species since only one prey is taken per searching bout.
Menge (1974) showed that these snails exhibited strong selection for

4

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specific prey early in the tidal phase. As the tide rose and the
available searching time decreased, selectivity decreased (less
preferred prey were taken when encountered). Here again, the optimal
diet changes as foraging time changes . Thus, the relative value of any
given prey type also changes as a function of foraging time.

It is often useful to think of foraging behavior (as illustrated by
the data from Freed, 1981, Jaeger et al. 1981 and Menge 1974) as
hierarchically organized (Dawkins 1976; also see Mesarovic et al.
1970, Hassell and Southwood 1978, and Gass and Montgomerie 1981). In
other words, the total set of behaviors that an animal exhibits can be
viewed as organized into levels. The decisions made in each level are
constrained by information from higher levels. We can dissect foraging
behaviors and explicitly model foraging only if we confine the models to
work within the constraints imposed by higher order effects. The
strength of this organization lies in the fact that foraging models,
when viewed as specific levels in a hierarchical design, become more
robust than non-hierarchical foraging models. Equation (6) specifies
the optimal diet when the foraging bout length is set at some time
interval [0,t]. This interval may be interpreted as the boundary of the
foraging model. Decisions regarding diet choice are based on factors
that exist only within these boundaries. At a broader level, the length
of the foraging bout is affected by constraints on the total time
available for foraging (t). In the above examples, the constraints are
predation pressure, intraspecif ic interactions and environmental
effects. Many other studies have pointed to similar factors affecting
foraging behavior and foraging time (Baker et al. 1981; Barnard 1980;
Garaco 1979, 1980; Hervey 1969; Milinski and Heller 1978; Norberg

145

1981). The effects of these constraints can themselves be modeled; the
output of such time-constraint models would define the boundaries of the
foraging models. The foraging model (viewed in terms of one level in a
hierarchy) need not be supplanted by another set of models that
specifically describe a new set of decision rules for the forager when
other factors impinge on foraging. Rather, this hierarchical
organization of models generates predictions that should describe
foraging behavior under a wide range of conditions and, therefore, may
be of more heuristic value than specific models based on unique
conditions.

Summary

Predictions generated from optimality models are inescapably based
on a number of assumptions. The predictive value of these models is
often determined by the degree to which the behavior of an organism fits
the underlying assumptions of the model. I analyzed optimal diet choice
by relaxing two sets of assumptions made in previous optimality models.
(1 ) Foraging-bout length fthe uninterrupted time devoted just to
foraging), generally treated as infinitely long, was shown to affect
optimal diet choice. For many foragers, foraging-bout length may be
considerably shortened by the presence of predators, or by physical or
social features of the forager's environment. A model was derived which
incorporates a short bout length into the decision of diet choice. The
model predicts that animals should become more catholic in their diet
choice as the amount of uninterrupted foraging time decreases. This
prediction appears to be supported by three studies from the literature.

1^-'

146

Jaeger et al. (1981) showed that salamanders incorporated more lower
ranked prey (small flies) when they were either on the territory of a
conspecific or on no territory as compared with prey choice when they
were on their own territory. In this case, foraging time was
uninterrupted when the salamanders were feeding selectively, but
continuously interrupted by submissive behavior and marking behavior
when no diet choice was exhibited. Freed (1981) showed that wrens
foraging for nestlings spent less time per foraging bout when a predator
was in the nesting area than when no predator was in sight. The
reduction in foraging bout time correlated with a reduction in prey size
fed to the young. The foraging time of some intertidal snails was shown
to be confined by the length of the low tide cycle (Menge 1974). As the
end of the low tide drew near, the snails decreased diet selectivity.
Thus, as the remaining time available for foraging decreased, the
predator exhibited a lower degree of prey selection. (2) Variance in
prey encounter interval was shown to affect the utility of classical
optimal diet models in predicting the optimal diet. Charnov's ( 1976b)
model is shown to over-estimate the net rate of energy intake when mean
encounter rate varies about some fixed level. Predictions from
Charnov's model are incorrect over some ranges of prey encounter rates
due to this over-estimation. I show that as variance in prey encounter
rate increases, the time over which the forager estimates prey encounter
rate will have a strong effect on the ability of the forager to maximize
the net rate of energy intake. Foragers that forage on patchily
distributed prey should use a shorter amount of time to estimate prey
density than foragers that prey on evenly dispersed prey. Thus, animals
that are capable of reducing the time required to estimate prey density

147

(for example, visually hunting predators in areas of high prey density)
should alter their diet in response to local variation in prey density.

For this type of forager, as variance in prey encounter rate increases,

fluctuations in the number of prey types in the diet will increase. As

a result, there should be an increase in the degree of partial prey

preference exhibited by the forager with increasing variance in prey
encounter rate.

CHAPTER V
OPTIMALITY, HIERARCHIES, AND FORAGING

Introduction

The conditions under which animals forage are generally complex,
even in the laboratory. The animal's perception of its environment, its
internal state (sensu Sibly and McFarland 1976), and numerous
environmental factors may influence their behavior. This complexity has
made some authors doubt the validity of optiraality models as a tool in
the study of behavior (Simon 1956; Schluter 1981; Zach and Smith
1981). The issue is an important one, since if this complaint is •• . ,

correct, then the approach should be abandoned in favor of a technique
(or techniques) that will be more useful in predicting the behavior of :-^'
animals in nature. At one level the approach is clearly useful, and it
has been used successfully in each of the four previous chapters of this
dissertation. However, none of the existing optimality models ,^ :;

adequately address the complexity of foraging behavior. I think there
are two aspects to this problem. First, there is the question of
generality: how robust are the predictions generated from optimality ''"■•:■'■
models? Second there is the problem of measurability: if foraging '■

decisions are based on environmental conditions, is it possible to »)

'i
measure these conditions adequately? I argue that hierarchical modeling

can be used to solve both problems: (l) hierarchies are useful in

148 ;.-^.-'

149

deriving generalizations about foraging decisions (or any other
decisions), and (2) the use of hierarchical models may aid in decreasing
the number of variables that need to be measured, and thereby reduce the
problem of measurability. I begin this chapter by briefly discussing
the major questions that are addressed when using optimality models, and
then briefly describe characteristics of hierarchical design. After
defining the question and the technique, I can then discuss generality
and measurability in foraging studies. The first four chapters of this
dissertation will be viewed in light of this discussion.

Optimality

In the introduction I suggested that optimality is a tool that can
be used to address specific questions concerning behavior. However, the
question is not whether animals are foraging optimally, but whether our
perception of the salient features that influence foraging is
sufficiently broad to predict an animal's behavior. Maynard Smith
(1978) discussed this issue in detail. We can use optimality theory in
three ways: (1 ) to formulate our perception, (2) to generate testable
predictions from this formula, and (3) to re-evaluate our perception and
add to it where it appears to be inadequate. Step three yields new,
testable hypotheses. Where does this fit in the study of behavior?

Tinbergen (1969) has suggested that the complete study of behavior
involves four major topics. The first two topics include an analysis of
the proximal mechanisms in behavior: (l ) the causation or control of
behavior, and (2) the development of behavior in the individual. The

y 'iit

150

se:oad two topics deal with ultimate factors in behavior: (3) the
adaptive significance of behavior in relation to an animal's
environment, and (4) the evolution or phylogenetic origins of behavior.
Optimality theory is particularly appropriate to the study of the third
topic, the adaptiveness of behavior. Although causal factors are
important, I argued in Chapter III that the study of optimal behavior
patterns should be made independently of the causal mechanisms, since
selection should act at the level of the phenotype (the behavior) and
not on its component parts. With optimality we are interested in how
the animal adaptively responds to its environment, and the relationship
between these adaptations and behavior. Natural selection is assumed.
This assumption allows us to compare the response of an animal in a
given situation with an expected response based on the selective
pressures that we feel are important. A poor fit to predictions
suggests that we are either incorrect or incomplete in our evaluation of
the selective pressures.

The mechanics of using optimality theory involve the construction
and testing of mathematical or graphical models. The model is
constructed around a maximization parameter, or currency (Pyke et al.
1977). The use of a given currency assumes that maximizing the net
accrual rate of that currency will maximize fitness. Some models have
been constructed that address the minimization of a cost function (of.
Sibly and McFarland 1976). However, minimization of net cost can be
considered identical to a maximization of net benefit (McCleery 1978).
Although most foraging models have used energy as a currency (e.g.
Krebs et al. 1977; Goss-Custard 1981; Hughes 1979; Pyke 1978), some
studies have shown that energy may not be appropriate for some animals

151

(Rapport 1981; Westoby 1978; Greenstone 1979). These latter studies
also indicate that more than one currency may be necessary to describe
the maximization parameters (also see McCleery 1978; Sibly and
McFarland 1976).

In this dissertation and in all other studies of optimal foraging
behavior, the measurement of the maximization parameter is determined in
absolute terms. For example, a joule of energy expended in prey capture
is considered to be equivalent to a joule spent traveling to a resource
patch or spent during mating behavior. Odum (1982) has suggested that
the importance of any absolute unit of currency will correlate with the
hierarchical position of the subsystem in which the currency is spent
(see "Hierarchy" below). For example, in the ecosystem one joule of
puma has a higher "embodied" energy than a joule of grass. This is
because the number of joules entering the system (in this case from
solar input) required to generate a joule of puma is vastly larger than
that required to make a joule of grass. In behavioral systems, a joule
spent in early spring searching for mates after a winter hibernation may
represent a larger investment than a joule spent searching for food the
previous fall. This will be true if the ratio of energy expenditure (or
the cost of the behavior) to fitness associated with that expenditure is
lower when searching for mates than it is when searching for food. Due
to relative differences in currency between subsystems, Odum (1982)
suggests that energy should be expressed in terms of embodied energy and
not in absolute terms. The use of embodied energy may be critical in
comparing costs and benefits of hierarchically distant subsystems, but
is less important when the analysis deals with any single level as is
the case with most optimality models. This issue is discussed further
below (see "^laximum Power and Foraging Hierarchies").

-..,. 152

If several maximization parameters are required to describe
behavior, two alternatives are available. (l) All parameters can be
combined into a single model of behavior. This requires that the costs
and benefits associated with all parameters be measured in the same
units. ?or example, the male smooth newt, Triturus vulgarly, performs a
long and elaborate courtship on the bottom of a stream (McFarland 1977;
Houston et al. 1977). During parts of this courtship sequence he builds
up an oxygen debt and must surface to breath. The male must correctly
time his courtship and breathing in order to pass his three
spermatophores successfully to the female. There are three different
behavioral components in the courtship sequence, two of which include
different segments of the display, and the third is the rate of
spermatophore transfer. These behavioral components were combined into
a single model with several parameters that predict oxygen debt at any
given time during the courtship. The model was based on the assumption
that the newt should maximize the probability of fertilization. Thus
each parameter was couched in terms of its effect on the fertilization
probability. (2) Instead of a single model, each parameter or set of
similar parameters can be treated separately in a series of nested
models. Each submodel represents a separate behavioral decision. This
is the hierarchical approach. The single model approach can be thought
of as a subset of the hierarchical approach, since only one behavior or
decision is addressed. In the newt example, the regulation of courtship
activity as a whole could be examined with a higher level model that
regulated the category of behavior that was exhibited by the animal,
such as mating, feeding, or inactivity. One advantage of the

153

hierarchical approach is that each submodel can use a different
currency. For example, the influence of predator pressure can be
modeled through the use of time budgets. Diet choice can be modeled in.
terms of calories derived from foraging during the time allotted for
foraging behavior. Thus the response to predator pressure can be
examined in terms of time, while diet choice can be couched in terms of
energy (or nutrients). I used this approach in Chapter IV (also see
Lucas 1983), but I only modeled a single hierarchical level (diet
choice) by treating decisions concerning time budgets as a simple
constraint on diet choice.

Since hierarchies have been used for a wide variety of research
fields, including ethology (Dawkins 1976; Tinbergen 1969), ecosystem
dynamics (Odum 1983), evolution (Gould 1982; Lewontin 1970), foraging
patch utilization (Charnov and Orians 1973; Gass and Montgomerie 1981;
Hassel and Southwood 1978), and the expression of sex ratio patterns
(Frank 1983), it is important to explain briefly the type of
hierarchical design I am using.

Hierarchy

Mesarovic et al. (1970) treated hierarchical systems as a series of
input/output subsystems, each at a given level of organization. The
input can be thought of as cause, the output as effect. In an
optimality model, the inputs are the variables, the subsystem is the
model (expressed in the appropriate currency), and the output would be
the behavior that resulted from the solution to the model. Mesarovic et

;^"'

154

al. (1970) suggested that for the hierarchical design to be useful, the
functioning on any level must be as independent as possible of the
functioning on other levels. Thus each l-,vel must be represented by a
unique model, although levels will be connected by input constraints or
feedback loops (Fig. 5.1 )• Each subsystem in some way controls or
intervenes in the input of the system below it and may receive feedback
information from that lower system, although there may be no feedback
loops in some subsystems. In foraging, subsystems can use either a
maximization parameter or a constraint parameter. Maximization
parameters in the wren foraging example include such factors as energy,
or the risk of predation. The type of subsystem that represents
maximization parameters can be thought of as a specific level of
behavior. More than one maximization parameter may be included in a
single model, as in the case of energy and nutrient co-requirements from
the diet (e.g. Pulliam 1975; Rapport 1971). Constraint parameters
include factors that will modify or limit the expression of the behavior
associated with lower-level subsystems. Thus, the constraint subsystems
will always lie above a subsystem in which maximization parameters are
evaluated. Examples of constraint parameters include tidal cycles or
daylight cycles that restrict foraging time, and weather patterns that
may influence migration or social behavior. Including constraint
parameters in subsystems changes the original definition of the
hierarchy proposed by Mesarovic et al. (1970), since these subsystems
are not input/output systems in the context of behavior. However, this
expansion of definition will give a model more flexibility, as I show
below.

^'^' ^

155

SYSTEM

input

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LEVEL N
SUBSYSTEM

intervention

input

input

input

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SUBSYSTEM

intervention

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/ output

/ cutout

Figure 5.1. A simple hierarchical system (after Mesarovic et al. 1970)

'•■•^1

156

As stated above, adjacent subsystems may or may not include
f ee Iback loops. A feedback loop involves the processing of information
from lower levels in the formulation of decisions at higher levels.
These higher-level decisions will then constrain the lower-level
processes. Thus, constraints on lower-level processes will be affected,
in part, by the result of previous constraints on those levels. For
simplicity, I will call a feedback subsystem a subsystem that receives
information feedback from a level below it (as in level 2 in Fig. 5.1),
and a non-feedback subsystem is a subsystem that receives no information
from the level below it (as in level 1 in Fig 5.1). By this definition,
the bottom layer of any hierarchical system will always be a
non-feedback subsystem, since it does not receive information from a
system below it.

Maximum Power and Foraging Hierarchies

The optimization principles of behavioral systems proposed here
assume that the evolution of decision rules will proceed such that
animals will maximize the net accrual of limiting resources or
currencies. By maximizing the net harvest rate of these resources, an
animal should be able to optimize investment in activities that maximize
fitness. The decomposability of systems into a hierarchically nested
set of subsystems, each of which can be treated as an optimality model,
is implicit in the use of hierarchical design in this dissertation.
Thus optimization principles apply to the whole as well as to each of
the parts. The optimization of time budgeting implies an optimization
of behavior within each behavioral bout. The optimization of diet

157

choice implies the optimization of prey utilization. The optimization
of patch choice implies the optimization of searching behavior once in
the patch. Higher level decisions impose constraints on lower level
decisions, and these constraints are to some extent conditional on the
behavior exhibited at lower levels. Information ("performance feedback"
in Fig. 5.1) is passed from lower levels to higher levels through the
behavioral output of the lower levels. The constraints imposed on lower
levels can be thought of as feedback from this information flow.

Another type of systems approach has been used in the analysis of
ecological and economic systems. Odum (1982, citing Lotka) proposed
that the design of systems should follow the maximum power principle.
In other words, systems should develop designs that maximize the flow of
useful energy. If the maximum power principle is consistent with the
optimization principle, then the two fields have developed convergent
paradigms, which may lend credence to both. If they are mutually
exclusive, then two possibilities exist. One of the paradigms must be
incorrect, or there may be no emergent properties that define all
systems.

On the surface, the definitions of optimality and maximum power are
identical. The maximization of currency intake by an organism is the
same as the maximization of the flow of "useful" energy, assuming that
any currency can be converted to energy equivalents. However, the two
approaches appear to differ over two major issues. The resolution of
these differences is a worthwhile goal. One major issue is control.
Optimization principles treat control as unidirectional. Higher level
subsystems constrain lower level systems. For ecosystems, Odum (1982)
states that lower level subsystems (for example producers) will benefit

158

if they increase the feedback from higher-level subsystems (for example
consumers). Here feedback is the flow of energy from higher- level to
lower-level subsytems which influences the energy accrual properties of
the lower-level subsystem. Odum (1982) has shown that this feedback
mechanism will generally increase the ability of the lower-level
subsystem to draw, or accrue, energy from a source. This viewpoint is
reflected in the "maximum power" definition of net energy, which is the
productive output (used oy higher subsystems) minus the feedback from
higher subsystems. The "optimality" definition of net energy would be
the intake of energy from lower systems minus the energy spent (or fed
back) to acquire that intake.

I define control as the capacity for constraint. In Chapter IV,
decisions concerning time allocation set the boundary for, and therefore
controlled, diet choice decisions. Here control is based only at the
higher level. If control is generally based at all levels of a system,
including control of higher-level subsystems by lower-level subsystems
as implied above for ecosystems, then the modeling of behavioral
decisions would be difficult without knowing the relationship between
production and feedback. This relationship requires a model of the
higher-level subsystem. Since there will always be a higher-level
subsystem, multilevel control is an intractable problem in behavioral
systems. On the other hand, if control is mediated through feedback
from high-level systems to lower-level systems (as suggested by
Mesarovic et al. 1970), then optimality and the maximum power principle
may be functionally equivalent. Further work is clearly called for.

A second issue that may differ between the approaches is the issue
of embodied energy. Should the energetics of foraging be modeled using

. JMTS"-

159

absolute or relative measures of currency? The answer lies in the use
of the currency. If currency flow is compared between different
subsystems, embodied energy may well be the most correct unit of
comparison. However, if the flow or accrual rate of currency is to be
determined through simulation or model dynamics, then absolute currency
measures are required. Here embodied energy is implicit in the dynamics
of the model.

Hierarchical models of behavior, as presented here, involve a
series of nested decision models about how the animal should behave.
Each model is based on the appropriate currency or currencies. Of
course, the setting up of a model does not mean that it will be of any
use in the study of behavior. Before I discuss the role I think
hierarchical models can play in the study of behavior, I will first
review the contributions of earlier, non-hierarchical models. This is
not intended to be a thorough literature review, since both optiraality
and optimal foraging have been reviewed elsewhere (Cody 1974; Maynard
Smith 1978; Pyke et al. 1977; Schoener 1971). I then discuss these
examples in the light of the generalities we are attempting to draw from
studies of foraging behavior.

Non-Hierarchical Foraging Models

There are numerous types of non-hierarchical models of foraging
behavior, each of which addresses a specific aspect of foraging. Pyke
et al. (1977) suggest that most optimal foraging models fall into four
major categories: (I) choice of which food types to eat, (2) choice of

160

patch type to forage in, (3) allocation of time to different patches,
and (4) patterns and rates of movement while foraging. In their review
of optimal foraging studies, Pyke et al. (1977) note that these four
decisions are assumed to be independent. However, the categories are
not independent since they naturally fall under a hierarchical
organization (Gass and Montgomerie 1981; Hassel and Southwood 1978;
Charnov and Orians 1973)- ?or example, decisions about patch choice
should reflect the type of prey expected to be taken from that patch
(diet choice) and the amount of time that will be allocated to the
patch. Thus all categories can be treated as a single, hierarchical
model.

All four categories listed above have been treated in the
literature in a similar manner, so I will focus on one of the four
categories, diet choice. The most widely used diet choice model is a
variation of MacArthur and Pianka's (1966) original optimal foraging
model. As Pyke et al. (1977) note, the same basic model has been
(independently) derived by at least nine authors (Schoener 1969, 1971;
MacArthur 1972; Charnov ■f976b; Charnov and Orians 1973; Timin 1973;
Maynard Smith 1974; Pearson 1974; Pulliam 1974; Werner and Hall 1974;
Estabrook and Dunham 1976). The diet choice model makes the following
predictions (see Chapter IV for a more detailed discussion of Charnov' s
[1976b] version): (I) prey should be ranked according to the ratio of
net energy derived from the prey divided by the time required to handle
that prey, (2) the forager should never exhibit partial prey preference
(i.e., prey types that are a part of the diet should be taken whenever
they are encountered), (3) the inclusion of any given prey type to the
diet should be independent of the encounter rate of that prey type.

161

Prey types should be added to the diet until the net energy per unit
handling time from a prey type falls below the net energy per unit
foraging time derived from a diet of all higher ranking prey. The
models were tested (mostly under laboratory conditions) on a number of
different predators (Werner and Hall 1974; Krebs et al. 1977; Charnov
1976b; Zach and Falls 1978; Dunstone and O'Connor 1979; Hughes 1979;
Palmer 1981; Goss-Custard 1977) and were found to be generally
consistent with the predictions, although partial prey preference
appeared to be exhibited more frequently than expected. ¥hat do these
studies tell us?

The studies cited above illustrate a number of features about
foraging behavior. They suggest that the optiraality approach can be
useful in studying diet choice under simple conditions. Thus foraging
behavior can be treated as a selected trait, just as physiological
(e.g., see HcNab 1980) or morphological (e.g., see Alexander 1982)
traits can be. This is an important step because it allows us to use
optimality to understand behavioral adaptations. The studies also give
us some insight into the decision-making processes of animals, which is
a primary focus of this field of research. The fact that the
maximization of net energy appears to be a useful maximization parameter
is also important because it suggests that energy may be a good
parameter to start with in future studies. This by no means implies
that it will work in all cases, but the utility of this maximization
parameter can be tested.

The fact that so many researchers have come up with the same model
independently, and that the model appears to predict, at least
qualitatively, the behavior of a variety of foragers, from insects to

162

mammals, suggests that the predictions are robust and have a high degree
of generality. However, several studies have shown that the predictions
will not hold for certain systems. In the following examples, each
factor will change at least one of the predictions: (l) learning will
change handling times and capture efficiencies (Murdoch 1969; McNair
I98O), (2) foragers nay need to test habitats to monitor prey
availability (Xrebs et al. 1977), (3) foragers often require some finite
time to recognize prey (Charnov and Orians 1973; Hughes 1979), (4) a
forager's diet choice may be constrained by nutrient requirements or
toxicity of food (Pulliam 1975; Rapport 1981; Janzen and Freeland
1974), (5) the presence of predators may influence diet choice (wilinski
and Heller 1978; Sih 1980b; Lucas 1983), and (6) foraging decisions
nay be influenced by variability in prey quality or patch q^uality (Real
1980; Caraco 1979, 1980). These studies suggest that the list of
factors originally included in the first diet choice models must be
expanded. I should point out that the optimality approach was used to
generate these new factors, and thus provided additional insights into
the factors that influence foraging behavior. The studies listed above
suggest that the original models are inappropriate under certain
circumstances, but not that the foragers are foraging non-optiraally.
Thus the original formulations will work under some circumstances and
not under others. How do we treat these additional factors? They could
be treated as special cases, and therefore we might conclude that the
original models can produce robust predictions, but that the predictions
are invalid in the special cases. On the other hand, we can add the new
factors to an overall model for which there are no special cases. I
suggest the latter approach. We should be able to generate a model that

163

embodies the essence of foraging behavior under a wide variety of
conditions. This model should be far more predictive than any simple
model of specific cases of foraging behavior. The best type of model
that could be used is a hierarchical one. Hierarchical models will
include the generality that is required and the flexibility needed to
study foraging behavior in different sorts of animals and under
different conditions. The original models were useful in their
simplicity; this feature should not be discarded. Using hierarchical
models, the simple models can be treated as special cases of the overall
model. As a result, simple models can be used under the conditions for
which these models are appropriate.

Hierarchy and Optimality Models

The lack of generality in the original diet choice models could
have been predicted, given the simplicity of the models. This point
brings up a slightly different issue: where should we look for
generality in the first place? The approach to all optimality problems
is to derive cost/benefit functions that are appropriate to the currency
under consideration (Pyke et al. 1977). Thus, one generality of all
optimal foraging models is that any given system can be studied using a
cost/benefit function. Unfortunately, this generality does not go very
far, since it makes no predictions about the type of decisions we should
expect to see from a forager. The fact that the diet choice model was
used under so many conditions and for so many types of organisms,
suggests that researchers can use these models as general predictors of

164

behavior across a diversity of animal groups. The same can be said for
the use of the marginal value theorem and its derivatives in the study
of patch choice (Charnov i976a; Parker and Stuart 1976; Parker 1978;
Sih 1980a; Cook and Gockrell 1978; Giller 1980; Orians and Pearson
1979; Krebs 1978; Cowie 1977). However, the work presented in
Chapters II, III, and IV suggests that the diet choice models and the
marginal value theorem may be much less general than was once thought.
In some cases, the predictions may have been so general, especially when
treated qualitatively, that they were upheld even when the systems were
clearly inappropriate for the models. Partial prey consumption is a
case in point (see Chapter II ) . This leads us back to the same
question: What models will give predictions that are generally
applicable to foraging behavior and that will therefore aid in a broad
understanding of this behavior? The generality must enable us to gain
insight about behavior, and not lead us to false conclusions. If the
predictions generated from a model are so general that they cannot be
rejected, even if the model is inappropriate, then the model has no
power and should be discarded.

The generalities we seek may be more in the framework or hierarchy
than in the specific models. This can be illustrated using an excellent
example from sex ratio theory. Sex ratio theory is an attempt to
predict the variance in sex ratio patterns seen in nature. In fact, sex
ratios have been shown to range from strongly male biased to strongly
female biased. Sex ratios should reflect differences in investment
pattern required to produce sons and daughters. Fisher (1958)
originally proposed that the sex ratio produced by a female should be
the inverse of the investment ratio. Sex ratios should be nroduced such

i-i\w-jir.''

165

that the female invests equally in both males and females. This theory-
seems to predict the sex ratio pattern in a nur.ber of groups, from
insects to mammals (Frank 1983; Charnov 1982). However, under certain
demographic conditions, the sex ratio is more female biased than
predicted. The fitness of a female within a deme is generally highest
if that female invests equally in males and females (assuming no local
mate competition; Hamilton 1967). However, the fitness between demes
within a population is generally higher if the females in that deme
produce mostly females, since demes with a female-biased sex ratio will
produce more of the dispersing female offspring than a deme whose sex
ratio is 50 percent males. Thus, there is a difference between the sex
ratio patterns expected from the two levels of selection, within deme
and between demes. If deme size is large enough, the effect of the
lowest level will override the effect at the higher level and the sex
ratio will be explained by Fisher's equal investment theory (again
assuming no local mate competition). When deme size is very small and
little breeding occurs between demes, the effect of inter-demic factors
becomes more important and the sex ratio becomes more female biased.
Conflicts between levels may also arise when subgenomic elements such as
mitochondria or viral infections influence sex ratio. Here there may be
a conflict between the equilibrium sex ratio at the level of the
subgenomic element and the equilibrium sex ratio at the level of the
individual. Which model produces the most generality? One could argue
that each model (equal investment, group selection, and subgenomic
selection) will work under specific conditions, and therefore will be
generally applicable under those conditions. On the other hand, if each
of the levels of selection is incorporated into a hierarchical

166

frame-rfork, then the framework itseli" will te generally applicable under
all conditions. if there are no subgenomic particles and dene sise is
large, the framework collapses to the original investment ratio theor;;-.
If deme size is small and interdemic genetic exchange is low, the
framework collapses to the group selection model. This feat^^re of
ccllapsibility is especially compelling, since it may allow us to
examine specific levels somewhat independently of other levels.

Collapsibility vrill be most applicable when the model is collapsed
on non-fesdback subsystems (see Hierarchy discussion above). ?his is
because the constraints imposed by higher levels on the non-feedback
subsystem can be treated as a constant in the model. Also the effect of
decisions at lower levels are not incorporated into the decision
process, therefore complex time components associated with feedback
loops are lacking. Chapter IV is a good example of this. In this
chapter I showed that diet choice should reflect the degree to which
foraging bout length is constrained. The time constraints model (see
Chapter IV) can be used irrespective of the factors that constrain bout
length. Thus, the framework could be collapsed and treated as a single
constraint within which diet choice decisions must be made. I cited
studies where tidal cycle, predator pressure, and intra-specif ic
pressures constrain bout length, ;/et the influence of each factor on
diet choice is identical: as foraging bout length decreases, the diet
should become more catholic. The regulation of foraging bout length is
a decision about time sharing: how much time should the forager devote
to each category of behavior (i.e. foraging, mating, etc.)? A study of
this decision would be more complex, because time sharing is a feedback
subsystem. To study this behavior properly, the dynamics of at least

■ -"A

167

two levels of decision need to te modeled. For example, ^-ith the
time-sharing decisions wrens make when foraging for nestlings, the wren
must decide to forage or stay and provide predator defense for her
young. The two factors are not mutually exclusive, since feeding the
young provides a predator deterrent. This level of decision was treated
as a "black box" in my study. This decision is a feedback subsystem,
because the amount of time required for foraging should be regulated by
the net benefit derived from that foraging time. Thus, the study of
time sharing in wrens requires at least two submodels to characterize
the system. It may need more depending on the structure of the
hierarchy.

Most behavioral systems can be characterized by a hierarchy of
decisions or effects. If we want to study the system as a whole, then
we must study the entire hierarchy if we are to understand the behavior.
As illustrated above, if we are interested in specific portions of the
hierarchy, then the framework can be collapsed, depending on the type of
subsystems characterizing the decision of interest. In chapter I, I
showed that the structure of an antlicn's pit appears to maximize
capture probability. The morphology of the pit included a layer of fine
sand on the pit walls, which was shown to retard the escape of prey.
Although there are a number of physical characteristics of sand that
would tend to generate this type of design, the antlion clearly exhibits
a number of behaviors that tend to make use of this

particle-size/prey-escape relationship. Antlions regulate both particle
velocity and trajectory angle. They also retain small particles inside
the pit until the basic shape is attained, after which they line the pit
with this fine sand. In addition to particle size distribution, the pit

168

is also characterized by two other features, slope and diameter.
Particle size distribution is non-hierarchical in that there are no
constraints imposed on this characteristic of pit morphology and the
functioning of the pit. The more fine sand on the walls, the better it
will function. Pit diameter is quite different. Here the energy
invested in enlarging the pit should reflect the expected gain from that
investment. The decision is characterized by a feedback subsystem. The
decision cannot be addressed without a full knowledge of both the
energetics of pit construction and the return from that investment. The
same may hold true for pit slope. A simple study of pit morphology is
possible due to the non-feedback nature of the decisions associated with
that level of organization. A study of the regulation of pit diameter
and slope will require a more thorough study to yield the same answer.
The decision structure associated with pit-construction behavior would
have to address the entire hierarchy, from pit morphology to pit
diameter and its relationship with net benefit.

When testing any model, the model gives us a framework within which
we can view a system. The model also directs the experimental approach
that is taken to study the system. In studies of foraging behavior, the
specific models that have been formulated have been useful in some
studies, but misleading in others. These mistakes appear to be caused
be the restrictive nature of the single model approach. The use of
systems-oriented, hierarchical models appears to be a much more powerful
method of analysis. Once the system has been characterized, the
hierarchy can be collapsed to address specific features of the system.
Thus, a systems-level approach will enhance generality, since it can be
used to address a broad spectrum of problems. It should increase our

169

ability to understand any system because it forces us to view a number
of features of the system under study. It •/fill also allow us to
simplify the question. This reduces the complexity of the question and
increases our ability to measure the parameters that are important in
each subsystem.

170

CONCLUSIONS

In evaluating the utility of optimality modeling, two issues must
be addressed. The first is whether or not the basic assumptions
associated with the technique are consistent with observations. Of
course, the most basic assumption is that animals are capable of making
decisions that approach the theoretical optima. The second issue is a
heuristic one. Does the technique provide insight into animal behavior
that would otherwise be overlooked using other approaches?

This dissertation covers three different temporal stages of
foraging behavior, (l) the preparation for prey capture (specifically
trap construction by antlion larvae), (2) the choice of prey, and (3)
handling of prey once diet choice decisions have been made (specifically
partial prey consumption). Optimal foraging models were used to
evaluate each aspect. The results of the study indicate that foraging
decisions are invariably constrained by factors that are unique to each
stage of foraging. In each case when the constraints were built into
foraging decision models, the behavior exhibited by the organisms under
study was consistent with the predictions from the models.
Trap-constructing behavior is consistent with the prediction that
antlions should maximize the net energetic return from foraging through
the manipulation of trap design. However, trap design is constrained by
the physical properties of the material with which it is built. Thus
the study of trap-constructing behavior can only be accomplished through

171

the characterization of physical constraints imposed on this behavior.
Diet choice can be similarly constrained by features of the animal's
environment. In systems where foraging time is limited through either
biotic or abiotic factors, optimal diet choice must be modeled by
explicitly including these factors in the model- Predictions from such
models agree with the observed diet choice of a variety of animals.
Time constraints may also be important in decisions concerning the
utilization of prey by foragers that only partially consume prey.
Predation modes (e.g. search or ambush methods) will impose unique
constraints on the ability of animals to utilize their prey. Here
again, if these constraints are included in foraging decision models,
the output of the models agrees with the behavior exhibited by foraging
animals. From the studies reported in this dissertation, it can be
concluded that observations of animal behavior have been consistent with
the assumptions and predictions from optimality models, but only if
constraints are explicitly included in the models.

Several models were reviewed in this dissertation which failed to
incorporate the factors listed above. As a result, the authors of these
studies concluded that the animals used in the experiments were not
foraging optimally. If we ignore the fact that this conclusion is
inappropriate to the technique (see Introduction and Chapter V), the
inability of these earlier optimal foraging models to predict behavior
suggests that these models were not robust enough to use under a
diversity of foraging conditions. Since any given model provides a
framework within which the researcher can formulate predictions, these
models seem to restrict the perception of behavioral systems as much as
they provide insight into behavioral adaptations. The use of optimality

*5^«r"

172

models to study animal behavior appears to be valid, but the application
of this technique to f oraging-behavior studies appears to be limiting
our perception of behavioral systems. The non-hierarchical nature of
most optimal foraging models is undoubtedly a major contributing factor
in this limitation. I propose that models of behavior should be
constructed using a hierarchical design. Unlike previous foraging
models, hierarchical models require a synoptic perception of behavioral
systems, and therefore are not restrictive in their predictive value.
The utility of this approach is demonstrated in Chapter IV, where a
single model was used to analyze foraging behavior of three
taxonomically disparate organisms under extremely different
environmental constraints.

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BIOGRAPHICAL SKETCH

Jeffrey Robert Lucas was born in Januarj-, 1953. He received his
B.S. degree at the Florida Institute of Technology and his M.S. degree
at the University of Florida. During his tenure at the University of
Florida, Jeff worked on a project funded through the Florida Power
Corporation which evaluated the effect of heated effluents on the
productivity of estuarine bays. He also worked as a teaching assistant
in the Department of Zoology. The fact that he met and married Lynda
Peterson should not be overlooked. He has received a NATO Postdoctoral
Fellowship to continue his antlion project with Dr. John Krebs at
Oxford. Tally ho.

182

I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.

H. Jane Brockmann, Chairperson
Associate Professor of Zoology

I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.

C^^t'V.-V^vvvt^ ll< f^^

Carmine Lanciani
Professor of Zoology

I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.

yjdiiUil ^

/

Brian K. McNab
Professor of Zoology

I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.

Frank Nordtie
Professor of Zoology

I certify that I have read this study and that in my opini-n it

conforms t: acceptable standards if scholarly presentation and is fully

adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.

-^CC"^^

Howard T. Odum

Graduate Research Professor of

Environmental Engineering Sciences

This dissertation was submitted to the Graduate Faculty of the
Department of Zoology in the College of Liberal Arts and Sciences and to
the Graduate Council, and was accepted as partial fulfillment of the
requirements for the degree of Doctor of Philosophy.

April, 1983

Dean for Graduate Studies and
Research

&

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UNIVERSITY OF FLORIDA

3 1262 08554 0077

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