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A consideration of the concepts of generality and complexity, as used in experimental components analysis Cuff, Wilfred R. 1972

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A CONSIDERATION OF THE CONCEPTS OF GENERALITY AND COMPLEXITY, AS USED IN EXPERIMENTAL COMPONENTS ANALYSIS by W i l f Cuff B. Sc., U n i v e r s i t y of Manitoba, 1966 M. Sc., U n i v e r s i t y of Western Onta r i o , 1967 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of ZOOLOGY We accept t h i s t h e s i s as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA August, 1972 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e Head o f my D e p a r t m e n t o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f Z o o l o g y  The U n i v e r s i t y o f B r i t i s h C o l u m b i a V a n c o u v e r 8 , Canada 1972 Date i fruly ABSTRACT i Two features of experimental compoents a n a l y s i s ( H o l l i n g , 1966) are considered. A method of w r i t i n g general f u n c t i o n s i s proposed and v e r i f i e d w i t h some observations from hydra. An a n a l y t i c a l method of studying c e r t a i n features of a s i m u l a t i o n model of att a c k i s then proposed and te s t e d . The method proposed to a t t a i n g e n e r a l i t y of funct i o n s i s a s t r a t e g i c one. O p e r a t i o n a l l y , one need only determine a st r a t e g y used by many animals, uncover some opposing v a r i a b l e s which may have been important i n i t s development, and p r e d i c t on the b a s i s of these a n t a g o n i s t i c forces some c h a r a c t e r i s t i c response of the strat e g y . I t i s suggested that t h i s c h a r a c t e r i s t i c response i s a broadly a p p l i c a b l e as i s the strat e g y . The dynamic response of a number of modes of searching through volume i s pr e d i c t e d to be of shorter d u r a t i o n the l a r g e r the hunger thr e s h o l d to which the mode responds. This i s found to be the case f o r hydra. A s i m i l a r hypothesis, that the steady states of these searching modes be d i r e c t l y r e l a t e d to the hunger thresholds of the modes, does not give as c l e a r r e s u l t s . I t i s suggested that an a d d i t i o n a l s t r a t e g y must be considered to give a b e t t e r p i c t u r e . These r e s u l t s are then used to w r i t e a model f o r the time which an animal spends searching f o r food. The s e n s i t i v i t y a n a l y s i s of the attack s i m u l a t i o n model attempts to for m a l i z e the conventional method: going through the block diagram and p i c k i n g out what seems to be important. A Boolean expression i s w r i t t e n to describe a l l p o s s i b l e paths through the model. This expression i s used to t a b u l a t e the va r i o u s forms of the equation of the dependent v a r i a b l e , the attack r a t e . These forms of the attack r a t e are used to i l l u s t r a t e how one might analyze a complex, non - l i n e a r model. i i TABLE OF CONTENTS Page ABSTRACT i LIST OF TABLES i v LIST OF FIGURES v i ACKNOWLEDGEMENTS v i i i GENERAL INTRODUCTION 1 PART I THE PROBLEM OF GENERALITY INTRODUCTION 2 THE PROPOSAL 6 HYDRA STUDIES: IDENTIFICATION OF THE SEARCH MODES AND PRELIMANARIES ON THE MODEL 14 HYDRA STUDIES: A MODEL OF VOLUME SEARCHED 30 HYDRA EXPERIMENTATION: THE APPROACH 43 HYDRA EXPERIMENTATION: TENTACLE LENGTHENING 49 HYDRA EXPERIMENTATION: WALKING 63 HYDRA EXPERIMENTATION: FLOATING 72 A TEST OF THE HYPOTHESIS 77 PART I I A TIME SPENT SEARCHING FUNCTION FOR HYDRA INTRODUCTION 79 METHODS AND RESULTS 80 i i i DISCUSSION FOR PARTS I AND I I 94 Searching by Animals 94 S t r a t e g i c a l Models 101 PART I I I SENSITIVITY ANALYSIS OF A MODEL OF ATTACK INTRODUCTION 108 The Approach 108 TECHNIQUES FOR WRITING LOGICAL EQUATIONS I l l REDUCTION OF THE LOGIC OF THE ATTACK MODEL 118 DEDUCTIONS 124 DISCUSSION FOR PART I I I 140 GENERAL DISCUSSION 142 LIST OF SYMBOLS 144 BIBLIOGRAPHY 145 APPENDICES I. Some observations on hydra i n Beaver Creek .... 151 I I . P a r t i a l e v a l u a t i o n of the equations of the model 165 I I I . E f f e c t of parameter values on the conclusions of PART I I 168 IV. Flow graph of the a t t a c k model 171 LIST OF TABLES Page i v Table I . Parameter estimates f o r the f o l l o w i n g equation: L ( t ) - L ( t ) = L ( t m a x ) m i n l+exp(s)/exp(r*t) 58 Table I I . The r e s u l t s of four runs of a t h r e e - l e v e l , nested ANOVA 59 Table I I I . Distance moved per time i n t e r v a l by Chlorohydra v i r i d i s s i m a 65 Table IV. Percentage of hydra which moved per hour, f o r successive hours a f t e r hour of f i r s t step 68 Table V. A f a c t o r i a l A n a l y s i s of Variance Table: the e f f e c t of food and hydra d e n s i t y on the time which hydra take to begin f l o a t i n g 74 Table VI. The va r i o u s forms, w i t h c o n t r o l l i n g l o g i c , of the f u n c t i o n of time spent i n d i g e s t i v e pause (TD) included i n the att a c k model 126 Table V I I . The various forms, w i t h c o n t r o l l i n g l o g i c , of the f u n c t i o n of time spent searching (TS) included i n the model of attack 127 Table V I I I . The various forms, w i t h c o n t r o l l i n g l o g i c , of the f u n c t i o n of time spent pursuing prey (TP) included i n the att a c k model 129 Table IX. The various forms, w i t h c o n t r o l l i n g l o g i c , of the f u n c t i o n of time spent e a t i n g the prey (TE) included i n the model of at t a c k 130 Table X. The various forms of T l . The at t a c k r a t e (A) i s defined as 24/TI 131 Table X I . The parameters of the attack r a t e , by type of f u n c t i o n 134 Table X I I . The t o t a l i t y of types of hunger equations outputted at the end of the i - l o o p of the at t a c k model. C o n t r o l l i n g l o g i c i s given 137 V Table I of Appendix I Density of Hydra carnea (number/cm ) i n Beaver Creek 155 Table I I of Appendix I R e l a t i o n s h i p between the d e n s i t y (numbers/ quadrat) of Hydra carnea i n Beaver Creek and the depth of the water above the quadrat (A) and the surface water v e l o c i t y (B) 158 Table I I I of Appendix I Percentage of hydra and percentage of substrate type, by substrate types, f o r each of f i v e s i t e s 160 Table IV of Appendix I Percentage of hydra w i t h one or more buds and percentage of hydra w i t h one or more t e s t e s as a c o r r e l a t e of the d e n s i t y of the hydra 162 V I LIST OF FIGURES Page Figure 1. Various types of volume searched by a hydra. The e f f e c t of prey movement i s included 32 Figure 2. A number of i n d i v i d u a l Chlorohydra graphs of t e n t a c l e length as a f u n c t i o n of time of food d e p r i v a t i o n 53 Figure 3. Mean t e n t a c l e and s t a l k length as a f u n c t i o n of time of food d e p r i v a t i o n 56 Figure 4. The time that hydra take to egest food-remains as a f u n c t i o n of the s i z e of t h e i r l a s t meal .. 62 Figure 5. The time that Chlorohydra v i r i d i s s i m a polyps take to walk f o r the f i r s t time, as a fu n c t i o n of the s i z e of their°last meal 71 Figure 6. The time that hydra take to detach and begin to f l o a t as a f u n c t i o n of the s i z e of t h e i r l a s t meal 75 Figure 7. A simulated r e s u l t of the searching r a t e of a hydra which has been pre-fed w i t h 3 Artemia n a u p l i i , as a f u n c t i o n of food d e p r i v a t i o n . VYO.O mm/minute, VPFO.50 mm/minute, and DMT=2.0 mm 86 Figure 8. A simulated r e s u l t of the searching r a t e of a hydra which has been pre-fed w i t h 3 Artemia n a u p l i i , as a f u n c t i o n of food d e p r i v a t i o n . VY=VPF=100 mm/minute, and DMT=300 mm 87 Figure 9. A simulated r e s u l t of the searching r a t e of a hydra which has been pre-fed w i t h 3 Artemia n a u p l i i , as a f u n c t i o n of food d e p r i v a t i o n . VY=500.0 mm/minute, VPF=100 mm/minute, and DMT=0.0 mm 88 Figure 10. A simulated r e s u l t of the searching r a t e of a hydra which has been pre-fed w i t h 3 Artemia n a u p l i i , as a f u n c t i o n of food d e p r i v a t i o n . VY=VPF=10,000 mm/minute and DMT=150 mm 90 Fig u r e 11. A simulated r e s u l t of the searching r a t e of a hydra which has been pre-fed w i t h 3 Artemia n a u p l i i , as a f u n c t i o n of food d e p r i v a t i o n . VY=10,000 mm/minute, VPF=100 mm/minute, and DMT=0.0 mm 91 v i i Page F i g u r e 12. Median a c t i v i t y of c h i c k s , r a b b i t s , guinea p i g s , and hamsters as a f u n c t i o n of days of depr i v -a t i o n (from Campbell ejt a l , 1966) 96 Figure 13. A FORTRAN I F ... statement from the flow graph of Appendix IV U 2 Figure 14. An equivalent form f o r the c o n d i t i o n a l statement of Figure 13 1X2 Figure 15. A l o g i c a l f l o w graph, w i t h operators A and B ... 112 Figure 16. A l o g i c a l flow graph, with operators Q, D, and C 114 Figure 17. A l o g i c a l flow graph, w i t h operators Q, D, C, and E 114 Figure 18. The l o g i c f o r the flow diagram of Appendix 3V .. 119 Figure 19. The l o g i c f o r the DEFG par t of the flow graph of Appendix IV 120 Figure 20. The l o g i c f o r the flow diagram of Appendix TV. The DEFG and HI p a r t s are modified from the form given i n Figure 18 123 Figure 1 of Appendix I A. map of the f i e l d study area 152 Figure 1 of Appendix IV A flo w graph of the att a c k model 172 ACKNOWLEDGEMENTS v i i i I am g r a t e f u l t o Dr. C.S. H o l l i n g f o r h i s advice during t h i s study. Suggestions by P. L a r k i n , N. G i l b e r t , and C. Wehrhahn are app r e c i a t e d . A number of other people have been of s e r v i c e at v a r i o u s stages. My thanks t o T. Gossard, R. Harger, D. McPhail, and N. Wilimovsky. GENERAL INTRODUCTION 1 The o r i g i n a l i n t e n t of t h i s study was to replace the s e c t i o n of the predation model of H o l l i n g that describes competition (see G r i f f i t h s and H o l l i n g , 1969) w i t h a s e c t i o n that included more explanatory content. The techniques of experimental components a n a l y s i s ( H o l l i n g , 1966) were to be a p p l i e d to the process of competition. In c o n s i d e r i n g experimental components a n a l y s i s and i t s bedpartner, s i m u l a t i o n , I became aware of two inadequacies. F i r s t , t h i s approach produces such l a r g e and i n t r i c a t e models that complexity of the model becomes a problem i n i t s own r i g h t . I t i s a d i f f i c u l t task to r e l a t e i n d e t a i l H o l l i n g 1 s model of predation to Beukema's (1968) model of s t i c k l e b a c k predation to Ware's (1971) model of t r o u t predation. Secondly, H o l l i n g 1 s method of w r i t i n g f u n c t i o n s which describe the a c t i v i t y of many species i s not as g e n e r a l l y u s e f u l as one might wish. In the end i t was decided to concentrate on methods to achieve both g e n e r a l i t y and s i m p l i c i t y using predation as the focus. The problem of j o i n t l y a c h i e v i n g both are fundamental quests i n science. I t i s not my presumption to f u l l y answer such d i f f i c u l t questions i n t h i s t h e s i s , but any step that can be made i n that d i r e c t i o n i s s i g n i f i c a n t . PART I THE PROBLEM OF GENERALITY INTRODUCTION 2 H o l l i n g (1966) suggested that a model should have the a t t r i b u t e s of r e a l i s m , p r e c i s i o n , wholeness, and g e n e r a l i t y . Whether these goals were t o guide d e c i s i o n s concerning v a r i a b l e s , f u n c t i o n s , or s t r u c t u r e of systems was not made c l e a r . I examined these a t t r i b u t e s at each l e v e l . The a t t r i b u t e s of r e a l i s m and g e n e r a l i t y are a p p l i c a b l e at the l e v e l of the v a r i a b l e . H o l l i n g (op. c i t . ) achieved r e a l i s m and g e n e r a l i t y at t h i s l e v e l by invoking the concept of component and by c l a s s i f y i n g components by t h e i r degree of u n i v e r s a l i t y . The word component tends t o mean membership i n a c l a s s i f i c a t i o n scheme; however, the choice of component i n f l u e n c e s d i r e c t l y the v a r i a b l e s used i n the model. When h i s study of the process of pr e d a t i o n began, H o l l i n g (op. c i t . ) f i r s t chose a dependent v a r i a b l e and then l i s t e d a l l processes which could i n f l u e n c e t h i s v a r i a b l e . These were termed components and separated i n t o b a s i c (shown by a l l species) and s u b s i d i a r y (shown by some species) . Each component was then subdivided again and again. This procedure was continued u n t i l each component was expressed i n terms of a set of r e a l i s t i c v a r i a b l e s . Each set r e f l e c t e d the s u b s i d i a r y or b a s i c nature of the component from which i t was d e r i v e d . At the l e v e l of s t r u c t u r e , the a t t r i b u t e s of wholeness and g e n e r a l i t y apply. H o l l i n g (op. c i t . ) achieved the t e c h n o l o g i c a l p o t e n t i a l f o r wholeness by using s i m u l a t i o n , r a t h e r than a n a l y t i c , models. G e n e r a l i t y was achieved by usin g the u n i v e r s a l attack c y c l e as the core of the model. 3 At the l e v e l of the f u n c t i o n , the a t t r i b u t e s of r e a l i s m , p r e c i s i o n , and g e n e r a l i t y apply. By " f u n c t i o n " i s meant the r e l a t i o n s h i p between the dependent and the independent v a r i a b l e s . From t h i s p o i n t of view, r e a l i s m has been achieved by conducting experiments f o r the form of the r e l a t i o n s h i p and by seeking an explanation f o r t h i s r e l a t i o n s h i p from the l i t e r a t u r e . P r e c i s i o n has been achieved by combining p r e c i s e experimentation w i t h t h e o r e t i c a l d i v e r s i o n s and by t r a n s l a t i n g b i o l o g i c a l explanations from E n g l i s h t o the more rig o r o u s language of mathematics. Two methods of ach i e v i n g g e n e r a l i t y of f u n c t i o n s have been attempted. F i r s t , where i t was p o s s i b l e , H o l l i n g (1966) considered b i o l o g i c a l processes i n p h y s i c a l terms. When d e a l i n g with the v i s u a l f i e l d of mantids, H o l l i n g (op. c i t . ) wrote the area of pe r c e p t i o n as a l i n e a r f u n c t i o n of the square of the dis t a n c e t o i t s outer boundary. The p h y s i c a l b a s i s i s that area i s a f u n c t i o n of some di s t a n c e squared. Secondly, g e n e r a l i t y of f u n c t i o n s was obtained by making s i m p l i f i e d explanations of the causative mechanism. H o l l i n g (1966; page 68) suggested that the v a l i d i t y of t h i s method would be confirmed or re-futed as f u r t h e r evidence accumulated. Some has been accumulated i n the case of the hunger equation. Dethier (1969) found that b l o w f l i e s , Phormia r e g i n a , do not continue t o eat u n t i l the gut i s f u l l of food. The f l i e s stop e a t i n g upon advice based upon sensory a d a p t a t i o n . Hubbell (1971), i n h i s study of the isopod A r m i d i l l i u m , found that gut ca p a c i t y i s not a p h y s i c a l q u a n t i t y but a f u n c t i o n of the o v e r a l l h e a l t h of the isopod. Beukema (1968), i n h i s study of the s t i c k l e b a c k Gasterosteus aculeatus, found that the p a t t e r n of pr e s e n t i n g and wi t h h o l d i n g food was of some s i g n i f i c a n c e i n d i c t a t i n g the form of the hunger equation. The work of Beukema and of Hubbell suggest that the causative e x p l a n a t i o n might 4 be incomplete. I cannot determine from t h e i r papers whether t h i s incompleteness i s t r i v i a l or not . The work of Dethier does not f i n d gut c a p a c i t y to be an important determinant of hunger i n b l o w f l i e s . I t i s no longer a simple problem of incompleteness. These examples suggest the sort of problems that t h i s method of formulating f u n c t i o n s might e n t a i l . What i s the most important causative feature? I s there a most important cause? This i s the inadequacy t o which I r e f e r r e d i n the GENERAL INTRODUCTION. I n H o l l i n g ' s model of p r e d a t i o n there are four e s s e n t i a l equations: time spent d i g e s t i n g prey (TD), time spent pursuing prey (TP), time spent searching f o r prey (TS), and time spent e a t i n g prey (TE) . The f u n c t i o n f o r the time spent d i g e s t i n g prey (TD) i s based on a causative e x p l a n a t i o n . I t i s a rewording of the hunger equation which, i n t u r n , i s based on an a b s t r a c t i o n of the gut as a s i z e - l i m i t e d bag. The fu n c t i o n s f o r the time searching f o r and pursuing prey (TS and TP) are both based mainly on p h y s i c a l e x p l a n a t i o n s . The exception i s the f u n c t i o n r e l a t i n g the s i z e of the r e a c t i v e f i e l d t o hunger. T h i s f u n c t i o n i s based on a s i m p l i f i e d e x p l a n ation of the p h y s i o l o g i c a l s t r u c t u r e of the eye. The f u n c t i o n f o r time spent e a t i n g a prey (TE) incor p o r a t e s the causative e x p l a n a t i o n . Thus each of the four equations of the attack c y c l e incorporates the causative e x p l a n a t i o n . The f u n c t i o n s f o r time spent d i g e s t i n g prey (TD), time spent pursuing prey (TP), and time spent e a t i n g prey (TE) are a l l e x p l i c i t f u n c t i o n s . Each f i t s i n t o the complete s i m u l a t i o n model as a small module, complete w i t h i n i t s e l f and with a w e l l defined boundary. On the other hand, the f u n c t i o n f o r time spent searching f o r prey (TS) i s an i m p l i c i t one and i s solved i t e r a t i v e l y i n the s i m u l a t i o n model of p r e d a t i o n . I t i s solved using two i n t e r n a l loops; these loops c o n t a i n eight i n t e r n a l choice statements (the FORTRAN statement: I F a n-^,^,^ r e s u l t i n g i n seven p o s s i b l e f u n c t i o n s f o r time spent searching. Some o these f u n c t i o n s are used as p a r t s of the arguments of three separate choice statements. The e f f e c t of these choice statements i s , i n two cases, t o determine the f u n c t i o n f o r time spent pursuing prey (TP) and, i n the other case, to introduce a major m o d i f i c a t i o n i n the searching f u n c t i o n (TS) i t s e l f . The f u n c t i o n f o r time spent searching (TS) i s simulated i n such a way that i t can be modified only w i t h great d i f f i c u l t y . The other three f u n c t i o n s are included i n the s i m u l a t i o n model as d i s t i n c t modules and so are e a s i l y m o d i f i a b l e i n s p e c i f i c cases. Thus i t was decided that the time spent searching v a r i a b l e (TS) would be an appropriate one f o r t e s t i n g a s t r a t e g i c method of w r i t i n g general f u n c t i o n s . This method w i l l be introduced i n the next s e c t i o n . An a d d i t i o n a l attempt w i l l be made t o w r i t e an e x p l i c i t f u n c t i o n f o r time spent searching (TS) . THE PROPOSAL 6 The i n i t i a l formulation by H o l l i n g (1966) of the time spent search-i n f u n c t i o n , as w e l l as a second unpublished v e r s i o n , are derived from where VD i s the v e l o c i t y of the predator during searching; VY i s the average v e l o c i t y of the prey; RD i s the r e a c t i v e distance of the predator; NO i s the prey d e n s i t y ; and NA i s the number of prey attacked i n time t . This equation i s w r i t t e n more g e n e r a l l y as where VS i s the volume searched by the predator. (The bracketted part of I I was expanded fr.om area to volume by Watt, 1968.) This volume was modified f o r the e f f e c t s of prey movement. I f time t can be separated from the r i g h t hand side of t h i s equation, then i t becomes i d e n t i c a l l y the time spent searching f o r prey, TS. The job of w r i t i n g an e x p l i c i t TS f u n c t i o n i s reduced to one of f i n d i n g a VS f u n c t i o n from which t may be separated. In attempting to f i n d such a f u n c t i o n f o r the volume searched by a predator, I reasoned from the fact that a l i m i t e d amount of g e n e r a l i t y has been obtained by using c a u s a t i v e , or t a c t i c a l , arguments. Yet a l a r g e number of t a c t i c s can be combined under the umbrella of a more l i m i t e d number of s t r a t e g i e s . Thus i t seemed val u a b l e to explore some pr o p e r t i e s of the t a c t i c a l - s t r a t e g i c a l concepts. F o l l o w i n g economic and m i l i t a r y etymology, I w i l l use the word t a c t i c as a synonym f o r method and s t r a t e g y as a synonym f o r goal. The concept of strategy has a s u b t l e r e l a t i o n s h i p to that of t a c t i c . I t i s p o s s i b l e f o r a s i n g l e process to be viewed e i t h e r as a strategy or as a t a c t i c . A t r i v i a l example w i l l serve to c l a r i f y what i s meant. Dr. S. Hubbell has invented a monopoly-type game of " e c o l o g i c a l II NA = VS(t ,VD,VY,RD)*NO 1 2 7 warfare". Each p l a y e r becomes a species who, during the game, attempts to s u r v i ve longer than a l l other c o n t e s t a n t s . The p l a y e r who survives the longest wins. The goal, or strategy, of each p l a y e r i s t o stay i n the game as long as p o s s i b l e . Consider the case where one contestant shares a t r o p h i c l e v e l with two other p l a y e r s . He i s a superi o r competitor i n every respect except that the other p l a y e r s have a habi t of k i l l i n g h i s young by stepping on them. The contestant can foresee that h i s recruitment i s too low t o in s u r e s u r v i v a l and that he i s going " e x t i n c t " . H i s immediate concern i s t o f i n d a way t o p r o t e c t the young from the feet of h i s competitors. This i s a goal, a s t r a t e g y . Thus there are at l e a s t two s t r a t e g i e s : to stay i n the game as long as p o s s i b l e and t o keep the feet of the competitors o f f the heads of the young. But the second str a t e g y i s one way of i n s u r i n g that the f i r s t w i l l be accomplished. This makes a st r a t e g y a t a c t i c . One way of d i s t i n g u i s h i n g between the two types of s t r a t e g i e s i n t h i s example i s t o consider the f i r s t as the u l t i m a t e s t r a t e g y and the second as a proximal s t r a t e g y . Thus i t i s necessary t o t a l k of s t r a t e g i e s at v a r i o u s l e v e l s . Since s t r a t e g i e s can be viewed as t a c t i c s , i t i s c l e a r that t a c t i c s can a l s o be defined at v a r i o u s l e v e l s . These l e v e l s can be viewed as h i e r a r c h i c a l . The u l t i m a t e s t r a t e g y of e c o l o g i c a l systems i s often thought t o be s u r v i v a l of the spe c i e s . S u r v i v a l i s ensured by p o p u l a t i o n processes such as reproduction, d i s p e r -s a l , and so on. These processes, i n t u r n , would not develop unless there had developed e a r l i e r a process of energy i n t a k e (predation i n a broad sense). There are three l e v e l s mentioned here. Each of these processes can be thought of as a s t r a t e g y : unless an animal takes i n energy i t w i l l d i e , unless i t disperses i t may e l i m i n a t e 8 i t s e l f by d e n s i t y dependent processes, unless i t reproduces i t w i l l go e x t i n c t i n one generation, and unless i t sur v i v e s i t w i l l not be part of the e c o l o g i c a l system. Yet the i n d i v i d u a l a c t s of energy accumulation, reproduction, d i s p e r s a l , and s u r v i v a l are concrete processes - t a c t i c s . The v a r i o u s subprocesses that combine t o make up a process c o n s t i t u t e the elements of another k i n d of h i e r a r c h i c a l scheme. For example, the process of p r e d a t i o n has been subdivided i n t o search, approach, capture, and p r e p a r a t i o n - i n g e s t i o n subprocesses (De R u i t e r , 1 9 6 7 ) . Each of these subprocesses as discussed i s a t a c t i c . This i s not the only p o s s i b l e view. For example, l a t e r i n the t h e s i s v a r i o u s search s t r a t e g i e s used by animals w i l l be considered. I n t h i s type of h i e r a r c h i c a l scheme the components at one l e v e l work together t o produce the process at the next l e v e l . The main p o i n t s of the l a s t few paragraphs a r e : 1): any-process can be viewed e i t h e r as a t a c t i c or as a st r a t e g y ; 2) the t a c t i c a l -s t r a t e g i c a l concept does apply at v a r i o u s l e v e l s on n a t u r a l l y o c c u r r i n g h i e r a r c h i c a l schemes. Other authors have already used s t r a t e g i c a l concepts i n w r i t i n g general f u n c t i o n s . Space l i m i t s t h e i r c o n s i d e r a t i o n here. The subject w i l l be taken up i n the d i s c u s s i o n . Some data are a v a i l a b l e which r e l a t e t o the subject of w r i t i n g s t r a t e g i c a l f u n c t i o n s . They i n d i c a t e that the responses of species tend to produce behaviour which seems t o the observer t o be that expected i f the animals use behaviour that i s optimal i n t h e i r current h a b i t a t s ( B r e t t , 1 9 6 5 ; Milsum,1 9 6 6 ). Without going i n t o examples, I w i l l assume that t h i s i s t r u e . The s t r a t e g i c a l method proposed f o r w r i t i n g general f u n c t i o n s f o l l o w s from t h i s assumption. The method i s f i r s t t o define a strategy, then t o d e l i m i t a few general opposing f o r c e s , and f i n a l l y t o look at the 9 e f f e c t s of the forces on the str a t e g y and to attempt to see c e r t a i n features which seem " d e s i r a b l e " . I t i s suggested that these features w i l l represent general methods f o r a c h i e v i n g a given goal. I t i s assumed that animals searching f o r food attempt to cover as much volume as i s needed i n order to catch enough prey to r e t u r n hunger to some l e v e l . One such l e v e l might be the "eat t h r e s h o l d " ( H o l l i n g , 1966). This assumption i s the str a t e g y chosen f o r study i n the t h e s i s . One general force a c t i n g upon t h i s s t r a t e g y i s s t a r v a t i o n . A response to t h i s , and only t h i s , f a c t o r would be a volume searched during some time i n t e r v a l (a searching rate) that i s always greater than the volume occupied by those prey that are necessary to r e t u r n the hunger of the predator to the eat t h r e s h o l d . A lower searching r a t e would r e s u l t i n a hungry predator. The hungrier the predator the more l i k e l y i t i s to starve to death. I t i s d i f f i c u l t to say how much greater the a c t u a l volume covered per time i n t e r v a l i n the future w i l l be over that needed i n current h a b i t a t s and under current f l u c t u a t i o n s i n prey abundance. However, during a period of e x c e p t i o n a l l y low prey abundance, the higher the volume covered per time i n t e r v a l , the more l i k e l y w i l l be s u r v i v a l . ( I t i s assumed that the animal i s not able to sidestep the i s s u e ; the a n i -mal can't leave the area during a period of low prey abundance.) Thus there should be a tendency f o r search r a t e s to increase as e v o l u t i o n proceeds. However, as the volume covered per time i n t e r v a l increases the animal becomes l e s s f i t under periods of high prey abundance. An animal that wastes time r e a c t i n g to prey i t w i l l not eat or e a t i n g prey i t cannot use, and so runs the r i s k of d e s t r o y i n g i t s food supply or of becoming more v u l n e r a b l e to i t s predators, i s not an e f f e c t i v e competitor. 10 This animal, being l e s s f i t than ot h e r s , would tend to be replaced by organisms w i t h a lower search r a t e . This replacement would tend to depress over time the average searching r a t e of the species. ( I t may be u s e f u l to emphasize that the argument i s based on an e v o l u t i o n a r y time s c a l e , not a short time s c a l e . I t supposes that each organism i n h e r i t s some searching r a t e . Over a long time period some searching r a t e s w i l l be more f i t than others. The f i t t e s t r ates w i l l s u r v i ve and replace the others, and so the average search r a t e of the species w i l l change. The r e d u c e d - e f f i c i e n c y force i s , from t h i s veiwpoint, a measure of the l i k e l i h o o d of an animal w i t h some searching r a t e being outcompeted by one w i t h a d i f f e r e n t search r a t e . This f o r c e can be viewed from a minimum of one generation.) One might expect that animals would tend to use a searching r a t e that l i e s somewhere between those extremes produced by separate c o n s i d e r a t i o n of the s t a r v a t i o n and the r e d u c e d - e f f i c i e n c y f o r c e s . Let hunger be measured as the amount of food required to r e t u r n an animal to i t s "eat t h r e s h o l d " ( H o l l i n g , 1966). When measured f o r mantids, hunger was found to be d i r e c t l y r e l a t e d to the time of food d e p r i v a t i o n (op. c i t . ) . I have suggested that the volume searched per time i n t e r v a l i s c l o s e l y a s sociated w i t h hunger. The hypothesized c o n t r o l i s the antag-o n i s t i c a c t i o n of two f o r c e s . I t f o l l o w s from t h i s argument that the search r a t e i s d i r e c t l y r e l a t e d to the time of food d e p r i v a t i o n . This statement does not seem to be p r o f i t a b l y t e s t a b l e as stated. I t becomes more so when we consider that many animals use more than one searching method (see the d i s c u s s i o n ) . Two and sometimes three methods are used. Each method has i t s own responsiveness to hunger and each has i t s own searching r a t e f u n c t i o n . The magnitude of the s t a r v a t i o n force i s d i r e c t l y r e l a t e d to the time of food d e p r i v a t i o n . The l i k e l i h o o d of an animal s t a r v i n g to death 11 increases w i t h the length of time i t i s deprived of food. The magnitude of the r e d u c e d - e f f i c i e n c y force i s assumed to be independent of the time of food d e p r i v a t i o n . This i m p l i e s that any method of searching that i s i n i t i a t e d at low hunger l e v e l s w i l l be molded more s t r o n g l y by the r e d u c e d - e f f i c i e n c y than by the s t a r v a t i o n f o r c e . Any method of searching that i s i n i t i a t e d at high hunger l e v e l s w i l l be molded more s t r o n g l y by the s t a r v a t i o n than by the r e d u c e d - e f f i c i e n c y force. For each search mode, I assume that the f u n c t i o n r e l a t i n g volume searched per time i n t e r v a l to the time of food d e p r i v a t i o n w i l l take the f o l l o w i n g form: a minimum steady ra t e of searching to a c e r t a i n hunger l e v e l , then an i n c r e a s i n g r a t e of searching, and f i n a l l y a maximum steady r a t e of searching. (Henceforth, I w i l l use the jargon: dynamic and steady s t a t e responses. A dynamic response i s a n o n - s t a b i l i z e d response. A steady s t a t e response i s a s t a b i l i z e d response. The minimum and the maximum steady r a t e s of searching are steady s t a t e responses. The i n c r e a s i n g r a t e of searching i s a dynamic response.) Any search mode which begins i t s r i s e from the minimum rate - which i s " i n i t i a t e d " - at low times of food d e p r i v a t i o n w i l l possess a dynamic response operating over a long time i n t e r v a l . I f the d u r a t i o n of the dynamic response approaches zero, then too much food may be contacted too soon. Conversely, any searching mode which i s i n i t i a t e d a f t e r a long i n t e r v a l of food d e p r i v a t i o n w i l l possess a dynamic response operating over a short time i n t e r v a l . I f the d u r a t i o n of the dynamic response i s l a r g e , then the animal may not search through enough volume soon enough to prevent i t s death. The animal may starve to death only because the search mode does not reach i t s f u l l p o t e n t i a l at a low enough hunger l e v e l . In general, the d u r a t i o n of the dynamic response of a searching ra t e f u n c t i o n should be i n v e r s e l y r e l a t e d to the f u n c t i o n ' s t h r e s h o l d of i n i t i a t i o n . PART I of t h i s t h e s i s 12 i s d i r e c t e d towards t e s t i n g t h i s hypothesis. Two f u r t h e r p o ints need e l a b o r a t i o n . I n equation 12 the volume searched v a r i a b l e i s w r i t t e n as a f u n c t i o n of VD, RD, t , and VY. Yet, prey v e l o c i t y (VY) has been mentioned only i n d i r e c t l y . This i s because the volume to which I have r e f e r r e d thus f a r i s an e f f e c t i v e volume, where prey v e l o c i t y i s incorporated as a component of predator movement. In t h i s way prey are considered to be f i x e d i n space. Secondly, i n f o r m a t i o n about the values of the maximum search r a t e s i s a l s o necessary i n order to formulate a f u n c t i o n f o r the volume searched by a hydra. PART I I w i l l take up t h i s problem. In order t o t e s t the proposed r e l a t i o n s h i p i t i s necessary to choose a s u i t a b l e species w i t h which to work and to make a l i s t of i t s search modes. The hypothesis i s i n terms of volume searched per time i n t e r v a l , as a f u n c t i o n of time of food d e p r i v a t i o n . Thus i t i s necessary e i t h e r to measure an index of volume searched f o r each search mode or to model what seems to be the volume searched by each search mode. The l a t t e r approach i s chosen, as i t i s d i f f i c u l t to measure d i r e c t l y the volume searched by an animal. When the model i s evaluated, i t i s used to o b t a i n estimates of the searching r a t e f o r each search mode, as a f u n c t i o n of time of food d e p r i v a t i o n . I t i s then l e f t only to place the search modes on the food d e p r i v a t i o n a x i s , to measure the i n i t i a t i o n t h r e s h old of each search mode. A s u i t a b l e experimental animal was deemed one which searches i n a manner more or l e s s unaided by d i r e c t i o n a l cues from the prey. This type of searching i s expected to be predominant i n p r i m i t i v e organisms. They of t e n have a r e s t r i c t e d set of sensory organs w i t h which to detect the presence of prey. One of the simplest known m u l t i c e l l u l a r animals, the hydra, might be a s u i t a b l e experimental animal. Hydra are r e a d i l y c u l t u r e d and reproduce a s e x u a l l y , thereby a l l o w i n g 13 one to work w i t h a clone. They can a t t a i n a s u b s t a n t i a l r a t e of population increase by means of asexual reproduction. Some brown hydra i n my l a b o r a t o r y took l e s s than f i v e days to double t h e i r numbers. HYDRA. STUDIES: IDENTIFICATION OF THE SEARCH MODES AND PRELIMINARIES ON THE MODEL 14 In t h i s s e c t i o n the search modes are i d e n t i f i e d and the model which i s used to c a l c u l a t e a measure of the volume searched by each search mode i s introduced. A l i t e r a t u r e survey opens the s e c t i o n ; those behaviours which seem to have any r e l a t i o n s h i p to searching are i d e n t i f i e d and described. Our d e f i n i t i o n of search mode i s r e s t r i c t e d to one of volume searched per time i n t e r v a l . Thus, a f t e r a l l p o s s i b l e search modes are i d e n t i f i e d some are e l i m i n a t e d . Those that do not c o n t r i b u t e to searching through volume are e l i m i n a t e d . This process leads n a t u r a l l y i n t o a d i s c u s s i o n of the ways i n which volume i s searched by the various modes. I t i s thus appropriate to describe i n a p r e l i m i n a r y way the o v e r a l l s t r u c t u r e of the model, presented i n d e t a i l i n the next s e c t i o n . Some f i e l d observations were a l s o conducted. The reader not f a m i l i a r w i t h hydra may wish to f a m i l i a r i z e himself w i t h a broad d e s c r i p t i o n of various aspects of the ecology of a hydra population by reading the r e s u l t s of a f i e l d study as presented i n Appendix I . Some aspects of the d i s t r i b u t i o n and abundance of Hydra carnea as they occurred i n Beaver Creek (Stanley Park, Vancouver) are described. These r e s u l t s are r e l a t e d to some observations on the searching of Chlorohydra  v i r i d i s s i m a and a t e n t a t i v e l i f e h i s t o r y f o r m u l a t i o n i s constructed f o r hydra i n Beaver Creek. Various aspects of these r e s u l t s from the f i e l d w i l l be r e f e r r e d to from time to time; however, the r e s u l t s have been put i n an appendix so as not to i n t e r f e r e w i t h the flow of t e s t i n g the hypothesis of the r e l a t i v e d u rations of the dynamic responses of the search modes. We r e t u r n to i d e n t i f i c a t i o n of the search modes, and begin w i t h the 15 l i t e r a t u r e survey. Hydra were f i r s t reported on by Leeuwenhoek i n a l e t t e r to the Royal S o c i e t y i n 1703 (Ewer, 1949). However, the f i r s t s e r i o u s s t u d i e s were done by Abraham Trembley around 1740. Trembley published the f i r s t d e t a i l e d d e s c r i p t i o n of what may be our f i r s t search mode, locomotion. Locomotion became the object of study again, about 150 years l a t e r . Trembley 1s o r i g i n a l observations were v e r i f i e d and extended. Such authors as Jennings (1906), Wagner (1905), and Wilson (1891) agreed w i t h Trembley that a hydra moves by a t t a c h i n g i t s e l f to the substrate a l t e r n a t e l y by i t s foot and by i t s a n t e r i o r end. The hydra f i r s t bends i t s a n t e r i o r end to the substrate. I t attaches the t e n t a c l e s to the substrate and detaches i t s foot. The foot i s brought up and set down c l o s e to the t e n t a c l e s . In t h i s way the animal moves forward almost i t s own l e n g t h , very much as a c a t e r p i l l a r or leech does. Trembley noted t h i s form of locomotion to be the usual one. This mode was described by Jennings (1906) as: "In the commonest method the animal places i t s f r e e end against the substratum, r e l e a s e s i t s f o o t , draws the l a t t e r forward, reattaches i t , and repeats the process, thus looping along l i k e a measuring worm". This mode of locomotion has become known as "stepping" (Ewer, 1949) or " l o o p i n g " (Ewer, 1947b). Trembley a l s o described a second, l e s s frequent type of locomotion. The foot i s not merely brought up to the t e n t a c l e s but i s swung over them and attached w e l l i n advance, so that the animal moves forward almost twice i t s own length. T h i s mode of locomotion has become known as "somersaulting" (Ewer, 1947b). Looping and somersaulting have tended to be described separately (Ewer, 1947b). Ewer found no c l e a r c u t d i s t i n c t i o n between these two types of locomotion. Trembley had noted t h i s l a c k of c l e a r c u t d i s t i n c t i o n but m i s l e a d i n g d i s t i n c t i o n s had crept i n t o the textbooks by 1947. Ewer found the l a t t e r form to be only an extreme form of the former. He l e t P i designate the p o s i t i o n of a hydra's foot and A designate the p o s i t i o n at which the a n t e r i o r end of the hydra i s set down. The foot i s then .. l i f t e d and the body contracted to a l i t t l e lump on the top of the t e n t a c l e s . The foot i s then set down at any point on a c i r c l e of small diameter P2, w i t h A i t s centre. I f the body i s not pro p e r l y contracted then the foot i s set down at a considerable d i s t a n c e from A, but again at any point on a c i r c l e , P3, of l a r g e r diameter than P2. The d i f f e r e n c e between the two forms of locomotion i s thus the degree to which the body c o n t r a c t s during the r e p o s i t i o n i n g of the foot . Ewer discovered that the hydra placed i t s foot anywhere on the locus of c i r c l e P2 or P3. E a r l y analogies to c a t e r p i l l a r s , leeches, and measuring worms are somewhat p r o s a i c . F o r t u n a t e l y , Ewer used a composite name f o r t h i s behaviour, the word "walking". I n t h i s t h e s i s we a l s o use the word "walking". Hydra w i l l take "steps". Walking has some obvious r e l a t i o n s h i p s to searching. I t w i l l be included as a t e n t a t i v e searching mode. The d e t a i l e d r e l a t i o n s h i p between walking and volume searched w i l l be given i n the next s e c t i o n . Wagner (1905) describes a form of locomotion where the foot i s not detached but simply i s p u l l e d towards the t e n t a c l e s . To my knowledge t h i s behaviour has not been observed by anyone other than Wagner. For example, Ewer (1947b, 1949) deals i n considerable depth w i t h w a l k i n g , but does not mention t h i s form. My observations on H. l i t t o r a l i s and Chlorohydra  v i r i d i s s i m a have not confirmed i t s presence. This form of locomotion w i l l be mentioned again i n l a t e r s e c t i o n s only because there i t w i l l be shown that foot attachment i s not relevant to volume searched. This behaviour i s t r e a t e d as j u s t another step. Jennings (1906) noted that a hydra had been observed to a t t a c h i t s e l f by i t s t e n t a c l e s , r e l e a s e i t s f o o t , and then use i t s t e n t a c l e s l i k e l e g s . I have never seen t h i s behaviour i n my observations of hydra and have never seen i t r e f e r r e d to i n any other p u b l i c a t i o n . I t i s d i f f i c u l t to imagine how t h i s behaviour operates. To move i n some d i r e c t i o n , the hydra must move a l l t e n t a c l e s i n the d i r e c t i o n of movement. Tentacle c o n t r a c t i o n s have been studied from an e l e c t r o -p h y s i o l o g i c a l viewpoint by Rushforth and Hofman ( i n press; a l s o see abst r a c t by same authors, 1966). The neural c o r r e l a t e s of t e n t a c l e c o n t r a c t i o n s do not suggest the degree of s o p h i s t i c a t i o n necessary to move a l l t e n t a c l e s i n the d i r e c t i o n of movement. Ewer (1947b) shows that only 10 or fewer nematocysts are used to anchor each t e n t a c l e during locomotion and that a period of 10 seconds of m i l d pressure i s necessary to discharge the small g l u t i n a n t s . I t i s conceivable that the i n v e s t i g a t o r to which Jennings r e f e r r e d saw a hydra which made repeated attempts to anchor i t s t e n t a c l e s f i r m l y to the substrate. This form of locomotion w i l l not be considered f u r t h e r as a p o t e n t i a l search mode because i t has only been reported once, because I could not observe i t i n my hydra, because current i n f o r m a t i o n does not suggest the existence of a mechanism s o p h i s t i c a t e d enough to c o n t r o l such a behaviour, and because an a l t e r n a t e explanation of the behaviour i s a v a i l a b l e . "A s t i l l d i f f e r e n t form of locomotion has been described, i n which the animal i s sa i d to g l i d e along on i t s f o o t ; how t h i s i s brought about i s not known." (Jennings, 1906). Jennings i s presumably r e f e r r i n g to the work of Wagner (1905). Wagner noted that a Chlorohydra moved by g l i d i n g on i t s f o o t . He w r i t e s that the movement i s very slow and n o t i c e a b l e only on very c l o s e observation, but contends that considerable d i s t a n c e s can be covered i n t h i s way. To my knowledge t h i s behaviour has not been seen by anyone other than Wagner. I n f a c t , B r i e n (1960) notes that the region of the basal d i s c does not p a r t i c i p a t e i n d i g e s t i o n and has a slow metabolic r a t e . Gastrodermal c e l l s i n a sta t e of d i s -i n t e g r a t i o n are continuously being evacuated through the aboral pore of the bas a l d i s c . The epidermal c e l l s of the basal d i s c s i m i l a r l y d i e and are sloughed o f f . I t i s d i f f i c u l t to see how such a region could c o n t r o l the behaviour described by Wagner. This i s considered s u f f i c i e n t evidence to e l i m i n a t e t h i s behaviour as being e i t h e r non-existant or not very frequent. Nevertheless, i t w i l l be considered as a p o s s i b l e search mode since i t w i l l be shown that t h i s mode does not c o n t r i b u t e to searching through volume. In summary, there are two p o s s i b l e search modes i n the area of locomotion. They w i l l be r e f e r r e d to as w a l k i n g and g l i d i n g . We tu r n to another set of behaviours, c o l l e c t i v e l y r e f e r r e d to as "spontaneous movements" (Jennings, 1906). A hydra which has not been r e c e n t l y fed does not remain s t i l l , but r e o r i e n t a t e s i t s s t a l k i n "a sort of rhythmical a c t i v i t y " (Jennings, 1906). A f t e r remaining i n a c e r t a i n p o s i t i o n f o r a given time, the hydra c o n t r a c t s . The hydra then bends to a new p o s i t i o n , and re-extends i t s s t a l k . I n t h i s new p o s i t i o n the hydra remains f o r a few minutes, then i t c o n t r a c t s , changes i t s p o s i t i o n and again extends. Jennings suggests that thereby "the animal thoroughly explores the region about i t s place of attachment and l a r g e l y increases i t s chance of o b t a i n i n g food". These movements have been r e f e r r e d to as "column c o n t r a c t i o n s " by more recent authors (Passano and McCullough, 1963; Rushforth, 1971). Reis (1953) challenged the observations of the e a r l i e r workers regarding s t r i c t temporal r h y t h m i c i t y of column c o n t r a c t i o n s i n hydra. Using Pelmatohydra o l i g a c t u s , he f a i l e d to f i n d any t r a c e of a temporal rhythm. By ig n o r i n g time and concentrating on sequences of movement, he d i d discover "rhythmic" behaviour. About 757o of the behaviour sequences took one of three forms. Suppose we l e t a l e t t e r designate the increase i n s t a l k length of a hydra when i t elongates from a contracted p o s i t i o n . For example, the l e t t e r b could designate some increase i n s t a l k length. Then the three sequences which appeared most fr e q u e n t l y were: aaabbbccc, abcdabcd, abba. In the f i r s t sequence a larg e number of elongations of d i f f e r e n t magnitude are repeated success-i v e l y i n s e r i e s . In the second sequence expansions of equal extent do not occur s u c c e s s i v e l y i n a block, but patterns (abed) are repeated. In the t h i r d sequence patterns are a l s o repeated, but i n reversed order, thereby producing a m i r r o r image e f f e c t . Arliythmical sequences occurred l e s s than 107o of the time. Thus Reis showed that the sequence of column co n t r a c t i o n s do f o l l o w d e f i n i t e p a t t e r n s , but that these patterns are not spaced r h y t h m i c a l l y i n time. Reis' r e s u l t s have not had widespread e f f e c t on the b e l i e f of rh y t h m i c i t y of column c o n t r a c t i o n s . His r e s u l t s have been overpowered by the disc o v e r y of a r h y t h m i c i t y i n e l e c t r i c a l a c t i v i t y , c o r r e l a t i n g to a degree w i t h column c o n t r a c t i o n s (Passano and McCullough, 1963). Two regions of rhythmical "pacemaker systems" of e l e c t r i c a l a c t i v i t y c o n t r o l column c o n t r a c t i o n s . While t h e i r paper i s not p r i m a r i l y concerned w i t h s t r i c t behavioural r h y t h m i c i t y , Passanoand McCullough note that "polyps u s u a l l y show re g u l a r c o n t r a c t i o n b u r s t s . Every 5-10 min they shorten i n t o a t i g h t b a l l , and then re-elongate t h e i r column and t e n t a c l e s " . Rushforth (1971) quotes these authors i n saying that c o n t r a c t i o n s of the body column and t e n t a c l e s i n hydra i s a " r h y t h m i c a l l y r e c u r r i n g behavioral' event. I n f a i r n e s s to Rushforth i t must be noted that h i s i n t e r e s t s , l i k e those of Passano and McCullough, are mainly at the p h y s i o l o g i c a l , and not at the be h a v i o u r a l , l e v e l . We conclude that patterned sequences of contraction-and-elongation do occur i n hydra but not i n f i x e d temporal patterns. Roughly speaking, Passano and McCullough (1963) f i n d the frequency of column c o n t r a c t i o n s to be once per 5-10 minutes. Rushforth and Hofman ( i n press) f i n d a ra t e of 4.1 c o n t r a c t i o n s per 15 minutes (SE=0.2, n=20) f o r H. l i t t o r a l i s , of 4.4 c o n t r a c t i o n s per 15 minutes (SE=0.3, n=20) f o r H. p s e u d o l i g a c t i s , and of 9.3 c o n t r a c t i o n s per 15 minutes (SE=0.4, n=20) f o r H. p i r a r d i . Another kind of spontaneous movement i s that of t e n t a c l e c o n t r a c t i o n (Wagner, 1905). Rushforth and Hofman ( i n press) w r i t e : " In unstimulated hydra, spontaneous a c t i v i t y of the t e n t a c l e s c o n s i s t p r i m a r i l y of s i n g l e c o n t r a c t i o n s of i n d i v i d u a l t e n t a c l e s , or bu r s t s of c o n t r a c t i o n s of one or more t e n t a c l e s " . Rushforth ( i n press) shows that there i s an increased frequency of such c o n t r a c t i o n s before the c o n t r a c t i o n of the body column of hydra. Rushforth and Hofman ( i n press) show that there are 1.5 t e n t a c l e c o n t r a c t i o n s per 15 minutes f o r H. l i t t o r a l i s , 1.6 per 15 minutes f o r H. p s e u d o l i g a c t i s , and 7.9 per 15 minutes f o r H. p i r a r d i . With 15 observations per mean, standard e r r o r s are 0.2, 0.4, and 0.4 r e s p e c t i v e l y . To the spontaneous movements of t e n t a c l e and s t a l k c o n t r a c t i o n s we add two a d d i t i o n a l observations from Wagner (1905). "The extended Hydra may a l s o change the d i r e c t i o n of i t s long a x i s without a general contrac-t i o n , by mere f l e x i o n of the expanded body. Sometimes a change from one oblique p o s i t i o n to another i s brought about by f i r s t swaying to the v e r t i c a l and then to the new oblique p o s i t i o n . Quite as o f t e n , however, i t occurs through circumnutation around the attached foot. In t h i s case there appears f i r s t a c o n t r a c t i o n of the ectoderm on one side near the f o o t . This c o n t r a c t i o n then t r a v e l s towards the hypostome i n s l i g h t l y s p i r a l form. The Hydra, i n t h i s manner, slowly swings around, the body curved i n t o a complete loop or even beyond." This i s the only published account of these two behaviours. I t i s i n t e r e s t i n g that i n three years of hydra-watching, I saw the swaying behaviour once. I t i s t h i s and other reasons that make me suspect that the frequent l a c k of confirma t i o n of many behaviours observed by Wagner i s a c r e d i t to h i s patience and h i s powers of observation. H i s novel a d d i t i o n s to the r e p e r t o i r e of hydra behaviour patterns are probably r e a l , but r a r e . We add the swaying and circumnutation behaviours to the l i s t of p o s s i b l e search modes. To summarize, there are four p o s s i b l e search modes under the t i t l e of spontaneous movement. We now t u r n to another of Trembley's observations that hydra are o f t e n found hanging from the surface f i l m of the water. F l o a t i n g occurs when hydra "produce a bubble of gas beneath t h e i r pedal d i s k . When the bubble i s s u f f i c i e n t l y l a r ge i t f l o a t s to the surface of the water, w i t h the Hydra hanging from i t . The bubble t y p i c -a l l y b u r s t s at the surface, l e a v i n g the Hydra suspended by i t s pedal d i s k from the surface f i l m " . "The f l o a t i n g Hydra may remain hanging from the surface f o r sev e r a l days." This quote i s from Lomnicki and Slobodkin (1966). Two methods of suspension of the hydra from the surface f i l m are described by Wagner (1905). In one case a large a i r - b u b b l e i s attached to the bas a l d i s c , the bubble apparently keeping the hydra a f l o a t . In the other case, the t y p i c a l one according to the above quote, the basal d i s c of the hydra r e s t s at the base of a c a p i l l a r y depression of the surface f i l m . The basal d i s c , however, i s above the surface f i l m and dry. Wagner notes that sometimes i t i s easy to di s l o d g e a hydra from - 22 the surface, but at other times i t i s extremely d i f f i c u l t . He does not say which method of suspension makes i t easy to d i s l o d g e the hydra. Wilson (1891) a s c r i b e s f l o a t i n g behaviour to oxygen d e f i c i e n c i e s i n the lower reaches of the water. His conclusions were based on c i r c u m s t a n t i a l evidence; he d i d not measure oxygen concentration (Ewer, 1947a). Ewer studied young buds and found a g r a v i t y , but no oxygen, response. He does caution the reader against i n t e r p r e t i n g h i s data as a demonstration "that Hydra has no r e a c t i o n to oxygen concentration". Lomnicki and Slobodkin (1966) may have provided a b e t t e r explanation to Wilson's observations when they showed that the " f r a c t i o n of Hydra that w i l l f l o a t i s r e l a t e d to the degree to which they are crowded by other hydra". The mechanism of f l o a t i n g may be r e l a t e d to the f i n d i n g that water "conditioned" by the presence of other hydra w i l l e l i c i t f l o a t i n g . Nevertheless, these authors a l s o showed a r e l a t i o n s h i p between hunger and the tendency to f l o a t . Thus f l o a t i n g may w e l l be a search mode. This takes us to "a remarkable c y c l e of behavior i n hungry yellow Hydras. Hydras u s u a l l y remain, as we have seen, i n the upper l a y e r s of the water, on account of the oxygen there found. But when the Crustacea on which the animals feed have become scarce, so that l i t t l e food i s obtained, Hydra detaches i t s e l f , and w i t h t e n t a c l e s outspread si n k s slowly to the bottom. Here i t feeds upon the d e b r i s , o f t e n gorging i t s e l f w i t h t h i s m a t e r i a l . I t then moves towards the l i g h t , and at the l i g h t e d side again upward to the surface. Here i t remains f o r a time, then s i n k s again and feeds upon the m a t e r i a l at the bottom. This c y c l e may be repeated i n d e f i n i t e l y , r e q u i r i n g u s u a l l y some days f o r i t s completion." (Jennings, 1906; from Wilson, 1891). This s t o r y i s p u z z l i n g from a number of aspects: f i r s t , r e a c t i o n s of hydra to oxygen concentration have not been confirmed a f t e r the i n i t i a l p o s t u l a t i o n i n 1891. Second, hydra have not been noted to be able to detach themselves. T h i r d , the work of Loomis (1955) and of Lenhoff (1961) have shown the "feeding r e f l e x " of the hydra to be chemically induced by the presence of the chemical g l u t i a t h o n e . As pointed out p r e v i o u s l y , the r e a c t i o n of hydra to oxygen con-c e n t r a t i o n has not been adequately demonstrated. In recent times Lomnicki and Slobodkin (1966) have shown r e l a t i o n s between f l o a t i n g and crowding of hydra and hunger. Ewer (1947a) has shown a r e l a t i o n between upward movement i n detached buds and g r a v i t y . Regarding hydra detaching themselves from the surface f i l m , we note that Lomnicki and Slobodkin (1966) suggest that hydra detach and sink when a wave r i d e s over t h e i r f o o t . Wagner (1905) d i d note that at times hydra are r e l a t i v e l y easy to detach from the surface. A wave might be s u f f i c i e n t . In Appendix I of t h i s t h e s i s evidence i s presented to suggest that hydra quit f l o a t i n g when they bang i n t o something s o l i d . The r e s u l t s of t h i s process are seen i n the l a b o r a t o r y when hydra, known to have been f l o a t i n g , are found attached to the container w a l l at the l e v e l of the water surface. Loomis (1955) notes that Trembley, i n 1744, observed that hydra feed e x c l u s i v e l y (emphasis h i s ) on l i v i n g animals. Hyman (1940; see Loomis, 1955) showed that hydra w i l l feed on a wide v a r i e t y of species - but only on the l i v i n g members, not on dead specimens of the same species. Even Wilson (1891) noted that Greenwood, i n a paper published i n 1888, found hydra to be e s s e n t i a l l y carnivorous. F i n a l l y , Wilson notes that the length of the c y c l e may vary from one day to s e v e r a l weeks and that i n order to see t h i s c y c l e i t i s necessary to observe large numbers of animals. Wilson observed groups of hydra from f i v e hundred to a thousand strong. Wilson observed t h i s behaviour i n H. fusca, but could not f i n d i t i n H. v i r i d i s (Chlorohydra). One i s thus tempted to dispute Wilson's observation, but the behaviour was confirmed by Welch and Loomis (1924) f o r H. o l i g a c t i s Hydra began t h i s behaviour a f t e r 5 days of s t a r v a t i o n and almost a l l were "muck feeding" (Welch and Loomis, 1924) at the end of 10 days. Thus we consider t h i s as another p o s s i b l e search mode. The f o l l o w i n g search modes have been suggested: two forms of locomotion (walking and g l i d i n g ) , four forms of spontaneous movements (column c o n t r a c t i o n s , t e n t a c l e c o n t r a c t i o n s , swaying, and circumnutation behaviour), f l o a t i n g , and "muck feeding". Some of these search modes a i d i n l o c a t i n g prey by moving the hydra from one l o c a t i o n to a d i f f e r e n t one. Others c o n t r i b u t e to l o c a t i n g prey by searching through volume. Some do both. The hypothesis tes t e d i n the t h e s i s deals only w i t h the second aspect, searching through volume. Thus for any one of the eight search modes to be included they must r e s u l t i n some r e a c t i v e volume being traversed. In order to a s c e r t a i n the relevance of these behaviours to the hypothesis i t i s u s e f u l to construct an i d e a l i z e d view of the searching hydra. I n t h i s regard we f i n d a sentence i n Wagner (1905): " In Hydra v i r i d i s the t e n t a c l e s , a l s o moderately expanded, extend o b l i q u e l y out-ward and forward, forming the framework of a sort of funnel w i t h the hypostome at the bottom". A l s o , Jennings (1906) r e f e r s to the "normal" p o s i t i o n of an unstimulated hydra: foot attached, head not attached, comparative s t r a i g h t n e s s of the body, and t e n t a c l e s outspread. The hypothesis to be te s t e d concerns an animal t r y i n g to l o c a t e prey organisms. We assume that other f a c t o r s do not i n t e r f e r e w i t h the animal t r y i n g to f i n d prey. Thus the normal p o s i t i o n i n hydra i s a s u i t a b l e i d e a l i z e d s t a t e from which to construct a model. The funnel i s a u s e f u l geometric i d e a l i z a t i o n from which to begin. We use only the 25 top part of the funnel - the cone. The b a s i c volume searched by the hydra i s taken to be the volume contained i n the cone of t e n t a c l e s by a hydra i n the normal p o s i t i o n . The volume of "volume searched per time i n t e r v a l " i s a r e a c t i v e volume - the volume w i t h i n which a prey's chance of being captured d i f f e r s from u n i t y only by the p r o b a b i l i t y of a s u c c e s s f u l r e c o g n i t i o n (SR) of the prey by the predator, of a s u c c e s s f u l p u r s u i t (SP) of the prey by the predator, and of a s u c c e s s f u l s t r i k e (SS) at the prey by the predator. This d e f i n i t i o n i s suggested by the work of H o l l i n g , 1966. The volume of the cone of t e n t a c l e s i s such a r e a c t i v e volume f o r hydra. Yet, small prey can s l i p through the t e n t a c l e s undiscovered or a prey can enter the cone from the top, move almost to the hypostome, tu r n around and leave without being captured. I t should be noted that hydra feed on many d i f f e r e n t species - nematode and an n e l i d worms, crustacea, i n s e c t l a r v a e , arachnids, and even such v e r t e b r a t e s as tadpoles and newly-hatched f i s h (Loomis, 1955). To a small f i s h the whole cone i s the r e a c t i v e volume, but to a small crustacean the i n s i d e r e gion of the cone i s not r e a c t i v e volume. The model may be of such a nature as to apply only to a s p e c i f i c prey species, or i t may be of a l e s s p r e c i s e nature. The hypothesis to be teste d regarding the dur a t i o n of the dynamic responses of the search modes i s q u a l i t a t i v e ; i t i s necessary only to show that the dur a t i o n of the dynamic response f o r one mode i s greater or l e s s than that f o r another. I f the volume of the cone overestimates the a c t u a l volume searched by one mode, i t w i l l do l i k e w i s e f o r a l l other modes. This f o l l o w s from the fa c t that the cone w i l l be shown to be an i n t e g r a l part of the volume measure of a l l the search modes. The reader should thus bear i n mind that our attempts do not r e l a t e to a d e s c r i p t i o n of the searching behaviour of hydra but only to t e s t i n g a q u a l i t a t i v e hypothesis. The stimulus needed f o r nematocyst f i r i n g i s not constant, but a f u n c t i o n of time of food d e p r i v a t i o n (Ewer, 1949; Wagner, 1905). T h i s i s of no concern here because we deal only w i t h r e a c t i v e volume, as defined a few paragraphs ago. V a r i a t i o n i n nematocyst f i r i n g r e l a t e s to v a r i a t i o n i n SS, the p r o b a b i l i t y of a s u c c e s s f u l s t r i k e . Thus v a r i a t i o n i n nematocyst discharge i s not d e a l t w i t h i n the t h e s i s . V a r i a t i o n i n e l i c i t a t i o n of the feeding r e f l e x (Lenhoff, 1961) i s not considered f o r the same reason. The b a s i c model of volume searched i s that of a cone enclosed by the t e n t a c l e s of a hydra standing i n the normal p o s i t i o n . Many people have noted that t e n t a c l e s change length (Jennings, 1906; Hegner, 1933). I t i s now c l e a r that changing t e n t a c l e length may change the volume covered by hydra. Thus t h i s behaviour, not mentioned i n the l i t e r a t u r e survey, i s a component of volume searched by hydra. I t i s a search mode Tentacle length can e a s i l y be shown to be a f u n c t i o n of time of food d e p r i v a t i o n . I f a hydra i s fed to s a t i a t i o n , i t c o n t r a c t s i t s t e n t a c l e s and s t a l k . The s t a l k becomes a round b a l l and the t e n t a c l e s shorten i n t stubs. The hydra lengthen t h e i r t e n t a c l e s over a matter of hours. We now r e t u r n to the study of the eight p o s s i b l e search modes i d e n t i f i e d p r e v i o u s l y . The f i r s t form of locomotion, w a l k i n g , has the cone of t e n t a c l e s moving through space as the hydra moves i t s t e n t a c l e s from the normal p o s i t i o n to the s u b s t r a t e , and as the hydra l i f t s i t s t e n t a c l e s back to the normal p o s i t i o n . I t i s thus a sear&mode. (Note that i t i s i r r e l e v a n t whether or not the foot i s detached from the substrate a f t e r the t e n t a c l e s have been attached. Thus the type of walking where the foot i s not detached does not have to be considered s e p a r a t e l y from the usual kinds.) The second form of locomotion r e s u l t s i n a slow g l i d i n g of the hydra i n normal p o s i t i o n . The volume of a moving cone must be considered w i t h regard to the v e l o c i t y of the prey movement. By l e t t i n g prey be considered f i x e d i n space and then by l e t t i n g the hydra's v e l o c i t y of movement be converted to r e l a t i v e v e l o c i t y , we can i n t e r p r e t volume searched by hydra. R e l a t i v e v e l o c i t y can be approximated, f o r any d i r e c t i o n of prey movement, by the square root of the sum of the prey v e l o c i t y squared and the predator v e l o c i t y squared ( H o l l i n g , 1966; a f t e r Skellam, 1958). Thus the e f f e c t of a slowly moving cone i s approximately that of a non-moving cone. The g l i d i n g movement i s thus not worthwhile i n c l u d i n g , e s p e c i a l l y i n l i g h t of i t s observation by only one person. We do not consider i t as a search mode. Spontaneous movements may be search modes. Column c o n t r a c t i o n s do not r e s u l t i n volume covered because when the animal i n normal p o s i t i o n i s about to contract i t s s t a l k , i t a l s o c o n t r a c t s i t s t e n t a c l e s . Thus when the animal i s i n contracted p o s i t i o n , i t s t e n t a c l e s are withdrawn. This i s shown i n the drawing of the behaviour given i n Jennings (1906) and Hegner (1933). I have noted t e n t a c l e withdrawal i n my observations of hydra. Column c o n t r a c t i o n s serve to move the r e a c t i v e cone to a d i f f e r e n t s p a t i a l p o s i t i o n , w i t h the p o s s i b i l i t y that the prey d e n s i t y i s higher i n the new p o s i t i o n . Tentacle c o n t r a c t i o n s w i l l not be d e a l t w i t h because they serve only to modify s l i g h t l y the shape of the r e a c t i v e volume from the cone to some other form. They occur only at a r a t e of once per .10 minutes f o r H. l i t t o r a l i s - one of the species used i n the experimental s e c t i o n of the t h e s i s . My observations suggest that each t e n t a c l e c o n t r a c t i o n l a s t s only a short time; the t e n t a c l e i s q u i c k l y withdrawn and, i n a few seconds, lengthened. F i n a l l y , we come to the spontaneous movements c a l l e d swaying and circumnutation behaviour. These behaviours are so infrequent i n the hydra which are studied here, Hydra l i t t o r a l i s and Chlorohydra  v i r i d i s s i m a , that none of them could be found to study. As noted p r e v i o u s l y , I saw swaying only one time i n three years of observation. These behaviours are not included because they are not used very o f t e n by hydra. F l o a t i n g seems to cover volume i n a number of ways. Just a f t e r hydra detach, they f l o a t to the surface w i t h t e n t a c l e s extended. When hanging from the surface, hydra do so w i t h t e n t a c l e s extended (my o b s e r v a t i o n ) . Hydra f l o a t i n g i n t h i s manner do take prey. Thus hydra cover r e a c t i v e volume by f l o a t i n g to the surface and by f l o a t i n g along the surface. F l o a t i n g w i l l be included as a search mode. L a s t l y , we consider "muck feeding". A relevant aspect of t h i s behaviour i s that hydra feed on d e b r i s on the bottom. Our hypothesis does not r e l a t e to where the animals feed, or on which kinds of foods.. Volume i s covered only by the animal g e t t i n g to the surface and f a l l i n g back down again. But the hydra e i t h e r use p r e v i o u s l y described modes of locomotion or f l o a t i n g to do so. Thus t h i s behaviour need not be considered outside of locomotion and of f l o a t i n g s t u d i e s . This argument may be unnecessary as Wilson (1891) could not f i n d t h i s behaviour i n Chlorohydra, our main l a b o r a t o r y hydra. N e i t h e r could I . This behaviour w i l l not be considered, outside of i t s p o s s i b l e e f f e c t s on walking and f l o a t i n g . The search modes have been chosen i n a somewhat a r b i t r a r y f a s h i o n . To be more o b j e c t i v e would r e q u i r e an even more d e t a i l e d survey of hydra behaviour than that given. I t i s important to n o t i c e , t h e r e f o r e , that the hypothesis does not r e q u i r e that a l l search modes are studied. I t r e q u i r e only that f o r those s t u d i e d , the greater the i n i t i a t i o n t h r e s h o l d of the search mode the shorter i t s dynamic response. We emphasize again that the o b j e c t i v e i s to t e s t the hypothesis, not to study hydra behaviour f o r i t s own sake. Tentacle lengthening, w a l k i n g , and f l o a t i n g have been chosen as being worthy of study. In the next s e c t i o n a model i s constructed which describes the volume components of each of these three search modes. I t i s more t e r s e and d e t a i l e d than that model al r e a d y described. HYDRA STUDIES: A MODEL OF VOLUME SEARCHED In the previous s e c t i o n the various search modes that are used by hydra were i d e n t i f i e d . I t was concluded that concern w i l l focus on only the three search modes of t e n t a c l e lengthening, walking, and f l o a t i n g . In t h i s s e c t i o n a model i s constructed to t r a n s l a t e c e r t a i n aspects of these behaviours i n t o a measure of volume searched by a hydra i n some time i n t e r v a l . Tentacle Lengthening. The volume d e l i m i t e d by the t e n t a c l e s of a hydra i s that of a cone w i t h the hypostome region of the hydra d e s c r i b i n g the apex of a cone and the t i p s of the t e n t a c l e s d e s c r i b i n g the c i r c l e at the base of the cone. Once a prey moves i n t o t h i s cone, i t s chances of being captured are r a t h e r l a r g e . I t i s suggested that the e f f e c t of t e n t a c l e lengthening i s to increase the volume of the cone. The d e f i n i t i o n of a r i g h t cone i s TT r 2 h 13 3 where r i s the radius of the cone, and h i s the height. I f we know the mean t e n t a c l e l e n g t h , L ( t ) , and the angle,o(, at the apex of the cone, then r i s s i n (o</2)*L(t) 14 and h i s cos (cx/2)* L ( t ) 15 S u b s t i t u t i n g 14 and 15 i n 13, one gets an equation of the volume searched by a hydra at time t : 2 3 TT s i n (cx/2)*cos(cx/2)*L ( t ) 16 3 The equation describes the a c t u a l volume searched by the hydra at time t . As stated p r e v i o u s l y , we do not want the a c t u a l but the e f f e c t i v e volume searched - the a c t u a l volume searched, modified by the e f f e c t s of prey movement. This m o d i f i c a t i o n of 16 f o l l o w s . The cone of the hydra i s assumed not to move through space at t h i s stage. Thus i n any i n t e r v a l of time T, a prey moves i n the d i r e c t i o n of the cone a distance equal to the product of T and the v e l o c i t y of the prey towards the cone. (Since the hydra i s assumed not to move at t h i s stage, we do not have to study r e l a t i v e v e l o c i t i e s . ) I t makes no computational d i f f i c u l t y to l e t the prey be f i x e d i n space and to l e t the cone move towards the prey. By doing t h i s the e f f e c t of prey movement on the cone of t e n t a c l e s can be included by extending the cone through space i n some  d i r e c t i o n and by l e t t i n g the t e n t a c l e s lengthen as the cone moves (Figures IA and IB) . I t i s too l i m i t i n g to s p e c i f y a c e r t a i n d i r e c t i o n of prey movement, r e l a t i v e to the cone. We thus w r i t e two equations f o r each search mode: a minimal e f f e c t of prey movement and a maximal e f f e c t . Then i t i s p o s s i b l to determine the volume searched f o r any angle of prey movement r e l a t i v e to predator movement by choosing between the two extremums given. We now determine the angles of prey movement which r e s u l t i n extremum values of e f f e c t i v e volume searched. I t can be shown that extension of the cone i n that d i r e c t i o n i l l u s -t r a t e d i n Figure IA gives a minimal e f f e c t of prey movement on volume searched by the hydra; and that extension of the cone i n that d i r e c t i o n i l l u s t r a t e d i n IB gives the maximal e f f e c t of prey movement on the volume searched by the hydra. The magnitude of the e f f e c t depends d i r e c t l y on the area of the cone which i s extended. This area i s t r i a n g u l a r i n Figure IA and c i r c u l a r i n F i g ure IB. This means that i f one can show that the t r i a n g u l a r area i s a minimal area and that the c i r c u l a r area i s a maximal one, then Figure 1. Various types of volume searched by a hydra. The e f f e c t of prey movement i s i n c l u d e d . The d i r e c t i o n of prey movement i s shown by the arrows. Figures IB and IC overlap only f o r purposes of economy of space. I n f i g u r e IB, the cone i s p u l l e d upwards and the t e n t a c l e s allowed t o lengthen as the cone extends upwards. I n f i g u r e IC, a s e r i e s of cones move sideways. The f i r s t p art i s the demi-cone of f i g u r e IB. For more d e t a i l on f i g u r e IC, see t e x t . 33 equivalent volume r e l a t i o n s h i p s can be assumed. (This i s only s t r i c t l y true i f the incremental volume due to t e n t a c l e lengthening i s the same f o r models of Figures 1A and IB. I t w i l l be shown that the area of the c i r c l e i s 5 times as l a r g e as that of the t r i a n g l e . Thus, i g n o r i n g the d i f f e r e n c e s i n incremental volume i s probably not important f o r our purposes.) The cone can be extended i n three ways: from the top ( c i r c u l a r a r e a ) , from the side ( t r i a n g u l a r a r e a ) , and from any angle between these two views. The l a s t view i s of a t r i a n g l e w i t h height l e s s than that of the t r i a n g l e viewed from the side and of a d e m i - e l l i p s e whose major a x i s i s equal to the diameter of the c i r c l e and whose semi-minor a x i s i s l e s s than the radius of the c i r c l e viewed from the top of the cone. Consider the equation of a r i g h t cone: 2 2 2 x + y - z = 0 17 z ~ ~2 a c Equation 17 can be changed to an equation f o r a c i r c l e at the base of the cone by l e t t i n g z = h ( t ) . From t h i s equation parameters a and c can be estimated. The equation i s 2 2 2 2 x + y = h ( t ) * a 18 2 c where h(t) i s the height of the cone of t e n t a c l e s . The parameter h(t) i s dependent upon estimates of <=< and L ( t ) f o r i t s e v a l u a t i o n (see 15). The angle oc has been measured to be equal to 112+25 (SD) degrees (n=102) f o r Chlorohydra. Let L ( t ) = 1 mm. Then the parameter h(t) i s equal to 0.5592. From 18 and 14 we can w r i t e 0.5592*a/c = s i n ( 5 6 ) * L ( t ) = 0.8290 2 Let c be a r b i t r a r i l y set to value of 1 and so a=1.48. Thus c =1 and 2 a =2.20. These parameter estimates are a l l we need i n order to w r i t e the equation which describes the cone of t e n t a c l e s as f o l l o w s : x H- y - z = 0 2.20 Now consider the three cases. An equation of the area extended in Figure 1A i s obtained by letting the x-coordinate (or the y-coordinate) have zero value: If only that part of the coordinate system for z^O is considered, then 110 describes two straight lines which meet at the origin. These lines may be visualized as a triangle centered along the z-axis, with vertex at the origin. Let each line terminate at the same z-coordinat at (z,+ y) . At this point we have Vz^+(+ y ) 2 = L.(t) . Then the z-coordinate represents the height of the cone of tentacles and the y-coordinate represents the radius of the cone. This means, by 110 that r(t)/1.48 = h(t). The area of the triangle, r(t)*h(t), may thus be written as 1.48* h 2 ( t ) . The second case to be considered i s that shown in Figure IB. An appropriate equation results from 19 by letting z=h(t); the equation i s written as 18. Since this i s an equation of a circ l e we 2 2 2 may write r (t) = 2.20 h (t) . The area of the circle, Trr (t), may 2 2 thereby be written as a function of h(t): 2.20 rrh (t) or 6.91 h (t) . Consider the final case. The largest conceivable area i s that of a demi-ellipse whose area i s equal to one half of that of the ci r c l e 2 at the base of the cone, vis., 3.45 h (t), and a triangle whose area 2 is equal to that of the triangle ofllO, vis., 1 . 48 h (t) . The sum 2 of these areas i s equal to 4.93 h ( t ) . This area i s less than that of the circ l e alone. This means that the extension of the cone in that direction illustrated in Figure IB gives the maximum effect of prey movement on volume searched by the hydra. z = Y_ 1.48 35 The ratio of the area of the c i rc le to the triangle is about 5:1. This ratio suggests that mixes of areas of triangle and demi-ellipse w i l l be larger than the area of a triangle alone. Thus the direction of prey movement i l lus t ra ted in Figure IA w i l l be considered to be that producing a minimal effect of prey movement. This completes a rather-long diversion of thought away from the model bui lding . It i s s t i l l lef t to describe the volume covered as in Figures IA and IB. Consider IA. At time t0+nT-T the hydra occupies the volume 2 3 TT sin ( ° < / 2 ) * c o s(<W2)* L (tQ+nT-T) 111 3 and at time tG+nT i t occupies the volume 2 3 TT sin (o< /2)*cos(<* 12)* L (t +nT) 112 3 These equations simply state that tentacle length changes over time. The volume between these two (demi-) cones i s due to prey movement and describable as: { to+nT dt [ L 2 ( t ) ] 113 „hiT-T An approximation to this integral which is i n line with the measurements available i s VY* sin(<W2)*cos(o< /2)*T 2 2 L (tQ+nT-T) + L (tD+nT) 114 2 Combining one half of each of equations H I and 112 with 114, we have an equation for the minimal volume covered by the predator between . time tD+nT-T and time tQ+nT: 36 T J s i n («/2)*cos(°c/2)* 6 VY*sin(oe/2)*cos (<*ID*T 2 L 3(t Q+nT-T) + L 3(t 0+nT) 2 2 L (t 0+nT-T) + L (t D+nT) 115 S i m i l a r i l y , we may describe the volume as portrayed i n Figure IB as T T sin 2 (o<./2)*cos(<^ ID* L 3 ( t +nT-T) + ^ * V Y * s i n 2 (<=t/2)* ntQ+nT dt 2 L ( t ) 116 'tQ+nT-T Making an approximation f o r the i n t e g r a l as p r e v i o u s l y , we get an equation f o r the maximal volume covered by the predator between time t Q+nT-T and time t Q+nT: 3 2 ^ sin 2(<x /2)*cos(ex/2)* L (t +nT-T) + Tr *VY* s i n (oc/2)*T* .3 ° 2 2 9 L (t -hiT-T) + L^(t Q+nT) 117 In summary, equations 115 and 117 give estimates of the minimal and the maximal values of the volume searched by the cone of t e n t a c l e s during the time t o+nT-T to the time t Q+nT. Walking. I t i s assumed that the e f f e c t of w a l k i n g i s t o move the cone of t e n t a c l e s through space f o r a c e r t a i n d i s t ance during a given time. Walking, described e a r l i e r i n the t h e s i s -occurs as f o l l o w s : the hydra begins t o move: i t bends over, attaches i t s t e n t a c l e s t o the substrate, and by c o n t r a c t i n g i t s s t a l k , p u l l s i t s foot f r e e of the s u b s t r a t e . Then i t r e s e t s i t s foot i n a new l o c a t i o n , attaches i t , p u l l s the t e n t a c l e s free by c o n t r a c t i n g i t s s t a l k , and then assumes again the upr i g h t p o s i t i o n . Let the amount of time spent moving during the time i n t e r v a l T be denoted as T m and the amount spent not moving and f i x e d by T . The term T i s measured as the time i t takes the hydra J u m 37 to bend the cone of t e n t a c l e s t o the substrate plus the time i t takes the hydra t o reassume th e normal p o s i t i o n a f t e r detaching. However, T u need not be measured experimentally as i t can be c a l c u l a t e d from the equation: T = T + T + T . T- i s the time of t e n t a c l e attachment to n m u a a the s u b s t r a t e . Consider the case where prey move towards the predator i n such a way as to give minimal e f f e c t t o the volume searched by the cone of t e n t a c l e s . During T t h i s w i l l be where the prey and predator move i n the same d i r e c t i o n , at the same speed. We now consider r e l a t i v e v e l o c i t y , equal i n t h i s case t o zero. This makes the e f f e c t i v e volume searched equal t o the a c t u a l volume searched (j_6) . However, i n order t o explore v a r i o u s f i e l d - l i k e s i t u a t i o n s i t i s not d e s i r a b l e t o s p e c i f y the magnitude of prey v e l o c i t y . Thus, the minimal case w i l l be considered the one where the prey and predator move i n the same d i r e c t i o n , w i t h speed u n s p e c i f i e d . The r e l a t i v e v e l o c i t y i s (IVY-VPLl), where VPL i s the v e l o c i t y of the w a l k i n g hydra. To describe t h i s case with a model, we w i l l d escribe the T m and the T^ p a r t s s e p a r a t e l y . During T u the s i t u a t i o n i s d e s c r i b a b l e by 115 with the term T replaced by T u- L a t e r i n the t h e s i s i t w i l l be shown that T U ~ T ; thus t e n t a c l e length was allowed to run from L(t Q+nT-T) t o L(t Q+nT) i n 115 . During T m the s i t u a t i o n i s a l s o describable by 115, but with the term T replaced by T m and VY replaced by the absolute value of (VY-VPL). There are other changes, r e s u l t a n t from the f a c t that Tm-~0 . The i n t e g r a l of 113 now has l i m i t s of t t o t+T m, with t being some time value w i t h i n T. But t may be any value w i t h i n T, from t Q+nT-T almost t o t Q+nT. Thus the i n t e g r a l was approximated by the product of a two-point-average t e n t a c l e length and the time i n t e r v a l T_. When t h i s i s done, the i n t e g r a l of 113 becomes the corresponding term of 114. A s i m i l a r argument f o r equations 111 and 112, r e l a t i v e t o t h e i r form i n 115, r e s u l t s i n the f i r s t term of 115 becoming At the time j u n c t i o n between T m and T u, the two models share a cone. Thus one of these cones i s e l i m i n a t e d i n the c o n s t r u c t i o n of the o v e r a l l model. The equation f o r the minimal e f f e c t i s thus: Next we consider the case where the prey moves towards the predator i n such a way as to give maximal e f f e c t . The degree of cone extension can vary g r e a t l y as a f u n c t i o n of the d i r e c t i o n i n which the prey approaches the moving hydra. Extension i s maximal i f predator and prey move d i r e c t l y towards each other. Suppose t h i s i s so. Then r e l a t i v e v e l o c i t y i s equal t o (VY+VPL) and the equation i s l i k e 118. However, the hydra moves with i t s side f i r s t . The minimal area of the cone i s thus p r o j e c t e d forward. At low VY values t h i s case would not be maximal - the increase i n the d i s t a n c e over which the minimal area i s p r o j e c t e d would not compensate f o r the f a c t that the area extended i s the minimal area. The movement that i s being discussed i s one with the cone moving towards the s u b s t r a t e . I t seems that prey do not o f t e n emerge from the s u b s t r a t e . T h i s makes the case d e s c r i b a b l e by a modifie d 118 one which one i s u n l i k e l y to f i n d i n the f i e l d . Therefore the equation of maximal e f f e c t w i l l be developed on the b a s i s of the maximal area extended, and not on the b a s i s of the maximal distance extended. The Tt/24* sin' 2(<v/2)*cos(<W2)* [L(t D+nT-T) + L(t D+nT)] 3 . o H-nT-T) + L (t +nT) > o 118 s i t u a t i o n to be modelled i s shown i n Figure IC. Hydra are v i s u a l i z e d as moving i n steps. The hydra stays i n some p o s i t i o n f o r time A t and then moves on t o the next p o s i t i o n . While a hydra occupies a c e r t a i n p o s i t i o n , prey are moving down onto the cone at a v e l o c i t y VY. This r e s u l t s i n a volume component due t o prey movement. This component i s c y l i n d r i c a l , of height V Y * A t . During T^ that part of the volume which i s covered by the cone of t e n t a c l e s i t s e l f ( a c t u a l volume covered) can be described by 115 with VY replaced by VPL and the term T replaced by T . Since T m - ^ 0 , and average t e n t a c l e lengths are most appropriate, m o d i f i c a t i o n i n term one of 115 i s again i n order. That part of the volume which i s due t o the e f f e c t of prey movement on the moving cone of t e n t a c l e s can be described as f o l l o w s : 2 r i 2 TT VY* s i n («=t / 2 ) * 6 t * L(t Q+nT-T) + L(t D+nT) + 4 L -t-HT m dt £ 2VY* sin(<*/2)*VPL* & t * L ( t ) 't 119 w i t h A t defined as T m [ 2 r ( t ) l = _2_ * s i n ( o c / 2 ) * L ( t ) 120 VPL* T VPL m By way of e x p l a i n i n g 119, the f i r s t term describes the demi-cylinders to the r i g h t and l e f t of Figure IC. Because T m i s s m a l l , I again constructed the model only as a f u n c t i o n of average t e n t a c l e l e n g t h . The second term of 119 describes the volume contained between the two d e m i - c y l i n d e r s . We can approximate the i n t e g r a l once again as the product of the mean t e n t a c l e length and T • From the a l t e r a t i o n s of .119, we get 3 r 13 o TT s i n (of / 2 ) * VY L(t Q+nT-T) +-L(t +nT) ' 4 VPL L ° [L 2(t o+nT-T) + L 2 ( t o + n T ) | . + 2 V Y * s i n ( c x / 2 ) * T * m 121 During T u the volume covered c a n b e described by 117, w i t h the term T replaced by T^. Tentacle length was allowed to run from L ( t +nT-T) t o L ( t +nT) i n 117. x o o As before, the T and T models share a cone. The same s o l u t i o n i s m u adopted. The equation f o r the maximal e f f e c t ' i s thus J Y s i n 2 ( c < / 2 ) * c o s ( < * / 2 ) + TT s i n 3 (<*/2)* VYl|"L(t +nT-T) + L(t 0+nT ) l 4 8 4 V P L J [ ° J + Tr s i n 2 ( o < / 2 ) * c o s ( ' X / 2 ) * L 3 ( t +nT-T) + | 2 V Y * T * s i n 2 ( < * / 2 ) + 6 L M . Ti*VY* T * s i n 2 ( c x / 2 ) + T m* VPL* sin (cv /2)*cos (°< / 2 ) I j L 2 (t +nT-T) 2 U T J L ° + L 2 ( t +nT)l 12 o J I n summary, equations 118 and 122 give the estimates of the minimal and the maximal values of the volume searched by the cone of t e n t a c l e s during the time t Q+nT-T t o t Q+nT, during which walking i s assumed t o take p l a c e . F l o a t i n g . I t w i l l be assumed that the e f f e c t of f l o a t i n g i s t o move the cone of t e n t a c l e s through space f o r a c e r t a i n d i s t a n c e during a given time. I t i s f u r t h e r assumed that at the beginning of some i n t e r v a l of time, the hydra detaches from the substrate and begins t o f l o a t . Let t h i s i n t e r v a l be designated as t Q + k T . For a l l subsequent i n t e r v a l s of time, the hydra i s assumed t o remain f l o a t i n g . When a hydra detaches, i t f l o a t s v e r t i c a l l y to some l e v e l and then h o r i z o n t a l l y along i t . T h is v e r t i c a l movement occurs only 41 during the hour described as t Q + kT. During t h i s i n t e r v a l the volume covered i s 9 2 TT*DMT*sin (<* / 2 ) * L (t + kT) 123 o where DMT i s the di s t a n c e of v e r t i c a l movement. The v e r t i c a l movement i s completed so q u i c k l y that the e f f e c t s of VY w i l l be small enough t o be ignored. The h o r i z o n t a l movement f i n d s the hydra f l o a t i n g upside down, but with the t e n t a c l e s extended and ready t o capture prey. The e f f e c t s of prey movement during t h i s time can be considered i n the usu a l way. Consider the case where the prey moves towards the predator i n such a way as t o give minimal e f f e c t . This w i l l be described by equation 115, with VY replaced by the absolute value of (VY-VPF). VPF i s the v e l o c i t y of hydra movement during f l o a t i n g . Consider now the case where the prey moves towards the predator i n such a way as t o give maximal e f f e c t . There are two p o s s i b l e s i t u a t i o n s , a lready discussed i n the walking s e c t i o n . The h o r i z o n t a l f l o a t i n g movement i s e n t i r e l y due t o water motion. I t does not seem l i k e l y that prey of a s i z e usable by hydra w i l l o f t e n swim upstream. They w i l l tend t o use the same currents that the hydra use. The maximal e f f e c t w i l l again be modelled on the b a s i s of the maximal area extended. The s i t u a t i o n i s that of Figure IC. The model i s a sum of two equations: 115 with VY replaced by VPF and 121 with VPL replaced by VPF and T m replaced by T. Combining 123 w i t h the, modified 115;, we get an equation f o r the minimal volume searched during i n t e r v a l T: TTsin (o</2)*DMT*L (t +kT) + Tr sin («c /2) *cos( °c /2)* L (t +nT-T) + 42 L (tQ+nT) + T*(|VY-VPFj)*sin(o</2)*cos(o</2)* L (t +nT-T) + o L (t +nT)l 124 o Combining 123 with the modified 121 and the modified 115, we get an equation for the maximal volume searched during the time interval T: 2 2 2 I" 3 TTsin (°t/2)*DMT*L (t +kT) + Tf sin (cx /2) *cos( of./2) * ! L (t +nT-T) + o 7 L o 3 L ( t +nT) I + T-o -> o r 2 T*VPF*sin(oc/2)*cos(oc/2)* L (t 2 L o +nT-T) + L (tQ+nT)J + TT sin (oc/2)*_VY* lh(t +nT-T) + L(tQ+nT) | + 2 VY*sin (o<:/2)*T* 4 VPF L ° ° 2 2 L (t +nT-T) + L (t +nT) o o 125 In summary, equations 124 and 125 give the estimates of the minimal and the maximal values of the volume searched by the floating cone of tentacles during the time t +nT-T to t +nT. o o HYDRA EXPERIMENTATION: THE APPROACH 43 In the previous s e c t i o n a set of equations f o r each of t e n t a c l e lengthening, walking, and f l o a t i n g has been developed. Each set of equations c o n s i s t e d of two equations: one d e s c r i b i n g the e f f e c t i v e volume searched by a hydra when prey moved i n such a d i r e c t i o n as to have minimal e f f e c t on the a c t u a l volume searched by the hydra, and the other d e s c r i b i n g the e f f e c t i v e volume searched by a hydra when prey moved i n such a d i r e c t i o n as to have maximal e f f e c t on the a c t u a l volume searched by the hydra. To t e s t the dynamic response hypothesis i t i s necessary to r e l a t e each of these equations to time of food d e p r i v a t i o n . The equations, as w r i t t e n , are not fu n c t i o n s of time of food d e p r i v a t i o n . I t i s thus necessary to make them funct i o n s of food d e p r i v a t i o n by expanding those terms of the equations that are s e n s i t i v e to food depriva-t i o n . I t i s also necessary to obt a i n estimates of the parameters of these equations. Only then i s i t p o s s i b l e to r e l a t e volume searched per time i n t e r v a l (the model) to time of food d e p r i v a t i o n . The hypothesis al s o r e q u i r e s that the i n i t i a t i o n t h r e s h o l d of each search mode be known. Measuring the v a r i a b l e s of the equations as fu n c t i o n s of time of food d e p r i v a t i o n , e s t i m a t i n g the parameters, and determining the i n i t i a t i o n t h r e s h o ld of each search mode i s the purpose of the f o l l o w i n g HYDRA EXPERIMENTATION se c t i o n s . The purpose of t h i s s e c t i o n i s to e x p l a i n c e r t a i n aspects of the approach which i s to be followed i n these s e c t i o n s . The HYDRA EXPERIMENTATION: TENTACLE LENGTHENING s e c t i o n begins w i t h a restatement of equations 115 and 117. The HYDRA EXPERIMENTATION: WALKING s e c t i o n begins w i t h a restatement of equations 118 and 122. The HYDRA EXPERIMENTATION: FLOATING s e c t i o n begins w i t h a restatement of equations 124 and 125. I say "restatement" because the equations are 44 presented i n a s l i g h t l y evaluated form. The b a s i c s of these a l t e r a t i o n s are given i n Appendix I I . The prime reason f o r beginning each experimental s e c t i o n w i t h a model i s to set the stage f o r the s e c t i o n . Experiments are conducted only to evaluate the equations. V a r i a b l e s are measured as functions of time of food d e p r i v a t i o n . Parameters are estimated. I n i t i a t i o n t hresholds are set. By the end of the s e c t i o n a l l necessary experimental observation i s completed; the model i s ready to y i e l d a l l i n f o r m a t i o n necessary to determine the d u r a t i o n of the dynamic response. The model i s ready to be placed on the time of food d e p r i v a t i o n a x i s . A f t e r the equations are introduced, the terms of the equations which are s e n s i t i v e to food d e p r i v a t i o n are i d e n t i f i e d . Then the bulk of each s e c t i o n i s given to determination of the r e l a t i o n s h i p between these v a r i a b l e s and time of food d e p r i v a t i o n . This job i s more d i f f i c u l t that i t appears at f i r s t glance. I t w i l l be shown that there i s only one v a r i a b l e s e n s i t i v e to food d e p r i v a t i o n i n each set of equations. This being the case, the d u r a t i o n of the dynamic response of the searching r a t e equations w i l l be equivalent to the d u r a t i o n of the dynamic response of the v a r i a b l e s i n these equations. For example, t e n t a c l e length i s the v a r i a b l e of equations 115 and 117. The d u r a t i o n of the dynamic response of 115 and 117 i s that of the t e n t a c l e length graph as a f u n c t i o n of food d e p r i v a t i o n . I t i s thus e s s e n t i a l that the d u r a t i o n of the dynamic response of the v a r i a b l e s be measured a c c u r a t e l y . Suppose some v a r i a b l e has an S-shape over time of food d e p r i v a t i o n . Now suppose we measure t h i s r e l a t i o n s h i p f o r a number of animals. Each animal gives the S-shaped r e l a t i o n s h i p . However, the f u n c t i o n f o r some animal begins i t s S-shaped r i s e at some a b s c i s s a value and the f u n c t i o n f o r another animal at a somewhat d i f f e r e n t a b s c i s s a value, and soon. Then, i f a l l data are p l o t t e d , means and variances c a l c u l a t e d , and a best f i t l i n e drawn, the d u r a t i o n of the dynamic response of t h i s graph would be l a r g e r than the r e a l d u r a t i o n ! Those animals whose f u n c t i o n i s pushed along the a b s c i s s a w i l l tend to delay the maximum; and those animals whose f u n c t i o n i s i n i t i a t e d e a r l i e r than the average w i l l prematurely p u l l the l i n e up from the minimal value. We w i l l encounter t h i s problem i n the f o l l o w i n g s e c t i o n s . This v a r i a b i l i t y e x i s t e d i n s p i t e of a number of attempts to minimize i t . Hydra were fed only w i t h newly hatched b r i n e shrimp (Artemia s a l i n a ) . Hydra stocks were fed 24 hours before experimental feeding. Only hydra of "average" s i z e were used. Yet t h i s v a r i a t i o n p e r s i s t e d . This v a r i a t i o n w i l l be shown to be due to the shape of the prey, r e l a t i v e to the way i n which they are ingested by the hydra. Again, i t i s not p r a c t i c a l to force the hydra to eat a prey i n a c e r t a i n manner and not convenient to r a i s e Chlorohydra on other prey organisms. Likewise i t i s too time consuming to a r t i f i c i a l l y i n j e c t n o n - l i v i n g food i n t o hydra, although the technique f o r doing so i s a v a i l a b l e (Claybrook, 1961). This i s e s p e c i a l l y so c o n s i d e r i n g the amount of r e p l i c a t i o n needed f o r the experiments of t h i s t h e s i s . I t was decided to accept the v a r i a t i o n and to f i n d a way to compensate f o r the v a r i a t i o n . The s o l u t i o n adopted was to overlap a l l functions by moving them along the time of d e p r i v a t i o n a x i s as r equired to meet a c e r t a i n c r i t e r i o n . This c r i t e r i o n i s that l e a s t v a r i a t i o n should occur i n the region where the graph r i s e s from the minimum to the maximum value. This c r i t e r i o n gives the best estimate of of the d u r a t i o n of the dynamic response. The s u b s t a n t i a l job of measuring those v a r i a b l e s s e n s i t i v e to hunger as f u n c t i o n s of time of food d e p r i v a t i o n i s followed by a b r i e f d e s c r i p t i o n 46 on e s t i m a t i n g the values of the parameters of the equations. Then emphasis s h i f t s to the second major job: to measure the i n i t i a t i o n t h r e s h old of the model. We have already noted that there e x i s t s only one v a r i a b l e i n each set of equations. Thus the i n i t i a t i o n t h r e s h o l d of each searching equation i s that of the v a r i a b l e . We measure the i n i t i a t i o n t h r e s h old of each v a r i a b l e over an a b s c i s s a of time of food d e p r i v a t i o n . An i n d i c a t o r of the approximate time of the t r a n s i t i o n from the minimum value of the v a r i a b l e to the i n c r e a s i n g range i s chosen. This i n d i c a t o r i s measured as a f u n c t i o n of the amount of food pre-fed to the hydra. One might expect that i t i s necessary only to measure the i n i t i a t i o n t h r e s h o l d from s a t i a t i o n . A range of pre-feeding l e v e l s was used to make c e r t a i n that the i n i t i a t i o n thresholds of the various search modes held f a i r l y constant p o s i t i o n s r e l a t i v e to the others. The hypothesis does not r e q u i r e a s t r i c t r e l a t i o n s h i p between the i n i t i a t i o n thresholds of the search modes. I t r e q u i r e s only that the search modes can be ranked and that t h i s ranking be the same over a l l pre-feeding l e v e l s . I f t h i s i s not p o s s i b l e , then one cannot r e l a t e the d u r a t i o n of the dynamic response to the i n i t i a t i o n t h r e s h o l d i n any general fashion. I t would be p o s s i b l e that the hypothesis would hold at some pre-feeding l e v e l s and not at others. Measurement of the i n i t i a t i o n t h r e s h old completes each s e c t i o n . The t e n t a c l e lengthening search mode i s followed by the walking search mode, which i s followed by the f l o a t i n g search mode. Then the data from these three s e c t i o n s are c o - r e l a t e d and the hypothesis t e s t e d i n A TEST OF THE HYPOTHESIS. I t may be u s e f u l to add a note on the degree of accuracy needed f o r the experiments. E a r l i e r i t was assumed that each search f u n c t i o n takes the f o l l o w i n g form: a minimum steady ra t e of searching to a c e r t a i n hunger l e v e l , then an increased r a t e of searching, and f i n a l l y a maximum ra t e of searching on a time of food d e p r i v a t i o n a x i s . The model rep r e s e n t i n g the search ra t e f u n c t i o n contains only one v a r i a b l e f o r each search mode. Thus the s p e c i f i c a t i o n s f o r the search f u n c t i o n are s p e c i f i c a t i o n s f o r the f u n c t i o n of the v a r i a b l e . When the v a r i a b l e i s measured over time of food d e p r i v a t i o n , a d e t a i l e d accurate d e s c r i p t i o n of the r e l a t i o n s h i p i s not needed. We have occasion to determine i n some d e t a i l a f u n c t i o n of t e n t a c l e length over time of food d e p r i v a t i o n . The reason, however, r e l a t e s to overlapping the data from i n d i v i d u a l animals i n such a way as to estimate the dynamic response as a c c u r a t e l y as p o s s i b l e . A l s o , measurement of i n i t i a t i o n thresholds i s based on the choice of a c r i t e r i o n of the approximate t h r e s h o l d value. Further accuracy i s not needed; i t w i l l be shown that the search modes are s u f f i c i e n t l y separated along the food d e p r i v a t i o n a x i s that crude estimates of the i n i t i a t i o n thresholds are a l l that are needed to rank the search modes. We now t u r n to a b r i e f d e s c r i p t i o n of the experimental methods. As the comments are a p p l i c a b l e to experiments on each of t e n t a c l e lengthening, walking, and f l o a t i n g , i t i s f e l t that they are w e l l placed as p r e l i m i n a r y comments. The species Chlorohydra v i r i d i s s i m a i s used most oft e n f o r the experimental work but Hydra l i t t o r a l i s i s a l s o used. Stocks of each species were begun from animals obtained from a b i o l o g i c a l supply house. Stocks were kept i n c i r c u l a r , disposable p e t r i p l a t e s of 9 cm i n diameter and of 1 cm i n depth. The hydra were maintained i n a medium which con s i s t e d of 3,765 mis of tap water modified by 17 mis from each of two s o l u t i o n s . The f i r s t of these c o n s i s t e d of 20 grams of NaHCO and 10 grams of "Versene" d i s s o l v e d i n tap water to make 1 l i t r e of s o l u t i o n . The second of these co n s i s t e d of 9.513 grams of CaCl^ d i s s o l v e d i n tap water to make 1 l i t r e of s o l u t i o n . Stocks were removed once d a i l y from the c o n t r o l l e d temperature chamber and fed. The food c o n s i s t e d of l i v e b r i n e shrimp n a u p l i i , hatched d a i l y at a temperature of 25-28 deg C. HYDRA. EXPERIMENTATION: TENTACLE LENGTHENING The equations which f o l l o w d e scribe the e f f e c t i v e volume covered by a hydra which stands upright w i t h t e n t a c l e s o u t s t r e t c h e d , but not walking or f l o a t i n g . They represent the lower and upper l i m i t s of the volume searched by a hydra w i t h i n a time i n t e r v a l , T, from t Q+nT-T to t Q+nT. The equation f o r the lower l i m i t i s as f o l l o w s : 0.20 L 3(t D+nT-T) + L 3(t Q+nT)j + 13.90VY L 2(t o+nT-T) + L 2(t Q+nT)j The equation f o r the upper l i m i t i s as f o l l o w s : 0.40 ^L3(to+nT-T)^j + 64.76VY 2 2 L ( t +nT-T) + L ( t +nT) v o o 126 127 The constant t i s time equal to zero, dependent upon the amount of food pre-fed to the hydra; n i s a constant which designates the number of T i n t e r v a l s which have passed; T i s an i n t e r v a l of time chosen to s u i t experimental p r a c t i c a l i t i e s , i n minutes; L(t Q+nT) i s the mean t e n t a c l e l e n g t h , i n m i l l i m e t e r s , at time t Q+nT; and VY i s the prey v e l o c i t y , i n m i l l i m e t e r s per minute, i n the d i r e c t i o n of the hydra. We deal f i r s t w i t h the v a r i a b l e of t e n t a c l e length. I t w i l l be studied as a f u n c t i o n of time of food d e p r i v a t i o n . S t a l k lengths w i l l be measured at the same time, even though they are not used i n the equations. They w i l l have a r o l e i n understanding the c o n t r o l l i n g mechanism behind t e n t a c l e lengthening. Any point can be l o c a l i z e d i n space by observing i t from two perpendicular d i r e c t i o n s . When viewing the point from one d i r e c t i o n the object i s to o b t a i n estimates of the x and y coordinates and when viewing i t from the other d i r e c t i o n the object i s to get estimates of the x and z coordinates. The point becomes l o c a l i z e d as a set of (x,y,z) c o o r d i n a t e s . The u s u a l distance formula can be used t o c a l c u l a t e the d i s t a n c e between any two p o i n t s . I measured t e n t a c l e lengths by noting the s p a t i a l p o s i t i o n of the hypostome and the p o s i t i o n of the d i s t a l part of each t e n t a c l e . The s t a l k length was measured by n o t i n g the p o s i t i o n of the foot and by using the measure of the hypostome p o s i t i o n already c o l l e c t e d . A p l e x i g l a s s container was constructed and f i l l e d w i t h the medium i n which hydra l i v e . A small p l e x i g l a s s stand was set i n t o t h i s c o n t a i n e r . The temperature of the medium was kept between 20 and 22 deg C. The container was set on a cement t a b l e that minimized v i b r a t i o n s . One camera was mounted d i r e c t l y over the stand (to measure the x and y coordinates) and another was mounted d i r e c t l y i n front of the stand (to measure the x and z c o o r d i n a t e s ) . Two 35 mm Nikon cameras, each mounted with bellows extension and equipped w i t h 135 mm lens, were used. The cameras were loaded with " T r i - X " f i l m whose stated ASA of 400 was e f f e c t i v e l y r a i s e d t o 1200 by developing the negatives i n "Acu-1" developer. T h i s allowed the use of a r e l a t i v e l y low l i g h t i n t e n s i t y of about 50 f o o t - c a n d l e s . A t y p i c a l green hydra, Chlorohydra v i r i d i s s i m a , was taken from a stock which had been fed 24 hours p r e v i o u s l y . This hydra was pre-fed a set number of newly hatched b r i n e shrimp n a u p l i i and introduced i n t o the c o n t a i n e r . The hydra was placed on the stand and allowed t o a t t a c h . I f the hydra d i d not a t t a c h w i t h i n 15 minutes, I proceeded as i f i t had. A p i c t u r e was taken from the top of the hydra at the same time as one was taken from the s i d e . Photographing at a r a t e of once per hour was s u f f i c i e n t t o capture the essence of the process. This means that the parameter T of equations 126 and127 i s t o be set at 60 minutes. The hydra was photographed f o r 12 hours at the r a t e of once per hour. The f i l m so produced was coded f o r s p a t i a l p o s i t i o n s with the use of a "Vanguard" Motion A n a l y z e r . T h i s machine magnifies an image a constant amount and allows one to measure two dimensional coordinates w i t h respect to some a r b i t r a r y reference p o i n t . The hypostome was chosen as the reference p o i n t f o r both t e n t a c l e and s t a l k length c a l c u l a t i o n s . This gave f o r each h o u r l y photograph a reading of the coordinates of the t i p of each t e n t a c l e and of the f o o t . This i n f o r m a t i o n was used to c a l c u l a t e s t a l k and t e n t a c l e lengths f o r each hydra, once per hour over a 12 hour i n t e r v a l . The d i s t a n c e c a l c u l a t i o n i n v o l v e d one minor complexity. Each h o u r l y record c o n s i s t e d of two photographs: one with a s e r i e s of (x,y) coordinates and the other with a l i k e s e r i e s of (x,z) c o o r d i n a t e s . There was no immediate way of determining which (x,y) t e n t a c l e went with which (x,z) t e n t a c l e . T h i s had to be done c o r r e c t l y i n order t o c a l c u l a t e d i s t a n c e s . A computer program was devised which, f o r each record, ordered by magnitude of x values each (x,y) coordinate p a i r from smallest t o l a r g e s t . I t d i d the same f o r each (x,z) p a i r . The program then matched the f i r s t (x,y) p a i r with the f i r s t (x,z) p a i r , the second (x,y) p a i r with the second (x,z) p a i r , and so on. When the p o s i t i o n s of c e r t a i n x coordinates were changed, the corresponding y or z coordinates were changed i n the same manner. C e r t a i n t e n t a c l e s were not h e l d out evenly when the p i c t u r e was taken. Furthermore, hydra do not possess a r i g i d s k eleton and p e r i o d -i c a l l y change t h e i r t y p i c a l t e n t a c l e and s t a l k lengths f o r short p e r i o d s by c o n t r a c t i n g a t e n t a c l e or by r e t r a c t i n g t h e i r whole body. Hourly photographs were taken when the hydra was not co n t r a c t e d . Unexpected data that was due to e i t h e r of these two sources was el i m i n a t e d from the c a l c u l a t i o n s . Only the mean t e n t a c l e length of the hydra i s used i n f u r t h e r c o n s i d e r a t i o n s of the data and i n the model. When the data were p l o t t e d , they i n d i c a t e d that t e n t a c l e length i s an S-shaped f u n c t i o n of time of food d e p r i v a t i o n (Figure 2). Although the form of the t e n t a c l e length graph appears to be the same from animal to animal, the time a x i s has considerable i n c o n s i s t e n c y . This i s e s p e c i a l l y so where the hydra were pre-fed w i t h two b r i n e shrimp n a u p l i i . The consequences of t h i s i n c o n s i s t e n c y have already been discussed. The reason f o r i t was explained b r i e f l y i n the previous s e c t i o n . We go i n t o more d e t a i l here regarding the reason f o r the in c o n s i s t e n c y . Some info r m a t i o n of the b i o l o g y of hydra w i l l help to e x p l a i n the reason. Hydra i s a two layered animal: a l a y e r of epidermal c e l l s and a l a y e r of endodermal c e l l s . Between these l a y e r s there i s a mass of g el and f i b r e s (mesoglea). Both the epidermal and the endodermal c e l l s have t h e i r bases drawn out i n t o a p a i r of processes running i n opposite d i r e c -t i o n s . L y i n g between the base of each c e l l and the mesoglea i s the c o n t r a c t i l e element, the myoneme. The t e r m i n a l branches from one epidermal c e l l contact the myoneme-containing t e r m i n a l branches from se v e r a l adjacent c e l l s and form a continuous c o n t r a c t i l e network. The myonemes of the epidermis l i e l o n g i t u d i n a l l y and form part of the system r e s p o n s i b l e f o r the c o n t r a c t i o n of the polyp. The'myonemes of the endodermis are organ i z e d i n a c i r c u l a r f a shion and are the antagonists of the epidermal myoneme Haynes el_ a l (1968) suggest that the f i b r e s of the mesoglea may be organized i n a s p i r a l p a t t e r n . The s p i r a l acts much l i k e a spring -i t can be stretched and i t can be compressed, but i t always holds i t s shape. This property of the mesoglea seems to be inv o l v e d i n the maintenance of the normal p o s i t i o n of a hydra. I t i s suggested that i n the unfed polyp the l o n g i t u d i n a l l y o r i e n t a t e d epidermal muscles are relax e d . The length of the polyp i s Figure 2. A number of i n d i v i d u a l Chlorohydra graphs of t e n t a c l e length as a f u n c t i o n of time of food d e p r i v a t i o n . In the top graph, data from four i n d i v i d u a l hydra are presented. Each hydra had been pre-fed w i t h two newly hatched b r i n e shrimp. In the bottom graph, data from three i n d i v i d u a l hydra are presented. Each hydra had been pre-fed w i t h six' newly hatched b r i n e shrimp. The l i n e s through the data represent my f e e l i n g f o r the p a t t e r n of the data. The reason f o r drawing t h i s p a r t i c u l a r locus i s discussed i n the t e x t . The darkened p o r t i o n s of the l i n e represent the time w i t h i n which the food-remains were egested. 53 • • T ime of F o o d D e p r i v a t i o n ( h o u r s ) determined by the c h a r a c t e r i s t i c s of the mesogleal s p i r a l . I t i s suggested that i n the fed polyp the c i r c u l a r muscles are relaxed. This gives the surface area necessary to surround the food i n the gut. As the c i r c u l a r muscles r e l a x the a n t a g o n i s t i c system of l o n g i t u d i n a l muscles c o n t r a c t s . The c o n t r a c t i o n of these muscles r e s u l t s i n the shortening of both the s t a l k and the t e n t a c l e s . As d i g e s t i o n proceeds the food packet becomes smaller and more densely packed. This allows the s p i r a l to r e c o i l and the c i r c u l a r muscles to contract a c e r t a i n amount. As the c i r c u l a r muscles c o n t r a c t , the antagon-i s t i c l o n g i t u d i n a l muscles r e l a x and the hydra lengthens. When the food packet i s f i n a l l y emitted, maximum t e n t a c l e length and s t a l k l e n g t h are r a p i d l y regained. I am suggesting that the r e l e a s e of the food packet i s the mechanism which produces body lengthening. Figure 2 shows that those hydra which ate two n a u p l i i each released the undigested food packet l a t e r along the S-shaped curve than d i d those which ate s i x n a u p l i i each. The smaller packet of the former case would, before e g e s t i o n , a l l o w the hydra considerable c o n t r a c t i o n of the c i r c u l a r muscles whereas the l a r g e r packet would allow l e s s . (The smaller packet would crumble f a s t e r than the densely packed, large food packet.) Hydra eat n a u p l i i head f i r s t , t a i l f i r s t , or side f i r s t . The amount of gut d i s t e n s i o n immediately a f t e r feeding could be q u i t e d i f f e r e n t , dependent upon the way that the n a u p l i i were eaten. I f two n a u p l i i were eaten i n such a way that the f i r s t prey was head down and the second t a i l down a very small packet could be achieved. Gut d i s t e n s i o n would be s l i g b at any one point. The t e n t a c l e length at time of food d e p r i v a t i o n equal to zero (Figure 2) would be q u i t e f a r i n t o the S-shaped graph. I f the two n a u p l i i were s t u f f e d i n t o the gut cross-wise at r i g h t angles to each other, the cross s e c t i o n a l area of the packet would be considerable. 55 The t e n t a c l e length at time of food d e p r i v a t i o n equal to zero would e i t h e r not have entered the S-shaped graph (eg., open c i r c l e s of Figure 2) or not be very f a r i n t o the graph. I f t h i s e x p l a n a t i o n i s c o r r e c t , then s t a n d a r d i z i n g the f u n c t i o n s of t e n t a c l e lengths any f u r t h e r might prove a considerable d i f f i c u l t y . Yet, as explained i n the previous s e c t i o n , the v a r i a b i l i t y of the t e n t a c l e length f u n c t i o n along the time a x i s has serious e f f e c t s on the e s t i m a t i o n of the d u r a t i o n of the dynamic response of the t e n t a c l e length search mode. The s o l u t i o n adopted i s to overlap a l l f u n c t i o n s by moving them along the time of d e p r i v a t i o n a x i s such that l e a s t v a r i a t i o n occurs i n the region of the point of i n f l e c t i o n . This i s a more s p e c i f i c c r i t e r i o n f o r overlap than that given i n the previous s e c t i o n ; and the reason f o r p o s t u l a t i n g the S-shaped graph i s to estimate i n an accurate way the d u r a t i o n of the t r a n s i e n t response. The r e s u l t a n t graph i s shown i n Figure 3 . The overlapping of s t a l k length data f o l l o w s the p a t t e r n a p p l i e d to the t e n t a c l e length data. The composite graph of s t a l k lengths i s shown i n F i g u r e 3 . S i m i l a r i t i e s e x i s t between s t a l k length and t e n t a c l e length data. The p h y s i o l o g i c a l mechanism proposed to e x p l a i n v a r i a t i o n i n the time a x i s of Figure 2 suggests that there i s a s i n g l e mechanism f o r the c o n t r o l of body length. The s i m i l a r i t i e s between the t e n t a c l e length and s t a l k l e n g t h data w i l l be studied to see how c l o s e l y the data conform to t h i s suggestion. A l o g i s t i c f u n c t i o n was f i t to each of the t e n t a c l e and the s t a l k l ength data (Figure 3 ). The n o n - l i n e a r l o g i s t i c f u n c t i o n was chosen i n preference to the s i m i l a r l i n e a r decaying exponential f u n c t i o n because I b e l i e v e the i n i t i a l slow r i s e i n the data i s r e a l . As a hydra d i g e s t s i t s food c e r t a i n amounts of packing can be expected. As t h i s packing occurs, the extent of the d i s t e n s i o n of the c i r c u l a r muscles w i l l be lessened and Figure 3 . Mean t e n t a c l e and s t a l k length as a f u n c t i o n of time of food d e p r i v a t i o n . The lower l i n e i s a l o g i s t i c f i t t o the t e n t a c l e length data. The upper l i n e i s a l o g i s t i c f i t t o the s t a l k length data. The means are ranged by one standard d e v i a t i o n of the mean. The sample s i z e v a r i e s from three at the extremes t o seven at the c e n t r a l phase. Time of Food D e p r i v a t i o n (hours) ON 57 the t e n t a c l e s and s t a l k w i l l become s l i g h t l y longer. The parameters of best f i t to the l o g i s t i c f u n c t i o n are given w i t h standard d e v i a t i o n s i n Table I , f o r each of t e n t a c l e and s t a l k length data. The parameter values f o r the two cases are very s i m i l a r , w i t h the major d i f f e r e n c e between t e n t a c l e and s t a l k length f u n c t i o n s o c c u r r i n g i n the maximum len g t h parameter. This d i f f e r e n c e i s not due to a p o s s i b l e d i f f e r e n c e between the r i s e f a c t o r s , since i t would produce the opposite e f f e c t . Thus the d i f f e r e n c e between maximum length parameters i s due to the length of time over which the t e n t a c l e s expand r e l a t i v e to the length of time over which the s t a l k expands. Figure 3 shows that t e n t a c l e leangh increases over a longer period of time than does s t a l k length. Since t e n t a c l e length i s c o n t r o l l e d by one f a c t o r ( l o n g i t u d i n a l muscles) r a t h e r than the two which c o n t r o l s t a l k length ( l o n g i t u d i n a l and c i r c u l a r muscles), t h i s observation f o l l o w s . 2 Furthermore, i t can be shown, by c a l c u l a t i n g r , that 737o of the v a r i a t i o n i n s t a l k length can be explained by v a r i a t i o n i n t e n t a c l e length. The variance found i n the t e n t a c l e length data was p a r t i t i o n e d i n t o three sources. This was done as a check on the experimental technique. I t was assumed that the variance could be due to t e c h n i c a l problems ass o c i a t e d w i t h the camera, motion analyzer, and so on; to v a r i a t i o n w i t h i n i n d i v i d u a l s , between t e n t a c l e s ; and to v a r i a t i o n between i n d i v i d u a l s . The v a r i a t i o n due to t e c h n i c a l e r r o r s was measured by p a r t i t i o n i n g the v a r i a t i o n between the two readings of the x-coordinate. The expected sums of squares f o r the t e s t s are given i n Table I I . Four analyses were conducted. Each corresponded to some tine of food d e p r i v -a t i o n . Compilations were made of the percentages of v a r i a t i o n explained by each of the three sources. These percentages are given i n Table I I . T his t a b l e shows that the t e c h n i c a l sources of e r r o r were adequately 58 Table I . Parameter estimates f o r the f o l l o w i n g equation: L ( t ) - L ( t . ) = L(tmax) mm m a x 1 + e x p ( s ) / e x p ( r * t ) The estimates were obtained by f i t t i n g t h i s equation t o data w i t h the use of a n o n - l i n e a r l e a s t squares r o u t i n e . PARAMETER TENTACLE LENGTH STALK LENGTH -2 "2 10 mm 10 mm Minimum length, kftmin) 58.00 159 ,00 Maximum length, ^ m a x ) 161.00 ± 4, .70 125.00 ± 7 .20 Lag f a c t o r , s 1.10 ± 0 .14 1.30 + 0.45 Rise f a c t o r , r 0.23 ± 0 .03 0.31 ± 0.11 1 : a r b i t r a r i l y f i x e d Table I I . The r e s u l t s of four runs of a t h r e e - l e v e l , nested ANOVA. Each t e s t contained readings from f i v e i n d i v i d u a l s . Tentacle length was recorded by x-coordinate, by t e n t a c l e , and by i n d i v i d u a l . Three t e n t a c l e s per i n d i v i d u a l and two x-coordinate readings per t e n t a c l e were used. TECHNICAL DETAILS OF ANALYSIS: SOURCE EXPECTED MEAN SQUARES I n d i v i d u a l s Tentacles Measuring Equipment PERCENTAGE OF VARIATION EXPLAINED: SOURCE DATA SETS MEAN OVER SETS ns 1 *** *** I n d i v i d u a l s 7.99 35.66 80.29 87.18 *** *** *** *** Tentacles 89.93 57.45 19.01 11.09 52.78 44.37 Measuring 2.08 Equipment ns ns ns ns 6.89 0.71 1.74 2.86 1 : p=0.088 ns : not s i g n i f i c a n t at p=0.05 *** : s i g n i f i c a n t at p=0.001 60 c o n t r o l l e d . An estimate of prey v e l o c i t y , VY i s a l s o necessary i n order to use equations 126 and 127. In Beaver Creek, prey v e l o c i t y was i d e n t i c a l to water v e l o c i t y . Thus estimates may be obtained from Appendix I . Thi s completes the f i r s t of two o b j e c t i v e s set f o r t h i s s e c t i o n . The attempt i s now made to f i n d the r e l a t i o n s h i p between the time of i n i t i a t i o n of equations 126 and 127, and the amount of food pre-fed to the hydra. The i n i t i a t i o n t h r e s h o l d of these equations i s measured i n terms of the t e n t a c l e l e n g t h graph. A s u i t a b l e i n d i c a t o r of the approximate time of l o g i s t i c growth i s chosen. This i n d i c a t o r i s r e l a t e d to amount of food pre-fed to the hydra. Some hours a f t e r e a t i n g , hydra emit a packet of undigested food-remains. The darkened areas of the graphs of Figure 2 represent the time range w i t h i n which t h i s egestion occurred. When hydra eat few n a u p l i i they egest the food-remains near the top end of the l o g i s t i c growth phase of t e n t a c l e length. When hydra eat to s a t i a t i o n , they egest the food-remains near the beginning of l o g i s t i c growth. E g e s t i o n always occurs during the phase of l o g i s t i c growth. Thus, t h i s e a s i l y measurable c h a r a c t e r i s t i c i s a s u i t a b l e i n d i c a t o r of the approximate time of l o g i s t i c growth. An experiment was devised to measure the time of egestion as a fu n c t i o n of the amount of food pre-fed to the hydra. Since Hydra  l i t t o r a l i s i s l a r g e r than Chlorohydra, i t emits a l a r g e r , more v i s i b l e food packet. Hydra were used i n the i n i t i a l stages of t h i s experiment. The form of the graph obtained was then checked w i t h some Chlorohydra observations. The brown hydra were fed at l e v e l s of 6, 11, 16, and 21 Artemia n a u p l i i per hydra and placed i n p e t r i p l a t e s of 5 cm i n diameter. The p e t r i p l a t e s c o n t a i n i n g hydra were placed w i t h i n a square arrangement on a cement t a b l e that minimized v i b r a t i o n s . The p l a t e s were banked on each of two sides by a neon l i g h t , g i v i n g a l i g h t i n t e n s i t y over the plates of 50 t o 100 f o o t - c a n d l e s . Temperature averaged 20 deg C. Hydra were checked at h o u r l y i n t e r v a l s f o r e g e s t i o n . The r e s u l t s of t h i s experiment are shown i n Figure 4. The data are presented i n two ways: f i r s t as a cumulative percentage of the t o t a l number of Hydra observed which have egested the food-remains (reached c r i t e r i o n ) and, i n the i n s e t , as the time t o e g e s t i o n . The lower set of p o i n t s i n the i n s e t describes the expected time that the f i r s t hydra w i l l take t o egest the food remains. The top set of p o i n t s describes the expected time that the l a s t hydra w i l l take to egest the food-remains. Note the 3 hour d i f f e r e n c e between the two l i n e s i n the i n s e t . T h i s r e s u l t i s i n t e r e s t i n g i n l i g h t of the problems encountered i n t r y i n g to set up Figure 3. Some data allowed me t o f i t an approximate a b s c i s s a f o r Chlorohydra. The graphs of the i n s e t give a t r u e r e f l e c t i o n of the time of l o g i s t i c growth only i f egestion occurs at the same p o s i t i o n on the f u n c t i o n , r e g a r d l e s s of the amount of food pre-fed to the hydra. But egestion does not occur at the same p o s i t i o n on the f u n c t i o n . As the amount of food pre-fed t o the hydra i n c r e a s e s , the p o s i t i o n of egestion s h i f t s back along the f u n c t i o n . Thus the l i n e a r r e l a t i o n s h i p shown i n the i n s e t underestimates the t r u e slope. Using the data of Figure 2 one can i n f e r that the maximum amount of s h i f t i n g of egestion along the l o g i s t i c f u n c t i o n i s of the order of 4 hours. This c o r r e c t i o n would r e s u l t i n the slope being revalued from 1.3 t o 2.3 h o u r s / n a u p l i i f o r Chlorohydra and from 0.2 to 0.4 h o u r s / n a u p l i i f o r Hydra. I w i l l now t u r n t o the model b u i l d i n g and the relevant measurements f o r the second searching behaviour, walking. Figure 4. The time that hydra take t o egest food-remains as a f u n c t i o n of the s i z e of t h e i r l a s t meal. I n the lower graph the l i n e s represent data from Hydra pre-fed at l e v e l s of 6, 11, 16, and 21 Artemia/hydra. I f the p o i n t s of the lower graph are assumed t o l i e along a s t r a i g h t l i n e , then each set of p o i n t s i n the i n s e t represents the 0% and the 100% l e v e l s of the a b s c i s s a . The p o i n t s i n the i n s e t are a t r a n s l a t i o n of the data below f o r Hydra l i t t o r a l i s (n= 69, 8, 55, 20). The open c i r c l e s represent data f o r Chlorohydra v i r i d i s s i m a (n= 15, 18, 12). 62 HYDRA. EXPERIMENTATION: WALKING The equations which f o l l o w d e scribe the upper and the lower l i m i t s of the volume searched by a hydra which steps at l e a s t once w i t h i n an i n t e r v a l of time from t Q+nT-T t o t Q+nT. The equation f o r the lower l i m i t i s 0.05 | L(t Q+nT-T) + L(t Q+nT ) j + 0.23*T *VY + 0 .23*T m*( |VY-VPL |) j L 2(t Q+nT-T) + L 2(t Q+nT)] 128 The equation f o r the upper l i m i t i s r -, r T 3 3 0.03 + 0.45* _ V Y l L ( t +nT-T) + L ( t +nT) +0.20 L (t +nT-T) L VPL j I -I ° + j^l.37*T m*VY + 1.08*TU*VY + 0.23*Tm*VPLJ j^L 2(t 0+nT-T) + L^(t G+nT) 129 Those terms not defined i n the previous s e c t i o n are defined as f o l l o w s : T i s the amount of time i n the i n t e r v a l T that i n v o l v e s hydra m J movement; T i s the amount of time i n the i n t e r v a l T that does not ' u i n v o l v e hydra movement; and VPL i s the v e l o c i t y of the movement of the cone of t e n t a c l e s , as occurs when the hydra i s walking. Parameters T and VPL w i l l be estimated. The term T can be m u e a s i l y c a l c u l a t e d once T m i s known. The terms L(t 0+nT) and VY were discussed i n the previous s e c t i o n . The frequency of stepping changes with time of food d e p r i v a t i o n . Thus T m i s b e t t e r w r i t t e n as T m ( t o + n T ) . The major job attempted i n t h i s s e c t i o n i s the measurement of T m as a f u n c t i o n of hunger. I t i s not experimentally convenient t o measure T (t +nT) i n terms J mN o of time. A distance measure i s more s u i t a b l e . The parameter T m i s equal t o the product of the average time t o complete one step and the number of steps completed. The v a r i a b l e of number of steps can be measured i n terms of distance i f the average distance moved per step i s known. The distance moved by a hydra i n one step was measured e x p e r i -m e n t a l l y . Two Chlorohydra polyps were each given two Artemia n a u p l i i . The hydra were then placed i n the experimental container described e a r l i e r . They were put on the small p l e x i g l a s s stand and allowed to a t t a c h . The p o s i t i o n of each hydra was recorded at i n t e r v a l s of 35 minutes. This time i n t e r v a l was chosen because the p r o b a b i l i t y of a hydra moving more than once w i t h i n i t i s low. Thus most movements w i l l be s i n g l e s t e p s . A frequency diagram of d i s t a n c e moved per 35 minutes w i l l d e f i n e a step . The p o s i t i o n of hydra was recorded with the a i d of a camera system placed d i r e c t l y over the hydra. This measuring equipment was i d e n t i c a l to that used f o r body length measurements, except that p i c t u r e t a k i n g at 35 minute i n t e r v a l s was f u l l y automated. The p o s i t i o n of each hydra was photographed both day and n i g h t . Hydra were kept i n the range of the camera by darkening a l l areas of the experimental container which were outside of the d e s i r e d area. The area w i t h i n which hydra moved was i l l u m i n a t e d at an i n t e n s i t y of 100 f o o t - c a n d l e s . I n each photograph the camera recorded the p o s i t i o n of each hydra. Estimates of the (x,y) coordinates f o r the p o s i t i o n of the hydra were obtained from the motion a n a l y z e r . Estimates of the distance moved per hydra were obtained from the usual d i s t a n c e formula. The r e s u l t s of these measurements are shown i n Table I I I (FREQUENCY per 35 minutes). The frequency c l a s s e s of 01-02 and of 00-01 mm were omitted i n s e t t i n g up t h i s t a b l e . Movement of l e s s than 2 mm was found to be due to experimental e r r o r . Most data centers around the mode of 05-06 mm. This observation r e l a t e s t o the previous observation that the 65 Table I I I . Distance moved per time i n t e r v a l by Chlorohydra v i r i d i s s i m a . Only those animals which moved were considered. DISTANCE MOVED FREQUENCY (mm) per 35 minutes per 60 minutes 02-03 3 3 03-04 11 14 04-05 9 9 05-06 16 33 06-07 10 26 07-08 6 12 08-09 2 3 09-10 0 1 10-11 0 0 11-12 0 0 12-13 1 0 13-14 0 0 14-15 3 0 15-16 1 0 16-17 0 1 17-18 0 0 18-19 0 0 19-20 0 0 20-21 0 0 21-22 0 0 22-23 1 1 T o t a l 63 103 66 maximum s t a l k l ength i s about 3 mm (Figure 3). The two types of walking discussed by Ewer (1947b) are v i s i b l e ; there i s a peak at each of the 03-04 and the 05-06 c l a s s e s . Those hydra movements of more than 6 mm might r e l a t e to my observation that hydra tend to elongate before walking. A step i s defined as a p o s i t i o n change of 3-8 mm i n length. I recog-n i z e that two p o s i t i o n changes of 3 mm or two of 4 mm would be counted as one step. However, when v i b r a t i o n s are minimized, Chlorohydra move at low r a t e s even when deprived of food f o r a long time. This makes i t u n l i k e l y that a hydra w i l l move more than once i n any of the time i n t e r v a l s used i n t h i s study. The distance moved per hour by i n d i v i d u a l Chlorohydra polyps was measured as, a f u n c t i o n of hunger. The i n t e r v a l T was chosen to be 1 hour because equations 128 and 129 are f u n c t i o n s of L ( t +nT) as w e l l as of o T ( t +nT) - and L ( t +nT) i s measured only at d i s c r e t e i n t e r v a l s of m o o J 1 hour's d u r a t i o n . Polyps were chosen from the Chlorohydra stock. They were fed and then deprived of food. Each polyp was fed w i t h three newly hatched n a u p l i i . Sets of four polyps which had been t r e a t e d i n t h i s manner were placed i n each of s i x p e t r i p l a t e s . The p l a t e s were 60 mm i n diameter. They were placed on a stand elevated from the top of a cement t a b l e that minimized v i b r a t i o n s . Once every hour the p e t r i p l a t e s were marked from the bottom f o r the p o s i t i o n of each hydra. This began immediately a f t e r feeding and continued f o r 12 successive hours. The p l a t e s were p l a s t i c and could be e a s i l y scratched w i t h a sharp probe. When a hydra moved from one p o s i t i o n to another, I marked the new p o s i t i o n and drew a l i n e from the former to the new p o s i t i o n . Records were taken i n the same way f o r another 12 hours, beginning the next morning - and again the next day. In t h i s way the paths of the hydra were traced during times of d e p r i v a t i o n of 1 to 12 hours, of 2 4 to 3 6 hours, and of 4 8 to 6 0 hours. A t o t a l of 9 6 hydra were followed i n t h i s manner. An a d d i t i o n a l set of 9 6 hydra were t r e a t e d as above, except that they were fed i n the evening, w i t h recording begun the next morning. This allowed observations of hydra behaviour during times of d e p r i v a t i o n of 1 2 to 2 4 hours, of 3 6 t o 4 8 hours, and of 6 0 t o 7 2 hours. A continuous 7 2 hours of recording was obtained by combining t h i s set of data with the other s e t . C e r t a i n aspects of the r e l a t i o n s h i p between the d i s t a n c e covered per hour by a hydra and the time of food d e p r i v a t i o n emerged from these data. The d i s t a n c e covered/hydra/hour r e t a i n s a minimum value u n t i l about 2 7 hours a f t e r f e e d i n g . To t h i s time, the p r o b a b i l i t y of a hydra walking during an i n t e r v a l of 1 hour i s 0 . 4 7 o . Then the d i s t a n c e covered/hydra/hour increases t o a l a r g e r , but v a r i a b l e , maximum. The p r o b a b i l i t y of a hydra walking during an i n t e r v a l of 1 hour i s 5 . 9 7 , . The standard d e v i a t i o n i s 3 . 5 7 , . The d u r a t i o n of the dynamic response between the minimum and the maximum value i s not c l e a r because of the v a r i a b i l i t y around the maximum response. P r a c t i c a l l y no hydra moved more than once i n any one hour i n t e r v a l : of 1 0 3 movements, 9 4 were of s i n g l e steps (Table I I I ) . This means that i f the dynamic response i s to be great e r than 1 hour (the i n t e r - r e c o r d i n t e r v a l ) , then the p r o b a b i l i t y of a hydra stepping at the r a t e of once per hour must increase as hunger i n c r e a s e s . Thus, a l l hydra data were standardized to the time of f i r s t step. The p r o b a b i l i t y of a hydra moving was c a l c u l a t e d f o r each h o u r l y i n t e r v a l a f t e r the time of the hydra's f i r s t step. Table IV gives the percentage of hydra which moved per hour f o r a l l hours i n which data from 2 0 or more hydra were a v a i l a b l e . The data do not i n d i c a t e a r e l a t i o n s h i p between the p r o b a b i l i t y of a hydra 68 Table IV. Percentage of hydra which moved per hour, f o r successive hours a f t e r hour of f i r s t step . HOURS AFTER FIRST STEP TOTAL NUMBER HYDRA OBSERVED NUMBER HYDRA MOVED PERCENTAGE HYDRA MOVED 1 2 3 4 5 6 7 8 9 17 18 19 20 21 22 23 24 25 26 27 28 29 77 62 69 75 54 34 28 29 24 21 26 32 34 30 37 48 52 35 •28 35 35 26 4 4 4 10 4 2 3 1 1 1 3 6 3 4 3 5 8 0 3 1 0 1 5 . 2 6 . 5 5 . 8 1 3 . 3 7 . 4 5 . 9 1 0 . 8 3 . 5 4 . 2 4 . 8 1 1 . 5 1 8 . 8 8 . 8 1 3 . 8 8 . 1 1 0 . 4 5 . 4 0 . 0 1 0 . 7 2 . 9 0 . 0 3 . 9 moving and the time a f t e r i t f i r s t steps. The dynamic response w i l l be chosen as l e s s than or equal to 1 hour. The data of Tables I I I and IV w i l l now be converted i n t o a time measure of T m ( t Q + n T ) . The p r o b a b i l i t y of movement to hour 27 i s 0.004. The p r o b a b i l i t y of movement r i s e s q u i c k l y 1 hour l a t e r to 0.059. Each movement i s one step. The time which a hydra takes tp bend over and a t t a c h i t s t e n t a c l e s to the substrate and then to p u l l up the t e n t a c l e s i s about 0.75 minutes per loop. Thus 0.00300 minutes i s the average T m(t Q+nT) f o r time l e s s than 27 hours. For time greater than 27 hours the average T (t +nT) i s 0.04435 minutes. ° m o The parameter T can be c a l c u l a t e d as the t o t a l time T minus T , v U m' the time spent moving. Since the time spent moving i s so s m a l l , an a d d i t i o n a l term w i l l be subtracted from T. This term i s T , the time during a step that the hydra has i t s t e n t a c l e s attached to the substrate. T has been measured to be about 1 minute per step. When modified f o r a r r the p r o b a b i l i t y of l o o p i n g , T& equals 0.004 minutes f o r time l e s s than 27 hours and 0.059 f o r time greater than 27 hours. Thus, f o r time l e s s than 27 hours, T u i s 59.9930 minutes and f o r time greater than 27 hours, T u i s 59.8968 minutes. The terms T and T are not accurate to s i x s i g n i f i c a n t f i g u r e s . However, u r n we keep these small d i f f e r e n c e s to see how such d i f f e r e n c e s can a f f e c t the volume searched equations. T^ might w e l l be l a r g e r i n the f i e l d than shown here. Wagner (1905) notes that v i b r a t i o n can e l i c i t v a r i o us behaviours. However, i n order to determine the e f f e c t of only hunger on T , a v i b r a t i o n - m i n i m i z i n g t a b l e was used f o r these experiments. The f i n a l parameter necessary to be estimated i s VPL. An approximate value i s 60 mm per minute. F i n a l l y , T ( t +nT) w i l l be r e l a t e d to the amount of food pre-fed to the hydra. The time of f i r s t step w i l l be the c r i t e r i o n which w i l l be r e l a t e d to the time of food d e p r i v a t i o n . Chlorohydra polyps were fed at l e v e l s of 1 , 3 , and 6 newly hatched n a u p l i i and then deprived of food. The polyps were placed i n p e t r i p l a t e s . These p l a t e s were set on the stand used i n the most r e c e n t l y described experiment. The p o s i t i o n of each hydra was marked by scratchin; the underside of the p l a t e . The number of hydra not standing on t h e i r mark was recorded each hour u n t i l a l l hydra had moved at l e a s t one time. The r e s u l t s are shown i n the customary manner i n F i g u r e 5 . There i s a 34 hour d i f f e r e n c e between the two l i n e s i n the i n s e t . T h i s completely f u l f i l l s the o b j e c t i v e s of t h i s s e c t i o n . I w i l l t u r n to the model b u i l d i n g and the analogous measurements of the f i n a l searching behaviour, f l o a t i n g . Figure 5. The time that Chlorohydra v i r i d i s s i m a polyps take t o walk f o r the f i r s t time, as a f u n c t i o n of the s i z e of t h e i r l a s t meal. I n the lower graph the l i n e s represent data from hydra pre-fed at l e v e l s 1, 3, and 6 Artemia per hydra. I f the p o i n t s of the lower graph are assumed t o l i e along a s t r a i g h t l i n e , then each set of p o i n t s i n the i n s e t represents the 0% and 100% l e v e l s of the a b s c i s s a . Each set of p o i n t s i n the i n s e t i s based on a sample of 24 hydra. c o 80H Z jo o o -c 0) E 2CH 1 -T-3 Number N a u p l i i I P r e - f e d 20 40 60 80 Time of Food D e p r i v a t i o n iBnours) HYDRA EXPERIMENTATION: FLOATING The equations which f o l l o w describe the upper and lower l i m i t s of the volume searched by a hydra from t 0+nT-T t o t 0+nT f o r a l l i n t e r v a l s of T a f t e r and i n c l u d i n g that i n which the hydra detaches and begins t o f l o a t . The equation f o r the lower l i m i t i s r 3 3 1 10.36DMT + 0.20[L (t +tiT-T) + L (t +nT)J + 13.91*( |VY-VPF|)* L 2(t o+nT-T) + L 2 ( t D + n T ) ] 130 The equation f o r the upper l i m i t i s 10.36DMT + 0.20 L 3(t Q+nT-T) + L 3 ( t G + n T ) ] + 13.91VPF* [ L 2 ( t o + n T - T ) + L 2 ( t 0 + n T ) l + 0.45*_VY* [ L(t D+nT-T) + L(t D+nT ) l 3 + 82.47*VY* VPF J L 2(t Q+nT-T) + L 2(t Q+nT)] l31 Those terms not defined p r e v i o u s l y are defined as f o l l o w s : DMT i s the dis t a n c e the hydra f l o a t s v e r t i c a l l y before beginning t o be swept h o r i z o n t a l l y . DMT i s equal t o zero f o r a l l time not equal to t Q+kT. VPF i s the v e l o c i t y of hydra movement during f l o a t i n g . The only unmeasured parameters of these equations are DMT and VPF. Neit h e r i s a f u n c t i o n of the time of food d e p r i v a t i o n . Much of the form of the f l o a t i n g f u n c t i o n i s c l e a r without the n e c e s s i t y of data c o l l e c t i o n . Before detaching, the hydra covers no volume through the use of the f l o a t i n g behaviour. Suddenly the hydra detaches and begins t o f l o a t . Thus the d u r a t i o n of the dynamic response i s l e s s than 1 hour (the i n t e r - r e c o r d i n t e r v a l ) . A f t e r detaching, the hydra continues t o f l o a t . I t covers volume as a f u n c t i o n of VPF. The assumption i s made that the hydra continues t o f l o a t i n d e f i n i t e l y , at constant VPF. The parameter DMT took values up t o 30 cm f o r Hydra carnea populations i n Beaver Creek. The parameter VPF, as 73 r e f l e c t e d by the water v e l o c i t y , ranged up t o 30 cm/second. This short e x p o s i t i o n completes the f i r s t of two o b j e c t i v e s which have been set f o r t h i s s e c t i o n . The form of the f l o a t i n g f u n c t i o n has been de s c r i b e d . The time of i n i t i a t i o n of t h i s f u n c t i o n w i l l now be r e l a t e d t o the amount of food pre-fed to the hydra. The c r i t e r i o n of i n i t i a t i o n i s detaching and b e g i n - f l o a t i n g . Detaching, with subsequent f l o a t i n g , has been hypothesized t o have a d u a l i t y of cause: food and d e n s i t y (Lomnicki and Slobodkin, 19 66). The j o i n t e f f e c t s of food and d e n s i t y were studied t o a s c e r t a i n the r e l a t i v e importance of each f a c t o r . A number of Hydra l i t t o r a l i s were pre-fed at l e v e l s of 1, 6, and 16 newly hatched n a u p l i i per hydra. The hydra were then placed at d e n s i t i e s of 1, 2, and 8 hydra per 20 cc medium i n p e t r i p l a t e s of 5 cm i n diameter. The p e t r i p l a t e s were set on a cement t a b l e that minimizes v i b r a t i o n and were i l l u m i n a t e d with a l i g h t i n t e n s i t y of 50-100 f o o t - c a n d l e s . Temperature averaged 20 deg C. The number of hydra f l o a t i n g was recorded at 12 hour i n t e r v a l s . The r e s u l t of t h i s experiment showed that d e n s i t y i s the minor f a c t o r (Table V ) . Thus a l l d e n s i t y data were combined. The cumulative percentage of hydra which were f l o a t i n g was c a l c u l a t e d as a f u n c t i o n of time of food d e p r i v a t i o n and of the amount of food pre-fed t o the hydra. The data are presented i n the us u a l fashion i n Figure 6. There i s a 32 hour d i f f e r e n c e between the two l i n e s of the i n s e t . I n Figure 4 i t was shown that the graph of the time t o c r i t e r i o n as a f u n c t i o n of the amount of food ingested was the same f o r both H. l i t t o r a l i s and C. v i r i d i s s i m a . The two graphs were overlapped when the maximum number of Artemia eaten by each species was l i n e d up together along the a b s c i s s a . Maximum i n g e s t i o n i s 32 newly hatched n a u p l i i f o r H. l i t t o r a l i s . For C. v i r i d i s s i m a i t i s s i x Artemia. The assumption 74 Table V. A f a c t o r i a l A n a l y s i s of Variance t a b l e : the e f f e c t of food and hydra d e n s i t y on the time which hydra take t o begin f l o a t i n g . SOURCE DEGREES OF FREEDOM SUM OF SQUARES MEAN SQUARE Food (F) 2 Density (D) 2 F X D 4 E r r o r 297 T o t a l 215 27,241 241 79 55, 590 83,151 13,621 121 20 269 51.00 p < 0.001 0.45 p >0.50 0.07 P >0.75 Figure 6. The time that hydra take t o detach and begin t o f l o a t as a f u n c t i o n of the s i z e of t h e i r l a s t meal. I n the lower graph, the l i n e s represent data f o r hydra pre-fed at l e v e l s of 1, 6, and 16 Artemia n a u p l i i / h y d r a . I f the p o i n t s of the lower graph are assumed t o l i e along a s t r a i g h t l i n e , then each set of p o i n t s i n the i n s e t represents the 0% and the 100% l e v e l s of the a b s c i s s a . I n the i n s e t the p o i n t s represent data c o l l e c t e d f o r Hydra l i t t o r a l i s (n= 24, 24, 24). The open c i r c l e s represent data f o r Chlorohydra v i r i d i s s i m a (n= 10, 10, 10). 7 6 i s made that the same r e l a t i o n s h i p holds f o r the f l o a t i n g c r i t e r i o n . A Chlorohydra a x i s was set to Figure 6 . Then some data was c o l l e c t e d t o t e s t t h i s assumption. The only unusual experimental c o n d i t i o n was that the hydra were subjected t o a d i u r n a l l i g h t c y c l e of 1 2 hours of l i g h t ( 5 foot-candles) and 1 2 hours of darkness. The r e s u l t s are shown as open c i r c l e s i n the i n s e t of Fi g u r e 6 . This completely f u l f i l l s the o b j e c t i v e s of t h i s s e c t i o n . I t a l s o completes the data necessary t o t e s t the searching r a t e h y p o t h e s i s . A TEST OF THE HYPOTHESIS The hypothesis to be t e s t e d i s that the d u r a t i o n of the dynamic response f o r some searching mode i s i n v e r s e l y r e l a t e d to the time of food d e p r i v a t i o n which i n i t i a t e s searching by that mode. Hydra use three searching modes: t e n t a c l e lengthening, walking and f l o a t i n g . The searching behaviour which i s i n i t i a t e d f i r s t on a time of food d e p r i v a t i o n a x i s i s t e n t a c l e lengthening. The searching f u n c t i o n appropriate to t h i s behaviour i s that f o r which equation 126 i s the lower l i m i t and f o r which 127 i s the upper l i m i t . I n these equations only t e n t a c l e length i s responsive to the time of food d e p r i v a t i o n . Thus, i t i s c l e a r from the form of 126 and 127 that the du r a t i o n of the dynamic response of these f u n c t i o n s i s the du r a t i o n of the dynamic response of the t e n t a c l e length graph (Figure 3). One can see from t h i s f i g u r e that the approximate d u r a t i o n i s 5 hours. The searching behaviour which i s i n i t i a t e d next on a time of food d e p r i v a t i o n a x i s i s walking. The lower and upper l i m i t s to t h i s f u n c t i o n are given by 128 and 129 r e s p e c t i v e l y . In these equations both t e n t a c l e length and the frequency of walking (T ) are responsive to the time of food d e p r i v a t i o n . However, as shown i n Figur e s 4 and 5, t e n t a c l e length reaches i t s maximum value before walking begins. Thus, i t i s c l e a r that the d u r a t i o n of the dynamic response of these fun c t i o n s w i l l be the du r a t i o n of the dynamic response of the T m f u n c t i o n . This d u r a t i o n , as noted i n the s e c t i o n c a l l e d HYDRA EXPERIMENTATION: WALKING i s l e s s than or equal to 1 hour. The searching behaviour which i s i n i t i a t e d l a s t on a time of food d e p r i v a t i o n a x i s i s f l o a t i n g . The lower and the upper l i m i t s to t h i s f u n c t i o n are given by 130 and 131 r e s p e c t i v e l y . In these equations only tentacle length i s responsive to the time of food deprivation. However, as shown i n Figures 4 and 6 , tentacle length reaches i t s maximum value before f l o a t i n g i s i n i t i a t e d . Thus, these functions are not sens i t i v e to time of food deprivation, other than being switched on at some hunger l e v e l . The duration of the dynamic response of these functions i s i n terms of a few seconds. I f my hypothesis i s correct, then the duration of the dynamic response for tentacle lengthening should not be less than that for walkin which should not be less than that for f l o a t i n g . We find that the duration of the transient response for tentacle lengthening i s of length 5 hours; for walking of length less than 1 hour; and for f l o a t i n g , of length much less than 1 hour. These, data do not contradict my assumption. I conclude that there may be some v a l i d i t y i n the use of s t r a t e g i c a l hypotheses i n the formulation of general functions. PART I I A TIME SPENT SEARCHING FUNCTION FOR HYDRA INTRODUCTION In PART I a r e l a t i o n s h i p between the du r a t i o n of the dynamic responses of various searching modes was formulated and te s t e d . The hydra behaved as pr e d i c t e d . The concern of t h i s s e c t i o n i s w i t h searching by hydra, and only i n d i r e c t l y w i t h s t r a t e g i c a l hypotheses. Much data has been c o l l e c t e d on searching by hydra. The data has been used only to t e s t a t r a n s i e n t response r e l a t i o n s h i p . However, enough data has been gathered i n order to formulate a complete f u n c t i o n f o r volume searched (VS) by hydra. To do so requ i r e s the a d d i t i o n a l knowledge of the steady s t a t e searching rates of the various searching modes used by hydra. When t h i s i n f o r m a t i o n i s combined w i t h the dynamic response i n f o r m a t i o n of PART I , i t w i l l a l low one to w r i t e an o v e r a l l searching rate f u n c t i o n . From t h i s i t i s p o s s i b l e to de r i v e a f u n c t i o n f o r time spent searching (TS) by hydra. This work was separated from that of PART I so as not to confuse the somewhat i n t e r r e l a t e d o b j e c t i v e s of PARTS I and I I . In the f i r s t PART the o b j e c t i v e was to see i f s t r a t e g i c hypotheses have value i n formulating general f u n c t i o n s . In the second PART the o b j e c t i v e i s to formulate a VS f u n c t i o n f o r hydra and to use t h i s f u n c t i o n i n w r i t i n g an e x p l i c i t time spent searching f u n c t i o n (TS). METHODS AND RESULTS 80 The r e l a t i o n s h i p between the steady s t a t e searching r a t e of a searching mode and the i n i t i a t i o n t h r e s h o l d of that mode w i l l be studied by e x p l o r i n g the behaviour of the searching model. The equations of minimal e f f e c t f o r body lengthening and looping w i l l be studied f i r s t . We may w r i t e equation 128, r e l a t i v e to 126, as f o l l o w s : 100 6.40 + 0.23*(lVY-60.0J) + 134.34VY 4.26 + 134.55VY 100 I I I This gives the percentage increase of the minimal volume searched per hour by a w a l k i n g hydra over one that does not move. Estimates of VPL, T„, T , and of maximum t e n t a c l e length were obtained from u* m ° Chlorohydra data. The d e r i v a t i v e of I I I , w i t h respect t o VY, shows a p o s i t i v e slope. The slope at VY=0 i s r a t h e r large but i t f a l l s q u i c k l y to l e s s than 1 7o-minute/mm by VY=2.91 mm/minute. For a l l l a r g e r VY values, equation I I I keeps almost the same v a l u e . At VY=3 mm/minute, i t has value of -2.84. Water v e l o c i t y i n Beaver Creek was observed to vary from 0 cm/sec t o 30 cm/sec (18,000 mm/minute). Thus, the e f f e c t of w a l k i n g on the minimal volume searched per hour i s s l i g h t l y negative over most of the water v e l o c i t i e s i n which hydra have been observed. The search-ing time not used when the t e n t a c l e s are attached (T ) t o the substrate cL i s not compensated f o r by the extended volume covered due t o the combined e f f e c t s of moving the cone through space (VY and VPL). Analogous to I I I , an equation f o r the percentage i nc re as e of the maximal volume searched per hour by a w a l k i n g hydra over one that does not move may be w r i t t e n as f o l l o w s : 100 19 .45 + 626.48VY 100 112 4.26 + 626.88VY The d e r i v a t i v e of t h i s equation, with respect t o VY, shows a negative slope. Although at VY= 0 there i s a large slope, by VY= 2.03 mm/minute the slope descends to l e s s than 1 7o-minute/mm. Thus over almost the whole range of VY values, the magnitude of equation 112 i s constant, at +1.157o. V a l k i n g at the r a t e of 0.059 times per hour i s not very b e n e f i c i a l to the hydra, at l e a s t i n terms of volume searched. The experiment from which the w a l k i n g r a t e was c a l c u l a t e d was . set up to minimize v i b r a t i o n s . T his was done so that the e f f e c t s of hunger alone could be determined. I n f i e l d s i t u a t i o n s with s u b s t a n t i a l water v e l o c i t i e s , I observed that hydra waved v i g o r o u s l y w i t h the changing motion of the water. I n the la b o r a t o r y , I noted that a common stimulus f o r walking was when the t e n t a c l e s of an upright hydra contacted the substrate:with some f o r c e . The stepping r a t e i n the f i e l d may be l a r g e r than that measured i n the l a b o r a t o r y . I t i s not unreasonab to suggest the p o s s i b i l i t y of w a l k i n g c o n t r i b u t i n g some s l i g h t amount t o the searching r a t e . Next the e f f e c t s of f l o a t i n g w i l l be examined. The percentage increase of the minimal volume searched by a f l o a t i n g hydra over one which does not move i s as f o l l o w s : 100 325.06 + 134.65* (IVY-VPF I) 4.26 + 134.55*VY 100 113 The parameter DMT was given the value of 30 mm. (Note that DMT occurs only during time t Q+kT. The f i r s t term of the numerator of 113 must be subtracted when time i s not equal to t Q+kT.) I f the assumptions that VY^VPF and VPF>2.3 are made, then the d e r i v a t i v e of 113 wit h respect t o VY has a p o s i t i v e slope. For values of VY near to VPF, the equation has a r e l a t i v e l y l a r ge slope. This slope f a l l s t o l e s s than 1 70-minute/mm by VY=28 mm/minute, f o r VPF= 10 mm/minute; by VY=99, f o r VPF=100; and by VY=223, f o r VPF=500. The stream s i t u a t i o n might w e l l be approximated when prey v e l o c i t y (VY) i s set equal t o water v e l o c i t y , which i s set equal t o the f l o a t i n g speed (VPF) of hydra. F l o a t i n g hydra have no apparent means of p r o p e l l i n g themselves. I n streams, independent means of locomotion of prey might w e l l be counteracted by water movement . Equation 113 was evaluated f o r VY=VPF=100 mm/minute. I t took the value of -97.97o. This value could be made l e s s negative only during the time i n t e r v a l , t Q-rkT, by a d j u s t i n g the value of DMT upward; however, the equation would always remain negative i n v a l u e . On the other hand, the value could be pushed to -1007o by l e t t i n g VY=VPF take l a r g e r v a l u e s . The p e r s i s t e n t negative e f f e c t of f l o a t i n g a r i s e s because the f l o a t i n g hydra l o s e s the prey v e l o c i t y component i t had when attached. That i s t o say, the volume searched by a hydra f l o a t i n g i n a stream i s the a c t u a l , not the e f f e c t i v e , volume searched. E s t i m a t i o n of the amount that the volume seached per time i n t e r v a l i s reduced by a f l o a t i n g hydra r e q u i r e s knowledge of the length of time which the hydra f l o a t s before r e a t t a c h i n g . Where the water ran q u i c k l y i n Beaver Creek, the hydra were swept not along the surface of the water but only a few m i l l i m e t e r s from the s u b s t r a t e . Thus one might expect that they do not f l o a t f o r long times before running i n t o some protuberance and r e - a t t a c h i n g . Another co m p l i c a t i n g f a c t o r has a r i s e n . I n Beaver Creek the f l o a t i n g hydra which I saw moved down the creek with t e n t a c l e s outspread. However, Slobodkin has seen hydra f l o a t i n g down a stream with t e n t a c l e s f u l l y withdrawn (communicated to me by B i l l C l a r k ) . When t h i s occurs, the volume searched by a f l o a t i n g hydra might be s u b s t a n t i a l l y reduced over the volume searched by a n o n - f l o a t i n g hydra. The lake s i t u a t i o n might be approximated when the absolute value of (VY-VPF) i s set equal t o the value of VY. This i s because i n a lake environment water v e l o c i t y and consequently VPF i s about zero. On the other hand, prey movement is often a f u n c t i o n of s e l f p r o p u l s i o n , r a t h e r than a f u n c t i o n of water movement. I n t h i s s i t u a t i o n the minimal volume searched per time i n t e r v a l remains i d e n t i c a l with that of the attached hydra. One exception i s the i n i t i a l jump i n searching r a t e during time t Q 4 k T . This a d d i t i o n becomes more s u b s t a n t i a l with increases i n DMT. The percentage increase of the maximal volume searched by a f l o a t i n g hydra over one which does not move i s as f o l l o w s : 100 4.26 + 10.36DMT + 798.31VY + 134.65VPF + 38.33* VY VPF 100 4.26 + 627.88VY 114 Simply by l o o k i n g at t h i s equation, one gets the impression that the e f f e c t of f l o a t i n g i s going t o be s u b s t a n t i a l . The constants modifying the VY values are much l a r g e r i n the numerator than i n the denominator. The d e r i v a t i v e of 114, with respect t o VY, has a negative slope. For values of VY which are not too l a r g e , the equation has a s u b s t a n t i a l slope. The r a t e of s t a b i l i z a t i o n i s about the same or s l i g h t l y f a s t e r than that of equation 113. As an example, l e t DMT= 30 mm and VPF= 100 mm/minute. These are modest estimates of the parameters as they occurred i n Beaver Creek. The slope of 114 f a l l s t o l e s s than 1 7>-minute/mm by VY= 47 mm/minute. I t i s noteworthy that the equation holds a r a t e s t a b i l i z e d value of +50?o. Enough i n f o r m a t i o n i s now a v a i l a b l e to draw some conclusions about 84 the r e l a t i o n s h i p between the maximum searching r a t e of some searching mode and i t s i n i t i a t i o n t h r e s h o l d . The minimal e f f e c t of walking on the volume searched i s s l i g h t l y negative. The minimal e f f e c t of f l o a t i n g i s e i t h e r no change (when the absolute value of (VY-VPF)^VY) or very negative (when VY=water velocity=VPF). On the other hand, the maximal e f f e c t of walking on the volume searched i s s l i g h t l y p o s i t i v e but the maximal e f f e c t of f l o a t i n g i s very p o s i t i v e . These r e s u l t s have been drawn from a d e t e r m i n i s t i c model. The conclusions also f o l l o w from a " s t o c h a s t i c " model; see Appendix I I I . I had expected the maximum rate of searching volume to be r e l a t e d to hunger i n a s i m i l a r manner as was hypothesized f o r the t r a n s i e n t responses. Although t h i s was so f o r the maximal e f f e c t data, i t was not so fo r the minimal e f f e c t data. The consequences are studied i n the d i s c u s s i o n . Simulation was used i n studying searching by hydra. Equations 126 to 131 were programmed; see Figure s 7 to 11 i n c l u s i v e . In a l l graphs the volume searched per hour by a hydra i s presented as a f u n c t i o n of time of food d e p r i v a t i o n . Volume searched i s i n the u n i t s of mm /hr. Each graph i s composed of three s e c t i o n s . Between 0 and 26 hours, equations 126 and 127 are p l o t t e d . These equations begin t h e i r r i s e from the minimum to the maximum value at about 6 hours a f t e r being pre-fed w i t h three Artemia n a u p l i i . T his i s so because those observations recorded i n Fi g u r e 4 show that a hydra pre-fed w i t h three n a u p l i i egests food-remains from 5 to 8 hours a f t e r feeding. I t i s impl i e d i n these f i g u r e s that at 26 hours, equations 126 and 127 are replaced by equations 128 and 129. This i s because walking i s considered a s u f f i c i e n t a d d i t i o n to the volume searched a c t i v i t y to be considered separately. One component of t h i s behaviour i s t e n t a c l e length. The placement of 128 and 129 i s r e l a t e d to the data of Figure 5. The dynamic responses of these 85 equations are shown as instantaneous. This c o n c l u s i o n formed part of the r e s u l t s of PART I. The maximum value of these equations does not show the v a r i a b i l i t y of the previous equations. This i s because the f i t t e d maximum value of t e n t a c l e length (Table I) was used, r a t h e r than the a c t u a l data as was done f o r equations 126 and 127. F i n a l l y , at hour 60, equations 130 and 131 began as 128 and 129 dropped out. The time of t r a n s i t i o n was according to data of Figure 6. The dynamic response was made instantaneous, again i n agreement w i t h data of PART I . In the various s i m u l a t i o n s , DMT and VY were v a r i e d . In Figure 7 the a c t u a l volume searched by a hydra (VY=0) i s portrayed. The d i f f e r e n c e between the minimal and the maximal e f f e c t s of any search mode i s s m a l l , r e l a t i v e to the d i f f e r e n c e s between modes. The parameter VPF took the small value of 0.50 mm/minute i n s t e a d of the value zero. This value was given i n order to avoid the problem of d i v i d i n g by zero i n the term VY/VPF of equations 130 and 131. A l a r g e r value of VPF would serve to increase the e f f e c t s of f l o a t i n g . DMT was given the small value of 2.0 mm. I f one were to describe t h i s graph w i t h a continuous f u n c t i o n , he might w e l l choose the exponential f u n c t i o n . The exponential f u n c t i o n has an exponential d e r i v a t i v e ; and one should note that the volume searched during the f l o a t i n g period i s l a r g e r than that during the w a l k i n g period by an amount much greater than the volume searched during the w a l k i n g period i s l a r g e r than that during the t e n t a c l e lengthening stage. In the next two f i g u r e s , searching by hydra i n a slow stream (Figure 8) and i n a lake (Figure 9) are shown. Prey are assumed to move slowly. In the stream, prey are allowed to move only as a f u n c t i o n of water v e l o c i t y , at 100 mm/minute. In the l a k e , prey movement i s set at 500 mm/minute. A feature of i n t e r e s t i s that even at the modest VY of 100, the e f f e c t of DMT i s masked; thus the large hump shown i n Figure 7 . A simulated r e s u l t of the searching r a t e of a hydra which has been pre-fed w i t h 3 Artemia n a u p l i i , as a f u n c t i o n of food depriv-a t i o n . VY=0.0 mm/minute, VPF=0.50 mm/minute, and DMT=2.0 mm. Because VY=0.0 the r e s u l t estimates the a c t u a l volume searched per hour, as a f u n c t i o n of time of food d e p r i v a t i o n . Minimum and maximum estimates of the volume searched f o r each search mode are given. These extremum estimates are based on the d i r e c t i o n of prey movement r e l a t i v e to the hydra. 3iS J . 3.0 1 9.E 4-B-C 7>5 7-0 .. S 5>0 LJ ' LU * 3 5.0 -L + O H • LJ 4<5 X 4<: fi 3-IjJ 3 . C 1 3 • > 1.5 1«C 0.5 4-H—H H—I—H H—(- 4-B 3 12 15 IS LiL 24 S7 730 733 :Ti Til 'S. 4S 43 ££L 24 57 BO 53 BB S3 TIME DEPRIVED DF FDDD (HDL'RS) Figure 8. A simulated r e s u l t of the searching r a t e of a hydra which has been pre-fed w i t h 3 Artemia n a u p l i i , as a f u n c t i o n of food depriv-a t i o n . VY=VPF=100 mm/minute, and DMT=300 mm. In a stream, prey movement may be p a s s i v e l y c o n t r o l l e d by water movement. F l o a t i n g by hydra i s e n t i r e l y the r e s u l t of water movement. Thus when VY=VPF, as here, the r e s u l t s estimate the searching ra t e of a stream d w e l l i n g hydra. Since the water moves at a v e l o c i t y of only 0.16 cm/sec, the stream i s a slow one. Minimum and maximum estimates of the volume searched f o r each search mode are given. These extremum estimates are based on the d i r e c t i o n of prey movement r e l a t i v e to the hydra. 87 3.5 ._ g.o ~ B . C . . + 7.5.. 3 B 3 12 IS OS 21 24 27 3D 7i3 H i 33 42 AB A3 51 54 57 EO 53 B5 B3 TIME DEPRIVED DF FOOD (HOURS) Figure 9. A simulated r e s u l t of the searching r a t e of a hydra which has been pre-fed w i t h 3 Artemia n a u p l i i , as a f u n c t i o n of food d e p r i v a t i o n . VY=500.0 mm/minute, VPF=100 mm/minute, and DMT=0.0 mm. In a lake s i t u a t i o n independent means of prey.locomotion may not be overridden by water v e l o c i t y . Thus when VY^VPF, as here, the r e s u l t s estimate the searching r a t e of a lake d w e l l i n g hydra. Minimum and maximum estimates of the volume searched f o r each search mode are given. These extremum estimates are based on the d i r e c t i o n of prey movement r e l a t i v e to the hydra. Figure 7 i s r e s t r i c t e d t o s i t u a t i o n s w i t h very small VY. The hump i s only b a r e l y v i s i b l e i n Figure 8. More importantly, these f i g u r e s show that with i n c r e a s i n g values of VY over zero the minimal e f f e c t s spread r a t h e r f a r from the maximal e f f e c t s f o r any searching mode. This d i f f e r e n c e tends to expand with time of food d e p r i v a t i o n . The volume searched drops t o the a c t u a l volume searched f o r minimal e f f e c t f l o a t i n g i n the stream s i t u a t i o n . The s i t u a t i o n f o r hydra searching i n a q u i c k l y flowing stream (Figure 10) and i n a lake (Figure 11) are shown. Prey are assumed to move q u i c k l y . Except f o r covering much more volume, the hydra search i n a s i m i l a r p a t t e r n . These r e s u l t s show that the exponential searching r a t e which was hypothesized f o r the a c t u a l volume searched graph (Figure 7) does not h o l d f o r the v a r i o u s e f f e c t i v e volume searched graphs (Figures 8 to 11). The primary searching mode, i n terms of e f f e c t i v e volume searched per time i n t e r v a l , i s t e n t a c l e lengthening. One might suggest, however, that the e f f e c t i v e searching r a t e i s i n i t i a l l y e xponential and then constant or s l i g h t l y i n c r e a s i n g . For reasons of s i m p l i c i t y , I w i l l consider only the exponential s e c t i o n of the searching f u n c t i o n . This assumption w i l l underestimate the maximal e f f e c t f l o a t i n g r a t e by. about 207o. On the other hand, f o r those s i t u a t i o n s where the e f f e c t i v e volume i s very c l o s e t o the a c t u a l volume, the exponential f u n c t i o n may be a p p l i e d over a l l three searching modes. Thus, we w r i t e a*t AVS = d VS = e 115 A t d t Figure 10. A simulated r e s u l t of the searching r a t e of a hydra which has been pre-fed with 3 Artemia n a u p l i i , as a f u n c t i o n of food d e p r i v a t i o n . VY=VPF=10, 000 mm/minute and DMT=150 mm. For reasons l i k e those given i n the legend of Figure 8, the s i t u a t i o n here i s f o r a stream d w e l l i n g hydra. As water v e l o c i t y i s about 17 cm/sec, the stream i s a f a s t one. Minimum and maximum estimates of the volume searched f o r each search mode are given. These extremum estimates are based on the d i r e c t i o n of prey movement r e l a t i v e to the hydra. 90 3-3 S- 5 3 5 3 12 IS IB 21 54 57 3D 33 :J5 33 -15 45 43 Til 54 B7 50 53 55 BS TIME DEPRIVED DF FDDD (HOURS) Figure 11. A simulated r e s u l t of the searching r a t e of a hydra which has been pre-fed w i t h 3 Artemia n a u p l i i , as a f u n c t i o n of food depriv-a t i o n . VY=10,000 mm/minute, VPF=100 mm/minute, and DMT=0.0 mm. For reasons l i k e those given i n the legend of Figure 9, the s i t u a t i o n here i s f o r a lake d w e l l i n g hydra. Prey move much more q u i c k l y i n t h i s f i g u r e (10,000 mm/minute) than i n Figure 9 (500 mm/minute). Minimum and maximum estimates of the volume searched f o r each search mode are given. These extremum estimates are based on the d i r e c t i o n of prey movement r e l a t i v e to the hydra. 3.5 ^ B . C . . 3 Fi 9 IH IS IB a 3 4 37 3D 3 3 : l i 3 3 4 3 4 5 43 ESL 5 4 57 5 0 B 3 BB 5 3 TIME DEPRIVED DF FDDD (HDUR55 The time which t e n t a c l e length takes to begin i t s r a p i d increase i s v a r i a b l e (Figure 4) . The time i s a f u n c t i o n of the amount of food pre-fed t o the hydra. Equation 115 can be made t o f i t a l l l e v e l s of pre-feeding by adding to the exponent a v a r i a b l e representing the amount of food pre-fed to the t e s t animals. T h i s v a r i a b l e would delay the f u n c t i o n without a f f e c t i n g i t s form when the f u n c t i o n i s f i n a l l y executed. I t i s assumed that when hydra are s a t i a t e d , the v a r i a b l e t o be added to the exponent equals zero. The value could be d i f f e r e n t from zero without a f f e c t i n g the subsequent argument. I t was chosen equal t o zero because t h i s value was simplest to work w i t h . Suppose a hydra gets a meal which i s s u f f i c i e n t only to r e t u r n the hunger of the animal t o some time of food d e p r i v a t i o n , TF1, but not t o s a t i a t i o n , time zero. The hydra does not search at r a t e exp (a*0) but at r a t e exp (a*TFl) . At some time l a t e r , say T, the animal w i l l be searching at the r a t e , exp (a*TFl + a*T) . This simply increments time by T u n i t s . The v a r i a b l e T i s a v a r i a b l e of time since l a s t meal, not of time since s a t i a t i o n . A f u n c t i o n of volume searched per u n i t time, from time of l a s t meal t o some time l a t e r , i s as f o l l o w s : By i n t e g r a t i n g 116 over I , from T=0 t o T=TS, we get an equation of the t o t a l volume searched by a predator from the time of l a s t meal to some time TS l a t e r . a*TFl a*T d VS = e d T e 116 a*TFl a*TS VS(TS) = _e e - 1 117 a I n t h i s equation VS(0)=0. The v a r i a b l e T F l i s w r i t t e n as a f u n c t i o n of hunger with the use of r e s u l t s from H o l l i n g (1966): HK T F l = l_ l n AD 118 HK-HO_ The parameter HO i s the hunger at time T F l . S u b s t i t u t i n g equation 118 f o r T F l i n 117, we get VS(TS) = a*TS e - 1 HK a/AD HK-HO From equation 12 (NA = VS*NO), i t f o l l o w s that TS = I l n a a/AD a*NA* (HK-HO) + 1 a/AD 119 NO*(HK) I f the terms f o r predator i n e f f i c i e n c y are inc l u d e d as a means of reducing prey d e n s i t y (op. c i t . ) , we have NO = NO*SR*SP*SS The parameter SR i s the success of prey r e c o g n i t i o n , SP i s the success of prey p u r s u i t , and SS i s the success of prey capture. When incorporated i n 119, we get a/AD TS = J_ l n a a*NA* (HK-HO) + 1 NO*SR*SP*SS*(HK) a/AD II10 93 DISCUSSION FOR PARTS I AND I I The data base i s the same f o r PARTS I and I I . The o b j e c t i v e s of PARTS I and I I are r e l a t e d i n some respects but are very d i f f e r e n t i n other r e s p e c t s . PART I attempts t o t e s t a method f o r w r i t i n g general f u n c t i o n s . PART I I attempts t o formulate an e x p l i c i t f u n c t i o n f o r time spent searching (TS) by hydra. A short review, of searching by animals w i l l open the d i s c u s s i o n . Emphasis w i l l be placed on the form of the volume searched f u n c t i o n . T h i s f u n c t i o n p l a y s a dominant r o l e i n determining the form of the f u n c t i o n f o r time spent searching (TS) . This review w i l l be followed by evidence concerning the dynamic response r e l a t i o n s h i p s of search-i n g modes f o r s e v e r a l s p e c i e s . A d i s c u s s i o n of o p t i m a l i t y and s t r a t e g models w i l l terminate t h i s s e c t i o n . I n PART I i t was mentioned that previous authors have used s t r a t e g i c hypotheses i n w r i t i n g general f u n c t i o n s . The work of these s c i e n t i s t s w i l l be reviewed i n the d i s c u s s i o n . Searching by Animals. I n t h i s t h e s i s , searching i n v a r i a b l y means searching through volume. Searching by changing l o c a t i o n and thereby changing prey d e n s i t y i s not considered. ( I t s omission i s not meant t o r e f l e c t on i t s importance r e l a t i v e t o searching through volume.) Searching and locomotion have of t e n been t r e a t e d as c l o s e l y r e l a t e d (De R u i t e r , 1967). T h i s r e l a t i o n s h i p was once elevated t o pedantic l e v e l s (Campbell et a l , 1966). The c l a s s i c a l n o t i o n was that locomobile a c t i v i t y i s d r i v e n by some d i s c o m f o r t i n g a c t i v i t y , c a l l e d d r i v e . Campbell ££. a_l (1966) compared the a c t i v i t y of a number of species as a f u n c t i o n of the time of food d e p r i v a t i o n . Various methods of measuring locomobile a c t i v i t y were used. Stabilmeter a c t i v i t y was measured. The r e s u l t s showed that c h i c k s increase t h e i r a c t i v i t y w i t h i n c r e a s i n g hunger; that hamsters move about at the same r a t e u n t i l 7 hours of food d e p r i v a t i o n and then move at a decreasing r a t e ; and that guinea p i g s and r a b b i t s show a c o n t i n u a l drop i n a c t i v i t y w i t h i n c r e a s i n g time of food d e p r i v a t i o n . However, when wheel a c t i v i t y was measured, i t was found that guinea p i g s and hamsters increased t h e i r a c t i v i t y f o r a whi l e (Figure 12) . The authors conclude that "the range and v a r i e t y of s t i m u l i which e l i c i t a c t i v i t y ... vary enormously from species t o species as a r e s u l t of e v o l u t i o n a r y devolopment" ( t h e i r page 127). T h i s c o n c l u s i o n i s important i n that i t warns against t a k i n g too narrow a view of searching. I t i s a l s o r e l e v a n t t o my t h e s i s as i t supports a major co n t e n t i o n : that t a c t i c a l approaches are u n l i k e l y t o y i e l d as much g e n e r a l i t y as s t r a t e g i c a l approaches. Pianka (1966) r e f e r s t o those animals which do not move when search i n g f o r prey as " s i t - a n d - w a i t " p r e d a t o r s . These types of animals have a l s o been r e f e r r e d t o as "ambush" predators ( H o l l i n g , 1966) . Examples such species are some hawks (Haverschmidt, 1962), l i z a r d s of the f a m i l y Iguanidae (Pianka, 1966), and dragonfly l a r v a e , mantids, c e r t a i n s p i d e r s , frogs, and chameleons ( P r i t c h a r d , 1965). Mantids e f f e c t i v e l y search through i n c r e a s i n g amounts of volume by in c r e a s i n g t h e i r d i s t a n c e of r e a c t i o n t o prey with i n c r e a s i n g hunger. I f the r e a c t i v e distance i s designated as r , then the area searched 2 by the mantid i s 0.8064 r Q ( H o l l i n g , 1966). The area searched as a fu n c t i o n of time of food d e p r i v a t i o n f o l l o w s from a d e s c r i p t i o n of Figure 12. Median a c t i v i t y of c h i c k s , r a b b i t s , guinea p i g s , and hamsters as a f u n c t i o n of days of d e p r i v a t i o n (from Campbell et al_ 19 66) . '96-> » FOOD DEPRIVED Ss »• - • WATER DEPRIVED Ss > 8 SATIATED S J Chicks Rabbits I 3 5 ; 9 II 13 Guinea Pigs 300 7 9 It 13 A 1 3 5 7 9 11 13 15 17 Hamsters 11 .3 15 17 3000 15000 1 o o S E >— 2500 1 A G U i n e a P i « S 12500 Hansters 2000 } / 1 10000 O S JL \ '7500 1500 1000 1 5000 • \ k i EEL A 500 - / 2500 \ \ * 0 . ~* i i • A 1 3 5 7 9 11 13 A 1 3 5 7 9 11 13 15 17 DAYS OF DEPRIVATION r as a function of time of food deprivation (TF). The area searched o then becomes 0.8064* GM2 [-HT+HK(l-exp (-AD*TF) )] 2 for hunger greater than the threshold HT (op. c i t . ) . For hunger less than HT, the area searched is zero. The form of the area searched function, from satiation to some time of food deprivation, might be suitably described by an 2 2 exponential function and then a linear function (= 0.8064* GM (HK-HT) ). This means that searching by mantids might be adequately described for some purposes by equation 115. The attempt to f i t the mantid data to equation 115 points to a shortcoming of the strategic method of writing general functions. The method does not incorporate a specific and testable explanation. Arguments are devised to predict certain characteristics of a relation-ship . However, in the case of the volume searched strategy the characteristics were not sufficient to permit choice among a number of descriptive equations. The form of a relationship can be pre-determined from strategic arguments. Realism of the parameters cannot be attained. This means that when the predicted equation is tested for a number of species, i t must be treated almost as an empirical equation. Data useful to our purpose on the searching biology of other s i t -and-wait predators is not available. Haverschmidt (1962), in describing some natural history of the Grey Hawk (Butea nitidus), notes that i t "hunts lizards mostly as a ' s t i l l hunter' from a perch". He also refers to the Laughing Falcon (Herpetotheres cachinnans) as a " s t i l l hunter". Kennedy (1956) notes a few observations made regarding the Rusty Lizard, Sceloporus olivaceus. In the morning and evening the lizard climbs onto a tree trunk and looks down, chasing after prey when they happen by. Pianka (19 66) notes that the Iguanid lizard, Callisaurus, 98 stands up on i t s f o r e l e g s , thereby i n c r e a s i n g "the area covered and maximizing the e f f i c i e n c y of u t i l i z a t i o n of the open spaces". Those animals which do move when searching f o r food might be r e f e r r e d to as a c t i v e p r e d a t o r s . D r o s o p h i l i a melanogaster, an a c t i v e searcher, has been studied by Haynes and S i s o j e v i c (1966) . They starved f l i e s from b i r t h through t o death and measured an index of the a c t u a l volume swept by the f l i e s by r ecording the number of contacts with an a r t i f i c i a l s pider ( t h e i r Figure 20). The data followed a monotonically i n c r e a s i n g f u n c t i o n which rose at an i n c r e a s i n g r a t e with time: at 0 hours of food d e p r i v a t i o n the number of contacts was l e s s than one; at 24 hours, equal t o one; at 48 hours, between one and two; at 70 hours, about 13; and at 72 hours, about 23. Death occurred at about 72 hours. Another d i p t e r a n , Phormia, has been e x t e n s i v e l y studied from a viewpoint relevant t o my studies (see Dethier, 1969; Green, 1964). Dethier (1969; page 254) c i t e s e a r l i e r work by Green to the e f f e c t that under c o n d i t i o n s of constant l i g h t and of s t a r v a t i o n , an emerging f l y i s more or l e s s quiescent during the f i r s t 72 hours. "At 72 hours a c t i v i t y begins i n earnest, increases t o a maximum at 84-94 hours, then begins t o wane, foreshadowing death." The r e s u l t s are s i m i l a r t o those obtained f o r D r o s o p h i l i a , except that Phormia do not d i e at the maximum r a t e of searching. My model of time spent searching (TS) i s l i k e l y t o be a p p l i c a b l e t o Phormia only during the f i r s t 80% of the s t a r v a t i o n response. A second searching mode used by Phormia i n v o l v e s the acceptance t h r e s h o l d of the f l y to v a r i o u s concentrations of sugar s o l u t i o n s (Dethier, 1969) . Lowering of the acceptance t h r e s h o l d of the feeding response i s i n e f f e c t i n c r e a s i n g the r e a c t i v e d i s t a n c e ; thus, i t i s analogous t o i n c r e a s i n g the v i s u a l d i s t ance of the mantid. Evans and Dethier (see Figures 24-27 of Dethier, 19 69) have measured changes i n glucose and fucose (6-desoxygalactose) thresholds as fu n c t i o n s of time of food d e p r i v a t i o n , f o l l o w i n g s a t i a t i o n on glucose (Figure 24), on fucose (Figure 25), on mannose (Figure 26), and on l a c t o s e (Figure 27). I n the f i r s t t e s t a l i n e a r decrease over time was found; i n the second and four t h t e s t s approximate negative exponentials were found; and i n the t h i r d t e s t a t h r e s h o l d that kept i t s value f o r a time and then f e l l q u i t e r a p i d l y was found. I n a l l cases the th r e s h o l d f e l l with time of food d e p r i v a t i o n . I t i s assumed that the acceptance t h r e s h o l d i s i n v e r s e l y r e l a t e d to the distance of r e a c t i o n . The volume searched i s a cubic f u n c t i o n of the r e a c t i v e d i s t a n c e . Thus we might expect that the volume searched i s an a c c e l e r a t i n g f u n c t i o n of time of food d e p r i v a t i o n . The t h r e s h o l d of i n i t i a t i o n f o r the f i r s t searching mode used by Phormia depends upon the type of food given t o the f l y . I t i s zero hours f o r the change i n glucose acceptance t h r e s h o l d when the f l y i s pre-fed w i t h glucose and 10-20 hours f o r the fucose t h r e s h o l d when the f l y i s pre-fed w i t h fucose. The i n i t i a t i o n t h r e s h o l d f o r the locomobile searching mode i s about 72 hours, when the f l i e s are fed to r e p l e t i o n on 1M sucrose. The acceptance t h r e s h o l d v a r i e s l e s s than one order of magnitude with hunger, whereas a c t i v i t y v a r i e s up t o three orders of magnitude with hunger. I t appears that the major searching component i s locomotion The o v e r a l l searching r a t e f u n c t i o n might w e l l be described by equation 115. However, t h i s b r i e f c o n s i d e r a t i o n of searching by Phormia by no means proves t h i s t o be the case. 100 A few pages back, I noted my expectation that the steady s t a t e searching r a t e be r e l a t e d to hunger i n a manner s i m i l a r to that hypothesized f o r the dynamic responses. This expectation was not v e r i f i e d i n the case of the minimal e f f e c t data f o r hydra. One explanation which may account f o r t h i s discrepancy f o l l o w s from the observation that hydra search not only be covering volume but a l s o by changing l o c a t i o n s . By changing l o c a t i o n s , the hydra may increase the d e n s i t y of prey organisms which they contact. Walking and f l o a t i n g may serve t h i s r o l e i n a d d i t i o n to that of searching through volume. I f two s t r a t e g i e s are i n t e r a c t i n g , then t h i s f a c t suggests the need to study more than one s t r a t e g y at a time. We now t u r n to the question of the dynamic response r e l a t i o n s h i p . I d e a l l y , i t should be checked f o r a wide v a r i e t y of species other than hydra. U n f o r t u n a t e l y , although much has been w r i t t e n about searching i n a v a r i e t y of species, i t i s u s u a l l y not p o s s i b l e to c a l c u l a t e the d u r a t i o n of the dynamic responses. One exception i s the work on the b l o w f l y by D e t h i e r and co-workers ( D e t h i e r , 1969). They found that the d u r a t i o n of the dynamic response of the glucose t h r e s h o l d , f o l l o w i n g i n g e s t i o n of 2.0 M glucose, i s about 60 hours. They found that the d u r a t i o n of the dynamic response of the fucose t h r e s h o l d , f o l l o w i n g i n g e s t i o n of 1.0 M fucose, i s about 25 hours. They tested a number of combinations of sugars - eg., pre-fed mannose, te s t e d fucose t h r e s h o l d - and got a number of responses. D e t h i e r c i t e s a paper by Green (1964) i n which i s given a f i g u r e of the a c t i v i t y of Phormia a f t e r being fed v a r i o u s amounts of sucrose. The d u r a t i o n of the dynamic response of a c t i v i t y as a f u n c t i o n of time of food d e p r i v a t i o n i s 40 to 50 hours. D e t h i e r notes i n h i s paper that the threshold search, mode i s i n i t i a t e d at lower hungers than the a c t i v i t y mode. Comparison of the glucose threshold to the sucrose a c t i v i t y data shows 101 that the dynamic . response of the former i s 60 hours long and that the dynamic response of the l a t t e r i s 40 t o 50 hours long . T h i s i s as p r e d i c t e d . Comparison of the fucose t h r e s h o l d t o the sucrose a c t i v i t y data shows th a t the dynamic response of the former i s 25 hours long and that the dynamic response of the l a t t e r i s 40 t o 50 hours long. This i s not as p r e d i c t e d . I t i s unfortunate that the same sugars were not used i n conducting both the t h r e s h o l d and the a c t i v i t y t e s t s . S t r a t e g i c a l Models. I t was noted i n the s e c t i o n r e f e r r e d t o as THE PROPOSAL that previous attempts have been made at using s t r a t e g i c a l , or r e l a t e d , concepts i n model b u i l d i n g . Space p r o h i b i t e d t h e i r d i s c u s s i o n i n that s e c t i o n . The s t r a t e g i c a l approach i s inherent i n the philosophy of many b i o l o g i s t s : that nature i s purposive. This view has been used i n asking the question: i f some h y p o t h e t i c a l behaviour can be defined -and an example of i t found i n nature - can statements be made about i t which w i l l apply t o the animals which use t h i s behaviour? Such b i o l o g i s t s as Rashevsky (1961) and Rosen (1967) s t r e s s the importance of o p t i m a l i t y p r i n c i p l e s i n b i o l o g i c a l systems. S t r a t e g i c a l models as proposed i n t h i s t h e s i s express ideas that have been p r e v i o u s l y pointed out by these authors. Rosen (1967) w r i t e s that " i n order to f i n d an optimal s o l u t i o n t o a p a r t i c u l a r design problem, three t h i n g s ... are r e q u i r e d : (a) t o determine the c l a s s of a l l p o s s i b l e s o l u t i o n s to the problem, (b) t o a s s i g n to each such s o l u t i o n a c e r t a i n number, which represents the cost i t i n v o l v e s , and (c) t o search among the set of costs t o f i n d that which i s l e a s t " . Although Rosen does w r i t e o p t i m a l i t y models, he does not de a l with s t r a t e g i c models. The d i f f e r e n c e between these two approaches has t o do w i t h the way i n which the problem i s formulated. Rosen uses a t a c t i c a l f o r m u l a t i o n . He describes a problem i n a mechanistic way and then a p p l i e s o p t i m a l i t y p r i n c i p l e s t o t h i s . d e s c r i p t i o n . Bioengineers have used o p t i m i z a t i o n technologies i n the biomedical area. Experimental observation has been shown t o r e l a t e w e l l t o theoret i c a l p r e d i c t i o n s based on the m i n i m i z a t i o n of performance c r i t e r i a based on energy (power consumption) or on f o r c e ( t o t a l muscular e f f o r t ; average i n s p i r a t o r y f o r c e ) . Some examples are given i n Milsum's (1966) book. The examples are a l l at the p h y s i o l o g i c a l l e v e l of b i o l o g i c a l complexity. To my knowledge, one of the e a r l i e s t attempts at w r i t i n g not only o p t i m a l i t y , but s t r a t e g i c models of e c o l o g i c a l systems i s by MacArthur and Pianka (1966). This c o i n c i d e d w i t h a paper by Emlen (1966) on the same s u b j e c t . These models are c a l l e d s t r a t e g i c because each of them de f i n e s a goal or s t r a t e g y f o r the animal; both papers optimize time and energy budgets f o r a predator feeding on a range of prey t y p e s . MacArthur and Pianka consider only time, i n terms of searching and^pursu in g time of prey i n a f i n e - g r a i n e d environment and i n terms of hunting and t r a v e l l i n g time i n a coarse-grained environment. Emlen proposed an a b s t r a c t model which suggests that a predator w i l l continue t o eat i n c r e a s i n g types of prey so long as the energetic r e t u r n s are p o s i t i v e . Thus, the o r i g i n a l approaches t o s t r a t e g i c models t r i e d t o p a r t i t i o n prey types to a predator feeding o p t i m a l l y i n terms of energy and time v a r i a b l e s . T h i s approach has been continued up t o and i n c l u d i n g a recent work: Schoener (1971). I n h i s review of s t r a t e g i c models of f e e d i n g , Schoener expands the range of optimal models from the previous area of optimal d i e t t o three other areas - optimal f o r a g i n g space, optimal 103 feeding p e r i o d , and optimal foraging-group s i z e . The s t r a t e g i c models reviewed by Schoener present a number of p r e d i c t i o n s i n areas of e c o l o g i c a l concern. Since Schoener 1s (1971) review i s q u i t e up-to-date, we w i l l work from h i s a r t i c l e . The most w e l l thought out area i s that of optimal d i e t , w i t h some s t r a t e g i c work completed i n the area of optimal f o r a g i n g space. No formal theory has been worked out fo r optimal p l a c i n g of feeding periods over the a c t i v i t y c y c l e and only q u a l i t a t i v e hypotheses e x i s t f o r optimal foraging-group s i z e . W i t h i n the area of optimal d i e t , p r e d i c t i o n s c l u s t e r i n t o c a t e g o r i e s of "range of items eaten" by an animal; "food s i z e and distance from the predator"; and "optimal kinds of feeders". W i t h i n the area of optimal f o r a g i n g space, p r e d i c t i o n s c l u s t e r i n t o c a t e g o r i e s of "home range s i z e " , "patch u t i l i z -a t i o n " , and "home range overlap and defense". We di s c u s s the major deductions i n each of these c a t e g o r i e s : 1) Range of items eaten - Models by MacArthur and Pianka (1966), Emlen (1966), Schoener (1971), and others a l l p r e d i c t that the lower the absolute abundance of food, the greater the range of items (prey types) taken. Schoener (1971) reviews supporting evidence found i n f i s h , weaverbirds, b l a c k b i r d s , s w i f t s , and Conus molluscs. MacArthur and Pianka (1966) suggest that i f some kinds of items are reduced d i f f e r e n t i a l l y , then the range of items taken e i t h e r increases or remains the same. Using h i s model, Schoener reaches the same conclu-s i o n . Oriens and Horn (see Schoener, 1971) have some i n d i r e c t evidence f o r t h i s contention. F i n a l l y , Schoener 1s model p r e d i c t s that an increase i n the energy requirements of an animal has the same e f f e c t as a decrease i n food d e n s i t y on s e l e c t i v i t y . Schoener notes that b i r d s s e l e c t l a rge prey from the h a b i t a t to feed young, and that the number of small prey brought 104 to the nest may increase w i t h brood s i z e . Hunger i s proposed as the short term mechanism whereby animals monitor small temporal p e r t u r b a t i o n s i n food a v a i l a b i l i t y . 2) Food s i z e and distance from the predator - Schoener's models p r e d i c t that "Food s i z e s should decrease w i t h decreasing predator s i z e , but a s y m p t o t i c a l l y " ; that "A predator at a given d i s t a n c e should take prey both l a r g e r and smaller than any at a greater d i s t a n c e , but the large s i z e l i m i t should d e c l i n e l e s s w i t h i n c r e a s i n g distance than the small s i z e l i m i t should i n c r e a s e " ; and that " D i s t r i b u t i o n s of prey s i z e s eaten by a predator from a uniform size-abundance d i s t r i b u t i o n of a v a i l a b l e prey should be more n e g a t i v e l y skewed ... i f the predator pursues i t s prey over greater d i s t a n c e s or i s r e l a t i v e l y l a r ge than i f the predator pursues i t s prey l e s s or i s r e l a t i v e l y s m a l l " . Schoener presents some supporting evidence. 3) Optimal kinds of feeding - D i s c u s s i o n i s l i m i t e d to the r e l a t i v e m e r i ts of s p e c i a l i s t versus g e n e r a l i s t feeders, and of large versus small animals. P r e d i c t i o n s are not e x p l i c i t l y s tated. 4) Home range s i z e - I t i s pr e d i c t e d t h a t , i f "home range s i z e i s p r o p o r t i o n a l to maximum di s t a n c e t r a v e l l e d f o r food f o r a vantage area", then " r e l a t i v e l y e f f i c i e n t pursuers should have l a r g e r home ranges than others of the same s i z e " . The suggestion i s made that t h i s may be why mammals have smaller home ranges than s i m i l a r l y s i z e b i r d s . 5) Patch u t i l i z a t i o n - In t h i s s e c t i o n a "compression hypothesis" has a r i s e n : "when competitors d i f f e r e n t i a l l y reduce food d e n s i t y , the range of patches u t i l i z e d should s h r i n k because some patches are then worse than o t h e r s , but range of food types w i t h i n patches should not decrease and i n fa c t may expand". 6) Home range overlap and defence - I f i n v a s i o n r a t e i s p r o p o r t i o n a l 105 only to the d e n s i t y of the invaders, then i f i t i s economical to defend a t e r r i t o r y at a given food d e n s i t y , i t i s a l s o so at a l l lower food d e n s i t i e s . I f i n v a s i o n d e c l i n e s to a l e v e l p r o p o r t i o n a l to food d e n s i t y , then an i n i t i a l lowering of food d e n s i t y w i l l favor switching from a l a c k of defense to defense, w h i l e a f u r t h e r lowering w i l l r e s u l t i n switching back to no defense. A l l of the p r e d i c t i o n s describe c e r t a i n consequences of an i n d i v i d u a l organism, w h i l e l o o k i n g f o r food and w h i l e defending i t s t e r r i t o r y . The deductions are r e a d i l y i n t e r p r e t e d i n terms amenable to f i e l d obser-v a t i o n . They are i n spheres of e c o l o g i c a l concern - predation, home range. Thus these examples, u n l i k e those i n Milsum's (1966) book, are at the e c o l o g i c a l l e v e l of b i o l o g i c a l complexity. The s t r a t e g i c approach used by previous authors has concentrated h e a v i l y on energy and time, on c o s t - b e n e f i t types of economic analyses. The approach tends u l t i m a t e l y to be c h a r a c t e r i z e d i n a f u n c t i o n of cost per u n i t e f f o r t to achieve some goal. Yet " c o s t " i s a m u l t i - f a c e t e d concept and t h i s feature has lead to d i f f i c u l t i e s . I t i s necessary i n Schoener's (1971) model to determine the energetic costs of p u r s u i t , and of handling and e a t i n g the prey. I t i s a l s o necessary to know the number of c a l o r i e s that the animal can e x t r a c t from a food item a f t e r having eaten i t . These are d i f f i c u l t determinations, and some rather s u b s t a n t i a l s i m p l i f y i n g assumptions (op. c i t . ) have been made i n an attempt to express these q u a n t i t i e s . This t h e s i s suggests that organisms not only act e f f i c i e n t l y i n the form of an i n t e g r a t e d u n i t , but that a l l b i o l o g i c a l processes are, more or l e s s , well-adapted. I t i s suggested that s t r a t e g i e s can be defined at various l e v e l s . Low-level s t r a t e g i e s are suggested as one way to avoid the nebulous formulations of previous s t r a t e g i c models. 106 Thus, even though searching i s a component of the energy/time performance c r i t e r i o n of Schoener, i t i s considered i n t h i s t h e s i s as an independent u n i t . I t i s suggested that searching " f i t n e s s " can be considered indepen-d e n t l y of the o v e r a l l f i t n e s s of the animal. I t was suggested i n the i n t r o d u c t i o n to PART I that i f one could guess the goal of some p a r t i c u l a r t a c t i c ( s ) and l i s t some of the major forces which have shaped i t s developnent, then v a r i o u s types of behaviour could be imagined. Some types would be more e f f i c i e n t i n reaching the goal than others. The assumption was made that i f the goal as defined were the only one which the species had to deal w i t h , then the most e f f i c i e n t type of behaviour would be the one used. I t was f u r t h e r assumed that the d i f f e r e n c e between predicted behaviour and observed behaviour r e l a t e s to the degree t o which the species has had to compromise o p t i m a l i t y i n one area f o r increased e f f i c i e n c y i n a d i f f e r e n t one. A s p e c i f i c example was taken and the method was t r i e d . By not c o n t r a d i c t i n g the p r e d i c t i o n , t h i s t e s t gave some assurance that compromise w i l l not cloud the simple p r e d i c t i o n s made on the b a s i s of a s i n g l e strategy. In review, we have shown how s t r a t e g i c hypotheses arose from optim-a l i t y c o n s i d e r a t i o n s . We have noted that s t r a t e g i c models have concen-t r a t e d at the l e v e l of the i n d i v i d u a l . I t has been suggested that one need not work only at t h i s l e v e l . Now we t a l k b r i e f l y about some i m p l i c -a t i o n s of t h i s approach. The n o t i o n behind my work i s that e c o l o g i c a l processes might be p r o f i t a b l y studied by d e a l i n g w i t h t h e i r p a r t s . This i s , of course, a view o f t e n expressed i n experimental components a n a l y s i s . By showing that s t r a t e g i c models e x i s t and (may) have some v a l i d i t y at a number of b i o l o g i c a l l e v e l s , t h i s t h e s i s has provided another way of formulating general f u n c t i o n s f o r use i n a model b u i l t under the d i c t a t e s of experim-e n t a l components a n a l y s i s . This approach works at various l e v e l s of b i o l o g i c a l complexity. I t s aim i s to provide a d e t a i l e d t h e o r e t i c a l view of predation, u s e f u l at the e c o l o g i c a l l e v e l . Yet the approach delves r e a d i l y to the p h y s i o l o g i c a l l e v e l . Hunger ( p h y s i o l o g i c a l l e v e l and prey d e n s i t y (population l e v e l ) are both v a r i a b l e s i n the same mode ( H o l l i n g , 1966). For the s t r a t e g i c approach to be u s e f u l i n w r i t i n g f u n c t i o n s f o r experimental components a n a l y s i s , i t must be u s e f u l f o r the hunger v a r i a b l e as w e l l as the prey d e n s i t y v a r i a b l e . Thus the i m p l i c a t i o n s of my work are c l o s e l y bound up i n those of experimental components a n a l y s i s . PART I I I SENSITIVITY ANALYSIS OF A MODEL OF ATTACK-INTRODUCTION In t h i s p a r t , an attempt i s made t o deal with the problem of complexity. I n the GENERAL INTRODUCTION i t was noted that experimental components a n a l y s i s produces such l a r g e and i n t r i c a t e models that complexity of the model becomes a problem i n i t s own r i g h t . The technique t o be proposed t o deal with t h i s problem i s most e a s i l y a p p l i e d i f a l l f u n c t i o n s i n the s i m u l a t i o n model are e x p l i c i t ; and PART I I was aimed at making the time spent searching f u n c t i o n (TS) e x p l i c i t . A f r e q u e n t l y used method of l e a r n i n g about the p r o p e r t i e s of a s i m u l a t i o n model i s by doing a " s e n s i t i v i t y a n a l y s i s " . A model i s simply run i n enough d i f f e r e n t ways u n t i l an i n t u i t i v e understanding of the model i s obtained. A group of computer based search techniques i s a v a i l a b l e t o speed up the process (Watt, 1968). This approach i s time consuming. I t leads t o d i f f i c u l t i e s i n comparing one s i m u l a t i o n model t o another s i m u l a t i o n model. T h i s s e c t i o n w i l l address i t s e l f t o f i n d i n g an a n a l y t i c a l way of studying the s e n s i t i v i t y of a complex flow graph. The attempt w i l l r e l a t e q u i t e s p e c i f i c a l l y t o the model of att a c k as given i n Appendix IV. Only a b r i e f d i s c u s s i o n of the breadth of a p p l i c a b i l i t y of the method w i l l be attempted. The Approach. Engineers have d e a l t w i t h computer s i m u l a t i o n of complex systems longer than b i o l o g i s t s . A survey was conducted of t h e i r techniques as they r e l a t e t o the problem stu d i e d here. I t was found that a b a s i c approach used by c o n t r o l systems engineers i s r e d u c t i o n of the block diagram t o equation form (Watkins, 1969). The equations are u s u a l l y d i f f e r e n t i a l or d i f f e r e n c e equations. There i s now a la r g e body of theory r e l a t i n g t o these forms of equations. Consequent-l y , engineers have been able t o f i n d common ground between many d i v e r s e types of s i m u l a t i o n models. Block diagram manipulation techniques are l i n e a r techniques. Engineers o f t e n manufacture t h e i r p a r t s t o f i t l i n e a r models. B i o l o g i s t s are not so f o r t u n a t e . A great deal of n o n - l i n e a r i t y occurs i n b i o l o g i c a l systems. Some n o n - l i n e a r i t i e s can be d e a l t w i t h i n such a way as t o make block diagram manipulation techniques a p p l i c a b l e . I t i s p o s s i b l e t o l i n e a r i z e around s i n g u l a r p o i n t s (Watkins, 1969) . I t i s a l s o p o s s i b l e t o deal i n c e r t a i n ways with flow graphs which are l i n e a r i n a l l but a few elements (Chua, 19 69; K a i n , 1962; and others) . Unfortunately, block diagrams f o r b i o l o g i c a l systems tend t o be mostly n o n - l i n e a r , with a few l i n e a r elements. More than t h i s , b i o l o g i c a l models i n c o r p o r a t e an a d d i t i o n a l k i n d of problem. This problem has been c a l l e d a " t h r e s h o l d e f f e c t " ( H o l l i n g , 1965). Our f u n c t i o n s are, at best, "piecewise continuous". Engineers have d e a l t w i t h an analogous s i t u a t i o n by d e a l i n g w i t h switches. They have developed c e r t a i n t r i c k s , but not much theory, t o deal with switches. The language used i s Boolean Algebra (see Wickes, 19 68; Hu, 1968). I assume that i t would be d e s i r a b l e t o reduce b i o l o g i c a l flow graphs t o equation form. The model of a t t a c k given i n Appendix IV i s being considered i n t h i s PART. I t i s assumed that i t would be d e s i r a b l e t o w r i t e an e x p l i c i t equation f o r the a t t a c k r a t e (A),, 1 lo-om whose p r e d i c t i o n s are i d e n t i c a l t o those from the s i m u l a t i o n model. There are 10 c o n d i t i o n a l statements i n the flow graph of the a t t a c k model. I f each combination of these c o n d i t i o n a l statements were t o produce a d i f f e r e n t a t t a c k r a t e f u n c t i o n , then there would be 2 ^ d i f f e r e n t f u n c t i o n s . T h i s assumes no feedback e x i s t s i n the model and that each c o n d i t i o n a l statement has only two outflowing paths. (Such a c o n d i t i o n a l statement would be an I F (a) ni,n2,n.j , where any two n^ took the same value.) There i s a great deal of feedback i n the a t t a c k model. Thus the p o s s i b i l i t y i s very r e a l of f a r more than 2 ^ d i f f e r e n t f u n c t i o n s . Each combination of c o n d i t i o n a l statements does not produce a d i f f e r e n t a t t a c k r a t e f u n c t i o n . The number i s s t i l l l i k e l y t o be l a r g e . Yet almost a l l a t t a c k r a t e f u n c t i o n s r e s u l t from changes i n the i n t e n s i t y of the independent v a r i a b l e , hunger. These changes do not a f f e c t the parameters or the s t r u c t u r e of the f u n c t i o n . Only those changes which r e s u l t i n s t r u c t u r a l m o d i f i c a t i o n i n a f u n c t i o n w i l l be considered. A l l f u n c t i o n s which are s t r u c t u r a l l y d i f f e r e n t w i l l be t a b u l a t e d . From such t a b l e s i t should be p o s s i b l e to e x t r a c t some knowledge about the s e n s i t i v i t y of the s i m u l a t i o n model. An attempt was made t o go through a l l the paths of the block diagram, w r i t i n g the v a r i o u s forms of the a t t a c k r a t e f u n c t i o n as they arose. I n v a r i a b l y , I became confused. I t was d i f f i c u l t t o determine when a l l the paths had been covered. Thus, an expression was developed which denoted a l l the p o t e n t i a l paths through which flow can move through the block diagram. Then i t became a simple job t o go through a l l the paths, w r i t i n g the v a r i o u s forms of the a t t a c k r a t e f u n c t i o n as they arose. TECHNIQUES FOR WHITING LOGICAL EQUATIONS Consider a block diagram, such as the one i n Appendix IV, only i n terms of i t s c o n d i t i o n a l statements. Then a resemblance to the c l a s s i c a l l i n e a r flow graph can be seen. Operators, nodes, and paths are a l l i d e n t i f i a b l e . The operators are not constants, but are i n e q u a l i t i e s . Each c o n d i t i o n a l statement has been decomposed i n t o a number of i n e q u a l i t i e s , corresponding to the number of paths flowing from i t . We l e t each i n e q u a l i t y be viewed as a sw i t c h . When the i n e q u a l i t y corresponds to the a c t u a l s i t u a t i o n , the switch i s considered to be c l o s e d . When i t does not, the switch i s considered t o be open. Using Boolean Algebra, we may designate a closed switch by value one, and an open switch by value zero. I n t h i s way the operators of the " l o g i c a l flow graph" become constants. For example, i n Appendix I v there i s a c o n d i t i o n a l statement, reproduced i n Figure 1 3 . This statement can be decomposed i n t o two separate i n e q u a l i t i e s : T P > 0 and T P ^ . 0 . When these i n e q u a l i t i e s are represented as switch-operators, the s i t u a t i o n may be redrawn as i n Figure 1 4 . The s t a t e TP <C 0 has been designated by the switch-operator A. The s t a t e TP >0 has been designated by i t s complement f u n c t i o n , A. (Technical terms are used as explained i n Wickes (1968) and Zehna and Johnson (19 62).) The l o g i c a l flow graph of Figure 14 i s d e s c r i b a b l e by the expression, A + A. I t describes the t o t a l i t y of paths through which flow may occur through t o the a-node. The expression should be read as f o l l o w s : f o r flow to reach the a-node, e i t h e r the path through A or through A must be used. The "or" r e f e r r e d to i s the Boolean-OR. We w i l l a l s o have occasion to r e f e r Figure 13. A FORTRAN I F ... statement from the flow graph of Appendix IV . Figure 14. An equivalent form f o r the c o n d i t i o n a l statement of Figure 13. Figure 15. A l o g i c a l flow graph, with operators A and B. 113 to the Boolean-AND. The r e s t of t h i s s e c t i o n w i l l be used t o introduce enough c o m p l e x i t i e s to a l l o w manipulation of the flow graph of Appendix IV. Consider the case drawn i n Figure 15. This case i s l i k e that of Figure 14, except that there are a couple of repeated u n i t s . Flow through t o the a-node i s d e s c r i b a b l e by the expression (A + A)•(B + B), where the elevated dot i s one symbol f o r the Boolean-AND. Often no AND symbol w i l l be used; t h i s i s equivalent to the way which m u l t i p l i c a t i o n symbols are o f t e n omitted. The form of t h i s expression i s c a l l e d the "standard product form" (Wickes, 1968). Another form of t h i s expression i s AB + AB + AB + AB. I t i s c a l l e d the "standard sum form". The meaning of an expression i s o f t e n more v i s i b l e from the standard sum form. The above expression s t a t e s c l e a r l y that flow may get t o the a-node v i a the AB path, the AB path, the A3 path, or the AB path. Because much use i s made of complement f u n c t i o n s i n t h i s expression, i t i s p o s s i b l e f o r flow to f o l l o w only one of these paths at a time. The word "equation" has been f a s t i d i o u s l y avoided i n the above d i s c u s s i o n . Boolean algebra has been used t o define expressions, not equations. I t i s not my purpose to enumerate these expressions. They are t o serve only as a short-hand d e s c r i p t i o n of the t o t a l i t y of. paths through the model. The expressions should be t r e a t e d more as sentences than as equations, more as symbolic language than as mathematics. Case 2 introduces feedback. I t i s drawn i n Figure 16. This flow graph, t o the a-node, i s d e s c r i b a b l e by the expression, QDC. The path QDC i s not r e a l i z e d because i f flow ever entered the C path, i t would never emerge t o the a-node. I n t h i s Figure Figure 16. A l o g i c a l flow graph, with operators Q, D, and C. Figure 17. A l o g i c a l flow graph, with operators Q, D, C, and E. the s t a t e of a switch-operator i s not allowed to change. Now suppose that operators D and C may change t h e i r s t a t e , as a f u n c t i o n of the number of times that flow passes around the feedback loop. Let the v a r i a b l e of i t e r a t i o n , or counter, be designated as i . Then we i n d i c a t e that C and D are fu n c t i o n s of i by w r i t i n g and . The set of i over which switch-operator C i s closed w i l l be c a l l e d the domain of C. A l i k e d e f i n i t i o n w i l l be used f o r . We now consider the l o g i c a l flow paths f o r the times when C i s c losed f o r i = l , 2 , 3 , ... . When C i s closed f o r i = l , the appropriate expression i s Q1\=^C^_^. This statement may be read as: flow enters the graph and passes through the Q operator, the D operator f o r i = l , and the C operator f o r 1=1 t o get t o the a-node. When C i s open f o r i = l (and so C i s closed f o r i = l ) and C i s closed "2 1_ f o r i=2, the appropriate expression i s Q T T , D. (C. ,. C. „. Boolean-i±=i y i = l i=2 AND's are i m p l i e d between each term. The term f o r D. introduces a new symbol, TT. I t i s used to AND a number of D^'s. This term would be equivalent t o w r i t i n g D. „D. „. Much use w i l l be made of t h i s i = l i=2 symbol i n the next few pages. Nevertheless, the above expression reads: flow enters the system, passing through a closed Q operator-switch, a closed D operator-switch, and a closed C operator during i = l . As flow re-entered the foreward path from the feedback loop, i was incremented t o value 2. Flow then passed through a closed D operator and a closed C operator to reach the a-node. When C i s open f o r i = l , 2 , 3 , ... but closed f o r i = i l , the appropriate expression i s Q n o n s e n s i c a l ; when t h i s i s the case, we de f i n e C t o be c l o s e d . Let the b-node of Figure 16 flow i n t o the path of the a-node at a l e v e l beyond the C operator. Where C and D are not fu n c t i o n s o f i , ' i l f i i - i i . £ i D i T T - c i = l i C. . I f i l = l , then the C term i s 1=11 the appropriate expression i s QDC + QD. Where C and D are functions of i , the appropriate expression i s f i l I " i l - l T T D . * i - i \ C. + QD i = i l x i = i 2 12-1 i = l D. 1 i 2 - l i = l T h i s expression s t a t e s that the flow t o the a-node f o l l o w s i f e i t h e r operators Q; D f o r a l l i ' s , such that U i $ i l ; C f o r a l l i ' s , such that 1-S i < i l ; and C f o r i = i l are closed, or operators Q; D f o r i = i 2 ; and D and C are closed f o r a l l i , such that 1 ^ i < i 2 . Two paths may thus be followed t o the a-node. However, many more functions than two may be outputted at the a-node, depending upon the values of i l and 12. That i s to say, i f flow goes out the QD path and i f the equation which i s outputted at the a-node (and not shown, of course, i n the l o g i c a l diagram of Figure 16) i s a r e c u r s i o n equation, then i t w i l l be s t r u c t u r a l l y modified with each c y c l e through the DC feedback loop. S i m i l a r dependencies occur when flow emerges from the feedback loop to the a-node v i a the QDC path. Now we t u r n t o c o n s i d e r a t i o n of the t h i r d and f i n a l case. T h i s i s where feedback loops occur w i t h i n other feedback loops. A simple example i s shown i n Figure 17. Note that i n t h i s case D. might be a f u n c t i o n of j as w e l l as of i . To i n d i c a t e t h i s , I w i l l w r i t e D. as D., . To be completely accurate, one should w r i t e both the range of the i and the j values f o r which D i s c l o s e d . However, as the range of the outer j loop w i l l be s p e c i f i e d i n other terms, i t s statement here would be redundant. Thus D and C . w i l l s p e c i f y j i J 1 the range of the i values only. E w i l l s p e c i f y the range of the j value j As a f i r s t step t o w r i t i n g an expression f o r the flow graph of Figure 17, we r e c a l l that the feedback loop, QCD has already been decribed. (The loop s p e c i f i e d as QDC i s simply that loop that contains the Q, D, and C operators.) The l o g i c f o r the QDC loop i s set as an operator-switch, X . Then the flow graph of Figure 17 becomes j i X E j i J = l 1 j l - l _ T T E j = l U I f we s u b s t i t u t e f o r X i n t h i s expression, we get the f o l l o w i n g expression: i l . J j l - l _ TT" E. 12.-1 " J T T " D.. J i i = l f i 2 - i j i = l J i D . + j , i = i 2 . J J i = l J i i l . - l J _ I n t h i s expression, 12 and i l are w r i t t e n as fu n c t i o n s of j by w r i t i n g i2.: and il. . This means that the number of times through the feedback J loop depends upon the number of times that flow has gone through the outer feedback loop. These are the only techniques used i n w r i t i n g a complete expression, f o r the t o t a l i t y of paths which flow may take through- the attack model. REDUCTION OF THE LOGIC OF THE ATTACK MODEL The f i r s t step i s to set up the l o g i c a l flow graph f o r the model of Appendix IV. This i s given i n Figure 18. I n c o n s t r u c t i n g the Figure, I have assigned i n d i v i d u a l I F ... statements l a b e l s such as described i n the previous s e c t i o n . I t has been necessary t o define three counters or v a r i a b l e s of i t e r a t i o n : i , j , k. The feedback loop that i s designated by the counter j i n v o l v e s only the D, E, F, and G operators. I t w i l l be r e f e r r e d t o as the DEFG loop. The HI loop i s designated by counter k, while the ABCDEFGIJ loop i s designated by counter i . Consider the DEFG loop. There are three paths i n t o t h i s loop, v i s . , B, BC, and BC. They need not be considered at t h i s stage. Likewise, the output from t h i s loop w i l l not be considered at t h i s time. The i n t e r n a l s e c t i o n of the DEFG loop can be c o l l a p s e d to give the flow graph of Figure 19. I n t h i s graph i s a f u n c t i o n of operators D, E, and F. Assume that flow can emerge from the path ...G-at j=j2 or from the path ...D at j = j l . Output from the loop i s thus d e s c r i b a b l e as f o l l o w s : j ? - l _ J2 j l - l _ " j l - l T T G , j-1 J T T or. j = l J + D j = j l T r G. j-1 \ j = l J The o<^ are s t r u c t u r a l l y the same i n each term, but have d i f f e r i n g domains. I f we elaborate on these terms, we get the f o l l o w i n g expression: Figure 18. The l o g i c f o r the fl o w diagram of Appendix 119 Figure 19. The l o g i c f o r the DEFG part of the flow graph of Appendix IV. 120 J=j2 J2-1 IT J2 _ n IT D j = 1 J J2 • TT E. j=l ' TT E. j=l J J2 TT F. j=l J J2 _ ~TT F. j=l J + j = j l j l - l _ T T G j=l : j l - l D 1-1 J j l - l _ TT E. j=l 3 j l - l TT F. J - l 3 j l - l _ E i i = 1 j j l - l _ ~TT F j=l The expression completely describes the l o g i c of the DEFG loop. Note the complex, E + EF + EF above. The domains are from 1 to j2 for the f i r s t appearance and from 1 to j l - l for the second. The f i r s t appearance of t h i s complex should be read: for j=l eit h e r E, or EF, or EF must be closed; and for j=2 again; and for j=3 again, e t c . The point i s that i f E, say, i s closed for j=l, i t need not be closed for a l l larger j . We now consider the HI loop. Because of the way i n which the DEFG loop was treated, there i s only one path into the HI loop. This i s v i a the above expression. In the HI loop only the H operator i s a function of the i t e r a t i o n v ariable, k. The operator I i s a constant. There are two paths from the HI loop. The I co n t r o l l e d path i s described by the expression I . ^ h e operator-switch H i s closed only f o r k=l because I i s a constant - for any program, I i s either open or closed. The other path i s - a p p r o p r i a t e l y described by the [kl-1 expression Hk=kl T T H. k-1 ^ I . This i s not e n t i r e l y true, for when k l = l , the state of I i s i r r e l e v a n t . To show t h i s , I l e t the condition _ k l - l _ " T , without the subscript k. An expression for . k=2 J I be written as the HI loop follows by OR'ing the two statements: "kl-1 " k l - l _ " H k - i 1 + H ^-1 k=kl T T H k k=l T T I k=2 A simpler flow graph of the attack model may now be drawn (Figure 2 0 ) . That part of the flow graph within the i-loop may be written without d e f i n i t i o n s of domains as (A + A)(BC + BC + B ) p . 1 j k The problem of nested loops i s met here for the f i r s t time. Recall that the method needed to deal with t h i s problem was developed as the last step of the previous section. Applying the rules developed there, we get the expression: i = i l i l _ " TT B [ i = l 1 i l - 1 T T i = l i l TT C i = l i A1* MuV* - " i l _ " + TT A. i = l 1 11 i l B i l _ TT B. i = l 1 i l TT C i = l 1 This expression i s a complete d e s c r i p t i o n of the l o g i c contained i n the flow graph of Appendix IV. A more useful form would be with following Boolean expression \ i and 0 ., written out i n f u l l . When t h i s i s done we have the 1 lk i = i l i l - 1 i = l H I + H i,k=l i , k = k l i k l . - l 1 l T T H.. k-1 l k r k i - i i i T T I k=2 " i l - " i l _~ • i i ' i l _ _ < TT B. + TT B. Li-i -TT C. + T T B. 1=1 TT C. i = l 1 > i l 1 " i l T T A T + T T A i Li-i J i = l F i g u r e 20. The l o g i c f o r the fl o w diagram of Appendix IV. The DEFG and HI p a r t s are modified from the form given i n Figure 18. DEDUCTIONS The l o g i c - e x p r e s s i o n developed i n the previous s e c t i o n w i l l be usei to d i r e c t the search f o r the va r i o u s forms of the attack r a t e f u n c t i o n The method which w i l l be used to f i n d the various forms of the attack r a t e r e l a t e s t o the way i n which the l o g i c - e x p r e s s i o n has been w r i t t e n I t i s i n the standard product form, where each "term" i s surrounded by braces. There are f i v e of these terms: an A term, a BC term, a DEFG term,an HI term, and a J term. The v a r i o u s paths of the J term w i l l be checked against the block diagram of Appendix IV i n order to determine t h e i r relevance with respect t o changing the form of the attack r a t e f u n c t i o n . I f t h i s term i s not relevant then i t w i l l be omitted from f u r t h e r consider-a t i o n s . We w i l l proceed t o the HI term. This procedure w i l l be continued through to the A term. Whenever a d i s t i n c t form of the attack r a t e f u n c t i o n i s found, the form and i t s accompanying l o g i c w i l l be recorded and t a b u l a t e d . This e x p l a i n s the procedure. The use of t h i s procedure w i l l now be considered. The attack r a t e i s a l i n e a r combination of the TD, TS, TP, and TE f u n c t i o n s . Each of these func t i o n s w i l l be considered separately and then together. Consider the TD v a r i a b l e . The J term regulates only the hunger, the independent v a r i a b l e . I t does not a f f e c t TD s t r u c t u r e . A study of the block diagram r e v e a l s that the next term, the HI loop, does a f f e c t the form of the TD f u n c t i o n . There are two paths out of the HI loop (see the two OR'ed terms of the l o g i c - e x p r e s s i o n ) . The f u n c t i o n generated when flow passes as H^ ^=^1 i s the same as that which enters the HI loop. C a l l t h i s TD as TD(k=l). On the other hand, the second path through the HI loop generates a f u n c t i o n that depends on the value of k l . according to the f o l l o w i n g equation: 1 TD(k=l) + (kl.-l)*RECAD*LN HK 1 [HK-HTE, The obvious task i s to determine the va r i o u s forms of TD(k=T). We thus move to the next terra of the l o g i c - e x p r e s s i o n , the DEFG term. But TD i s not considered here and so we go on to the BC term. I t i s a l s o not r e l e v a n t . The A term, however, i s of importance. Here we f i n d that TD(k=l) can take two forms: 0 and RECAD*LN j"HK-HO 1 . Each [HK-HTEJ of these two funct i o n s must be combined with those of the HI loop t o produce four forms of the TD f u n c t i o n . The l o g i c which c o n t r o l s the generation of each form of TD i s r e a d i l y obtained by AND'ing the appropriate HI l o g i c with the appropriate A l o g i c , as i m p l i e d by the standard product form of the l o g i c - e x p r e s s i o n . From t h i s we obtain a l i s t of f u n c t i o n s and the l o g i c appropriate to each f u n c t i o n (TableVL) . I n t h i s t a b l e the HO v a r i a b l e has been changed to H; t h i s v a r i a b l e i s meant t o designate some undefined value of hunger. (The v a r i a b l e H used i n one place may or may not equalan H used i n some other place.) We now t u r n t o the TS f u n c t i on. TS i s f i r s t defined i n the DEFG loop. I n t h i s f u n c t i o n , SP i s not a constant. SP i s defined i n var i o u s forms i n both the DEFG and the BC terms. Regardless of where SP i s defined, i t always has two forms. When these forms are included two forms of the TS f u n c t i o n emerge. These forms, along with the c o n t r o l l i n g l o g i c , are given i n Table VTI.. A l s o i n t h i s t a b l e a s l i g h t n o v e l t y was introduced: we w r i t e D. . . i i v i • This says that operator D, as a f u n c t i o n of i and j , i s closed f o r some i values and f o r some j value equal t o j l . , such that j l ^ i s greater than one. 126 Table VI. The v a r i o u s forms, with c o n t r o l l i n g l o g i c , of the f u n c t i o n of time spent i n d i g e s t i v e pause (TD) inc l u d e d i n the attack model. FUNCTION GENERATED CONTROLLING LOGIC RECAD*LN f H K - H "[ [HK-HTE] A., . T H. , I i = i l i , k = l 2 . 0 3. RECAD*LN i = i l i , k = l [HK-H "] + ( k l [HK-HTEJ X -1)*RECAD*LNj HK (k1 -1)*RECAD*LN/ HK i l i f [HK-HTE f J L ] [HK-HTE J A H i = i l i , k = k l . A i = i l H i , k = k l . k l . - l T T H k=l r K l . - l i i k k=l H., i k [kl -1 I T 1 I k=2 r'ki-i J k=2 127 Table VIL- The v a r i o u s forms, wit h c o n t r o l l i n g l o g i c , of the f u n c t i o n of time spent searching (TS) in c l u d e d i n the model of a t t a c k . FUNCTION GENERATED w i t h CONTROLLING LOGIC 1 LN a a/AD a*NA*(HK-H) + 1 a/AD _NO*SR*SS*(HK) D ( B + B. ., C I i , j = l { i = i l i = i l i = i l j i , J = J 2 i J 2 1 - i TT G. . J-1 L J J 2 , IE. -• + E. . -, F ,\ + D = .-, . ^ i , J = j 2 . - l i , j = j 2 i " l i , j = j 2 i " 1 J i , j = j l ± ; 9 j l ± > l j l i - l _ TT D. . 3 1 . - 1 1 _ r r G. . j - i l J IE. . . + E , F ^ V i . j - j i i - i i , j = j i i - i i . j - j i i - i y 2 . I LN a a*NA*(HK-H) a/AD + 1 a/AD NO*SR*SS*(HK) *EXP^-AM n 1 KA*HONP*(L+EYE)*(H-HTP) -DS ] n: VP D. . B C. . + G. . „ i , J = l i = i l i = i l i , J = j 2 i J 2 . - 1 ' j 2 i _ ' TT D. . [j-1 l > . E i , j = j 2 i - l F i , J = j 2 . - l p v 1 1 + D i , j = j l . ; a j l . > i TT D. . _j=l l J _ T T G. . j-1 1 J. E. . .n i F. , .- _ i . J - j l i - l i . j - j l i - 1 We now t u r n t o the TP f u n c t i o n . As with the previous f u n c t i o n , TP i s determined i n the DEFG and the BC paths. There are two forms, given i n Table V I I I . We now t u r n t o the TE f u n c t i o n . As fo r a l l previous f u n c t i o n s , the J term does not a f f e c t the form of TE. The HI complex does. I f th e H ± k = 1 I path i s used, then TE = -RECAD* LN HK-RECAD/AKE HK-RECAD/AKE-Hj I f the other path i s chosen, then a number of p o s s i b l e f u n c t i o n s seem to be generated according to the f u n c t i o n : (k-l)*RECAD*LN HK-RECAD/AKE + AKE*W + (k-l)*RECAD* LN HK-RECAD/AKE HK-RECAD/AKE-H HK-RECAD/AKE-H J The f i r s t and t h i r d terms cancel out, l e a v i n g only AKE*W. Thus there are only two forms of TE, shown with the l o g i c appropriate t o each i n Table IX. The v a r i o u s forms and c o n t r o l l i n g l o g i c have now been described f o r each of TD, TS, TP, and TE. The l o g i c may be combined i n t o s e t s , to give the v a r i o u s types of attack r a t e f u n c t i o n s . The r e s u l t s are shown i n Table X. I n t h i s t a b l e TS1 r e f e r s to equation 1 of Table V I I , whereas TP2 r e f e r s to equation 2 of TableVLTI, e t c . There are eight types of fun c t i o n s of the attack r a t e included i n the s i m u l a t i o n model of Appendix IV. I t i s towards the development of Table X that the work has been d i r e c t e d . I f the procedure developed i s t o be u s e f u l , then i n s i g h t i n t o the operation and c o n t r o l of the att a c k r a t e should be a v a i l a b l e through the use of in f o r m a t i o n presented i n t h i s t a b l e . The f i r s t four f u n c t i o n s of Table X d i f f e r i n l o g i c a l c o n t r o l from the l a s t four by the behaviour of the HI loop. (More s p e c i f i c a l l y , 129 Table V I I I . The va r i o u s forms, with c o n t r o l l i n g l o g i c , of the f u n c t i o n of time spent pursuing prey (TP) included i n the attack model. FUNCTION. GENERATED CONTROLLING LOGIC @ TX same as f o r TS1 2. TX + ,/KA*HONP*(L + EYE)*(H-HTP) - DS same as f o r TS2 YTT VP @ Since t h i s t a b l e was constructed, i t has been discovered that the l o g i c f o r TP i s not e x a c t l y the same as that f o r TS. In term 2 of TS1, the expression, E + EF, holds f o r j = j 2 . , and not f o r j = j 2 . - l . In term 2 of TS2, the expression, EF, holds f o r j=i2^ and not f o r j = j 2 ^ - l . 130 Table IX. , The v a r i o u s forms, with c o n t r o l l i n g l o g i c , of the f u n c t i o n of time spent e a t i n g the prey (TE) included i n the model of a t t a c k . FUNCTION GENERATED CONTROLLING LOGIC -RECAD*LN HK-RECAD/AKE LHK-RECAD/AKE-H AKE*W H I i , k = l H i , k = k l i k l . - l T T H k=l i k k l -1 rf1 T k=2 131 T a b l e X.. The v a r i o u s f o r m s o f T I . The a t t a c k r a t e (A) i s d e f i n e d a s 2 4 / T I . FUNCTION GENERATED 1 . RECAD*LN ("(HK-H) (HK-RECAP/AKE-H)"]+ TS1 + TX [(HK-HTE)(HK-RECAD/AKE) J 2 . RECAD*LN [ (HK-H) (HK-RECAD/AKE-H)"] + TS2 + TP2 [_ (HK-HTE) (HK-RECAD/AKE) J 3 . -RECAP*LN F~HK-RECAD/AKE "1 + TS1 + TX [HK-RECAD/AKE-HJ 4 . -RECAD*LN [HK-RECAD/AKE *] + TS2 + TP2 LHK-RECAD/AKE-HJ 5 . RECAD*LN [HK-H *[ + (K 1 -1)*RECAD*LNF HK 1 + AKE*W + TS1 + TX IHK-HTEJ 1 1 [HK-HTEJ 6 . RECAD*LN [~ HK-H 1 + (KL. -1)*RECAD*LNJ" HK ~] + AKE*W + TS2 + TP2 LHK-HTEJ 1 1 [HK-HTEJ 7 . RECAP* (k 1 - 1 ) * L N f HK ~\ + AKE*W + T S l + TX |_HK-HTEJ 8 . RECAP*(kl, - 1)*LN(~ HK 1 + AKE*W + TS2 + TP2 1 1 (JHK-HTEJ CONTROLLING LOGIC 1 . A . H. . . I ID. . , (B + B C. ... ^ + . . . see T S l l o g i c i = i l i , k - l j i , j = l \±=n i = i l i = i 1 J 5 2 . A . , H . , , 1 J D . . B . C. . N + . . . see TS2 l o g i c i = i l i , k = l \ i , J = l i = i l i = i l 3 . 1 H. , I /D (3 + B C ) + . . . i = i l i , k = l ( i , j = l l i = i l i = i l i = i l / 4 . A H. , ,1) D. . ,B C. + . . . i = i l i , k = l j i , j = l i = H i = i l 132 Table X continued. 5- A. . H i=il i,k=kl± 6. A. H. , . , 1=11 i,k=kli 7. A H i=il i,k=kli i=il i,k=kl^ "kl.-l 1 rklrl ] 1 TT~ k=l Hik TT I k=2 "kl.-l I 'kl-1 " i 'I 1 Hik _ TT I _k=l [k=2 J D |B + B. ..C A i , j=l ^ i = i l 1=11 i = i l ) D B. ..C , + .. i,j=l i = i l i = i l 'kl-1 TT H k l . - l TT H ik .k=l ik ' k l i ! T T I k=2 Tkl -1 i _ TT I Lk=2 J D (3 + B. . C. . \ i5J=T V i=il 1 = 1 1 L = l l y D B C ' + . i,j=l i=il i=il + . @ The foot-note to TableVEEI (page 106) changes this table. The forms of TI are not 8, as given, but 16. Table X was constructed under the assumption that TP logic and TS logic are identical. Thus TS1 logic occurred in association with TP1 logic and TS2 logic occurred in association with TP2 logic. The modifications necessary to correct this table are: to change T i l by letting TP2 replace TPl, and then adding this equation as a new form of TI; to change TI2 by letting TPl replace TP2, and then adding this equation as a new form of TI; and so on through TI8. The rest of this section does not incorporate this change. The section i s valid in so far as i t goes. The effects of the additional 8 forms are ignored, of course. 133' the f i r s t d i f f e r s from the f i f t h , the second d i f f e r s from the s i x t h , etc.) The l o g i c d e s c r i b i n g that path through the HI loop that generates TE2 (Table IX ) may be s t a t e d as f o l l o w s - i f f o r some captured prey, the predator i s hungry enough to eat the complete prey a l l at once; or i f not, eat a l l i t can, l e a v i n g t o d i g e s t the food, but r e t u r n i n g t o the carcass to eat again and again u n t i l the prey i s completely consumed and then going away f o r good. Thus there are two d i s t i n c t t a c t i c s i n c l u d e d i n t h i s model. This i s an i n t e r e s t i n g c h a r a c t e r i s t i c of the model since i t i n c r e a s e s the breadth of a p p l i c a b i l i t y by i n t r o d u c i n g v a r i o u s cases, c o n t r o l being channeled to the appropriate case by a switch (I) which i s not a f u n c t i o n of any v a r i a b l e of i t e r a t i o n . T h is c h a r a c t e r i s t i c has been noted, among other reasons, t o i l l u s t r a t e how one can begin to break up an otherwise complex model i n t o cases. Logic-a l sets of a t t a c k r a t e equations 1,2,5, and 6 could be compared t o the set of 3,4,7, and 8 equations. I n t h i s case the d i v i s i o n i s on the b a s i s of types of f u n c t i o n s of time spent i n d i g e s t i o n (TD). I n the f i r s t set, the predators do spend some time d i g e s t i n g food; i n the second case, they do not. The animals operating under 1,2,5, and 6 equations might be l e s s prone t o p r e d a t i o n . The e f f e c t s of p r e d a t i o n on the predator might w e l l be studied from the f i r s t set of equations. One might study the p o s s i b l e e f f e c t s of changes i n parameter values on the value taken by the a t t a c k r a t e . I t might be of i n t e r e s t to know which parameters occur i n which f u n c t i o n s and which are h e l d i n common i n a l l f u n c t i o n s , as w e l l as the purpose the parameters f i l l i n the f u n c t i o n s . I n Table X"I the f i r s t two p o i n t s are answered. Only the parameters HK, AD, ARE, TX, a, NO, SR, and SS are p a r t s of a l l forms of the a t t a c k f u n c t i o n . I n f a c t , the l a s t four parameters r e l a t e only 134 Table XT. The parameters of the attack r a t e , by type of f u n c t i o n . An X designates the presence of the parameter i n the model of a c e r t a i n number. PARAMETER MODEL NUMBER PARAMETERS IN COMMON 1 2 3 4 5 6 7 8 HK X X X X X X X X X HTE X X X X X X AD (RECAD) X X X X X X X X X ARE X X X X X X X X X k l X X X X W 1 1 X X X X TX X X X X X X X X X a X X X X X X X X X NO X X X X X X X X X SR X X X X X X X X X SS X X X X X X X X X AM X X X X KA X X X X HONP X X X X L X X X X EYE X X X X HTP X X X X DS X X X X VP X X X X 1 9 0 1 0 1 1 1 1 1 9'- 7 8 6 1 9 1 9 0 number of 8 parameters t o that p a r t of the TS f u n c t i o n h e l d i n common 1 LN a a/AD a*NA*(HK-H) + 1 i/AD NO*SR*SS*(HK) The parameter SP takes two values, dependent upon the l o g i c a l c o n t r o l (TableVLT). Thus SP was not incorporated i n the above f u n c t i o n . HK and AD (=1/RECAD) parameters occur i n common places w i t h i n the f u n c t i o n RECAD* LN jf 1(HK,H)/f 2(HK,H)] The parameter TX holds a common p o s i t i o n i n the v a r i o u s f u n c t i o n s . However, AKE occurs i n more than one pla c e i n the v a r i o u s f u n c t i o n s . C o n t r o l of the value taken by the att a c k r a t e might be s u i t a b l y achieved by conce n t r a t i n g a t t e n t i o n on HK, AD, TX, a, NO, SR, and SS. The p o s i t i o n of each i s according t o the f o l l o w i n g general form of the a t t a c k r a t e f u n c t i o n : 24 TX + RECAD* LN f 1(HK,H)~ + 1 LN [f 2 (HK,H) a a/AD a*NA*(HK-H) + i a/AD _NO*SR*SS*(HK) The r a t e of d i g e s t i o n , AD, i s d i r e c t l y r e l a t e d t o the att a c k r a t e . The parameter TX, a hoarding parameter, has an inv e r s e r e l a t i o n s h i p w i t h the a t t a c k r a t e . The parameter a, the searching r a t e parameter, a l s o has a d i r e c t r e l a t i o n s h i p w i t h the a t t a c k r a t e . The parameter HK, the maximum hunger, has both d i r e c t and i n v e r s e r e l a t i o n s h i p s w i t h the attack r a t e . I t i s a l s o c l e a r that parameters a, TX, and AD have the greatest e f f e c t on the above f u n c t i o n . T h i s i s because a l l other parameters operate as arguments of log f u n c t i o n s . The value taken by the a t t a c k r a t e can a l s o be i n f l u e n c e d by the l e v e l of the independent v a r i a b l e , hunger. The t o t a l i t y of types of hunger fu n c t i o n s i n c l u d e d i n the block diagram are l i s t e d i n Table XI I . A l l types of hunger f u n c t i o n s were l i s t e d i n t h i s t a b l e because the number of them i s s m a l l . Hunger has a simple d i r e c t r e l a t i o n s h i p w i t h each of TE, TS, and TP. The longer the predator pursues a prey, the longer i t looks f o r a prey, and the longer i t takes t o eat the prey, the h u n g r i e r i t w i l l be a f t e r eating the prey. There i s a d i r e c t r e l a t i o n s h i p between AKE, the feeding r a t e , and hunger. There are both d i r e c t and i n v e r s e r e l a t i o n s h i p s between AD and hunger. I t i s not only necessary t o know how t o c o n t r o l hunger, but to know how hunger a f f e c t s the value of the attack r a t e . Looking back at Tables VI through IX, 0- we f i n d that hunger i s d i r e c t l y r e l a t e d t o TP2; i n v e r s e l y r e l a t e d t o TD1 and TD3, t o TS1, and to T E l ; both d i r e c t l y and i n v e r s e l y r e l a t e d to TS2; and not r e l a t e d t o TD2 and TD4, to T P l , and to TE2. Thus hunger can a f f e c t the a t t a c k r a t e i n numerous ways. I t i s concluded that the response of the system t o a d i r e c t i o n a l change i n hunger i s not e a s i l y p r e d i c t a b l e . I t has been shown that the value of the a t t a c k r a t e i s e f f e c t i v e l y m odified by a l t e r i n g the values of the AD, a, and TX parameters. Yet the parameter AD has been shown to have both d i r e c t and i n v e r s e r e l a t i o n s h i p s with hunger. Thus the parameters a and TX are the b a s i c ones that w i l l e l i c i t a p r e d i c t a b l e response that w i l l show reg a r d l e s s of the s t a t e of the r e s t of the system. This concludes the attempt at f i n d i n g an a n a l y t i c a l way of studying the s e n s i t i v i t y of a complex flow graph t o changes or v a r i a t i o n i n i t s parameters. I n conclusion, we attempt t o use t h i s technique i n a n t i c i p a t i n g the form of simulated output and i n h e l p i n g to e x p l a i n 137 Table XEI. The t o t a l i t y of types of hunger equations outputted at the end of the i - l o o p of the attack model. C o n t r o l l i n g l o g i c i s gi v e n . FUNCTION GENERATED 1. 2. 0 HK-RECAD AKE HK-RECAD AKE HK-RECAD AKE HK-RECAD AKE -AD*TE 1-e AD(TP+TE) -AD*TE 1-e + f H O ( i - l ) - H K ) - A D ( T P + T E ) + (HTE-HK~) e -AD*TE1 . ^ - A D ( T S + T P + T E ) 1-e J + ( H O ( i - l ) - H K ) e -AD*TE] . - A D ( T S + T P + T E ) . 1-e j + ( H T E - H K J e CONTROLLING LOGIC 1. H I i, k = l 2. H i , k = k l i k l ± - l H k=l i k I D i> J=jl,-j l . - l j l . - l T T G L j - i TT" E + j = l i j .. see l o g i c - e x p r e s s i o n of previous s e c t i o n . A-term only of A^ 3. l o g i c as f o r 2, with A. ., replaced by A i = i l K J i = i l 4. H i, k = k l , k l -1 I H. k=l i k J i»J=J2. 5. l o g i c as f o r 4, with A replaced by A i = i l i = i l use only the A part i = i l of the A-term, as above 138 c e r t a i n features of simulated output that are not c l e a r . H o l l i n g (1966) presents a p l o t ( h i s Figure 31) of the time spent i n the d i g e s t i v e pause (TD) as a f u n c t i o n of the number of f l i e s per square centimeter (NO). Using the t a b l e of the v a r i o u s forms of the TD f u n c t i o n , we can see that no TD f u n c t i o n i n c l u d e s NO as a parameter. Thus the e f f e c t s of NO on TD must occur through the hunger v a r i a b l e . The hunger generated at the end of the i - l o o p i s d i r e c t l y r e l a t e d t o TS (Table X I I ) . NO i s i n v e r s e l y r e l a t e d t o TS ( T a b l e V I I ) . Therefore, NO i s i n v e r s e l y r e l a t e d t o hunger, H. The v a r i a b l e H i s r e l a t e d t o TD as a log f u n c t i o n i n such a way that TD=0 when H=HTE. The v a r i a b l e TD, as a f u n c t i o n of NO, i s s t r i c t l y a lo g f u n c t i o n . The r e l a t i o n s h i p operates v i a hunger, through•changes i n TS. H o l l i n g mentioned (1966; page 62) that the magnitude of t h i s f u n c t i o n i s very s e n s i t i v e t o changes i n some of the parameters. I t i s c l e a r that parameters AD and HTE are the only two that a f f e c t TD d i r e c t l y . Others may operate through hunger and the parameter AKE i s the most obvious. "In the H. crassa - h o u s e f l y system, f o r example, the ' d i g e s t i v e ' pause d i d not appear even at the highest d e n s i t y that was simulated. This occurred mainly because the s i z e of the prey (W) r e l a t i v e t o the s i z e of the predator (HK) was very much smaller i n the H. crassa than i n the M. r e l i g i o s a one." ( H o l l i n g , 1966; page 62). This statement i s equivalent t o H^ k = k l - = l ? which i s part of the l o g i c p e r t a i n i n g t o TD4 (TableVI). The reason given i n the quote may w e l l be the main reason, but the l o g i c of TD4 sta t e s that A . ^ _ . Q i s a l s o necessary. That i s , the hunger of the predator j u s t a f t e r i t s preceeding meal had t o be greater than or equal t o the eat t h r e s h o l d , HTE. A p o s s i b i l i t y f o r e l i m i n a t i n g the d i g e s t i v e pause that was not mentioned above i s given i n TD2. This i s where the s i z e of the prey !39 i s irrelevant because the predator i s not very hungry and w i l l not return to the carcass when i t becomes hungrier. The last question to be considered is whether the technique can be of value in anticipating the form of simulated output. Qualitative aspects of simulated output seem to follow readily from Table X . It i s easy to say something about the behaviour of A as H changes; or to say something about the behaviour of A as RECAD, or TX, or AKE change. One can say whether a l l functions operate more or less in the same manner, or whether some operate very differently from the others. To be quantitative in one's predictions, however, is quite another matter. This i s because a generalized hunger term, H, has been used in preference to, say, hunger referred to that at the beginning of the i-loop. Each H term could, of course, be referred to this point, but at the expense of expanding Table X.. ., Even-then, in order to pradict output, one would have to go iteration by iteration, each time updating hunger in the equations of Table X. But then one would almost be doing what the computer is doing. It has been suggested that perhaps this technique, developed for studying analytically the sensitivity of a model, might i t s e l f be programmed. As an aid to following the internal working of this program i t might be useful to expand Table X", referring a l l hungers to the hunger at the beginning of the i-loop. I f during a simulation run these hungers were being continuously printed, one could use this knowledge along with the Table to understand and control the program. DISCUSSION FOR PART I I I 140 The approach which has been developed i l l u s t r a t e s that complex, h i g h l y n o n - l i n e a r models are not completely i n t r a c t a b l e to a n a l y t i c a l methods. By d e a l i n g w i t h q u a l i t a t i v e , as opposed to q u a n t i t a t i v e changes i n the model, i t i s p o s s i b l e to d e l i n e a t e the various forms of the output v a r i a b l e . This was so even when there was more than 2 ^ d i f f e r e n t p o s s i b l e output equations. From t h i s set of equations a set of parameters which serve a common r o l e i n a l l equations could be l i s t e d . Where the equations of the set d i f f e r e d g r e a t l y , i t was p o s s i b l e to compile subsets of s i m i l a r equations; the c o n t r o l l i n g l o g i c was c l e a r . I t was shown how the independent v a r i a b l e v a r i e s over the i t e r a t i o n s of the program. I t was a l s o p o s s i b l e to e x p l a i n simulated output - by a more organized form of the method u s u a l l y used. Because of t h i s o r g a n i z a t i o n i t was p o s s i b l e to know when the f u l l explan-a t i o n was at hand, and to know when there was more than one explanation. The method i s u s e f u l mainly i n that i t t e l l s the i n v e s t i g a t o r when a l l cases have been considered. I f one simply moves haphazardly through the model seeking e x p l a n a t i o n s , he may w e l l overlook some important d e t a i l s . This p o s s i b i l i t y i s l e s s l i k e l y using my method because, i n the a t t a c k model, the l o g i c - e x p r e s s i o n developed f o r the block diagram of Appendix IV and Table X de s c r i b e the t o t a l i t y of cases. Because of the organized form that t h i s method produces, v a r i o u s s i m u l a t i o n models may be compared and contrasted. I t i s p o s s i b l e to compare t h i s model of a t t a c k w i t h that by Ware (1971) and that by Beukema (1968). T h i s comparison i s , however, another task and i s not attempted i n t h i s t h e s i s . The method i s unable to deal w i t h the i t e r a t i v e nature of s i m u l a t i o n models: the value of the dependent v a r i a b l e i n one i t e r a t i o n depends upon the value of the independent v a r i a b l e of the l a s t i t e r a t i o n ; but the value of the independent v a r i a b l e during that i t e r a t i o n depended upon the value of the dependent v a r i a b l e of the previous i t e r a t i o n . T h is seemed to be the major problem i n determining the exact form of simulated output from the a t t a c k model of Appendix IV. GENERAL DISCUSSION 142 This study attempted to deal w i t h two inadequacies of experimental components a n a l y s i s . I t i s , perhaps, appropriate to end t h i s t h e s i s w i t h an a p p r a i s a l of the methods proposed and t e s t e d . The problem of g e n e r a l i t y was studied from a s t r a t e g i c , r a t h e r than from a t a c t i c a l , viewpoint. An hypothesis was formulated regarding the du r a t i o n of the dynamic responses of va r i o u s searching modes. Hydra were used to t e s t t h i s hypothesis. Searching behaviours were i d e n t i f i e d through the j o i n t use of the l i t e r a t u r e and my own observations on Chlorohydra v i r i d i s s i m a and Hydra l i t t o r a l i s . Those behaviours which changed i n frequency over time of food d e p r i v a t i o n were regarded as searching behaviours. I t was a l s o necessary that those behaviours could be i n t e r p r e t e d i n terms of searching through volume. Three search modes were i d e n t i f i e d : t e n t a c l e lengthening, w alking, and f l o a t i n g . I t was shown that over a l l hunger l e v e l s , the t y p i c a l response fo r hydra i s to lengthen the t e n t a c l e s f i r s t , then to begin to walk, and l a s t l y to detach and begin to f l o a t . Furthermore, i t was shown that the d u r a t i o n of the dynamic response of a search mode i s i n v e r s e l y r e l a t e d to the hunger l e v e l at which the mode i s i n i t i a t e d . These r e s u l t s on hydra suggest that the s t r a t e g i c hypothesis may have some usefulness i n w r i t i n g general f u n c t i o n s . The i n i t i a l hypothesis was based on an argument which was independent of the s p e c i f i c c h a r a c t e r i s t i c s of hydra. An e x p l i c i t f u n c t i o n f o r time spent searching (TS) was w r i t t e n . T his f u n c t i o n was formulated i n an attempt to s i m p l i f y the i m p l i c i t TS f u n c t i o n by H o l l i n g (1965, 1966). Both equations r e l y on the parameters HK, AD, SP, SS, and SR. The equation developed i n t h i s t h e s i s adds the parameter c a l l e d "a" while H o l l i n g ' s equation adds the parameters VR, AKRGM, HTS, HTO, and AF. I n H o l l i n g ' s (19 66) paper the r e s u l t s of some simulations are presented. H i s Figure 31 presents a graph of the time spent searching, as a f u n c t i o n of the d e n s i t y of prey (NO). On the whole i t takes a - c * t form reminiscent of that produced by the expression e . Most of the change takes p l a c e over small NO v a l u e s . Thus i n the equation 1110, we might ignore the second term of the argument of the log term, t o get 1/a [ l n ( f ) - l n ( g * N O ) ] , where f= a*NA*(HK-HO)& and a/AD g= SR*SP*SS*(HK) . I f l n ( f ) > ln(g*NO), the r e s u l t a n t of the d i f f e r e n c e between a constant TS and a log f u n c t i o n resembles the graph simulated by H o l l i n g . The problem of model complexity was approached simply by i d e n t i f y i n g the v a r i o u s k i n d s of equations of the outputted v a r i a b l e that were included i n the s i m u l a t i o n model. These forms of the outputted v a r i a b l e were l i s t e d . From t h i s l i s t c e r t a i n deductions were made. A serious problem with t h i s approach i s that of omission; the i n t e r e s t i n g problem of how t o deal with the i t e r a t i v e f eature of s i m u l a t i o n models could not be d e a l t w i t h . LIST OF SYMBOLS The l o g i c a l symbols of PART I I I w i l l not be i n c l u d e d . These symbols may be checked by r e f e r r i n g t o Appendix I I . For PARTS I and I I , the symbols are as f o l l o w s : a - a parameter a s s o c i a t e d w i t h the r a t e of searching AD - r a t e of d i g e s t i o n DMT -distance of v e r t i c a l movement during f l o a t i n g HK -maximum hunger l e v e l HO -hunger l e v e l j u s t a f t e r a prey i s consumed t 0+kT -time i n t e r v a l i n which hydra detach and begin t o f l o a t L -mean t e n t a c l e length NA -number of prey attacked NO -prey d e n s i t y n -a time counter f o r d i f f e r e n c e equations RD - r e a c t i v e d i s t a n c e t - u n s p e c i f i e d time fco -time zero f o r d i f f e r e n c e equations T -time i n t e r v a l f o r d i f f e r e n c e equations TF -time of food d e p r i v a t i o n timed from a c o n d i t i o n of complete s a t i a t i o n TD -time taken i n a d i g e s t i v e pause a f t e r a prey i s eaten TE -time spent e a t i n g prey TP -time spent pursuing each prey TS -time spent searching f o r each prey Tm -time i n T i n which hydra move T u -time i n T i n which hydra are f i x e d and not moving T a -time i n T i n which hydra have t e n t a c l e s attached t o the substrat VD -average v e l o c i t y of predator during searching VY -average v e l o c i t y of prey VPL -average v e l o c i t y of the looping hydra VPF - v e l o c i t y of the f l o a t i n g hydra VS -volume searched per time i n t e r v a l BIBLIOGRAPHY 145 Beukema, J . 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Rushforth, N.B. i n press B e h a v i o r a l and e l e c t r o p h y s i o l o g i c a l studies i n Hydra: I I I . I n t e r a c t i o n s between the column c o n t r a c t i o n and t e n t a c l e c o n t r a c t i o n pacemaker systems. B i o l . B u l l , i n press. Rushforth, N.B. and F. Hofman 1966 B e h a v i o r a l sequences i n the feeding response of Hydra l i t t o r a l i s . B i o l . B u l l . 131:403-404 ( A b s t r a c t ) . Rushforth, N.B. and F. Hofman i n press B e h a v i o r a l and e l e c t r o p h y s i o l o g i c a l s t u d i e s of hydra: IV. An a n a l y s i s of feeding behavior. B i o l . B u l l , i n press. Schoener, T.W. 1971 Theory of feeding s t r a t e g i e s , p. 369-404. In; Annual Review of Ecology and Systematics. Volume 2. R.F. Johnson (Ed.). Annual Reviews Inc., Palo A l t o , U.S. of A. Wagner, G. 1905 On some movements and r e a c t i o n s of hydra. Quart. J . Microscop. S c i . 48:585-622. Ware, D.M. 1971 The predatory behaviour of rainbow t r o u t (Salmo  g a i r d n e r i ) . Ph. D. Thesis. U n i v e r s i t y of B r i t i s h Columbia. Watkins, B.O. 1969 I n t r o d u c t i o n to c o n t r o l systems. Macmillan Company, New York. 625p. Watt, K..E.F. 1968 Ecology and resource management. McGraw-Hill Book Company, New York. 450p. Welch, P.S. and H.A. Loomis 1924 A l i m n o l o g i c a l study of Hydra o l i g a c t i s i n Douglas Lake, Michigan. Anat. Record 29:129. Wickes, W.E. 1968 Logic design w i t h i n t e g r a t e d c i r c u i t s . John Wiley and Sons, L t d . , New York. 249p. Wilson, E.B. 1891 The h e l i o t r o p i s m of hydra. Amer. Nat. 25:413-433. Zehna, P.W. and R.L. Johnson 1962 Elements of set theory. A l l y n and Bacon, Inc., Boston. 194p. 151 APPENDIX I SOME OBSERVATIONS ON HYDRA IN BEAVER CREEK A f i e l d study was i n i t i a t e d i n an attempt to determine the d i s t r i b u t i o n and abundance of a hydra population. Emphasis was placed on the feeding b i o l o g y of the population. As the data are o b s e r v a t i o n a l , the r e s u l t s are only t e n t a t i v e . This program was judged necessary i n order that l a b o r a t o r y s t u d i e s could be r e l a t e d to circumstances a r i s i n g i n a n a t u r a l h a b i t a t . The study area was the o u t l e t creek from Beaver Lake. This lake i s l ocated i n Stanley Park, Vancouver (Figure 1). The hydra were studied i n v a r i o u s parts of the creek, but always between the lake and the Pipe L i n e Road. Very few hydra were found i n the r e s t of the stream. Beaver Creek i s approximately 15-30 cm deep. The creek i s exposed to f l u c t u a t i o n s i n depth, as i t i s supplied w i t h water which enters v i a a s p i l l w a y from Beaver Lake. The mainstream flow of the creek i s around 30 cm/sec. Near the shore, where populations of hydra were r e a d i l y observable, flow v a r i e d from l e s s than 2 cm/sec to the mainstream flow. Bottom type i s v a r i a b l e : mud, sand, g r a v e l , gravel covered by filamentous algae, i n t r i c a t e i n t e r l a c i n g of a t u b u l a r weed, and human refuse were a l l observed. The period of study was two weeks i n l a t e August. Afternoon water temperature was between 19 and 20 deg C. Taxonomy of Beaver Creek Hydra. Using the taxonomic key of Forrest (1959), I i d e n t i f i e d the hydra c o l l e c t e d from Beaver Creek as Hydra carnea, L. Agassiz. The most important features of t h i s species are: a)Nematocysts: those types of nematocysts c a l l e d penetrants were measured to be between 11 and 15 microns. They are between 9 and 19 gure 1. A map of the f i e l d study area. The enlarged portion of th map shows Beaver Creek more clearly. The darkened arrows indicate the location of the substrate or core, samples. The open arrows indicate the location of the grid samples. 152 1 microns i n length for H. carnea. The large ( s t r e p t o l i n e ) glutinants are of narrowly oval form of 10 microns i n length. Those of H. carnea are of 10 microns i n length. The small (stereoline) glutinants are of oval form but, unlike the large forms, are pointed at one end and of 7 to 8 microns i n length. Those of H. carnea are about 9 microns long. The volvents were measured to be between 6 and 7 microns long. Those of H. carnea are between 6 and 8 microns i n length. As s p e c i f i e d for H. carnea, the Beaver Creek animals showed three or four transverse or obliquely-transverse c o i l s of thread within the large g l u t i n a n t s . b) Tentacle arrangement: during tentacle formation on a bud of_H. carnea, the f i r s t protuberance occurs on one side of the hypostome. This protuber ance i s followed by the growth of other t e n t a c l e s . A t y p i c a l arrangement i s with the older tentacles on one side of the hypostome, the younger ones on the other side, with some tendency for a l t e r n a t i o n of longer and shorter t e n t a c l e s . These patterns were shown i n the f i e l d hydra. c) Number of t e n t a c l e s : a sample of 2,317 hydra from the f i e l d showed 457, with f i v e tentacles, 457° with s i x tentacles, and 57, with four t e n t a c l e s . Animals with one, two, three, seven, eight, or nine tentacles were recorded infrequently. d) Gametes: testes of the f i e l d species were small, mammiform, and with n i p p l e s . None of the hydra specimens" which were c o l l e c t e d had eggs attached to them. However, two eggs were c o l l e c t e d from substrate _samples These eggs were spherical, with small spines emerging from the theca. This i s t y p i c a l of H. carnea. Some Observations on the D i s t r i b u t i o n and Abundance of Hydra carnea  i n Beaver Creek. Preliminary observation had indicated that hydra 154 occurred i n a great many places along the stream. The hydra were often found i n such d e n s i t i e s that i t was impossible t o estimate t h e i r d e n s i t i e s d i r e c t l y . Thus the h a b i t a t , w i t h adhering hydra, was sampled. Measure-ments were obtained from the preserved hydra. To t h i s end a syringe of 16 mm diameter was modified t o produce a bottom sampler. The t i p of the syringe was removed, l e a v i n g a c y l i n d e r with i t s plunger. In using t h i s sampler, I set i t over a s e c t i o n of the bottom and pushed i t i n t o the s u b s t r a t e . This r e s u l t e d i n a core of substrate at the bottom of the c y l i n d e r . R e t r a c t i n g the plunger moved the core f u r t h e r up i n t o the c y l i n d e r . The core sample was then t r a n s f e r r e d t o a c o l l e c t i n g b o t t l e , and preserved i n a f o r m a l i n s o l u t i o n . This technique allowed me t o sample known areas of the substrate i n each of s i x l o c a t i o n s . The l o c a t i o n s were s i t u a t e d at v a r i o u s d i s t a n c e s from the s p i l l w a y (see black arrows of Figure 1 ). F i v e samples were taken at each l o c a t i o n . They were not chosen at random, but s t r a t i f i e d i n such a way as t o i n d i c a t e the ranges i n hydra d e n s i t y present. I simply looked around each l o c a t i o n f o r various d e n s i t i e s of hydra. (Also, some bottom substrate was c o l l e c t e d and examined i n the l a b o r a t o r y . These c o l l e c t i o n s were made where the water was too deep, or the current too s w i f t , or the bottom of u n s u i t a b l e t e x t u r e t o c o l l e c t i n the previous manner. The o b j e c t i v e was to a s c e r t a i n only the presence or absence of hydra. I n each of f i v e samples taken, hydra were found. There were many hydra i n two samples but only a few i n the r e s t of the samples.) The data from t h i s sampling e x e r c i s e i s presented i n terms of o numbers of hydra per cm of substrate (Table I ) . The data i s presented such that l o c a t i o n 1 of Table I corresponds t o the most upstream black Table I . Density of Hydra carnea (numbers/cm ) i n Beaver Creek. The data are given f o r groups of f i v e samples per l o c a t i o n , with l o c a t i o n s numbered from the most upstream to the most downstream (see Figure 1) . Sample s i z e i s the t o t a l number of hydra counted i n a l l f i v e samples per l o c a t i o n . LOCATION DENSITY SAMPLE SIZE (number/cm ) 1 2. 12 12 16 33 150 2 29 51 58 80 88 612 3 0 3 16 42 110 342 4 17 19 24 28 132 442 5 12 57 75 84 136 729 6 5 6 8 13 16 96 arrow of Figure 1; l o c a t i o n 2 of Table I corresponds to the next black arrow of Figure 1; etc. Thus lo c a t i o n 6 of the table corresponds to the most downstream black arrow of Figure 1. I t i s r e a d i l y observable that no straightforward r e l a t i o n s h i p e x i s t s between the density of hydra i n a l o c a l i t y and the distance of that l o c a l i t y downstream from the spillway. However, i t has been noted that few hydra were found downstream from the Pipe Line Road. A large amount of v a r i a t i o n i n density of hydra e x i s t s i n i n d i v i d u a l sections of the stream (see data from l o c a t i o n 5). Next the d i s t r i b u t i o n of H. carnea within small areas was studied. A g r i d of 1 foot to each side was constructed. It was subdivided i n t o 2 576 quadrats, each of them being 1.6 cm i n area. A hundred quadrats were then selected at random, under the r e s t r i c t i o n that each one-quarter of the g r i d had to contain one-quarter of the selected quadrats. In using the gri d , I placed i t over an area of the bottom of the stream and counted the number of hydra i n each of the 100 selected quadrats. For each quadrat the composition of the substrate, the depth of the water, and the surface water v e l o c i t y were also noted. This procedure was followed i n each of f i v e d i f f e r e n t s i t e s (see Figure 1; open arrows). The s i t e s were chosen i n such a way as to i l l u s t r a t e the various types of h a b i t a t s : 1) an even water depth of 30-50 mm; with a fast water flow of 30 cm/sec; and with mixed gravel and filamentous a l g a l substrate; 2) an uneven water depth of 30-60 mm; with a medium water flow of 10 cm/sec; and a mixed mud and weed substrate; 3) an uneven water depth of 6-35 mm; with a va r i a b l e water flow of 2-30 cm/sec; and with mixed mud, s i l t , weed, and gravel bottom; 4) an uneven water depth of 20-60 mm; with a fast water flow of 30 cm/sec; and with mixed mud and weed substrate; and 5) an even water depth of 20-37 mm; with a slow water flow of 4 cm/sec; and with mixed mud and weed s u b s t r a t e . The order of data p r e s e n t a t i o n i n Table I I i s from the most upstream s i t e t o the most downstream s i t e . I t i s v i s i b l e i n Figure 1 that the area designated as l o c a t i o n 1 i s the same area as s i t e 1. Likewise i t may be observed that l o c a t i o n 2 was taken i n the same area as s i t e 2 and that l o c a t i o n 6 was taken i n the same area as s i t e 5. L o c a t i o n 4 was taken more or l e s s i n the same area as s i t e 3. L o c a t i o n 3, l o c a t i o n 5, and s i t e 4 were a l l taken i n unique areas of the creek. The d e n s i t y of hydra was c o r r e l a t e d with the depth of the water, the water v e l o c i t y at the surface, and the substrate type. The data are analyzed separately f o r each s i t e . The s i t e s were not chosen at random. They were chosen d e l i b e r a t e l y to i l l u s t r a t e the v a r i e t y of h a b i t a t s p r e s ent. Thus an appropriate e r r o r d i s t r i b u t i o n cannot be assumed f o r data lumped from a l l s i t e s . The data suggest that over a range of depths from 8 to 30 mm, hydra d e n s i t y i s d i r e c t l y r e l a t e d t o water depth ( s i t e 3 of Table IIA) . At greater depths the r e l a t i o n s h i p may be negative ( s i t e s 1,4, and 5 of Table I I A ) . In three of the f i v e s i t e s water v e l o c i t y was uniform over the whole g r i d . Thus c o r r e l a t i o n s were run f o r the data from the remain-i n g two s i t e s . The r e s u l t s are given i n Table I I B . They suggest that over low v e l o c i t i e s (7-10cm/sec) there i s a negative r e l a t i o n s h i p between d e n s i t y of hydra and water v e l o c i t y , and that over higher v e l o c i t i e s (2-30 cm/sec) there i s no r e l a t i o n s h i p . A r e l a t i o n s h i p was found between the d e n s i t y of hydra and the substrate type. For each quadrat the number of hydra per quadrat and the substrate type were recorded. The dominant substrate types were mud, mud and weeds, weeds, weeds and g r a v e l , g r a v e l , g r a v e l and Table I I . R e l a t i o n s h i p between the density (numbers/quadrat) of Hydra carnea i n Beaver Creek and the depth of the water above the quadrat (A) and the surface water v e l o c i t y (B). A. SITE RANGE OF REGRESSION COEFFICIENTS CORRELATION COEFFICIENTS DEPTHS (mm) SLOPE INTERCEPT 1 30-45 -0.004 +00.76 -0.0098 n S 3 08-30 +0.087 -00.36 *** +0.3792 4 31-52 -0.545 +29.03 -0.6618 5 20-37 -0.077 +04.51 -0.1506 n S B. SITE RANGE OF VELOCITIES REGRESSION COEFFICIENTS CORRELATION COEFFICIENTS (cm/sec) SLOPE INTERCEPT 2 07-10 -1.006 +15.41 -0.4326 3 02-30 +0.006 +00.75 +0.0887 n S ns: not s i g n i f i c a n t at p=0.05 s i g n i f i c a n t at p i=0.001 algae, and algae. The number of hydra i n a given quadrat was c h a r a c t e r i z e d by the substrate type of that quadrat. Then i t was p o s s i b l e t o c a l c u l a t e the percentage of hydra by substrate type. For comparative purposes, c a l c u l a t i o n s were made of the percentage of va r i o u s substrate types. As 100 quadrats were counted at each s i t e , these c a l c u l a t i o n s were done simply by counting the number of quadrat types. This l a t t e r c a l c u l a t i o n was then considered t o be the percentage of hydra by substrate type, i f the hydra were d i s t r i b u t e d e n t i r e l y on the b a s i s of substrate type. The r e s u l t s are given i n Table I I I . The a c t u a l numbers of hydra found are given t o the l e f t of each p a i r of numbers while the expected numbers are given t o the r i g h t of each p a i r of numbers. I t i s c l e a r from Table I I I that the number of hydra expected on the mud bottom i s c o n s i s t e n t l y h i g her than the number a c t u a l l y present. The opposite s i t u a t i o n i s shown f o r weedy s u b s t r a t e s . The mud and weed substrate data show the intermediate s i t u a t i o n , w i t h the data from three l o c a t i o n s behaving as f o r the weed covered substrates and the data from one l o c a t i o n behaving as f o r the mud s u b s t r a t e . The frequency d i s t r i b u t i o n of hydra w i t h i n each s i t e was s t u d i e d . o — I n d i c e s of d i s p e r s i o n (s /Y) were c a l c u l a t e d ( P i e l o u , 1 9 6 9 ) . The expected value of t h i s r a t i o i s one when i n d i v i d u a l s are dispersed at random. I t i s gre a t e r than one when i n d i v i d u a l s are d i s t r i b u t e d c o n t a g i o u s l y . For s i t e s one through f i v e c o e f f i c i e n t s took values of 3 . 1 8 , 5 . 0 3 , 2 . 0 9 , 1 0 . 6 0 , and 4 . 3 8 r e s p e c t i v e l y . These i n d i c e s suggest that the hydra are d i s t r i b u t e d c o n t a g i o u s l y . One would expect on the b a s i s of the data given i n Table I I I that the clumping might be due t o the d i s t r i b u t i o n of the weeds w i t h i n the g r i d s . The d i s t r i b u t i o n s were analyzed f o r only the weed covered Table I I I . Percentage of hydra and percentage of substrate type, by substrate types, f o r each of f i v e s i t e s . Blanks i n the t a b l e should be read as zeros. 160 SUBSTRATE SITE TYPE 1 2 3 4 5 CU CU <u CU a !>> CX EH EH EH EH rt rt rt rt S-i • 5-1 • J-i H • * T> X. T3 Xi TJ X X) X >, 3 !>. 3 3 >> 3 3 CO W CO W co « co tn CO 8-8 6~8 5^ B-2 £-2 B*S B~2 if? B-2 mud 00-01 00-01 07-22 10-36 40-74 mud and weeds 19-31 21-12 42-25 46-18 weeds 71-55 40-24 39-32 04-02 weeds and g r a v e l 19-17 g r a v e l 07-14 02-01 00-01 g r a v e l and algae 76-62 algae 15-20 00-01 o t h e r 02-03 08-11 13-24 09-07 10-06 Number of hydra per s i t e 59 641 112 543 265 161 quadrats at s i t e s two, three, and four. The other s i t e s d i d not contain enough data. The i n d i c e s were 4.63, 1.59, and 3.62 f o r s i t e s two, three, and four r e s p e c t i v e l y . The d i s t r i b u t i o n s s t i l l seem to be contagious. However, the degree of clumping i s c o n s i s t e n t l y l e s s when only the weedy substrates are considered than when a l l substrate types are considered. Some informat i o n on the reproductive a c t i v i t y of hydra was c o l l e c t e d from the core samples. Table IV gives the percentage of hydra w i t h reproductive organs, c o r r e l a t e d w i t h hydra d e n s i t y . These data suggest 2 that budding i s low at d e n s i t i e s of 0 to 24.5 hydra/cm and that i t i s 2 a l s o low at d e n s i t i e s greater than 75 hydra/cm . At intermediate d e n s i t i e s budding i s about twice as frequent. Sexual reproduction 2 appears to be low below 50 hydra/cm and high above t h i s d e n s i t y . Some Observations on the Searching of Chlorohydra v i r i d i s s i m a as  Performed i n the Laboratory. Hydra carnea i s not used i n these l a b o r a t o r y s t u d i e s as i t i s d i f f i c u l t to c u l t u r e i n a r t i f i c i a l medium. Chlorohydra v i r i d i s s i m a i s used. This species performs r e a d i l y the usual behavioural r e p e r t o i r e of hydra (Hegner, 1933). The behaviour patterns of the hydra were i d e n t i f i e d by feeding the hydra to s a t i a t i o n and then d e p r i v i n g i t of food and watching what i t d i d . Each hydra was assumed to be s a t i a t e d when a f r e s h l y k i l l e d b r i n e shrimp, Artemia, placed near i t s mouth, was not eaten. The behaviours whose frequency of occurrence changed w i t h hunger were considered searching behaviours. When fed to s a t i a t i o n the hydra c o n t r a c t s i t s t e n t a c l e s and s t a l k . The s t a l k becomes a round b a l l and the t e n t a c l e s , stubs. During the f i r s t few hours of food d e p r i v a t i o n , the hydra lengthens i t s t e n t a c l e s and s t a l k . L a t e r , the hydra begins to move: i t bends over, attaches i t s t e n t a c l e s to the sub s t r a t e , and by 162 Table IV. Percentage of hydra with one or more buds and percentage of hydra with one or more t e s t e s as a c o r r e l a t e of the d e n s i t y of the hydra. One standard d e v i a t i o n i s a l s o presented. Results were derived from the core samples of the substrate of Beaver Creek. DENSITY NUMBER NUMBER PERCENT PERCENT (numbers/ HYDRA CORES BUDDING WITH c m 2 ) SAMPLED SAMPLED GAMETES 00-24.5 365 16 6.85+8.96 0.85+2.13 25-49.5 263 4 14.61+6.83 0.75+0.90 50-74.5 332 3 15.66+8.18 1.45+1.32 75-99.5 650 4 8.76+7.89 2.82+1.83 100 up 749 3 9.58 + 2.92 1.59 + 1.11 c o n t r a c t i n g i t s s t a l k , p u l l s i t s foot f r e e of the substrate. Then i t r e s e t s i t s foot i n a new l o c a t i o n . A f t e r i t s foot i s attached, the hydra p u l l s i t s t e n t a c l e s f r e e by c o n t r a c t i n g i t s s t a l k , and then assumes again the upright p o s i t i o n . U s u a l l y a few hours l a t e r the hydra detaches from the substrate and f l o a t s upside down to the surface. Here i t r e -mains f l o a t i n g w i t h foot on the surface of the water and s t a l k and t e n t a c l e s extending down i n t o the water. A T e n t a t i v e L i f e H i s t o r y Formulation f o r Hydra. The f i e l d and l a b o r a t o r y data give a q u a l i t a t i v e p i c t u r e of the behaviour of hydra. I t i s assumed that Hydra carnea data can be taken as a complement of the Chlorohydra v i r i d i s s i m a data. My observations suggest that t h i s i s not an unreasonable assumption. Hydra which have not eaten r e c e n t l y stand w i t h outspread t e n t a c l e s and elongated s t a l k . When a prey comes by and brushes against a t e n t a c l e nematocysts discharge, h o l d i n g the prey to the t e n t a c l e s . The t e n t a c l e s are r e t r a c t e d and the prey i s eaten (Hegner, 1933). I f the prey i s large enough to reduce the hunger l e v e l of the hydra, the hydra shortens i t s t e n t a c l e s and s t a l k . Searching begins a few hours l a t e r when the t e n t a c l e s and s t a l k begin to lengthen. Provided that no prey chances along, the hydra begins to move about and e v e n t u a l l y detaches and f l o a t s somewhere i n the water column ( l a b o r a t o r y o b s e r v a t i o n s ) . I have observed i n Beaver Creek some hydra f l o a t i n g from the surface of the water. However, many hydra were swept along the creek j u s t s l i g h t l y above the bottom. As the hydra f l o a t downstream w i t h the water perhaps they r e a t t a c h when they run i n t o some protuberance (such as a weed bed, or a s t i c k ) up from a r e l a t i v e l y f l a t substrate (such as mud). Such a mechanism would be i n agreement w i t h the data of Table I I I . High d e n s i t i e s of hydra were observed i n the f i e l d (Table I ) . Such d e n s i t i e s might r e s u l t from a l i m i t e d number of hydra budding at a high r a t e . Low-medium d e n s i t i e s of hydra do show r a p i d budding a c t i v i t y (Table I V ) . As d e n s i t i e s become medium-high, the rate of asexual reproduction f a l l s w h i l e the r a t e of sexual reproduction increases (Table I V ) . 165 APPENDIX I I PARTIAL EVALUATION OF THE EQUATE)NS OF THE MODEL Equations 115, 117, 118, 122, 124, and 125 are repeated i n the t e x t i n a p a r t i a l l y evaluated form as equations 126, 127, 128, 129, 130, and 131 r e s p e c t i v e l y . Here the b a s i c s of the e v a l u a t i o n f o r each equation are provided. Expression 126 was derived from 115. We w r i t e 0.20 = (3.1416)(0.8290) 2(0.5592) = Trsin 2(56)*cos (56) 6 6 where 0.20 i s the evaluated part of the f i r s t term of 126 and | l T s i n 2 ( 5 6 ) * c o s ( 5 6 ) J /6 i s an equivalent form from 115. T h i s e x p l a i n s only one h a l f of the enumeration of 115. We a l s o w r i t e 13.90 = (0.8290)(0.5592)(60.0) = s i n (56)*cos(56)*T 2 2 where 13.90 i s the evaluated part of the second term of 126 and £sin (56)*cos (56)*T^J/2 i s an equivalent form from 115. This completely describes the e v a l u a t i o n of 115, as given i n 126. This technique w i l l be followed f o r the other equations. Expression 127 was derived from 117. We w r i t e 2 2 0.40 = (3.1416)(0.8290) (0.5592) = TTsin (56)*cos (56) 3 3 and 2 2 64.76 = (3.1416)(0.8290) (60.0) = TY s i n (56)*T 2 2 Expression 128 was derived from 118. We w r i t e 2 2 0.05 = (3.1416)(0.8290) (0.5592) = TTsin (56)*cos (56) 24 24 and 0.23 = (0.8290)(0.5592) = s i n (56)*cos (56) 2 2 E x p r e s s i o n 128 was d e r i v e d f r o m 1 2 2 . We w r i t e 0 . 0 3 = ( 3 . 1 4 1 6 ) ( 0 . 8 2 9 0 ) 2 ( 0 . 5 5 9 2 ) = TT s i n 2 ( 5 6 ) * c o s (56) 48 48 and 3 3 0 . 4 5 = ( 3 . 1 4 1 6 ) ( 0 . 8 2 9 0 ) = TT s i n (56) 4 4 and 0 . 2 0 = ( 3 . 1 4 1 6 ) ( 0 . 8 2 9 0 ) 2 ( 0 . 5 5 9 2 ) = T T s i n 2 ( 5 6 ) * c o s (56) 6 6 and 1 . 3 7 = ( 2 . 0 ) ( 0 . 8 2 9 0 ) 2 = 2 s i n 2 ( 5 6 ) and 2 2 1 . 0 8 = ( 3 . 1 4 1 6 ) ( 0 . 8 2 9 0 ) = T T s i n (56) 2 2 and 0 . 2 3 = ( 0 . 8 2 9 0 ) ( 0 . 5 5 9 2 ) = s i n ( 5 6 ) * c o s (56) 2 2 E x p r e s s i o n 130 was d e r i v e d f r o m 1 2 4 . We w r i t e 1 0 . 3 6 = ( 4 . 8 0 ) ( 0 . 8 2 9 0 ) 2 ( 3 . 1 4 1 6 ) = L 2 ( t Q + k T ) * s i n 2 ( 5 6 ) * TT and 2 2 0 . 2 0 = ( 3 . 1 4 1 6 ) ( 0 . 8 2 9 0 ) ( 0 . 5 5 9 2 ) = TT s i n ( 5 6 ) * c o s (56) 6 6 and 1 3 . 9 1 = ( 6 0 . 0 ) ( 0 . 8 2 9 0 ) ( 0 . 5 5 9 2 ) = T * s i n ( 5 6 ) * c o s ( 5 6 ) 2 2 E x p r e s s i o n 131 was d e r i v e d f r o m 1 2 5 . We w r i t e 2 2 _ 2 1 0 . 3 6 = ( 4 . 8 0 ) ( 3 . 1 4 1 6 ) ( 0 . 8 2 9 0 ) z = L ( t Q + k T ) * T T * s i n (56) and 2 2 0 . 2 0 = ( 3 . 1 4 1 6 ) ( 0 . 8 2 9 0 ) ( 0 . 5 5 9 2 ) = Tf s i n ( 5 6 ) * c o s (56) 6 6 and 167 1 3 . 9 1 = ( 6 0 . 0 ) ( 0 . 8 2 9 0 ) ( 0 . 5 5 9 2 ) = T * s l n ( 5 6 ) * c o s (56) 2 2 and 3 3 0 . 4 5 = ( 3 . 1 4 1 6 ) ( 0 . 8 2 9 0 ) = T t s i n (56) , 4 4 and 2 2 8 2 . 4 7 = ( 2 . 0 ) ( 0 . 8 2 9 0 ) ( 6 0 . 0 ) = 2 * s i n ( 5 6 ) * T T h i s s e r i e s o f e q u a l i t i e s c o m p l e t e l y expands a l l o f t h e c o n s t a n t s o f e q u a t i o n s 1 2 6 - 3 1 b a c k t o t h e i r o r i g i n a l f o r m s . APPENDIX I I I EFFECT OF PARAMETER VALUES ON THE CONCLUSIONS OF PART I I In PART I I the model i s studied as a f u n c t i o n of time of food d e p r i v a t i o n . Parameters VY, VPF, and DMT are v a r i e d to determine t h e i r e f f e c t s on the volume searched per time i n t e r v a l . However, parameters o«? , T , T m, VPL, and L ( t e n t a c l e length) are f i x e d i n advance. Yet these parameters are b i o l o g i c a l q u a n t i t i e s , and e x h i b i t v a r i a b i l i t y . The v a r i a b i l i t y i n parameters can a f f e c t the conclu-sions regarding the steady s t a t e search r a t e s . We attempt to de-termine the e f f e c t of v a r i a b i l i t y i n parameters «V , T , T , VPL, u' m' ' and L on the f o l l o w i n g conclusions. 1. The minimal e f f e c t of walking on the volume searched by a non-moving hydra i s s l i g h t l y negative; 2. the minimal e f f e c t of f l o a t i n g on the volume searched by a non-moving hydra i s e i t h e r no change (when the absolute value of (VY-VPF)*VY) or very negative (when VY=water velocity=VPF); 3. the maximal e f f e c t of walking on the volume searched by a non-moving hydra i s s l i g h t l y p o s i t i v e ; 4. the maximal e f f e c t of f l o a t i n g on the volume searched by a non-moving hydra i s very p o s i t i v e . Parameter T u was measured i n the text to be very c l o s e to T (=60 minutes). Since T = T u + t m , the parameter T^ i s very c l o s e to zero. The walking equations (I18,I22) V reduce to the t e n t a c l e lengthening equations (115,117) when Tu=T; and T u was never more than 0.1 minutes from 60 minutes (T). No evidence was c o l l e c t e d to suggest that T ever wanders f a r from T. Thus we dismiss the e f f e c t s of v a r i a t i o n i n T and T on the conclusions drawn i n PART I I . m u Parameter VPL operates only during T m. Since T m i s so s m a l l , i t i s u n l i k e l y that VPL w i l l have much of an e f f e c t on the conclusions. I t s e f f e c t s are considered to be unimportant. I t i s suggested that the e f f e c t of walking on volume searched by a non-moving hydra i s minimal regardless of expected v a r i a t i o n i n parameters T m, -T , and VPL. Conclusions 1 and 3 are not g r e a t l y changed by v a r i a t i o n i n these parameters. We now consider the e f f e c t s of and L on conclusions 2 and 4. Consider c o n c l u s i o n 2. We compare equation 115 to 124. I n 115, parameter T=60 and VY may take values up to 18,000 mm/minute. I t i s thus f r e q u e n t l y the case that term one i s much l e s s important than term two. We w r i t e 115 = VY*sin(^/2)*cos(<x/2)*T* r 2 2 7 2 [ J L (t Q+nT) + L (t Q+nT-T) J . The major e f f e c t s of 124 are due to T, VY, and VPF. Term one i s non-zero only f o r t ^ t Q+kT. Since we are comparing steady s t a t e s , we ignore t h i s term. Term two i s ignored because i t contains n e i t h e r T, nor VY, nor VPF. Thus 124 ^ T*sin ( o </2)*cos ( c i</2)*(|VY-VPF|)*[ L 2 ( t +nT) + L ^ ( t +nT-T) 2 *— o ^ j The r e l a t i v e e f f e c t of 124 and 115 i s 124/115. Since the L c o n t a i n i n g terms are the same i n each of 124 and 115, i t i s c l e a r that L cannot change the conclusions. Likewise f o r t h e ^ terms. 2 Now consider c o n c l u s i o n 4. Equation 117 =Tf_*sin (<^/2)*T*VY* ^ L 2 ( t Q + n T ) + L 2(t Q+nT-T ) J . Term one and two of 125 are not considered f o r the same reasons that they were not considered f o r 124. Term four does not c o n t a i n T, as do terms three and f i v e . However, i t contains VY, as VY/VPF. When VY=VPF, t h i s r a t i o equals one and term four i s small r e l a t i v e to terms three and f i v e . In f a c t , i t i s only important when VPF becomes i n c r e a s i n g l y l e s s than 1 mm/minute. This i s a very small v e l o c i t y 170 and so term four i s e l i m i n a t e d . Thus 125 T*VPF*sin<>, / 2 ) * c o s ( / 2 ) * 2 r 2 2 1 2 F 2 ? L ( t +nT) + L (t +nT-T) + 2 VY*sin (<W2)*T* L ( t +nT) + L (t +nT-T) L o O L_ O O The r a t i o 125/117 must hold more or l e s s constant as rx and L change f o r the c o n c l u s i o n 3 to remain v a l i d . Since the same L c o n t a i n i n g term can be f a c t o r e d out of each of 125 and 117, L cannot a f f e c t the conclusions. T h i s c o n c l u s i o n does not hold f o r parameter oL , since term two of 125 has<x. as s i n (°£/2) to term one's s i n ( <x_ /2) *cos(o<r/2) . I f VPF tends to zero, then co n c l u s i o n 4 i s unaltered. This i s a l s o the case i f VY>^>VPF, sinc e then the major volume covered by 125 i s due to term two, as above. The term sin(oc/2) *cos(oc/2) has a r e l e v a n t property. I t has a maximum value at oc/2 = 45 degrees and f a l l s monotonically from t h i s peak to value zero atoc/2 = 0 and at <x/2 = 90 degrees. Conclusion 4 can be i n v a l i d a t e d only by a decrease i n the value of 125/117. This can occur only i f cos (<y/2)*sin (oc/2) takes a smaller value. Since the standard d e v i a t i o n of o<. i s 25 degrees, i t must be 12.5 degrees f o r <v/2. The parameter ocV2 has to be l e s s than 34 degrees before the s i n (QC"/2)*COS (<=C/2) term can take a smaller value than i t has at <=cV2 = 56 degrees. This low a value of o<_ i s a rare event. Thus the 125/116 r a t i o can be decreased only as ^i/2 i s increased to 90 degrees. I t i s c l e a r from the expanded form of 125/116 that when cx/2 = 90, then the r a t i o w i l l tend to value of one, or approximately so. We conclude that parameters T , T , VPL, and L are formulated i n u m such a way that v a r i a t i o n i n them does not a f f e c t to any s i g n i f i c a n t degree the conclusions of PART I I . The parameter<may negate co n c l u s i o n 4 only under the f o l l o w i n g c o n d i t i o n s , and then only maybe. The c o n d i t i o n s are: that VPF i s not equal to zero, that VY i s not much l a r g e r than VPF, and that «v/2 approaches the value of 90 degrees. APPENDIX TV FLOW GRAPH OF THE ATTACK MODEL I n Figure 1 i s a flow graph of the f u n c t i o n a l response of a predator t o prey d e n s i t y ( H o l l i n g , 1966). I t has been modified s l i g h t l y , by r e p l a c i n g h i s equation f o r time spent searching (TS) by predators by one developed i n t h i s t h e s i s (equation 1110) . Figure 1. A flow graph of the attack mod WO - *0 H e e C A D c l . o / A 0 TP-Tp-t - rx : ~ 1(1 *• V K + Cho-Hk) *xp(-AovTs) S P l - l . — C V P <2> T e s T 6*|sp-api/ - 0 . 0 2 . 1 SPA AA1« Q C C A D Sp. (te Ho= WK-AAI-^A^-AAOEXPC-AD-TE) 173 «c i <c cr -z. J bi u» *g < 3 i Ul cc 3 0 l n it » 'I ID o O H 3 ^ ml <1 0 3 D u St 

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