Optimality, constraints, and hierarchies in the analysis of foraging strategies

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Optimality, constraints, and hierarchies in the analysis of foraging strategies
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Foraging strategies
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vii, 182 leaves : ill. ; 28 cm.
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Lucas, Jeffrey Robert, 1953-
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Animals -- Food   ( lcsh )
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Thesis:
Thesis (Ph. D.)--University of Florida, 1983.
Bibliography:
Includes bibliographical references (leaves 173-181).
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by Jeffrey Robert Lucas.
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Typescript.
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Vita.

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OPTIMALITY, CCllTF.AIITS, AND HIERARiCEIES IN THE
ANALYSIS OF FORAGING STRATEGIES







BY

JEFFREY ROBERT LUCAS


A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILL :'-.E OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY


UNIVERSITY OF FLORIDA


1983
















ACKNOWL EGI; EJiT.:


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

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. Frank Maturo 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. Thain:s to my committee, Drs. Jane Brockmann, Carmine

Lanciani, Brian McNab, 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 anonymous

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








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.















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 C.NSiUMPTION 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 EINOUNTE IN OPTIMAL DIET CHOICE....... 110

Introduction............................................. 110
The Cost Model........................................... 112
Discussion............................................... 140
Summary.................................................. 145










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, 1983


Chairperson: H. Jane Brockmann
Major Department: Zoology



Three phases of foraging behavior were analyzed: (1) 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 efficiency. 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 foraging-bout length. For a two-prey system, the forager

should specialize when M j>Eh /(E.hj-Eh), where i,j=high and

vi








low quality prey (respectively), E=energy, h=handling time. M.. is
13
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, M.. no longer measures the cost

of eating i. Here the predator should specialize when

P(Y.E.P(Y.)/E. where P(Y.)=probability that j arrives in a

foraging bout and P(Y.
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 ph=(-D t C-(g t C(D C+g max-D t ))
oph pe p max p pe max pe p
where gmax 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
densities, t0.5
densities, top is topl=(CDwe-(Dwegmax C) )/(-D we), where Dwe is
op opl e we max we we
waiting minus eating costs. At medium densities, t will always be
op
less than topl, and varies with prey encounter rate. Each model

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.















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 (1966) 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 I).

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 theory 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 omniscient 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: (1) 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 C,'rISTRUCTIO,:





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 (Trichoptera) (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, M. 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 (Levy 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 (0.1 to 1 mm diameter). Each antlion was kept at 240 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. Dark-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. (1) 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 individually 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




























































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.





9

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: (1)

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.






















500k 30 5
white 250-1
I. crudelis 1
c m --45.



Ir-
1~~ cnl
R!r











































L
i ..

0
S-





T -







2S 250 250 -00 500 O-1000 1000- 2000
SAND GRAIN SIZE (P)




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






























S *-I A.**& 65 mm

n*-120
so


% .
C-
Wo- 1

20

W 40

I--




oo I I 1oo1
125-250 250-500 500-4000 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.





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 34.35 0.0001
2. Slope 1 102.60 0.0001
3. Sand size *** 3 51.04 0.0001
4. Regression model
including all variables 5 58.06 0.0001
5. Error 475



Camponotus floridanus

Independent variables degrees of freedom F value Prob.>F

1. Pit diameter 1 127.88 0.0001
2. Slope 1 20.72 0.0001
3. Sand size *** 3 25.61 0.0001
4. Regression model
including all variables 5 45.08 0.0001
5. Error 235


* 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 450; 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.








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




























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








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 (Fk)
kC







(Bird et al. 1960):



Fk= T R2(0.5pV2)(24/Re)=6 uRV, (1)



where

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

R=particle radius,

p=fluid density=0.00123 g/cm for air at about 25 C and 660/0

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 2sin29 K'V sing
D(x)= ------- [1-(4/3) ---] (2)
g gR



where

K'=(7.16x10-5)c,

@=initial trajectory angle,
2
g=acceleration due to gravity=980 cm/s2

V =initial particle velocity,

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








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: (1) Distance (D(x)) will increase

monotonically with increasing initial velocity (Vo). (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 450

as particles become smaller. As sand particle size increases, D(x) is

maximal at 9=450. (3) As sand particle size increases, distance should

increase monotonically.

At intermediate Reynold's numbers (2
underestimated and must be modified as follows (Bird et al. 1960):



Fk= R2(O.5pV 2)(18.5/Re06), (3)



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

1982 for the derivation)


V 2sin29 K"V 1c(Q)
D(x)=- 1 ------- (4)
g gR


where
K'=(7.79x10-5)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







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 (9)(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 1)

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 (Fk) 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.





















S --- MAKING PIT PREDICTED
00- MAKING PIT OBSERVED
+ CLEANING OBSERVED F vm 76




6000 .0
30 2,,
I I--- -a 1




100 0
400 o .,,' -II 1 J 1.2. .
goo T ,J .... j T






8 00 0



.00
oo- vm 9 1 IVm 8S










0 ,0 4 I 6.* 8. 1-T J
t doo V i o,,t .n i e
too -. J



-observd v s +/- 1 SD. Curvs repr vals pd by













(4) (see text). =best fit value for particle velocity during initial
construction. Velocities listed are cm/s.
Lyeo 50 -. --i,, 50 .. *-. .
Goo 7+ /' +

17 -- f



T ,+ tT" 13 + "I-1 .
Goo-1 V- 071 I VC 112





Figure 1.6. Observed and predicted values of diameters of sand grains











construction. Velocities listed are cm/s.








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 (9) 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 450, 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


































1000


too



5)


0
S700


C 00




400


TRAJECTORY
ANGLE
30* 0---0
40* .........
45* o---o
50 --
60' *--*


J r
J ;


7
DISTANCE (cm)


Figure 1.7. Distances sand grains of different radii travel under
varying trajectory angles as predicted by equation (4) (see text).
V was 100 cm/s.
o








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

out of the pit. Thus 9 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 9 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 9 was 470 with about a 120 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=450

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.






















,-CARDBOARD


^ i-" i i :
/ 8 : ,,I .


*" ; < 'A .A'. .,;


A I
-(AS IN FIGURE C).." .


(AS IN FIGURE I-C)


A POSITION OF ANTLION

B EDGE OF PIT

C = POSITION OF SAND
ON TAPE


ABC


AC


THE FOLLOWING DISTANCES ARE REQUIRED TO DETERMINE 0:
DISTANCE FROM ANTLION TO PIT EDGE
DISTANCE FROM PIT EDGE TO SAND ON TAPE
DISTANCE FROM ANTLION TO SAND ON TAPE


APPLYING THE LAW OF COSINES:

COS. 9 BC + 2 COS. 9 Z A2 2
S2Bn.AB 2 2 A G+aA
9 a 92+ (


eI a
THEREFORE TRAJECTORY ANGLE O&


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.


SCOTCH TAPE







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 9


Range of 9 for
each antlion


47.00 (0.7)


12.20 (2.3)


46.40 (2.3)


16.00 (2.8)

10


60.00 (4.5)


43.70 (8.6)

10


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







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 460

Finally the antlion cleans the pit by increasing 9 to about 600 (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 (460) 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 9, 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


























POSITION OF ANTLION IN PIT
(,-- 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.












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


--------------------------------------------------------------
Antlion mean grain size (standard deviation)
number ---------------------------------
Initial phase N Final phase N Z
---------------------------------------------------------------

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







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

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 (9), particle radius (R), particle density

(p ), and the particle coefficient. Methods for estimating OT 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
o
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









construction, enhancing the differential sorting on the walls due to the

angle of repose. By using a 460 trajectory 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








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 is

certainly exhibited in the earliest pit construction stages before there








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 1941).

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, N=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,








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 energetic (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 (1976) 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







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 theory 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).








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.

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 (1976a), 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: (1) 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.








Proximate Models


Models concerning the physiological constraints on foraging have

focused on two aspects of the predator, (1) gut size (The Gut-Limitation

Model) and (2) the maximal rate of ingestion (Digestion-Rate-Limitation

Model).



Gut-Limitation Nodel


The Gut-Limitation Model (GLM) was first proposed by Holling (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 Holling

(1966) in that two separate compartments (foregut and midgut) each








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 Holling (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.








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 (1980) 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.







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: (1)

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 (1980a) 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








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,

t =ingestion or feeding time,

T =total foraging time=t +t +t ,
T p s e
where t and t are constants and TT is a function of t Let the
p s T 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 (1)
C + t
e


where gmax 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 (1980a) 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. Michaelis-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. I also assume that capture








probability is unity, although this assumption will be relaxed later in

this paper. I define the costs of foraging as

C =cost of pursuit and capture per unit time,

C =cost of search per unit time.
s
Therefore,

C t =total cost of pursuit and capture,
PP
C t = total cost of search.
ss
The total benefit (E) per unit foraging time (T ) is (Sih 1980a)




E g(t )-C t -Ct g t Ct C t
= e p p 5s max e pp s s (2)
-------------------- ----- ------- (2)
T t +t +t (C+t )T TT TT
e p s e T T T

The optimal solution is obtained when


a (E/TT)
= 0. (3)
a t
e

From equation 2


t [(g C(t +t )+C t +Ct )/g ]05 (4)
eop max p s pp ss 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,































11-3N38


C
-(
Z
Q.









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










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 MVT. 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 McMurtry 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








E g(t ) C t C t
e pp ee

T t + t
T e p


g t Ct Ct
max e p e e ,
--------------- ------- --------- ( 5)
(C + t )(t + t ) (t + t ) (t + t )
e e p e p e p


where C is the cost of feeding per unit feeding time (t ) and the
e 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 )]05 (6)
p maxp max p,(
t -= -----p---- --------max--- ----------- (6)
ep Dtp-gmax
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















AMBUSH/HIGHEST DENSITY MODEL


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 graph.
e p








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 variety 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 (Cd) and the time required for the pause (td)


E g(t ) C t Cdtd C t
e P dd ee. (7)
m t + t + t
T e p d


The optimal feeding time is



t =[(g (t +t )+Cdt +C t )/g ]0.5 ()
eop max p d dd pp max


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

addition, increases in td and Cd should increase teop
d d op








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

(te) 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
p e w


I will also treat pursuit time as a constant.

If waiting costs (C ) and feeding costs (Ce) are negligable, then

the predator should hold on to its prey until it is entirely consumed.

Unfortunately, the Michaelis-Menten function asymptotes to gm at

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

























*




o

c

4.J 4.J
C




c c




c c


-H

IJj
O




w0

C)


= m




0 -
*u















r-H
.1 0
4O-J




0 F,

-t
0
C)4-1






N
3 U











I- r
,C
.j.




C) C
oc



cc



*i m
4e 1




i-i en
C *i-




53












>-





LU



z
SC/)




114 I 3
LU \
0 \ "z
-J
I




o0




lj13N38









E g(t )-C t -C t -C (T-t -t )
e ee pp w e p
-- = ------------- .-- ---P_-.
TT T


Thus,


E g t C t C t C (T-t -t )
S ax e e pp w e p (9)
-- ----------- ----- --E-- ------ (9)
TT T (C + te) T T T
lT e

Here
C(C -c )-[(c -c )g C]0(-5
t = --e------------^-? ----. (10)
eop (10)
eop- (Ce-C )


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 gmax and C ;

2) feeding time and percent consumption should be

negatively correlated with C and Ce;

3) 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.








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 p=inter-prey interval=t +te

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 net) is
net















CC,



II=


CL
>- >.






O < -
I --- [^ a


w w C
\W "-'





D c





0
\

CO 4-)O












LC
u-

E
4-









0
w -o
C4





0










LC
>. UJ ^








max e
E t Ct (11)
net C + t P P e e


The net benefit per unit foraging time will be


g (XT -t ) Ct C t
G(X) max IP p Pp ee (12)
G(X) = ------------ P-P-- ------ (12)
(C+XT-tp)(XT p) XTIp XTIp


If X

(XT -t )g Ct Ct
-- p gax_ p e_ ee
(C+XTIp-tp)(XT p) XTIp XT p



(YT p-t )gma Ct Ct
---------------- ---- (12)
(C+YT p-tp)(YTIp) YT p YTIp


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:

3) as prey density increases, handling time and percent

consumption will decrease until the "optimal" prey























C:J


-o 4 ai C- 4

4 CU OC


W -j 4-a 1
3 C C:U
UC 0



4 2 U "-
-*H *-



1) -4 r-a C





0 60 r







-H1
Ca 0) M
4.J i-4


O M C






0 W 41 .. H
c uC =
'J a Q- *


O co 4 -



u C-- *
41 .1-a

g 4-1i *-




E 1- r 1 4-4
a r) *K o-

4.) 41
()C LI-


C U >> -
iLlD
C WO W *


C C '0 3


C0 CE J 4

0 I CU 3






0b1 l 0


*H C C *CO
F-e am cc

*r* 0 If E II
















































'C -UCi- C4 a cC '
_ a


59

aau
3


0 /I

O/







i
-7-7








7--





















S'N C M cC -
I I I


Q--0 (X)3








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 prey 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








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

X =encounter rate of prey,

P =probability of capture while handling a prey,

P =probability of capture while empty handed,

P =Po-P w

N=total number of prey handled in a foraging bout

lasting a given length of time, TT,

T =total time predator devotes to a given prey=t +t .
d 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

number eaten while empty handed (No) must be calculated:

N = X NTP (13)
w w
N = o (TT-NT)Po (14)

The total number of prey eaten (N) is therefore

N=N +N = A NTP + X (TT-NT)PO (15)

Solving for N,


STTo TTPo (1
N= --------= --------- 16)
1+ ATd(P -P ) 1+ TdP
d o w d d

The benefit associated from each prey,


Kt
g(t ) = (17)
C+t
e








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


(Kt )( XTT P )
B = g(t )N = ----------- --- (18)
e (C+t)(i+ XTdPd)


The benefit gained per unit foraging time is


B (Kt )( XP )
-- = --------------. (19)
TT (C+t )(1+ XTdPd)
T Ce dd

The optimal solution to equation 19 is


C
t = (Ct + ---)0.5 (20)
eop p p
d
From equation 20, a predator should decrease handling time as the

difference in capture probability (P -P ) increases. This is because
o 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 1981) may switch from ambush to








searching predators as prey density fluctuates. Some predators may

continuously reach satiation (ex. mantids in Holling 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

antlion larvae fed at low feeding rates do not change the percent

consumption with changing feeding rates, as predicted by the GLM.








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
b) no satiation

intercapture interval
a) near satiation
b) no satiation

cost of pursuit (C )
P


cost of search

cost of eating


(C)

(Ce)


-/- -/- -/- o/o o/o -/- -/-
0/0 -/O -/- 0/o 0/0 -/- -/-


+/+
o/o


0/0 N/A 0/0 +/+ 0/0
0/0 N/A 0/0 +/+ 0/0


0/0 0/0 +/+ +/+ 0/0 0/0

0/0 o/o -/- N/A N/A N/A N/A

0/0 0/0 -/- -/- -/- 0/0


cost of waiting (C )

pursuit time (t )
a) near satiation
b) no satiation

extraction coefficient
a) near satiation
b) no satiation


N/A N/A


N/A N/A


+/+ +/+ +/+ +/+ o/o o/o +/+
0/0 0/0 +/+ +/+ 0/0 0/0 +/+


-/- -/- -/- -/- -/- o0/ -/-
0/0 o/o -/- -/- -/- 0/0 -/-


-/- N/A


* 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).








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 Notonecta is an

ambush predator (Gittelman 1974). Giller's (1980) results are predicted

by 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 MVT (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.

Holling's (1966) 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








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








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 1956). 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.









Digestion Rate Limitation (DRL) 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

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








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 measured 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 initial 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 (1 fly

per hr), or FS-48 (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 was 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









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



Th = bo + blFS + b2 + bAN + e ,



where Th = handling time,

FS= feeding schedule,

I = initial fly weight,

AN = antlion "name",

e = error term,

b -b = regression coefficients.



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








more efficiently. This information was used to build 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 l of partial prey

consumption under low prey density conditions. The model predicted that














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 B, 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.


If;EFEN1DEINT VARIABLES
2 model total
FS initwt lionnm* r df df


P>F for
F model


AS % ext -.0051 +.0254


-.0001 +.0001


.0001 0.26 79 552


.0001 0.74


2.1 .0001


79 552 17.2 .0001


extrate +.0001 +.0001


.0001 0.64 79 552 10.7 .0001


* Class variable, no slope estimatable


B)
dependant
variable


INDEPENDENT VARIABLES
2 model total
FS initwt lionwt r df df


AS % ext -.0234 +.0020 -.2898

T, -.0001 +.0001 -.0015


P>F for
F model


.03 3 552 6.0 .0006

.46 3 552 156.3 .0001


+.0020 .30 3 552 80.1 .0001


A)
dependent
variable


extrate +.0001 +.0001







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: (1)

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: (1) 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

must predict the biomass extracted at any given time during the handling







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 maximizes 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: (1) a Michaelis-Menten 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:












4C





0 .0 D
4 -

13 a
L .n 3 -


C] 0-




o.0 0 0
Co


OC O

0 C C. C 1
+













































Extracted weight is expressed in grams wet weight, time in minutes.
.weight <0.0007 gm; triangle: 0.0007 gm initial weight 0.0009 gm;
asq : i l w t >0.09 gm.
SATM

Ext ed w g s e


ei h 0 0 gn. gm i i e gh 0
e+ e+
3 C
3 -D + -








-1
-J


1 1





-J





o 3 6 9 12 15 1? 21 2i 27 30 23 36 3S 42 y5 LS

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 (0.0007 gm; triangle: 0.0007 gm square: initial weight > 0.0009 gm.

























SCO


0.0012-




3
0.001 1






0.00031






J















0.000o?]
0.0010I




0i
jI








0 0007"








0.C000CJ
j


















0
IC'
]1


4.
44
4-r


+ +
4 4-


5 10 15 20 25 30 35

TIME


Figure 3.2. Biomass extracted
handling times (time) when the
Symbols as in Fig. 3.1.


from fruit flies
feeding rate was


(extrwt) at various
eight flies per day.


C
n
a 9 m
CQ0 0 0
S 0 C2
C CC
D A

A
A A
A a L
a4A
A A

dsr- 4


F'eLcn u-i
'5-W=3




























0.0n00
-4







0.000E7
3


0.0003-


















0. o, C:;
o.o0000-
-.


0.000 -











-4
0.000-








0.0002-
-4
.4
i
























0.00006


A A A

A + A +
+ + +




S + 4- +
4- 4'- ,

' ^'-


5 10 15 20 25 30 35

TIME


Figure 3.3. Biomass extracted
handling times (time) when the
Symbols as in Fig. 3.1.


from fruit flies (extrwt) at various
feeding rate was 24 flies per day.


AI
0
A

0
0C
Aa
Ai a AA~
0


0 0


a


0
oB a




0 D 0
O

-c-
O A


&o D





6 'A


A B A


A
0 A
A
A




_+
L


-!


+ 4-+









~EESCH=Me


0.0010
4








0,0008




0.00071




0.0006









0.000 L1





0.000i
1
-i





































0.0001



0,0000-

-i
0.0003


0. 00 -


& a~ _p a
D A
O



0 0
LA .A A A


A^a %s a
A ^A CA


Ct aa^ ,
A A a a
A A A AA
A AA ^& A



T ^+ f- f
A A -? A
D A AA A


4--
+ +
*


5 10 15 20 25 30 35

TIME


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.


0 0










G (T) = XI(k +k2 k )(1-(k+k5 +k7W -(TL)) (1)



where G (T) = the gross gain, in terms of wet weight, extracted from
g
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, kl-k8 = constants,

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 600 C. The regression equation fit to these

data was used to convert wet weight to dry weight in the extraction rate

curves. Dry weight was converted to calories using the following

conversion for fruit flies (Cummins 1967; Jaeger and Barnard 1981):


1 gm dry weight = 5797 cal.








(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 240 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 Ce, 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. Hemmingsen 1960):









MR = aWb



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, I 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:



ln(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

02 is equivalent to 0.0048 cal (DeJours 1975).








(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 harndlinr and pursuit times). Thus net gain, G (T),

is



G (T) = G (T)-C L-C (T-L)-Cw(T -T) (2)
n p p e w T



where C = energetic cost of pursuit + pre-ingestion behavior,
P
C = energetic cost of eating,
e
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 by equation 1.

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


k k k
G (T) = I(k +k21 )(1-(k +kI +kW k)-c(T-L))
g 1 4 5

= k (1-k b-c(T-L)). (4)
a b


Thus







G (T) = k (1-k) Cd(T-L) Cw(T -L) CL
n a b d CT p


where C. = C -C .
a e w


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

8 G(T)/TT -k k -(T-L)n(k ))(-c) Cd
------- -------------- = 0 (5)
a (T-L) TT TT



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

get


Cd -c(T-L)
-- = k ln(kb)
Ck
a


Rearranging and taking the natural log of both sides


C
n( ----- ) = -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
-n(- ---
ck ln(kb)
T = ------ ------+ L (6)
op ln(k)c
In(kb)c







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
g
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 t'r antlion in terms of handling time

and percent utilization 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 3.2). 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 3.3). Estimates for eating and waiting costs

are also listed in Table 3.3.























()



C()
0







mcn
c) o
















>










0


C
'-1 1










CO
u

































UCL
3 *







































C)
L 0 c)





Q- C)
010


























),
CO
0 *n
*H i- 4J
4-1 4-1 ;<





4J 'O











































00
L


4


PN C TC E

pa~3oejxa s TIOTBO


II






88


















Ch (s 0"> 0



0 CD~OC 0 1^
3c O rM .- 1
I 9 I



3 > 0 C0
OO I




a ) I
0) I 0 UI
I 4 0 0 0l
I .1




0\ I

SI 0 0 0
I O o- U I
- O I I
I 0 O0 O 0 I
I I



,I) < -






0 I 0 0' 0
Io* m I
I 01
O I 0 v 'f

ffl A I 0> 0 0 i
'. I O.

I 0 0 I
to I c -I


0 I 0I O- 0c 0I
I 0> 0 0 I
IC t- *
4-' I L M
I I I I
I LrU t N I
I 0M *.
I 1 0 l


-a
oI C
E I 0 0. 0 1<
S I 0M 0- I
I X CM C<> 0M
a) I N- a- ^ i

0 0
I -
I 0 0 0
O I O X X XI
I M> N- 0
SI a
CM -PI
SI C\ 01
I 01 C 01M
0) 1 0 v 01I
I ** .
C O I *- OI
I I 0 0M *3-







Table 3.3. Fruit fly wet-weight/dry-weight regression equation and
metabolic cost equations for eating and waiting behavior in antlion
larvae.


Wet weight-dry weight regression

ln(dry weight) = -0.63 + 1.00 ln(wet weight)

dry weight = 0.23 (wet weight).
2
N=25, F=35, P<.0001, r = 0.602.


Metabolic Rate Equations

ln(MR) = 1.983 + .910 ln(W) .255 E .499 (E ln(W)).

metabolism measured as pl 02/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'411
e
waiting C = 0.0077 W'910
Cost calculated as l/min
* cost calculated as cal/min









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 percent 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 3.4). 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. Top = optimal handling time. Th =
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 %ext). M-0
ASext = difference between the observed percent extraction
(transformed) and the percent extraction predicted from the model for
the observed Th. 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 T Th PASiext OASexz

1/5 29.7 18.9*** 1.279 1.224**
std 1.5 4.1 .001 .053

1/7 34.4 21.9*** 1.230 1.231***
std 1.7 3.4 .001 .052

1/9 39.3 25.6*** 1.281 1.233***
s-d 2.2 5.1 .0004 .064

:/1! 46.2 3'.*** '.281 1.251**
s-d 3.2 5.3 .0002 .043


8/5 23.5 18.4** 1.205 1.!56ns
std 0.3 3.5 .001 .070

8/7 24." 21."*** 1.209 1.216ns
std 0.6 2.7 .002 .058

8/9 26.7 23.7** 1.216 1.217ns
std 0.7 4.3 .003 .050


%ext3p




.918


.918


.915


%extob

.884


M-OASext

-.015ns
.072


N initwt lionwt

31 .00052 .0278
.00005 .0094


.889 -.010ns 93 .00070 .0303
.054 .00005 .0118

.890 -.007ns 85 .00088 .0343
.0"5 .00006 .0139


-.01=* 48 .00110 .07=
.049 .00008 .0108


.872 .838 .026ns
.076


9 .00056 .0303
.00002 .0096


.875 .879 -.015ns 33 .00070 .0293
.060 .00007 .0064

.879 .880 -.O09ns 24 .00088 .0292
.050 .00005 .0064


8/11 33.8 27.4** 1.242 1.218ns .896 .881 .005ns 25 .00113 .0301
std 7.4 .53 .015 .066 .052 .00010 .0066


24/5 24.' *5. *** 1.192 1.189ns
std 1.3 2.3 .CC0 .064

24/'" 25.7 1?.2*** 1.195 1.168**
std 2.0 3.4 .001 .072

24/9 27.2 21.5*** '.202 !.'Q"ns
std 2.7 4.5 .00o .076


.863 .861


.865 .846


-.052* 12 .00055 .0350
.079 .00003 .0145

.CC3ns 61 .00069 .0355
.068 .00005 .0145


.S70 .86" -.016ns 51 .00089 .0530
.072 .00005 .0151


24'"! 29.4 24.2*** '.22C '.1* .882 .860 .012ns 35 .C0' 1 .0 38
st! 5." -'.i .01' .085 .066 .00009 .0173
............................................................................















Table 3.4---Continued.






..........-----------.--------------------- .---------- -----------

48/5 21.0 '6.'s .21 '.267ns .880 .91 -.067ns .00058 .0386
s3t 3.! 3.3 .C003 .055 .61 .30001 .009~

48/7 22.2 18.3- 1.222 1.228ns .883 .887 -.l09ns 32 .00070 .Cc05
3td .6 3. 3 .002 .066 .68 .CCCC6 .017

8/9 23.9 21.1 1.231 1.226ns .389 .886 -.003ns 30 .0CCC7 .0396
s- 0.6 .8. .004 .055 .060 .00005 .0132

48/11 30.5 24.6na 1.263 1.246ns .908 .898 .OO1ns 9 .30112 .068
std 8.5 1.6 .019 .062 .056 .0C010 .3090
----------------------------------------------------------------
as difference no- significant at ?>0.05
difference significant at 0.C5 S- difference significant at O.01 *- difference significant at P<.C01.









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