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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 godsend 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 nontraditional 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 PitConstruction 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 NonHierarchical 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 finesand layer is caused by physical properties of sand (angle of repose and sand trajectory), and by three components of pitconstruction behavior: regulation of trajectory angle and initial velocity, and presorting of thrown sand. (2) Two general models were derived that predict diet choice when foraging time is unconstrained and when external factors constrain foragingbout length. For a twoprey system, the forager should specialize when M j>Eh /(E.hjEh), 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 foragingbout 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. 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 MichaelisMenten uptake curve, at high densities optimal handling time was found to be t ph=(D t C(g t C(D C+g maxD 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 lowdensity 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 preyhandling behavior is adapted to stochastic preyarrival 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 dietchoice 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 nonoptimal (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 nonoptimally 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 nonoptimally, 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 IIV all use this approach. Where the models did not work, I reevaluated 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 highsloped 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 dumpywinged 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 lightcolored sand used throughout the study was obtained commercially as children's sterilized play sand. Darkcolored sand was mixed with lightcolored 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. PitConstruction 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 wedgeshaped 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 2501 I. crudelis 1 c m 45. Ir 1~~ cnl R!r L i .. 0 S T  2S 250 250 00 500 O1000 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 125250 250500 5004000 10002000 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: 125250, 250500, 5001000 and 10002000 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 noncontinuous 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'=(7.16x105)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'=(7.79x105)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 doublesided 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. 1T 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 + "I1 . Goo1 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 (300400 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 twosided 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* 00 40* ......... 45* oo 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 IC) 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). Aposition of antlion; Bedge of pit; Cposition of sand on tape. The following distances are required to determine 9: ABfrom antlion to pit edge; BCfrom pit edge to sand on tape; ACfrom 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 heartshaped 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 reestablish 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  * MannWhitney Utest 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), trapbuilding 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 trapconstructing 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 trapconstructing behavior. Trap biomechanics have been studied in only two groups of organisms, the orbweaving spiders (Denny 1976) and antlions (this study). Denny (1976) showed that the spider web conforms to the principle of leastweight structures, known as Maxwell's lemma. A trap built with this design will minimize the amount of material needed to function as a preycatching 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 orbweaving 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 orbweaving spiders suggest that trapconstructing 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 trapconstructing 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 finesand 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 trapconstructing 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 GutLimitation Model) and (2) the maximal rate of ingestion (DigestionRateLimitation Model). GutLimitation Nodel The GutLimitation 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 preyconsumption behavior. A second prediction from this model is that there should be a correlation between intercapture 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 predigested 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 digestiveenzyme 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 extraintestinal 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 GutLimitation Model, the DigestionRateLimitation 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 MichaelisMenten 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 MichaelisMenten 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. MichaelisMenten, 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, 113N38 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 interarrival interval. This implies that there should be no correlation between interprey 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 interarrival 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 interarrival 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. Interprey 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 Dtpgmax 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 AmbushPredator 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 interprey 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 interprey 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 MichaelisMenten function asymptotes to gm at infinity, thus assuming that the predator can always extract more from the prey. If feeding costs are nonnegligible, 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 rH .1 0 4OJ 0 F, t 0 C)41 N 3 U I r ,C .j. C) C oc cc *i m 4e 1 ii 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 (Tt t ) e ee pp w e p  =  . P_. TT T Thus, E g t C t C t C (Tt 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 (CeC ) 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 interprey interval sets the feeding times. In fact, the pursuit time (t ) plus the feeding time (t ) are equal to P e the intercapture interval. However, the predator should never hold on to a prey longer than the time predicted by the ambush/lowdensity model (once interprey interval drops below this threshold, the ambush/lowdensity model predicts predatory behavior). A few new terms must be defined: T p=interprey 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) =  PP  (12) (C+XTtp)(XT p) XTIp XTIp If X (XT t )g Ct Ct  p gax_ p e_ ee (C+XTIptp)(XT p) XTIp XT p (YT pt )gma Ct Ct   (12) (C+YT ptp)(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" interprey interval. Since G(X) decreases monotonically as the interprey interval increases above this optimum (see Fig. 2.5), the following predictions can be made: 1) at intervals larger than the "optimal" interprey 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 4a 1 3 C C:U UC 0 4 2 U " *H * 1) 4 ra C 0 60 r H1 Ca 0) M 4.J i4 O M C 0 W 41 .. H c uC = 'J a Q * O co 4  u C * 41 .1a g 41i * E 1 r 1 44 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 Fe am cc *r* 0 If E II 'C UCi C4 a cC ' _ a 59 aau 3 0 /I O/ i 77 7 S'N C M cC  I I I Q0 (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 =PoP 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 (TTNT)Po (14) The total number of prey eaten (N) is therefore N=N +N = A NTP + X (TTNT)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 interprey 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 intercatch 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 intercatch 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 DigestionRateLimitation (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' interprey 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 nonexistent 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 preycapture 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 preweighed (to +/ 0.00001 gm) fly per day (FS1) 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: FS8 (1 fly per 3 hr), FS24 (1 fly per hr), or FS48 (1 fly per 0.5 hr). For FS8, antlions were fed from 4 to 7 fruit flies in a row; for FS24 they were fed from 6 to 17, and for FS48 they were fed from 5 to 12 in a row. Each run (FS1 then FS8, FS24, or FS48) 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 (FS48) 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 DRLmodel. 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 interindividual 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; OwenSmith 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 prepupal 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. 3.13.4) 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 MichaelisMenten 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 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 4r + + 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 ui '5W=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, klk8 = constants, X = conversion from wet weight to calories, see below. The coefficients were estimated using a nonlinear 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 opentopped 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+preingestion 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 ratewas 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 + preingestion, and 'waiting' (the nonforaging time spent between prey harndlinr and pursuit times). Thus net gain, G (T), is G (T) = G (T)C LC (TL)Cw(T T) (2) n p p e w T where C = energetic cost of pursuit + preingestion 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(TL)) g 1 4 5 = k (1k bc(TL)). (4) a b Thus G (T) = k (1k) Cd(TL) 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 (TL)n(k ))(c) Cd   = 0 (5) a (TL) TT TT Replacing the left hand side of equation 5 with zero and rearranging we get Cd c(TL)  = k ln(kb) Ck a Rearranging and taking the natural log of both sides C n(  ) = c(TL) 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 41 41 ;< 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 wetweight/dryweight regression equation and metabolic cost equations for eating and waiting behavior in antlion larvae. Wet weightdry 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 arcsinsquareroot transforming the percent then subtracting the transformed values. A onesample ttest 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 overestimation 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). M0 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*** sd 2.2 5.1 .0004 .064 :/1! 46.2 3'.*** '.281 1.251** sd 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 MOASext .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 3.4Continued. ........... .  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. 