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An experimental component analysis of sexual reproduction : I. The egg production and egg fertilization… Gossard, Thomas W. 1973

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AN EXPERIMENTAL COMPONENT ANA L Y S I S OF SEXUAL REPRODUCTION I . The E g g P r o d u c t i o n a n d E g g F e r t i l i z a t i o n P r o c e s s e s , w i t h some C o n s i d e r a t i o n o f t h e M a t i n g P r o c e s s , f o r D r o s o p h i l a m e l a n o g a s t e r M e i g e n by Thomas W. G o s s a r d B . S c , U n i v e r s i t y o f C a l i f o r n i a a t D a v i s , 1968 A THESIS SUBMITTED I N P A R T I A L FULFILLMENT OF "THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n t h e D e p a r t m e n t o f Z o o l o g y ^ We a c c e p t t h i s t h e s i s a s c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF B R I T I S H COLUMBIA S e p t e m b e r , 1973 In presenting 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 of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the 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 reference and study. I f u r t h e r agree that permission f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n permission. Thomas W. Gossard Department of Zoology The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8, Canada Date 25 September 1973 ABSTRACT E x p e r i m e n t a l c o m p o n e n t s a n a l y s i s ( H o l l i n g 1966) i s u s e d t o d e v e l o p a c o m p u t e r m o d e l o f t h e f o u r p r o c e s s e s o f s e x u a l r e p r o d u c t i o n : m a t i n g , e g g p r o d u c t i o n , e g g f e r t i l i -z a t i o n , a n d o v i p o s i t i o n s i t e s e l e c t i o n . A g e n e r a l f u n c t i o n o f i n t e r a c t i n g p o p u l a t i o n s i s d e v e l o p e d , a n d i t s a p p l i c a t i o n t o m a t i n g a n d o v i p o s i t i o n s i t e s e l e c t i o n i s d i s c u s s e d . D a t a f r o m t h e l i t e r a t u r e o n m a t i n g a r e u s e d t o e s t i m a t e p a r a m e t e r v a l u e s f o r t h i s f u n c t i o n . A m o d e l o f e g g p r o d u c t i o n a n d e g g f e r t i l i z a t i o n i s d e v e l o p e d f r o m a n e x p e r i m e n t a l s t u d y o f t h e v g s t r a i n o f D r o s o p h i l a m e l a n o g a s t e r . E g g p r o d u c t i o n i s a c o m p l e x p r o c e s s c o n s i s t i n g o f f o u r c o m p o n e n t s a f f e c t i n g i n d i v i d u a l o v a r i o l e s : o v a r i o l e a c t i v a t i o n , o v a r i o l e p r o d u c t i o n , v i t e l l e g e n e s i s , a n d o v a r i o l e d e a c t i v a t i o n . T h r e s h o l d e f f e c t s a r e f o u n d t o e x i s t f o r a l l f o u r c o m p o n e n t s . E g g f e r t i l i z a t i o n i s a s i m p l e p r o c e s s i n v o l v i n g number o f s p e r m s t o r e d a n d a c o n s t a n t p r o b a b i l i t y o f s u c c e s s f u l f e r t i l i -z a t i o n . H o w e v e r , r e s u l t s i n d i c a t e t h a t b o t h e g g f e r t i l i -z a t i o n a n d e g g p r o d u c t i o n become more c o m p l e x b e y o n d t h e r a n g e o f t r e a t m e n t s u s e d h e r e . A s s u m p t i o n s , n o t s u p p o r t e d b y d a t a , a r e made f o r t h e p r o c e s s e s o f o v i p o s i t i o n s i t e s e l e c t i o n , a g i n g , m o r t a l i t y , a n d d e v e l o p m e n t . T h e s e a s s u m p t i o n s a r e c o m b i n e d w i t h t h e m o d e l s o f m a t i n g , e g g p r o d u c t i o n , a n d e g g f e r t i l i z a t i o n i n t o a s i n g l e t e n t a t i v e m o d e l f o r s e x u a l r e p r o d u c t i o n . S i m u l a t i o n s u s i n g t h i s m o d e l s u g g e s t p o s s i b l e e f f e c t s o f e c o l o g i c a l i m p o r t a n c e : a s i g m o i d r e l a t i o n s h i p b e t w e e n r e p r o d u c t i v e r a t e a n d d e n s i t y ; a n d a c h a n c e i n t a c t i c s w i t h i n c r e a s i n g m o r t a l i t y due t o p r e d a t i o n . i i i . TABLE OF CONTENTS Page ABSTRACT i LIST OF TABLES v i i LIST OF FIGURES X LIST OF EQUATIONS x i i i ACKNOWLEDGEMENTS x v i GENERAL INTRODUCTION 1 REVIEW OF INSECT REPRODUCTION MODELS 5 SEXUAL REPRODUCTION: PROCESSES AND COMPONENTS 7 PART I MATING INTRODUCTION 12 MODEL OF INTERACTING POPULATIONS 13 Development of the Model 13 Further Development of the Model 17 Components of the Model 20 Application of the Model to Reproduction i n Drosophila me lanogas t e r 25 PARAMETER ESTIMATION FOR MATING MODEL 32 Rate of Successful Search 32 Time Exposed and Handling Time 37 Ex p l o i t a t i o n , Interference, S o c i a l F a c i l i -t a t i o n and Motivation * 37 i v . Page Motivation: Density of Receptive Females and Males 38 CONCLUSION OF PART I 43 PART II EGG PRODUCTION INTRODUCTION . 44 FIRST EGG PRODUCTION EXPERIMENT .. 47 Introduction 47 Methods 47 Results 50 MATED FEMALES 56 Model Development 56 Parameter Estimation 65 VIRGIN FEMALES 74 Model Development 74 Parameter Estimation 76 SECOND EGG PRODUCTION EXPERIMENT . 79 Introduction 79 Methods 80 Results 83 Model Development . 84 Parameter Estimation 87 CONCLUSION OF PART II 93 V. Page PART III EGG FERTILIZATION INTRODUCTION AND METHODS 95 SPERM STORAGE 98 SPERM RELEASE 102 PARAMETER ESTIMATES 105 CONCLUSION OF PART III 108 A TEST OF THE PREDICTIVE POWERS OF MODELS DEVELOPED IN PARTS II AND III 110 OTHER PROCESSES 117 Aging 117 Morta l i t y 117 PART IV THE MODEL OF SEXUAL REPRODUCTION INTRODUCTION 120 CONSTRUCTION AND PROGRAMMING. 121 SIMULATION 127 RESULTS . 130 DISCUSSION 133 GENERAL CONCLUSION 143 BIBLIOGRAPHY 144 v i . Page APPENDICES I. E s t i m a t i n g f l y v e l o c i t y knowing the number of c o n t a c t s of f l i e s w i t h a 0.018 m. diameter c o n t a c t area 150 I I . P h y s i o l o g i c a l b a s i s o f the two a l t e r -n a t i v e egg p r o d u c t i o n f u n c t i o n s : Equations 23 and 24 152 I I I . E f f e c t of food i n t a k e on parameters of the egg p r o d u c t i o n f u n c t i o n 155 I n t r o d u c t i o n 155 Data f o r Podisus m a c u l i v e n t r i s .... 156 Data f o r D r o s o p h i l a melanogaster ... 162 D i s c u s s i o n 168 IV.• Symbols f o r equations i n P a r t IV 172 v i i . LIST OF TABLES Page Table 1. A disaggregation of the sexual re-productive process 9 Table 2. A comparison of d e f i n i t i o n s of para-meters common to both Holling's predation function and the general function f o r i n t e r a c t i n g populations ... 14 Table 3. Components of the predation process .... 21 Table 4. Possible components of the processes involving i n t e r a c t i n g populations 22 Table 5. Possible components of the mating process for Drosophila melanogaster .... 26 Table 6. Possible components of the o v i p o s i t i o n s i t e s e l e c t i o n process involving female Drosophila melanogaster and ov i p o s i t i o n s i t e s . . . 29 Table 7. A comparison of parameters common to both the general function for i n t e r -acting populations and the s p e c i f i c function for mating 31 Table 8. Relationship between f l y density and f l y v e l o c i t y 34 Table 9. Analysis of variance for l i n e a r r e-gression, Equation 11, using f l y v e l o c i t y data i n Table 8 35 Table 10. Analysis of variance f o r regression, Equation 16, using female r e c e p t i v i t y data i n Figure 5 42 Table 11. Two-way non-orthogonal analysis of variance of rate of egg production on 1) age of female and 2) whether female was i s o l a t e d or paired 51 Table 12. Relationship between age at copulation and 1) age at f i r s t egg laying, 2) maxi-mum egg production rate, and 3) age at maximum egg production rate 52 v i i i . Page Table 13. Relationship among parameters of the three egg production functions showing the sameness of the three equations 66 Table 14. Analysis f o r regression, Equation 25, using part of the egg production data i n Figures 7-11: i n d i v i d u a l estimates of parameter c f o r each treatment 68 Table 15. Analysis of variance for regression, Equation 22, using egg production data i n Figures 7-10: i n d i v i d u a l estimates of a and b f o r a l l treatments 70 Table 16. Analysis of variance for regression, Equation 22, using egg production data i n Figure 10 71 Table 17. Comparison of parameter estimates and c e r t a i n independent variables for the x egg production model 73 Table 18. Analysis of variance for regression, Equation 22, using egg production data i n Figure 11 77 Table 19. Analysis of variance for regression, Equation 25, using part of the egg production data i n Figure 14: single estimate of a and c for a l l t r e a t -ments . .. . . 89 Table 20. Analysis of variance for regression, Equation 25, using part of the egg production data i n Figure 14: i n d i -v i d u a l estimates of a and c for each treatment 90 Table 21. Analysis of variance for regression, Equation 22, using egg production data i n Figure 14: sing l e estimate of a, b, and c f o r a l l treatments 91 Table 22. Analysis of variance for regression, Equation 22, using egg production data i n Figure 14: i n d i v i d u a l estimates of a and b f o r each treatment . ... 92 i x . Page Table 23. Analysis of variance f o r regression, Equation 29, using f e r t i l e egg data i n Figures 17 and 18 106 Table 24. Comparison of parameter estimates f o r the egg f e r t i l i z a t i o n model 107 Table 25. Linear regression parameters f o r com-paring experimental observation from second experiment to p r e d i c t i o n of the egg production and egg f e r t i l i z a t i o n models 114 Table 26. Linear regression parameters for com-paring experimental observation from x f i r s t experiment to pre d i c t i o n of the egg production and egg f e r t i l i z a t i o n models 116 Table 27. Comparison of parameter estimates for the mortality function 118 Table 28. C l a s s i f i c a t i o n of population types for model simulation 129 Table 29. Relationship between food a v a i l a b i l i -ty and food intake for female Podisus  macu1iventr i s 158 Table 30. Relationship between food l e v e l and 1) age at f i r s t egg laying, 2) maximum egg production r a t e , and 3) age at maximum egg production rate 160 Table 31. Analysis of variance f o r regression, Equation 22, using egg production data i n Figure 27: i n d i v i d u a l estimates of b and c for each treatment 161 Table 32. Analysis of variance f o r regression, Equation 22, using egg production data i n Figure 28: i n d i v i d u a l estimates of a for each treatment 165 Table 33. Analysis of variance f o r regression, Equation 22, using egg production data i n Figure 28: i n d i v i d u a l estimates of b and c for each treatment ............. 166 Table 34. Comparison of parameter estimates and ce r t a i n independent variables for the "new"egg production function ... 167 X . LIST OF FIGURES Page Figure 1. Flow diagram of the re l a t i o n s h i p s among the processes of sexual repro-duction and of the r e l a t i o n s h i p of sexual reproduction to other e c o l o g i c a l processes 8 Figure 2. A p a r t i a l disaggregation of the generalized process of i n t e r a c t i o n between two populations 23 Figure 3. Flow diagram of the rela t i o n s h i p s among the components of mating 27 Figure 4. Flow diagram of the rela t i o n s h i p s among the components of o v i p o s i t i o n 30 Figure 5. E f f e c t of age of v i r g i n female Drosophila melanogaster on proportion of females accepting a courting male .... 39 Figure 6. Flow diagram of the re l a t i o n s h i p s among the components of egg production .. 45 Figure 7. E f f e c t of age o f female Drosophila melanogaster on r a t e of egg production for age at f i r s t egg laying of 5 days: f i r s t experiment 54 Figure 8. E f f e c t of age of female Drosophila me1anogaster on r a t e of egg production for age at f i r s t egg laying of 10 days: f i r s t experiment 55 Figure 9. E f f e c t of age of female Drosophila melanogaster on r a t e of egg production for age at f i r s t egg laying of 15 days: f i r s t experiment 57 Figure 10. E f f e c t of age of female Drosophila melanogaster on r a t e of egg production for age at f i r s t egg laying of 30 days: f i r s t experiment 58 x i . Page Figure 11. E f f e c t of age of v i r g i n female Drosophila melanogaster on egg pro-duction r a t e : f i r s t experiment 59 Figure 12. E f f e c t of age of female Drosophila melanogaster on rate of egg production for age at copulation of 1 to 5 days: second experiment 81 Figure 13. E f f e c t of age of female Drosophila melanogaster on rate of egg production for age at copulation of 10 days: second experiment 82 Figure 14. E f f e c t of age of female Drosophila melanogaster on egg production rate for various ages at copulation: second experiment 85 Figure 15. E f f e c t of age of v i r g i n female Drosophila melanogaster on egg produc-t i o n r a t e : second experiment 86 Figure 16. Flow diagram of the re l a t i o n s h i p s among the components of egg f e r t i l i z a -t i o n 96 Figure 17. E f f e c t of age at f i r s t egg laying on t o t a l number of f e r t i l e eggs l a i d by mated Drosophila melanogaster: f i r s t experiment . . 99 Figure 18. E f f e c t of age at copulation on t o t a l number of f e r t i l e eggs l a i d by mated female Drosophila melanogaster: second experiment 100 Figure 19. Proportion of eggs f e r t i l i z e d as a function of the percentage of the t o t a l number of f e r t i l e eggs l a i d 103 Figure 20. Plot of observed on expected f e r t i l e egg production rate f o r mated female Drosophila melanogaster: second experiment I l l Figure 21. Plot of observed on expected f e r t i l e egg production rate f o r mated female Drosophila melanogaster: f i r s t experi-ment 112 x i i . P a 9 e Figure 22. Flow diagram of the model of sexual reproduction 122 Figure 23. Flow diagram of the computer program for sexual reproduction 125 Figure 24. Simulation r e s u l t s for the sexual reproduction model 131 Figure 25. Three hypotheses for the e f f e c t of population density on the egg produc-t i o n of insects 136 Figure 26. Interaction of various fecundity and mortality curves 139 Figure 27. E f f e c t of age of female Podisus maculiventris on egg production rate for various food intake l e v e l s 159 Figure 28. E f f e c t of age of female Drosophila  melanogaster on egg production rate for various food intake l e v e l s 163 Figure 29. E f f e c t of age of female Podisus maculiventris on egg production rate for various food intake l e v e l s 171 x i i i . LIST OF EQUATIONS P a ? e Equation 1. Holling's predation function '.. 15 Equation 2. Modification of Holling's predation function 15 Equation 3. Preliminary function f o r in t e r a c t i n g populations 15 Equation 4. Preliminary function for in t e r a c t i n g populations 16 Equation 5. Function for i n t e r a c t i n g populations 16 Equation 6. Function for i n t e r a c t i n g populations 16 Equation 7. Function for i n t e r a c t i n g populations: modified to include interference 19 Equation 8. Mating function 28* Equation 9. Preliminary function for rate of successful search 32 Equation 10. Function for estimating f l y v e l o c i t y 33, 151 Equation 11. Preliminary f l y v e l o c i t y function ... 33 Equation 12. F l y v e l o c i t y function 36 Equation 13. Preliminary r e l a t i v e v e l o c i t y function 36 Equation 14. Relative v e l o c i t y function 36 Equation 15. Function for rate of successful search 37* •Equation included i n l i s t i n g of fragmentary equations (THE MODEL: CONSTRUCTION AND PROGRAMMING). xiv. Page Equation 16. Female r e c e p t i v i t y function 40* Equation 17. Function for density of receptive females 41* Equation 18. Function for density of receptive males 41* Equation 19. Function for age at f i r s t egg layi n g : f i r s t experiment 53,93* Equation 20. Function for age at maximum egg production rat e : f i r s t experiment .. 53,93* Equation 21. Preliminary egg production func-t i o n 61 Equation 22. Egg production function 63,93,155* Equation 23. Al t e r n a t i v e egg production function 64,152 Equation 24. Second a l t e r n a t i v e egg pro-duction function 64,153 Equation 25. Si m p l i f i e d egg production function 65 Equation 26. Log transformation of Equation 25 67 Equation 27. Function for age at f i r s t egg layi n g : second experiment 87 Equation 28. Function for age at maximum egg production rate: second experiment 87 Equation 29. Sperm storage function 98,108* Equation 30. Sperm release function 104,108* Equation 31. Function for determining propor-t i o n of females dying 119* Equation 32. Female s u r v i v a l function: aging 119* Equation 33. Male aging function 121* XV. Page E q u a t i o n 34. Female aging f u n c t i o n 121* E q u a t i o n 35. F u n c t i o n f o r t o t a l female d e n s i t y 121* E q u a t i o n 36. Female s u r v i v a l f u n c t i o n : p r e d a t i o n 123* E q u a t i o n 37. "New" egg p r o d u c t i o n f u n c t i o n .... 168 x v i . ACKNOWLEDGEMENTS I w o u l d l i k e t o t h a n k W i l f C u f f , N e i l G i l b e r t , M o n i c a G o s s a r d , Buzz H o l l i n g / P a t J e n k i n s o n , P e t e r L a r k i n , R o b i n L i l e y , a nd B i l l W e l l i n g t o n f o r t h e i r h e l p d u r i n g t h e c o u r s e o f t h i s s t u d y . GENERAL INTRODUCTION An i n d i v i d u a l organism may be v i s u a l i z e d as a system that combines a l l actions and interactions within the organism and between the organism and i t s environment. Interdependence of these actions and interactions precludes complete separation into d i s t i n c t groupings, but i t i s possible to approximate such groupings or processes. These processes can be general or s p e c i f i c . They can be broken i n t o smaller processes which themselves may be further subdivided. The desired l e v e l of understanding of the system determines the degree of t h i s disaggregation. There i s no need to d i s t i n g u i s h between a hierarchy of types of processes provided r e l a t i o n s h i p s among the actual processes are made c l e a r . Grouping actions and int e r a c t i o n s into processes i s the f i r s t step i n the modeling approach to studying complex problems. This approach has been applied to i n d i v i d u a l organisms (Holling 1966; Hubble 1971), groups of organisms (Paulik and Greenough 1966), and human s o c i a l systems (Holling and Chambers 1973) . Once i d e n t i f i e d , the action of each process can be summarized by regression equations describing observed re l a t i o n s h i p s between independent and dependent va r i a b l e s . The equations constitute a mathematical model which through simulation gives i n s i g h t i n t o the processes. During simu-l a t i o n , however, as a consequence of the regression approach, values of independent variables are li m i t e d to the range over which observations were made. Thus, these models give at best, high p r e d i c t a b i l i t y over the normal range of con-d i t i o n s but usually f a i l i n p r e d i c t i n g the outcome of any traumatic disturbance of conditions that c a r r i e s values beyond those observed. Experimental components analysis i s an a l t e r n a t i v e approach to modeling that overcomes the l i m i t a t i o n s of l i n e a r additive multiple regression equations. Experimental com-ponents analysis " i s based on the argument that e c o l o g i c a l processes can be broken into^components that are defined as responses to variables, these responses being simple func-t i o n a l r e l a t i o n s h i p s * " (Holling 1966). Having broken the processes into smaller processes, the components of each process are i d e n t i f i e d . Components are c l a s s i f i e d as basic i f "shared by a l l examples of the process" and subsidiary i f "present i n some situ a t i o n s and not i n others" (e.g. Table 1). Possible hypotheses with appropriate equations are constructed to explain the workings of each component. Experiments are then conducted to determine the v a l i d i t y of these hypotheses. Guidance for hypotheses comes from two sources: a p r i o r i knowledge and previous experiments. I n i t i a l l y , i n t u i t i o n and l i t e r a t u r e review suggest a l t e r n a t i v e hypo-theses with a l t e r n a t i v e equations for each component. *0r more formally, "...these responses having mono-tonic d i f f e r e n t i a l s " (Holling pers. comm.). 3. C a r e f u l l y d e s i g n e d e x p e r i m e n t s a r e c o n d u c t e d t o d i s t i n g u i s h among t h e s e a l t e r n a t i v e s . S e c o n d l y , e x p e r i m e n t a l r e s u l t s w i l l u s u a l l y show one h y p o t h e s i s t o be b e t t e r , p r o v i d e d i t i s m o d i f i e d . M o d i f i -c a t i o n o f t h i s h y p o t h e s i s w i l l s u g g e s t a new s e t o f h y p o t h e s e s and e q u a t i o n s . A g a i n , e x p e r i m e n t s c a n d i s t i n g u i s h among t h e s e a l t e r n a t i v e s . T h i s s e c o n d s t e p i s r e p e a t e d u n t i l t h e d e s i r e d r e s o l u t i o n o f t h e h y p o t h e s i s i s a c h i e v e d . F o r e x a m p l e , c u r s o r y k n o w l e d g e o f i n s e c t s m i g h t s u g g e s t t h e f o l l o w i n g h y p o t h e s e s a n d e q u a t i o n s . R a t e o f egg p r o d u c -t i o n (dE/dT) as a f u n c t i o n o f age (A) i s 1) c o n s t a n t , i n d e p e n d e n t o f a g e : dE/dT = a 2) r i s e s t o a maximum w i t h a g e : dE/dT = a - ( l - b A ) o r 3) r i s e s t o a maximum and t h e n d e c l i n e s w i t h a g e : dE/dT = a - ( l - b A ) - c A E x p e r i m e n t a t i o n m i g h t show t h a t t h e t h i r d h y p o t h e s i s i s m ost d e s c r i p t i v e b u t s u g g e s t s a new s e t o f h y p o t h e s e s . T T R a t e o f egg p r o d u c t i o n , a-(1-b )«c , i s e i t h e r : 1) i n d e p e n d e n t p f age a t c o p u l a t i o n : T = A o r 2) d e p e n d e n t on age a t c o p u l a t i o n ( T c ) : T = A - T c E x p e r i m e n t a t i o n w o u l d a g a i n be u s e d t o s e l e c t one o f t h e two h y p o t h e s e s , and t h e r e s u l t s m i g h t s u g g e s t new h y p o t h e s e s . Once a l l components of the various processes are analysed, r e s u l t i n g functions can be linked together into a simulation model. The model w i l l give i n s i g h t into the o r i g i n a l process provided that 1) the model describes the whole system and not just some of i t s parts; 2) the des-c r i p t i o n of each process i s r e a l i s t i c ; 3) values of dependent variables generated are precise estimates of the observed values; and 4) the model has general a p p l i c a t i o n to organisms other than those a c t u a l l y studied (Holling 1966 p. 6). Ho l l i n g (1966 p. 6-7) believes that experimental components analysis "provides at l e a s t the hope that the four q u a l i t i e s mentioned e a r l i e r w i l l be retained. Wholeness i s achieved since each step i s not considered as an end i n i t s e l f but i s combined with other steps so that progressively more and more of the process i s consi-dered. Reality i s ensured by the intimate union established between theory and experiment, with experiment d i c t a t i n g theory and theory suggesting experiments i n many small, successive steps. P r e c i s i o n i s assured because the demands of the mathematics forces us to cast the explanation i n a form precise enough to permit t e s t i n g of i t s adequacy. F i n a l l y , a degree of generality i s achieved by c l a s s i f y i n g each example by the u n i v e r s a l i t y of the components". REVIEW OF INSECT REPRODUCTION MODELS Past attempts at constructing insect reproduction models have had shortcomings (Watt 1968 p. 295-302; Conway 1969 p. 37). The models of Lotka (1923), V o l t e r r a (1928), F u j i t a and Utida (1953), and F u j i t a (1954) attempted i n t u i t i v e b i o l o g i c a l realism without reference to data. The models of Pearl and Parker (1922), Pearl (1932), and Anderson (1957) f i t t e d equations to data, but the equation's parameters did not represent b i o l o g i c a l processes or values. Both Watt and Conway t r i e d to overcome these problems by building "inductive-deductive" models based on the data of others with consideration for reproductive biology. More quickly constructed that experimental components models, inductive-deductive models can be an important t o o l i n management situations such as Conways. However, inductive-deductive models are l i m i t e d by t h e i r necessary reliance on published data. Such data i s often c o l l e c t e d with l i t t l e e f f o r t to develop a synthesized understanding of the process involved. This lack of syn-thesis i s i l l u s t r a t e d by experiments purporting to show ef f e c t s of interference between parasites. These experi-ments were conducted at densities f a r greater than those found n a t u r a l l y ; there i s no interference e f f e c t at r e a l i s t i c densities ( G r i f f i t h s and Holling 1969 p. 797-799). B i o l o g i c a l i n s i g h t and data are not enough to construct r e a l i s t i c and general functions. What i s needed i s the 6 . continual feedback between i n s i g h t and data generated by the experimental components analysis approach outlined above (GENERAL INTRODUCTION). Models of Drosophila melanogaster reproduction are of i n t e r e s t because t h i s was my experimental animal. The work of Pearl and Parker (1922) and Pearl (1932) has already been mentioned. Of more i n t e r e s t i s accomparative, between s t r a i n study of d a i l y egg production as a function of age ( F i t z - E a r l e , McMillan, Butler, and Robson 1969; McMillan, F i t z - E a r l e , Butler, and Robson 1970a; McMillan, F i t z - E a r l e , and Robson 1970b). The equation they develop i s i d e n t i c a l to one of the equations t h i s author derives below from d i f f e r e n t p h y s i o l o g i c a l considerations. Their study included copulation only at or near eclosion, but the present study includes e f f e c t s of copu-l a t i o n at d i f f e r e n t ages, as well as d i f f e r e n t food l e v e l s and the dynamics of male-female population i n t e r a c t i o n . This necessitated modification of the common basic equation and the addition of a mating model. Past insect reproduction models have evolved through time i n t o models that are extremely u s e f u l : Conway (1969) at the pest managerial l e v e l ; F i t z - E a r l e e t a l . (1969), McMillan et a l . (1970a,b), and F i t z - E a r l e (1971) f o r the study of reproductive inheritance. However, there i s s t i l l need for models that give an understanding of the reproductive process i t s e l f and are not tools of the applied ecologist or g e n e t i c i s t . SEXUAL REPRODUCTION: PROCESSES AND COMPONENTS The process of sexual reproduction has been subdivided into i t s i n t e r a c t i n g processes and components. Sexual repro-duction can be considered as having four c h a r a c t e r i s t i c s (Figure 1): 1) four external inputs: male density, v i r g i n female density, energy for eggs, and o v i p o s i t i o n s i t e density; 2) f i v e external processes that generate these inputs: aging, mortality, feeding, energy p a r t i t i o n i n g , and o v i p o s i t i o n s i t e creation; 3) one output density of i n f e r t i l e and f e r t i l e eggs l a i d ; 4) four component processes that modify inputs and generate the output: mating, egg production, egg f e r t i l i z a t i o n and o v i p o s i t i o n s i t e s e l e c t i o n (Table 1).* The density by age group of males and v i r g i n females re s u l t s from t h e i r i n t e r a c t i o n with the external processes of aging and mortality (Figure 1). These two densities i n t e r -act with the component process of mating to give density of mated females by age and age at copulation. This l a t t e r density also i n t e r a c t s with the processes of aging and mor-t a l i t y i n the same way as males and v i r g i n females. Energy a v a i l a b l e for eggs i s a product of the energy p a r t i t i o n i n g process. This process allocates energy form feeding and stored energy among the various energy require-*A more d e t a i l e d consideration of the four component processes i s given i n the sections dealing with each: Mating, Figure 3: Oviposition s i t e s e l e c t i o n , Figure 4; Egg production, Figure 6; Egg f e r t i l i z a t i o n , Figure 16. c Figure 1. Flow diagram of the re l a t i o n s h i p s among the processes of sexual reproduction and of the r e l a t i o n s h i p of sexual reproduction to other ecolo-g i c a l processes. Rectangles represent variables, hexagons represent processes. 8 . ^Feeding^ Energy Intake Energy P a r t i t i o n i n g Stored Energy O v i p o s i t i o n S i t e Density O v i p o s i t i o n S i t e Creation Male Density Energy f o r Eggs Aging and Mo r t a l i t y V i r g i n Female Density Mating^ SEXUAL REPRODUCTION Mated Female Density Egg Product U n f e r t i l i z e d Eggs Egg F e r t i l i z a t i o n F e r t i l i z e d Eggs Ovip o s i t i o n S i t e Selection Density of Eggs Laid ( F e r t i l e and I n f e r t i l e ) 9. Ta b l e 1. A d i s a g g r e g a t i o n o f the s e x u a l r e p r o d u c t i v e p r o c e s s . Processes and components c l a s s i f i e d as b a s i c : process or component always p a r t o f s e x u a l r e p r o d u c t i o n , or s u b s i d i a r y ( s u b s i d . ) : process o r component o p t i o n a l t o s e x u a l r e p r o d u c t i o n ( H o l l i n g 1966). • -Processes Components Occurrence Mating B a s i c Rate of s u c c e s s f u l s e a r c h a B a s i c B a s i c B a s i c Time exposed Handling time E x p l o i t a t i o n I n t e r f e r e n c e S u b s i d . Subsid. S u b s i d . Subsid. Subsid. S o c i a l f a c i l i t a t i o n L e a r n i n g M o t i v a t i o n Egg P r o d u c t i o n B a s i c O v a r i o l e a c t i v a t i o n . O v a r i o l e p r o d u c t i o n V i t e l l o g e n e s i s 3 O v a r i o l e d e a c t i v a t i o n b B a s i c B a s i c B a s i c Subsid. Egg F e r t i l i z a t i o n B a s i c Sperm storage Sperm r e l e a s e Subsid. B a s i c 10. Table 1. (continued) Processes Components Occurrence Oviposition S i t e Selection Subsid. Rate of successful search Time exposed Handling time E x p l o i t a t i o n Interference S o c i a l f a c i l i t a t i o n Motivation Learning Components modeled with data from l i t e r a t u r e . Components modeled with experimenter's data. Basic Basic Basic Subsid. Subsid. Subsid. Subsid. Subsid. 11. ments (e.g. r e s p i r a t i o n , egg production) and energy storage. The egg production process produces u n f e r t i l i z e d eggs by combining density of mated females with energy avail a b l e for eggs, thus, decreasing the store of available energy. Egg f e r t i l i z a t i o n unites stored sperm with u n f e r t i l i -zed eggs. This increases the store of f e r t i l i z e d eggs and decreases the store of u n f e r t i l i z e d eggs. Oviposition s i t e s e l e c t i o n determines density of eggs l a i d through i n t e r a c t i o n of mated females with o v i p o s i t i o n s i t e d . This reduces the store of f e r t i l i z e d and u n f e r t i l i z e d eggs. Mating, egg production and egg f e r t i l i z a t i o n were con-sidered basic as they are common to a l l sexually reproducing animals. Oviposition s i t e s e l e c t i o n by females was considered subsidiary, as some insects i n d i s c r i m i n a t e l y deposit t h e i r eggs wherever the female happens to be (Engelmann 1970 p. 193). 12. PART I MATING INTRODUCTION Mating i s considered to include a l l time consuming a c t i v i t i e s that are exclusive to male-female i n t e r a c t i o n leading to insemination. Thus mating would include court-ship, copulation, insemination, and any post copulatory refractory period. If mating i s considered to be one of many time consuming a c t i v i t i e s engaged i n by i n d i v i d u a l s , such as eating, feeding, o v i p o s i t i o n s i t e s e l e c t i o n , and predator avoidance, then i t may be possible to develop a single equation, or set of equations, that would be applicable to a l l these population i n t e r a c t i o n s . In the development of models for the processes of sexual reproduction, t h i s model of i n t e r a c t i n g populations could be used both for mating, the i n t e r a c t i o n between males and females, and for o v i p o s i t i o n s i t e s e l e c t i o n , the in t e r a c t i o n between females and o v i p o s i t i o n s i t e s . 13. MODEL OF INTERACTING POPULATIONS D e v e l o p m e n t o f t h e M o d e l . N o t i n g t h e e x p a n s i o n o f H o l l i n g ' s (1965, 1966) p r e d a t i o n m o d e l t o i n c l u d e a c o m b i n a t i o n o f p a r a s i t e s m a n d o v i p o s i t i o n s i t e s e l e c t i o n ( G r i f f i t h s and H o l l i n g 1 9 6 9 ) , N e i s h ( p e r s . comm.) s t a t e d t h a t t h e p r e d a t i o n m o d e l c a n be v i e w e d as a g e n e r a l m o d e l f o r r e s o u r c e u s e by an o r g a n i s m . M o r e o v e r , a s m o d i f i e d b e l o w , t h e p r e d a t i o n m odel c a n f u r t h e r be v i e w e d a s a g e n e r a l m o d e l f o r any i n t e r a c t i o n b e tween two d i s c r e t e p o p u l a t i o n s ; one o r b o t h o f w h i c h i s a n i m a t e . F o r c o n t i n u i t y , H o l l i n g ' s n o t a t i o n f o r h i s p r e d a t i o n m o d e l (1966 p . 11) i s u s e d f o r t h i s g e n e r a l m o d e l o f i n t e r -a c t i n g p o p u l a t i o n s ( T a b l e 2 ) . * I n t h e m o d e l o f i n t e r a c t i n g p o p u l a t i o n s t h e r e a r e no p r e c o n c e p t i o n s a b o u t r e l a t i v e r o l e s o f t h e two p o p u l a t i o n s (No a n d P, T a b l e 2 ) ; t h e y a r e i n t e r -c h a n g e a b l e . E i t h e r c o u l d be t h e p r e e a t o r o r p r e y , o r t h e m a l e o r f e m a l e d u r i n g m a t i n g , o r t h e c l e r k o r c u s t o m e r a t a g r o c e r y s t o r e c h e c k o u t . H o l l i n g ' s p r e d a t i o n m o d e l d o e s n o t a l l o w f o r t h e i n t e r -c h a n g e a b i l i t y o f t h e two p o p u l a t i o n s . The b a s i c p r e d a t i o n f u n c t i o n ( m o d i f i e d f r o m G r i f f i t h s a n d H o l l i n g 1969 p . 789) : * I n t h i s t e x t a l l l o w e r c a s e l e t t e r s d e n o t e c o n s t a n t s ( a , c , e, g, e t c . ) . U p p e r c a s e l e t t e r s u s u a l l y d e n o t e v a r i a b l e s LA, Co, d E / d T , Mr, e t c . ) , a l t h o u g h t h e y o c c a s i o n a l l y d e n o t e c o n s t a n t s (D, D r , Ds, and F ) . 14. Table 2. A comparison of d e f i n i t i o n s of parameters common to both Holling's predation function (Equation 1) and the general function for i n t e r a c t i n g populations (Equation 6). Parameter Predation (Holling 1966) Interacting Populations (Present Paper) Na Density prey attacked Density of interactions No Prey density Density of f i r s t population P Predator density Density of second population Rate of successful search Rate of successful search Tt Time prey exposed to predators Time two populations together Th Time spent handling prey Time per i n t e r a c t i o n 15. Na = A-No-(Tt-P-Th'Na) (1) assumes, for a constant predator density (P), that density of attacks (Na) increases with prey density (No). This increase i s negatively accelerated r e s u l t i n g i n density of attacks (Na) approaching a plateau for high prey density (No). Interchanging the two populations gives: Na = A«P« (Tt-Na-Th-Na) (2) which assumes a p o s i t i v e , l i n e a r r e l a t i o n s h i p between density of attacks (Na) and prey density (No). Holling (1966) has shown that such a r e l a t i o n s h i p does not e x i s t and that density of attacks (Na) approaches a plateau for high prey densities (No) as predicted by Equation 1. Holling's function, adequate as a model of predation, can not be used as a general function for i n t e r a c t i n g popu-la t i o n s because of these preconceptions about roles of the two populations. The simplest predation function (modified from Holling 1966 p. 9), and thus, the simplest function f o r i n t e r a c t i n g populations, assumes density of interactions i s a l i n e a r function of rate of successful search, time exposed, and density of both populations: Na = A'Tt-No-P (3) multiplying by Tt gives: 16. Na«Tt = A«(Tt-No)•(Tt-P) (4) where (Tt «No) and (Tt«P) are t o t a l time avai l a b l e to the populations for i n t e r a c t i o n . In terms of predation, (Tt«No) represents t o t a l time a v a i l a b l e for prey to be handled, and (Tt«P) represents t o t a l time avai l a b l e for predators to handle prey. If (Th-Na) represents time l o s t to both popu-l a t i o n s due to i n t e r a c t i o n s , then: Na-Tt = A-(Tt-No-Th-Na)•(Tt«P-Th-Na) (5) transposing gives: .. .7 _, Th'.Na. Th'-Na. , -. Na = A-Tt.No-P. * ( 1 - T t T p - ) ( 6 ) which i s the basic function f o r the model of i n t e r a c t i n g populations. This equation was derived i n a manner s i m i l a r to the d e r i v a t i o n of the predation function (Equation 1 above) by H o l l i n g (1966 p. 11). I t i s interchangeable with respect to the two populations (No and P) s e t t i n g no preconditions on r e l a t i v e roles of the two populations. The general function f o r i n t e r a c t i n g population (Equation 6) i s s i m i l a r to the function from which i t was derived (Equation 3). Equation 6 i s equal to Equation 3 m u l t i p l i e d by the expressions ( l - ^ [ ^ ) and (,l-^]p a) which represent the proportion of each population not presently engaged i n i n t e r a c t i o n . Holling's predation function (Equation 1) i s a s p e c i a l case of the general function for i n t e r a c t i n g populations 17. (Equation 6). Assume for predation that prey density i s much greater than predator density (No>>P). As both populations engage i n the same density of interactions (Na) with the same handling time (Th), the proportion of prey engaged i n i n t e r -action i s much less than the proportion of predators so Th•Na Th«Na engaged ( j^- No"<<:Tt P ^ " A s t' i e P r o P o r t ; L O n °f predators engaged i s at most one (^fp—£l)» the proportion of prey Th • Na engaged i s much less than one ( N - <<1). Therefore, the proportion of prey not engaged i s approximately one, i . e . , Th«Na . „ , ... , . , ,, Th.Nav . , Tt.No Substituting one for ( l ~ T t . ^ Q ) i n t n e general function (Equation 6) gives, with transposing, the predation function (Equation 1). Thus, Holling's predation function i s a l i m i t i n g case of the general function for i n t e r a c t i n g popu-l a t i o n s where density of one population i s much greater than that of the other. The general function applies only where the populations are approximately equal i n density (No - P) or where f l u c t u a t i o n s i n population density are such that f i r s t one and then the other has the higher density: P t = l < N o t = l ; N o t = 2 < P t = 2 ' Further Development of the Model. There are three further complexities to the general function for i n t e r a c t i n g popu-l a t i o n s (Equation 6): 1) i n t e r a c t i o n a f f e c t s upon either population ( e x p l o i t a t i o n ) ; 2) other interactions which compete for the av a i l a b l e time (interference), and 3) the 18. frequency d i s t r i b u t i o n of inte r a c t i o n s among ind i v i d u a l s of the two populations. Long term e x p l o i t a t i o n i s handled by d i v i d i n g the time the two populations are together (Tt) into short i n t e r v a l s , and adjucting eit h e r one or both population densities (No and/ or P) as i s appropriate at the end of each i n t e r v a l ( G r i f f i t h s and H o l l i n g 1969 p. 801-802). Short-term e x p l o i t a t i o n i s already included i n the general function (Equation 6); Th'Na. , ,, Th'Na, . . , . . . ^  ( l ~ T t < N o ) and ( l - _ - p — ) measure current involvement i n i n t e r -action and, thus, are indices of temporary reduction (exploitation) i n the two populations. Secondly, interference between two populations occurs when population growth or other a t t r i b u t e s of each population are d i r e c t l y i n h i b i t e d by the other (Odum 1971 p. 211). Thus, time spent by a predator attacking a prey i n t e r f e r e s with the a b i l i t y of both predator and prey to f i n d a mate. Broadly, interference, r e l a t i v e to the i n t e r a c t i o n i n question, comes about when one or both populations i s engaged i n a second i n t e r a c t i o n e i t h e r with a t h i r d population or with i t s e l f . The l a t t e r i s a second l i m i t i n g case of the general function of i n t e r a c t i n g population (Equation 6) occurring when No and P are the same population. I t describes i n t e r -ference within a population. I t i s assumed that the two inter a c t i o n s do not occur at the same time. If one population (P) engages i n a second time consuming i n t e r a c t i o n where N equals density of interactions and T equals 19. handling time, then N'T i s the time taken for the second i n t e r a c t i o n . This time (N«T) represents the interference e f f e c t of the second i n t e r a c t i o n on the f i r s t i n t e r a c t i o n and reduces the proportion of the population (P) l e f t for i n t e r a c t i o n with the other population (No) to ( 1 - T n ' N ^ t + p T * N ) . Thus, the general function f o r i n t e r a c t i n g populations (Equation 6) becomes: KT , m. »ir « n Th«Na N ,, Th-Na + T-Nv No = A-Tt-No-P. ( l - T t . N p ) • (1- rtTP~ } ( 7 ) F i n a l l y , the frequency d i s t r i b u t i o n of interactions among members of each i n t e r a c t i n g population i s complex. In b r i e f , there are two basic s i t u a t i o n s : i n d i v i d u a l s can i n t e r -act only once, or i n d i v i d u a l s can i n t e r a c t more than once. Individuals only capable of a single i n t e r a c t i o n (e.g. a prey can only be eaten once) are handled as outlined above i n the discussion of long-term e x p l o i t a t i o n . I f , however, individ u a l s are capable of repeated i n t e r a c t i o n s , then i t may be necessary to model the frequency of occurrence of the various multiple i n t e r a c t i o n s ( i . e . how many females never copulate, copulate once, twice, three times, e t c . ) . The general function for i n t e r a c t i n g populations (Equation 6) predicts t o t a l number of i n t e r a c t i o n s , not frequency d i s t r i b u t i o n of those i n t e r -actions. At the extreme, there are two possible e f f e c t s of multiple i n t e r a c t i o n s : 1) No e f f e c t of i n t e r a c t i o n a f t e r the f i r s t ; " or 2) e f f e c t of i n t e r a c t i o n s may be independent and a d d i t i v e . 20. I n n a t u r e , many e x a m p l e s o f m u l t i p l e i n t e r a c t i o n may be more compl e x and f a l l b e t w e e n t h e s e s i m p l e e x t r e m e s . A d d i t i o n a l i n t e r a c t i o n s w i l l h a v e some e f f e c t , b u t t h e e f f e c t m i g h t d i m i n i s h a t h i g h f r e q u e n c y w i t h s u c c e s s i v e i n t e r a c t i o n s ( G r i f f i t h s a n d H o l l i n g 1969 p . 7 9 6 ) . Components o f t h e M o d e l . H o l l i n g (1966) s e p a r a t e d t h e components o f p r e d a t i o n i n t o two t y p e s : 1) t h o s e d e a l i n g w i t h t h e f u n c t i o n a l r e s p o n s e t o p r e y d e n s i t y ; and 2) t h o s e d e a l i n g w i t h t h e f u n c t i o n a l r e s p o n s e t o p r e d a t o r d e n s i t y , ( T a b l e 3 ) . A more g e n e r a l a p p r o a c h i s t o c l a s s i f y components i n t o t h r e e t y p e s : 1) t h o s e i n v o l v i n g r e s p o n s e s t o i n t e r -a c t i o n b e t w e e n two p o p u l a t i o n s ; 2) t h o s e i n v o l v i n g r e s p o n s e s t o a c t i o n w i t h i n one p o p u l a t i o n ; and 3) t h o s e i n v o l v i n g r e s p o n s e s t o i n d i v i d u a l e x p e r i e n c e f r o m p a s t i n t e r a c t i o n s and a c t i o n s ( T a b l e 4, F i g u r e 2 ) . T h e r e i s no r e l a t i o n s h i p b e t w e e n H o l l i n g ' s component t y p e and i t s o c c u r r e n c e ( T a b l e 3 ) ; components c l a s s i f i e d as a r e s p o n s e t o p r e d a t o r d e n s i t y , o r t o p r e y d e n s i t y , c a n be e i t h e r b a s i c o r s u b s i d i a r y . A r e l a t i o n s h i p d o e s e x i s t f o r t h e new s y s t e m o f c l a s s i f i c a t i o n ( T a b l e 4 ) ; components c l a s s i f i e d a s a r e s p o n s e t o i n t e r a c t i o n b e t w e e n two p r i m a r y p o p u l a t i o n s , b u t n o t w i t h some t h i r d p o p u l a t i o n , a r e a l w a y s b a s i c w h i l e components c l a s s i f i e d a s r e s p o n s e s e i t h e r t o a c t i o n w i t h i n a p o p u l a t i o n o r t o i n d i v i d u a l e x p e r i e n c e f r o m p a s t a c t i o n s and i n t e r a c t i o n s a r e a l w a y s s u b s i d i a r y . 21. Table 3. Components of the predation process. Components c l a s s i f i e d as the functional response to eith e r prey or predator (pred.) density (Holling 1966). Component Type Occurrence Rate of successful search prey basic Time exposed prey basic Handling time prey basic E x p l o i t a t i o n pred. basic Interference between predators pred. subsid. S o c i a l f a c i l i t a t i o n pred. subsid. Hunger (motivation) prey subsid. Learning by predator prey subsid. I n h i b i t i o n by prey prey subsid. Avoidance learning by prey pred. subsid. T a b l e 4. P o s s i b l e components o f t h e p r o c e s s e s i n v o l v i n g i n t e r a c t i n g p o p u l a t i o n s . Components c l a s s i f i e d as b e i n g b e t w e e n : component i n v o l v e s i n t e r a c t i o n b e t w e e n p o p u l a t i o n s ; w i t h i n : component i n v o l v e s a c t i o n w i t h i n a p o p u l a t i o n ; o r e x p e r i e n c e : component i n v o l v e s i n d i v i d u a l e x p e r i e n c e f r o m p a s t i n t e r a c t i o n s and a c t i o n s . Component T y p e O c c u r r e n c e R a t e o f s u c c e s s f u l s e a r c h b e t w e e n b a s i c Time e x p o s e d b e t w e e n b a s i c H a n d l i n g t i m e b e t w e e n b a s i c E x p l o i t a t i o n : s h o r t - t e r m l o n g - t e r m b etween e x p e r i e n c e b a s i c s u b s i d . I n t e r f e r e n c e b e t w e e n a s u b s i d . S o c i a l f a c i l i t a t i o n w i t h i n s u b s i d . M o t i v a t i o n e x p e r i e n c e s u b s i d . L e a r n i n g : t h r e s h o l d e f f e c t s t e c h n i q u e e f f e c t s e x p e r i e n c e e x p e r i e n c e s u b s i d . s u b s i d . Si Component i n v o l v e s a s e c o n d i n t e r a c t i o n b e t w e e n one o f t h e two p o p u l a t i o n s i n v o l v e d i n t h e f i r s t i n t e r a c t i o n and some t h i r d p o p u l a t i o n . F i g u r e 2. A p a r t i a l d i s a g g r e g a t i o n o f t h e g e n e r a l i z e d p r o c e s s o f i n t e r a c t i o n b e t w e e n two p o p u l a t i o n s . R e s p o n s e t o i n t e r a c t i o n b e t w e e n two p o p u l a t i o n s G e n e r a l i z e d p r o c e s s -R e s p o n s e t o a c t i o n w i t h i n one p o p u l a t i o n -R e s p o n s e t o i n d i v i d u a l e x p e r i e n c e f r o m p a s t i n t e r a c t i o n s and a c t i o n s R a t e o f s u c c e s s f u l s e a r c h -P r o b a b i l i t y o f s u c c e s s R e a c t i v e d i s t a n c e ^ R e l a t i v e v e l o c i t y - T i m e e x p o s e d — - H a n d l i n g t i m e S h o r t t e r m ^_ e x p l o i t a t i o n - I n t e r f e r e n c e -_ S o c i a l f a c i l i t a t i o n • L o n g t e r m e x p l o i t a t i o n - M o t i v a t i o n — T h r e s h o l d e f f e c t • - L e a r n i n g T e c h n i q u e e f f e c t Certain of Holling's components (Table 3) have been combined i n l i g h t of the new c l a s s i f i c a t i o n of components (Table 4 ) . Learning by predators, i n h i b i t i o n by prey, and avoidance learning by prey have been lumped under the term "learning", which can have p o s i t i v e or negative reinforcement. Learning i s s p l i t into threshold and technique e f f e c t s : the former i n t e r a c t i n g with motivation, and the l a t t e r changing handling time. E x p l o i t a t i o n has been s p l i t into short-term e x p l o i t a -t i o n (the loss of i n t e r a c t i n g i n d i v i d u a l s from both populations due to handling time) and into long-term e x p l o i t a t i o n (temporary or permanent removal of individuals from either population due to e f f e c t s of the i n t e r a c t i o n ) . E x p l o i t a t i o n , interference, s o c i a l f a c i l i t a t i o n , and moti-vation (Holling's hunger) have been expanded to include both populations. A l l these changes are designed to make the components general by applying them equally to both popu-l a t i o n s . The eight components generally do not change i n terms of being basic or subsidiary i n the change from predation to general i n t e r a c t i o n of populations. Rate of successful search, time exposed, and handling time remain basic while interference, s o c i a l f a c i l i t a t i o n , motivation and learning are s t i l l subsidiary (Tables 3 and 4 ) . However, while short-term e x p l o i t a t i o n remains basic, long-term e x p l o i t a t i o n becomes subsidiary. Presence of long-t e r m e x p l o i t a t i o n w i l l d e p e n d on t y p e o f p o p u l a t i o n i n t e r -a c t i o n . I t o c c u r s i n a l l f o r m s o f p r e d a t i o n b u t n o t i n a l l c a s e s o f o v i p o s i t i o n s i t e s e l e c t i o n , p a r a s i t i s m , a n d m a t i n g . T h u s , i t i s c l a s s i f i e d a s s u b s i d i a r y . ( F o r e x a m p l e , a f e m a l e may o r may n o t m a t e more t h a n o n c e ) . A p p l i c a t i o n o f t h e M o d e l t o R e p r o d u c t i o n i n D r o s o p h i l a m e l a n o - g a s t e r . The two p o p u l a t i o n s i n t e r a c t i o n s a p p l i c a b l e t o s e x u a l r e p r o d u c t i o n , m a t i n g a n d o v i p o s i t i o n s i t e l o c a t i o n , u s e t h e t h r e e b a s i c c o m p o n e n t s p l u s a s u b s e t o f t h e f i v e s u b s i d i a r y c o m p o n e n t s w h i c h d e p e n d on t h e n a t u r e o f t h e two i n t e r a c t i n g p o p u l a t i o n s . W h i l e l o n g - t e r m e x p l o i t a t i o n a n d i n t e r f e r e n c e n e e d n o t be c o m p o n e n t s o f m a t i n g , t h e f o r m e r o c c u r s f o r f e m a l e D. m e l a n o g a s t e r , a n d t h e l a t t e r o c c u r s f o r m a l e s ( T a b l e 5 ) . S i m i l a r l y , s o c i a l f a c i l i t a t i o n n e e d n o t o c c u r i n a l l c a s e s o f m a t i n g , b u t i t d o e s o c c u r f o r b o t h f e m a l e a n d m a l e D. m e l a n o g a s t e r . M o t i v a t i o n i s i m p o r t a n t f o r f e m a l e s b u t , f o r t h e p r e s e n t , i t i s a s s u m e d n o t t o be f o r m a l e s . L e a r n i n g i s n o t e x h i b i t e d b y e i t h e r s e x . A t e n t a t i v e r e l a t i o n s h i p among t h e s e c o m p o n e n t s ( F i g u r e 3) shows t h e r e a r e t w o i n p u t s , v i r g i n f e m a l e d e n s i t y a n d m a l e d e n s i t y , a n d one o u t p u t , m a t e d f e m a l e d e n s i t y . G i v e n v i r g i n f e m a l e d e n s i t y , m o t i v a t i o n d e t e r m i n e s r e c e p t i v e f e m a l e d e n s i t y . T h r o u g h r a t e o f s u c c e s s f u l s e a r c h , m a l e d e n s i t y a n d r e c e p t i v e f e m a l e d e n s i t y g i v e d e n s i t y o f m a l e -26. T a b l e 5. P o s s i b l e components of the mating process f o r D r o s o p h i l a melanogaster. Components c l a s s i f i e d as b e i n g 6*,$>: component i n v o l v e s i n t e r a c t i o n between male and r e c e p t i v e female p o p u l a t i o n s ; 6*,6* and o,c^: component i n v o l v e s a c t i o n w i t h i n the male or r e c e p t i v e female p o p u l a t i o n ; or (j>: component i n v o l v e s i n d i v i d u a l e x p e r i e n c e by r e c e p t i v e females from p a s t i n t e r a c t i o n s and a c t i o n s . Component Type Rate o f s u c c e s s f u l s e a r c h Time exposed 6*,o. Ha n d l i n g time 6 E x p l o i t a t i o n : s h o r t - t e r m 6*,cj> long-term <p I n t e r f e r e n c e 6*, 6" S o c i a l f a c i l i t a t i o n <3,<S and £,o M o t i v a t i o n o Figure 3. Flow diagram of the relationships among the components of mating. Rectangles represent v a r i a b l e s , hexagons represent com-ponents . V i r g i n Female Density Male Density <( Motivation j Receptive Female Density MATING < Rate of Successful Search Density of Male-Female Pairs < 1 Handling_Time Time Exposed Mated Female Density 28. female p a i r s . Mated female density i s determined by density of male-female pairs acting through the components handling time and time exposed which also determine short-term e x p l o i t a t i v e e f f e c t s on v i r g i n females and males. The subsidiary components of i n t e r a c t i n g populations included i n D. melanogaster mating are not, except for motivation, shown (Figure 3) because they a f f e c t one or more basic components. Long-term e x p l o i t a t i o n of females i s included i n the feedback between handling time and v i r g i n female density. Interference by males e f f e c t s male-female pa i r creation through rate of successful search. Rate of successful search i s also affected by s o c i a l f a c i l i t a t i o n . F i n a l judgement on components of o v i p o s i t i o n s i t e s e l e c t i o n involving female D. melanogaster and o v i p o s i t i o n s i t e s (Table 6 and Figure 4) w i l l have to await further study. Relabelling the variables and parameters of the model of i n t e r a c t i n g populations (Equation 6) for the mating process (Table 7) gives: o 7> m i . TT / i " T r a'«Co, Tnt ' C O x / o x Co-Ac.Tt-Mr-W. ( 1 — _ ) . (!-___) (8) 29. Table 6 . Possible components of the o v i p o s i t i o n s i t e s e l e c t i o n process inv o l v i n g female Drosophila  melanogaster and o v i p o s i t i o n s i t e s . Components c l a s s i f i e d as being o , s i t e and 0 , 6 * : component involves i n t e r a c t i o n , r e s p e c t i v e l y , between mated females and ov i p o s i t i o n s i t e s , and between mated females and males; 0*9: component involves i n t e r a c t i o n within the mated female population; or <j>: component involves i n d i v i d u a l experience from past int e r a c t i o n s and actions of the mated females. Component Type Rate of successful search 5 , s i t e Time exposed 5 ,site Handling time 9 , s i t e E x p l o i t a t i o n : short-term (j>,site Interference £,<j> and 0 , 6 * Motivation o_ F i g u r e 4. F l o w d i a g r a m o f t h e r e l a t i o n s h i p among t h e c o m p o n e n t s o f o v i p o s i t i o n s i t e s e l e c t i o n . R e c t a n g l e s r e p r e s e n t v a r i a b l e s , h e x a g o n s r e p r e s e n t c o m p o n e n t s . Density of Mated Females by Number of U n f e r t i l i z e d and F e r t i l i z e d Eggs OVIPOSITION SITE SELECTION Rate of Successful Search 1 Density of Female-Site Contacts i < Motivation > Density of Ovipositing Females Handling Time Time "Expos e~d ~~ Density of Oviposition Sites by S u i t a b i l i t y Density of Eggs Laid F e r t i l e and I n f e r t i l e 31. Table 7. A comparison of parameters common to both the general function f o r i n t e r a c t i n g populations (Equation 6) and the s p e c i f i c function f o r mating (Equation 8 ) . Interacting Populations Mating Na Co Density of copulations A Ac Rate of successful search Tt Tt Time exposed (time males and females are together) Th Tm Handling time (courtship plus copulation time) No W Density of receptive females P Mr Density of males 32. PARAMETER ESTIMATION FOR MATING MODEL Rate of Successful Search. Rate of successful search i s a l i n e a r function of three v a r i a b l e s : 1) p r o b a b i l i t y (P) of a contact between a male and a receptive female r e s u l t i n g i n copulation; 2) r e a c t i v e distance of a male to a female (D), and 3) r e l a t i v e walking v e l o c i t y of males and females (V): Ac = P. (2D) -V (9)* Manning (1967) found female D. melanogaster to be either completely receptive or unreceptive. The mating function (Equation 8) deals only with receptive females whose density (W) i s determined below. Thus, the p r o b a b i l i t y of a successful contact i s assumed to be equal to one (P = 1.0). I t i s possible that at high population densities multiple contacts couid occur, two males-one female or one male-two females, which would cause P to be less than one. Multiple contacts would be handled as a second type of i n t e r -action (interference) as outlined i n the subsection on further development of the model. However, no data i s av a i l a b l e for D. melanogaster on t h i s p o s s i b i l i t y , so i t i s ignored. The reactive distance of D. melanogaster males i s less than one centimeter: D = 0.69 cm. + 0.05 (SE), N = 64 (Gpssard unpublished data). •Derived from an equation by Holling (1965 p. 54) Haynes and Si s o j e v i d (1966 p. 126-127) determined v e l o c i t y of D. melanogaster as a function of density (Table 8). The data i s i n the form of contacts per hour with a unit area on the side of a one gallon mason j a r . "The number of contacts made i s a function of v e l o c i t y , time, f l y density, and s i z e of the unit area, and i f every-thing but v e l o c i t y i s measured, then v e l o c i t y can be calculated using the expression derived" i n Appendix I (Holling 1966 p. 32): Nc - 3.14.R2-(Mr+Ms) V m = 2.R.T.(Mr+Ms) ( 1 0 ) where: Vm = average f l y v e l o c i t y i n m. per hr. Nc = contacts with uni t area per hr. r = radius of u n i t area i n m. Mr+Ms = density of males and females i n number per hr. per m. T = duration of experiment i n hr. A l i n e a r r e l a t i o n s h i p between density and v e l o c i t y i s suggested by the data (Table 8). This r e l a t i o n s h i p was con-firmed by an analysis of variance (Table 9) for the regression equation: Vm = e + d-(Mr+Ms) (11) Maximum and minimum values for v e l o c i t y are reasonable assumptions, but the f l y v e l o c i t y function (Equation 11) 34. Table 8. Relationship between f l y density (Mr+Ms) and f l y v e l o c i t y (Vm). Data are for Drosophila melanogaster from Haynes and S i s o j e v i c (1966). Analysis a f t e r H o l l i n g (1966): see Appendix I where R = 0.009 m. and T = 1.0 hr. Density was converted from f l i e s per gallon (N) to 2 f l i e s per m. (Mr+Ms) by assuming a surface area of 2 0.1502 m. per one gallon j a r . Density Density Contacts V e l o c i t y f l i e s / h r . f l i e s / h r . number meters per j a r per m.2 per hr. per hr. (N) (Mr+Ms) (Nc) (Vm) 4 26.5 3.28 6.85 -8 53.0 8.84 9.25 12 79.5 15.6 10.8 16 106.1 24.8 13.0 Table 9. Analysis of variance for l i n e a r regression, Equation 11, using f l y v e l o c i t y data i n Table 8. Source SS DF P Regression 18.92 1 0.289 % Re s i d u a l 5 0.11 2 T o t a l b 19.03 3 Parameter estimates: e = 5.0 and d = 0.0752 Residual for f i t t i n g mean f l y v e l o c i t y (Vm) = 9.98 m. per hr. implies only the l a t t e r (e). A second experiment by Haynes and Sisojevid (1966 p. 132) indicated that maximum v e l o c i t y (f) of D. me1anogaster i s approximately 13.0 m. per hr. (n = 6 ) . As no c u r v i l i n e a r trend was noted i n the data of Haynes and Sisojevid's f i r s t experiment, t h i s maximum v e l o c i t y (f = 13.0) was incorporated into the f l y v e l o c i t y function (Equation 11) by assuming a threshold at a density of 106.6 f l i e s per m. which corresponds to t h i s maximum v e l o c i t y : Vm = e + d-(Mr+Ms) [Mr+Mx < 106.6] (12) Vm = f [106.6 < Mr+Ms] The average r e l a t i v e v e l o c i t y (V) of two separate objects with v e l o c i t i e s Vs and Vr i s approximated by: 1 V = (Vs 2 + V r 2 ) Z (13) This approximation has a maximum error of 10% when Vs = Vr (Skellam 1958), which i s the case here because Haynes and SisojevicS d i d not determine separate v e l o c i t i e s for males and females. With Vm as the common v e l o c i t y for both males and females (Vm = Vs = Vr), Equation 13 becomes: 1 • . V = 22-Vm (14) = 1.41-Vm Combining the function for rate of successful search with the f l y v e l o c i t y function and Skellam's r e l a t i v e v e l o c i t y f u n c t i o n ( E q u a t i o n 9, 12, 14 r e s p e c t i v e l y ) g i v e s t h e e q u a t i o n f o r r a t e o f s u c c e s s f u l s e a r c h (Ac) a s a f u n c t i o n o f p r o b a b i l i t y (P) o f a s u c c e s s f u l c o n t a c t , r e a c t i v e d i s t a n c e ( D ) , a n d d e n s i t y o f f l i e s (Mr+Ms): A c = P.(2D).1.41.[e + d.(Mr+M s ) ] [Mr+Ms < 106.6] (15) A c = P-(2D)-1.41.f [106.6 < Mr+Ms] Tim e E x p o s e d a n d H a n d l i n g T i m e . T i m e e x p o s e d ( T r ) i s n o m i n a l l y t a k e n t o be 16 h r . , t h e l e n g t h o f t h e l i g h t p e r i o d i n b o t h e g g p r o d u c t i o n e x p e r i m e n t s ( P a r t I I ) . H a n d l i n g t i m e (Tm = 0.7 h r . ) i s t h e sum o f c o u r t s h i p a n d c o p u l a t i o n t i m e . ' Time t a k e n b y a m a l e a n d f e m a l e D. m e l a n o g a s t e r v g . f o r c o u r t s h i p (0.35+0.012 (SE) h r . , N = 71) a n d f o r c o p u l a t i o n (0.3 h r s . ) was d e t e r m i n e d , r e s p e c t i v e l y , b y M a n n i n g (1967) a n d by L e f e v r e a n d J o h n s o n (1962). E x p l o i t a t i o n , I n t e r f e r e n c e , S o c i a l F a c i l i t a t i o n a n d M o t i v a t i o n . L o n g - t e r m e x p l o i t a t i o n , a p e r m a n e n t e f f e c t o f i n t e r a c t i o n u p o n e i t h e r p o p u l a t i o n , o c c u r s f o r f e m a l e D. m e l a n o g a s t e r a s i t i s as s u m e d t h a t t h e y become i r r e v e r s i b l y u n r e c e p t i v e a f t e r c o p u -l a t i o n . T h i s l o n g - t e r m e x p l o i t a t i o n i s h a n d l e d i n t h e manner o u t l i n e d a b o v e ( F u r t h e r D e v e l o p m e n t o f t h e M o d e l ) . I n t e r f e r e n c e c a n o c c u r b e t w e e n m a l e D. m e l a n o g a s t e r ( B a s t o c k ' a n d M a n n i n g 1955 p. 98). A l t h o u g h i t c o u l d be h a n d l e d i n t h e . m a n n e r o u t l i n e d a b o v e , ( F u r t h e r D e v e l o p m e n t o f t h e M o d e l ) , 38. interference w i l l be ignored for the present. S o c i a l f a c i l i t a t i o n occurs since v e l o c i t y of both males and females i s proportional to t h e i r density, t h i s increases the p r o b a b i l i t y of contact, as outlined above (Rate of Successful Search). Motivation i s included i n the subsections on rate of successful search and on determination of receptive female density which follows. Motivation: Density of Receptive Females and Males. Density of receptive females (W) i s dependent on female density and age structure (Ms(A)*) and on female r e c e p t i v i t y (R) which i s i t s e l f a function of age (Manning 1967). Manning's data (Figure 5) suggest that f o r a population of female D. melanogaster i n d i v i d u a l s are unreceptive for a few days a f t e r eclosion. A f t e r t h i s period, a high constant propor-t i o n of females are receptive. Later, the proportion of receptive females declines to zero. Manning has l i n k e d the sudden r i s e i n r e c e p t i v i t y (Figure 5) to the release of juvenile hormone by the corpora a l l a t a . This i s the same hormone that causes the female's ovaries to develop. The hormone's release i s probably related to threshold age (Tmin) at which females can f i r s t lay eggs (as mentioned below under EGG PRODUCTION: F i r s t Experiment: Results). I t i s not s u r p r i s i n g that a threshold *Ms(A) represents the vector of v i r g i n female densi-t i e s (MS) f o r given ages (A). Figure 5. E f f e c t of age of v i r g i n female Drosophila  melanogaster (A) on proportion of females accepting a courting male (re c e p t i v i t y or R). Each point represents 16 or more females. Data are from Manning (1967). The s o l i d l i n e of p r e d i c t i o n i s generated by the female r e c e p t i v i t y function (Equation 16) using parameter estimates from Table 10. 40. (Tr) also exists for minimal age of r e c e p t i v i t y . Because there i s a delay for egg development between release of the hormone and laying of the f i r s t eggs, r e c e p t i v i t y (Tr<2.0 days) begins before f i r s t egg laying (Tmin = 4.7+0.5 days). The remaining trends i n the data (Figure 5) are possibly linked to other p h y s i o l o g i c a l events i n the corpora a l l a t a . The high l e v e l of r e c e p t i v i t y (g) may be due to continued production of juvenile hormone. The decline (h) i n proportion receptive appears to be exponential. Possibly the mechanism maintaining female r e c e p t i v i t y has a constant p r o b a b i l i t y of f a i l u r e as the female ages. A l t e r n a t i v e l y , the l e v e l of hormone production may gradually f a l l below a theshold. The following function summarizes these hypotheses: R =0.0 [A <_ Tr] R = g [ f : £ f ] (16) R = g . e - h ' ( A - T f ) [Tf< A] where: R = p r o b a b i l i t y of a v i r g i n female being receptive g = i n i t i a l high l e v e l of r e c e p t i v i t y h = rate of decline i n r e c e p t i v i t y from g Tr = minimum age for v i r g i n r e c e p t i v i t y Tf = age at s t a r t of decline i n r e c e p t i v i t y 41. This female r e c e p t i v i t y function was f i t t e d to Manning's data (Figure 5) by non-linear regression (Table 10). Non-linear regression gives a larger value for the regression sum of squares than does l i n e a r regression. As most parametric tests assume l i n e a r r e l a t i o n s h i p s , the true p r o b a b i l i t y values (P) are larger than those given. In Table 10, the P value given i s only a minimum estimate ( i . e . a biased estimate) of the actual P value. Thus, there i s a greater chance of committing a Type 1 error (rejecting a true hypothesis). Care must therefore be taken i n accepting the i n c l u s i o n of a d d i t i o n a l parameter estimates. This applies to a l l analysis of variance tables i n t h i s thesis except Tables 9 and 11. Using the female r e c e p t i v i t y function (Equation 16) to c a l c u l a t e r e c e p t i v i t y (R) of v i r g i n females of a given age (Ms(A)), the t o t a l density of receptive females (W) can be determined from: W = ZMs(A)-R (17) A l l males are t e n t a t i v e l y assumed to be receptive. Therefore, density of receptive males i s the same as t o t a l male density (Mr), which i s found by summing over a l l age groups (Mr(A)*): Mr = IMr(A) (18) *Mr(A) represents the vector of male densities (Mr) for given ages (A). Table 10. Analysis of variance f o r regression, Equation 16, using female r e c e p t i v i t y data i n Figure 5. P i s a biased estimate; see text. Source SS DF P Regression 1335 1 0.275 % R e s i d u a l 3 1564 15 Total 2899 16 Si Parameter estimates: g = 0.91; Tf = 22.0; h = 0.221 . 43. CONCLUSION OF PART I r In theory, the general mating function (Equation 8) i n conjunction with the function for rate of successful search (Equation 15) and the functions determining density of receptive females and males (Equation 16, 17 and 18) can predict number of females copulated during any given time period. However, u n t i l an experimental basis i s given to these functions, such theorizing i s of li m i t e d value. Nevertheless, these functions contain the important components of mating, show a tentative way of r e l a t i n g these components, and by so doing give guidance to a future experimental program. 44. PART II EGG PRODUCTION INTRODUCTION Egg production can be considered as having four c h a r a c t e r i s t i c s : 1) two external inputs: mated female density and energy f o r eggs: 2) one i n t e r n a l input: number of mature ovarioles; 3) one output: number of u n f e r t i l i z e d eggs; and 4) four components (Table 1) that modify the i n -puts and generate the output: ovariole a c t i v a t i o n , ovariole deactivation, ovariole production, and v i t e l l o g e n e s i s . For D. melanogaster, and some other i n s e c t s , there i s an i n i t i a l period when t h e i r ovarioles do not produce eggs. This stage ends (Figure 6) when ovariole a c t i v a t i o n stimulates the ovarioles to begin oocyte production. Ovariole production causes activated ovarioles to produce ovarian cystoblasts by d i v i s i o n of stem c e l l s . Once pro-duced, ovarian cystoblasts under go several d i v i s i o n s r e s u l t i n g i n one oocyte and a number of nurse c e l l s . (There are four d i v i s i o n s giving one oocyte and 15 func-t i o n a l nurse c e l l s i n D. melanogaster.). Oocytes then begin to increase i n siz e through addition of n u t r i t i v e material (vitellogenesis) u n t i l they become mature eggs (King 1970). Ovariole production i s not continuous as ovarioles can be deactivated, and possibly reactivated during a female's l i f e t i m e , often as a r e s u l t of mating Figure 6. Flow diagram of the rela t i o n s h i p s among the components of egg production. Rectangles represent v a r i a b l e s , hexagons represent components. 4 5 . EGG PRODUCTION No. M a t u r e O v a r i o l e s / O v a r i o l e \ \ A c t i v a t i o n / D e n s i t y o f M a t e d F e m a l e s by Age a n d A g e a t C o p u l a t i o n E n e r g y f o r E g g s No. A c t i v e O v a r i o l e s O v a r i o l e D e a c t i v a t i o n O v a r i o l e P r o d u c t i o n No. S e n i l e O v a r i o l e s No. O o c y t e s \ - • ^ V i t e l l o g e n e s i s y> I U n f e r t i l i z e d E g gs (David 1963). The r e l a t i o n s h i p between these four components and t h e i r r e l a t i o n s h i p to external factors constitutes the egg production process. Ovariole a c t i v a t i o n , ovariole production, and v i t e l l o -genesis are considered basic components of egg production. A l l ovarioles can pass from an ina c t i v e to an active phase. Once ac t i v e , a l l ovarioles can produce oocytes. Then v i t e l l o -genesis can transform the oocytes into mature eggs. Ovariole deactivation i s considered a subsidiary com-ponent of egg production because i t seems possible that a female's ovarioles could remain functional over her t o t a l l i f e s p a n . Two sets of experiments were conducted to study e f f e c t of age of female and age of female at copulation on egg production. These two variables were chosen because the mating model generates a population of mated females which i s subdivided into groups by age and age at copulation. Corresponding work with males was not conducted as other studies indicated that age of males has no e f f e c t on females' egg production or egg f e r t i l i z a t i o n (Butz and Hayden 1962). P a r t i c u l a r note was made of e f f e c t s of age and age at copu-l a t i o n on the components ovariole a c t i v a t i o n , ovariole pro-duction, and ovariole deactivation. 47. FIRST EGG PRODUCTION EXPERIMENT Introduction. I t was i n i t i a l l y assumed that, as age increased, egg production rate might show a negatively accelerated r i s e to a plateau. I t was f e l t that the e f f e c t of copulation would be variable during the r i s i n g phase and would become constant throughout the plateau phase. In other words, as older and older females are copulated, the change i n egg production might s t a b i l i z e . It was decided, therefore, to set ages at copulation closer together for younger females ( r i s i n g phase) than for older females (plateau phase). A d d i t i o n a l l y , the experimental methods should not constrain the female through a v a i l a b i l i t y of food or space, i . e . there should be no e f f e c t s of density on egg production. The t e s t of possible density e f f e c t s was done by varying the number of females i n a universe rather than by varying universe s i z e . Methods. Individual female p_. melanogaster, of a vg s t r a i n * , from one cohort were i s o l a t e d within four hours of eclosion i n separate p e t r i dishes (6.0 x 1.5 cm.) containing food (Cordon Blue formula*) covering the bottom to a depth of •Supplied by D.T. Suzuki, Dept. of Zoology, U.B.C., Vancouver 8, B.C., Canada. 48. 1.0 cm. Females were moved to new dishes every two to f i v e days. A f t e r t r a n s f e r a l , or a f t e r the female was found dead, the old dis h was examined once, and sometimes twice, a week for eggs, l a r v a , pupa, and imagos. The general condition of the medium was also noted. Single males were i s o l a t e d from b i r t h i n v i a l s (2.4 x 9.5 cm., food to a depth of 1.0 cm.) for f i v e to seven days. In a previous experiment males younger than two to three days d i d not adequately f e r t i l i z e females. Males were placed with females for 24 hours when females reached a predetermined age from eclosion (1, 3, 5, 10, 15 or 30 days). Males were placed with females for 48 hours for r e p l i c a t e s three to s i x , s t a r t i n g 24 hours sooner. Combined with co n t r o l females not mated, t h i s method resulted i n seven treatment l e v e l s : age at copulation Tc = V i r g i n , 1, 3, 5, 10, 15 and 30 days. There were i n i t i a l l y s i x r e p l i c a t e s at each treatment l e v e l . During the a n a l y s i s , however, the number of r e p l i c a t e s was reduced since some f l i e s died before copulation and the data for others were discarded i f a female f a i l e d to lay f e r t i l e eggs a f t e r mating or l a i d less than a tenth of the eggs l a i d by her treatment group. Environmental conditions were kept as constant as possible: 17-23°C and 16 hours l i g h t (08:00 to 24:00 P.S.T.) and 8 hours darkness f o r r e p l i c a t e s one and two; and 19-21°C and r e l a t i v e humidity 77-83% for r e p l i c a t e s three to s i x . 4 9 . The d i f f e r e n c e i n methods between r e p l i c a t e s one to two and three to s i x d i d not make any difference i n t h e i r respective r e s u l t s . Two of the females that were copulated at age 30 days had started laying eggs p r i o r to copulation. The data for these females p r i o r to copulation were included with the v i r g i n data. To determine i f o v i p o s i t i o n s i t e s e l e c t i o n , v i t e l l o -genesis, or s o c i a l f a c i l i t a t i o n were a f f e c t i n g egg production, paired female experiments were conducted i n conjunction with the above single female experiments. If egg production per paired female was not less than per single female, then paired females were not competing for e i t h e r food (v i t e l l o g e n e s i s ) or o v i p o s i t i o n s i t e s (oviposition s i t e s e l e c t i o n ) . Thus, neither factor would be l i m i t i n g for single females, and o v i p o s i t i o n s i t e s e l e c t i o n and v i t e l l o -genesis could, for the present, be ignored. S i m i l a r l y , i f a female of a pair d i d not lay more eggs than single females, then s o c i a l f a c i l i t a t i o n might not be a component of egg production or o v i p o s i t i o n s i t e s e l e c t i o n . However, food competition, o v i p o s i t i o n s i t e s e l e c t i o n and s o c i a l f a c i l i -t a t i o n could possibly occur such that t h e i r e f f e c t s cancel each other. In the paired female experiment two females were kept together i n a p e t r i d i s h , except for the mating period. The t o t a l egg count was s p l i t between both females. There were 50. two r e p l i c a t e pairs for each of three treatments: age at copulation (Tc) = v i r g i n , 1 and 3 days. In other respects methods were as above. Results. There was (Table 11) no difference i n egg produc-t i o n between paired and single females, implying no apparent e f f e c t s of competition for food (vitellogenesis e f f e c t ) , of competition for o v i p o s i t i o n s i t e s , or of s o c i a l f a c i l i t a t i o n on o v i p o s i t i o n s i t e s e l e c t i o n . Thus, r e s u l t s from both paired and single females could be combined, and, as mating was c o n t r o l l e d by allowing a single copulation* at a preset age, only three components of egg production and the two components of egg f e r t i l i z a t i o n (the f i v e components marked "c" i n Table 1) needed to be considered i n constructing a model from the present data.** There was a minimum age (Tmin = 4.7 days) before which females would not lay eggs.*** Females mated before the minimum age (Tc <^  Tmin) started laying a f t e r t h i s age. Females mated a f t e r the minimum age (Tmin < Tc) started lay i n g immediately a f t e r copulation (Table 12). *Female D. melanogaster are assumed to copulate only once, which may not be s t r i c t l y true (Manning 1967). ••Subsequently, the v i t e l l o g e n e s i s component was added to the egg production model (Appendix I I I ) . ** * E f f e c t s of the copora a l l a t a on r e c e p t i v i t y and egg production are discussed i n PARAMETER ESTIMATION FOR MATING MODEL: Motivation. 51. T a b l e 11. Two-way n o n - o r t h o g o n a l a n a l y s i s o f v a r i a n c e o f r a t e o f egg p r o d u c t i o n (dE/dT) on 1) age o f f e m a l e (Age) and 2) w h e t h e r f e m a l e was i s o l a t e d o r p a i r e d ( S i n g l e - P a i r e d ) . D a t a f o r D r o s o p h i l a m e l a n o g a s t e r f e m a l e s f r o m t h e f i r s t e x p e r i m e n t mated a t age 1 day (Tc = 1) . S o u r c e SS DF P S i n g l e - P a i r e d 109.1 1 23.0 % Age ( a d j . S i n g l e - P a i r e d ) 3940.2 15 0.0001 % S i n g l e - P a i r e d ( a d j . Age) 143.7 1 6.14 % Age 3905.6 15 0.0001 % R e s i d u a l 3235.7 81 T o t a l 7285.0 97 52. Table 12. Relationship between age at copulation (Tc) and: 1) age at f i r s t egg laying (Ts), 2) maximum egg production rate (max. dE/dT), and 3) age at maximum egg production rate (Tmax). Data are means ± 1 SE f o r female Drosophila melanogaster from f i r s t egg production experiment. Tc Number of Females T s a max. dE/dT Tmax V i r g i n _d 20.2+2.0 11.8±1.6 2 5 . 6 1 3 . 1 1 3 4.67±.06 24.412.0 15.512.1 3 5 4.6+0.1 20.0+2.3 16.711.2 5 6 5.0±0.8 27.H0.7 15.110.7 10 4 10.0±0.2 23.711.5 18.010.5 15 3 14 . 7 7 1.04 2 7 . 6 1 4 . 7 17.0+0.2 30 3 30.5+0.3 25.112.2 30.4+0.2 a F o r Tc = 1 to 5, average Ts = 4.7 1 0.5 . For Tc = 1 to 30, average max. dE/dT = 24.4 l 0.3 . °For Tc = 1 to 15, average Tmax = 16.4 + 0.2 • • ^Number of females equals 6 for Ts and 4 for max. dE/dT and Tmax. 53. This i s expressed mathematically by: Ts = Tmin [Tc < Tmin] (19) Ts = Tc [Tmin < Tc] where: Ts = age at f i r s t egg laying Tmin = minimum age at f i r s t egg laying Tc = age at copulation Maximum egg production rate and age at t h i s maximum (16.4 days) are independent of the female's age at copula-t i o n (Tc) except for v i r g i n s and females copulated at age 30 days (Tc = V i r g i n and 30). Since egg laying only s t a r t s at the time of copulation i t i s ine v i t a b l e that females copulated at age 30 days cannot reach maximum egg production rate at less than 30 days. Thus: Tmax = 16.4 [Tc v l 6 . 4 ] Tmax = Tc [16.4 < Tc] C2.Q). Where: Tmax = age at maximum egg production rate Tc = age at copulation Following copulation, egg production began eithe r at a low l e v e l (Ts = 5 and 10, Figures 7 and 8) or at the Figure 7. E f f e c t of age of female Drosophila  melanogaster (A) on rate of egg production (dE/dT) for age at f i r s t egg laying of 5 days (Ts = 5): f i r s t experiment. Each point represents the mean of 18 or less r e p l i c a t e s + 1 SE. The s o l i d l i n e of p r e d i c t i o n i s generated by the egg production function (Equation 22) using parameter estimates from Table 17. 54. > 0-0 50-0 1 0 0 - 0 AGE (DAYS) Figure 8. E f f e c t of age of female Drosophila  melanogaster (A) on rate of egg production (dE/dT) for age at f i r s t egg laying of 10 days (Ts = 10): f i r s t experiment. Each point represents the mean of 8 or less r e p l i c a t e s + 1 SE. The s o l i d l i n e of pr e d i c t i o n i s generated by the egg production function (Equation 22) using parameter estimates from Table 17. maximum r a t e (Ts = 15 a n d 3 0 , F i g u r e s 9 a n d 1 0 ) . I n b o t h c a s e s , o n c e maximum r a t e o f e g g p r o d u c t i o n was a c h i e v e d , e g g p r o d u c t i o n r a t e b e g a n t o d e c l i n e w i t h i n c r e a s i n g a g e . W h i l e commencement o f e g g l a y i n g t y p i c a l l y f o l l o w e d m a t i n g , i f m a t i n g d i d n o t o c c u r , v i r g i n f e m a l e s e v e n t u a l l y b e g a n l a y i n g e g g s s p o n t a n e o u s l y . T h e r e was more v a r i a b i l i t y i n age a t s p o n t a n e o u s e g g l a y i n g t h a n f o r commencement o f e g g l a y i n g b y m a t e d f e m a l e s . Maximum e g g p r o d u c t i o n r a t e was l e s s f o r v i r g i n s t h a n f o r m a t e d f e m a l e s . F o r v i r g i n s , age a t maximum e g g p r o d u c t i o n a n d a g e a t f i r s t e g g l a y i n g w e r e n o t s i g n i f i -c a n t l y d i f f e r e n t , ( T m a x ^ T s , T a b l e 1 2 ) . The g e n e r a l t r e n d o f t h e d a t a a p p e a r e d s i m i l a r t o t h a t f o r f e m a l e s c o p u l a t e d a t 15 a n d 30 d a y s (Tc = 15 a n d 3 0 ) ; e g g p r o d u c t i o n b e g a n a t a m a x i m a l l e v e l a n d t h e n d e c l i n e d w i t h i n c r e a s i n g a ge ( F i g u r e 11 c o m p a r e d t o F i g u r e s 9 a n d 1 0 ) . MATED FEMALES M o d e l D e v e l o p m e n t . A s a f i r s t a p p r o x i m a t i o n , assume 1) t h a t o v a r i o l e s p r o d u c e o o c y t e s i n d e p e n d e n t l y o f o t h e r o v a r i o l e s i n t h e same o v a r y ; 2) t h a t o v a r i o l e s c e a s e f u n c t i o n i n g w i t h a g e , p r o b a b l y i n d e p e n d e n t l y o f one a n d o t h e r ( o v a r i o l e d e a c t i v a t i o n ) ; 3) t h a t o v a r i o l e s become a c t i v e i n a s i m i l a r l y i n d e p e n d e n t manner ( o v a r i o l e a c t i v a t i o n ) ; a n d 4) t h a t a l l o v a r i o l e s h a v e t h e same Figure 9. E f f e c t of age of female Drosophila  melanogaster (A) on rate of egg production (dE/dT) for age at f i r s t egg laying of 15 days (Ts = 15): f i r s t experiment. Each point represents the mean of 5 or less r e p l i c a t e s + 1 SE. The s o l i d l i n e of pr e d i c t i o n i s generated by the egg production function (Equation 22) using parameter estimates from Table 17. 57. Figure 10. E f f e c t of age of female Drosophila  melanogaster (A) on rate of egg production (dE/dT) for age at f i r s t egg laying of 30 days (Ts = 30): f i r s t experiment. Each point represents the mean of 3 or less r e p l i c a t e s + 1 SE. The s o l i d l i n e of pr e d i c t i o n i s generated by the egg production function (Equation 22) using parameter estimates from Table 17. 58. Figure 11. E f f e c t of age of v i r g i n female Drosophila melanogaster (A) on egg production rate (dE/dT): f i r s t experiment. Each point represents the mean of 10 or less r e p l i c a t e s + 1 SE. The s o l i d l i n e of p r e d i c t i o n i s generated by the egg production function (Equation 22) using parameter estimates from Table 17. 3 > 0 . • + 1 1 1 1 1 — ' • 1 1 H • in LD 1 5 - 0 . UJ • \ L±J a 0 - 0 . i \f \f w \f 0 - 0 50-0 AGE (DAYS) 1 0 0 - 0 60. constant oocyte production rate (ovariole production) given an adequate supply of energy ( v i t e l l o g e n e s i s : Appendix IV). For ovariole deactivation, assume that over any time i n t e r v a l there i s a p r o b a b i l i t y (c) that a given ovariole i s s t i l l functioning. At the end of one time i n t e r v a l the proportion of ovarioles s t i l l functioning i s c. At the end 2 of two time i n t e r v a l s the proportion i s c«c = c . At the 3 end of three i t i s c « c c = c and at the end of N time i n t e r v a l s the proportion of ovarioles s t i l l functioning i s N c . If length of the time i n t e r v a l i s one day and a female's age i s A and the female began laying eggs at age Ts, then the time i n t e r v a l over which the ovarioles would stop functioning i s A - Ts and the proportion of t h i s female's ovarioles s t i l l A Ts functioning i s c . This implication that as a female ages she has fewer and fewer functional ovarioles i s supported by the observations of David and C l a v e l (1967). This loss of functional ovarioles would cause the observed decline i n egg production rate (Figures 7-10). Assuming for ovariole a c t i v a t i o n that the r i s e i n egg production rate (dE/dT) i s due to a s i m i l a r random event, then the p r o b a b i l i t y that over any time i n t e r v a l an ovariole would not s t a r t functioning i s b. Using the same argument as given above, at the end of A - Ts time i n t e r v a l s the propor-A — Ts t i o n of ovarioles s t i l l not functioning i s b ; and the proportion of ovarioles which have started functioning i s : 6 1 . 1 - b A " T s M u l t i p l y i n g t h e s e two p r o p o r t i o n s t o g e t h e r , ( o v a r i o l e s s t i l l f u n c t i o n i n g t i m e s o v a r i o l e s t h a t h a v e s t a r t e d f u n c t i o n -i n g ) , g i v e s p r o p o r t i o n o f o v a r i o l e s t h a t h a v e s t a r t e d f u n c t i o n i n g a n d a r e s t i l l f u n c t i o n i n g : (1 - b A ~ T s ) ' c A " T s A female's p o t e n t i a l o o c y t e p r o d u c t i o n r a t e (a) i s t h e p r o d u c t o f o v a r i o l e number t i m e s o o c y t e p r o d u c t i o n r a t e p e r o v a r i o l e . H o w e v e r , t h e s e l a t t e r a r e c o n s t a n t s ( D a v i d a n d C l a v e l 1 9 6 7 ) . T h u s , a c t u a l o o c y t e p r o d u c t i o n r a t e p e r f e m a l e i s t h e p r o d u c t o f a c o n s t a n t p o t e n t i a l o o c y t e p r o d u c t i o n r a t e (a) t i m e s p r o p o r t i o n o f o v a r i o l e s f u n c t i o n i n g : a n . A - T s . A - Ts a* (1 - b ) c The p a i r e d f e m a l e e x p e r i m e n t showed t h a t v i t e l l o g e n e s i s i s n o t l i m i t i n g ( T a b l e 1 1 ) . T h e r e f o r e , e g g p r o d u c t i o n r a t e (dE/dT) i s e q u a l t o o o c y t e p r o d u c t i o n r a t e : W here: dE/dT = 0.0 [A• <_ T s ] dE/dT = a>(1 - b A ~ T s ) - c A " 1 5 3 [ T s < A] dE/dT = r a t e o f e g g p r o d u c t i o n i n e g g s p e r d a y A = age o f f e m a l e f r o m e c l o s i o n i n d a y s Ts = age o f f e m a l e a t f i r s t e g g l a y i n g (21) a n d where t h e p a r a m e t e r s a r e r e l a t e d t o t h e components o f eg g p r o d u c t i o n : a i s t h e p r o d u c t o f i n d i v i d u a l o v a r i o l e p r o d u c t i o n t i m e s number o f o v a r i o l e s ; b i s t h e p r o b a b i l i t y o f an o v a r i o l e r e m a i n i n g i n -a c t i v e a n d , t h u s , i s r e l a t e d t o o v a r i o l e a c t i v a t i o n ; a n d c i s t h e p r o b a b i l i t y o f an o v a r i o l e r e m a i n i n g a c t i v e a n d , t h u s , i s r e l a t e d t o o v a r i o l e d e a c t i v a t i o n . However, t h e r e a r e t h r e s h o l d e f f e c t s i n t h e d a t a t h a t E q u a t i o n 21 d o e s n o t t a k e i n t o a c c o u n t . Maximum egg p r o d u c -t i o n r a t e i s a c o n s t a n t , i n d e p e n d e n t o f age a t c o p u l a t i o n ( T c ) , b u t age a t t h i s maximum (Tmax) i s o n l y c o n s t a n t f o r f e m a l e s mated b e f o r e 16.4 d a y s ( T a b l e 12 and E q u a t i o n 2 0 ) . A p p a r e n t l y , a l l o v a r i o l e s a r e n o t a c t i v a t e d u n t i l f e m a l e s a p p r o a c h t h e age o f 16.4 d a y s . I f a f e m a l e has n o t b e e n c o p u l a t e d b e f o r e r e a c h i n g t h i s a g e , t h e n a l l o v a r i o l e s r e m a i n p o t e n t i a l l y a c t i v e u n t i l s h e i s c o p u l a t e d a t w h i c h t i m e o v a r i o l e d e a c t i v a t i o n b e g i n s t o t a k e p l a c e . T h i s a f f e c t s E q u a t i o n 21 i n two ways. F i r s t , i t w o u l d d i s p l a c e t h e l i n e o f p r e d i c t i o n t o t h e r i g h t f o r c o p u l a t i o n a f t e r 16.4 d a y s (compare s o l i d l i n e s i n F i g u r e s 9 and 1 0 ) . T h i s i s done by c h a n g i n g t h e f u n c t i o n c o n t r o l l i n g d e a c t i v a -. . , _A - T s . _,A - Tmax t i o n f r o m C t o C S e c o n d l y , t h i s a s s u m p t i o n t h a t o n l y a f t e r c o p u l a t i o n d o e s o v a r i o l e d e a c t i v a t i o n b e g i n t o t a k e p l a c e , when a p p l i e d t o f e m a l e s m a t e d a t 30 d a y s , i m p l i e s t h e c o r o l l a r y a s s u m p t i o n t h a t o n l y a c t i v e , f u n c t i o n i n g o v a r i o l e s a r e s u b j e c t t o d e -a c t i v a t i o n . A s s u m i n g p a r a m e t e r a i s n e a r l y e q u a l t o t h e o b s e r v e d m a x i m a l e g g p r o d u c t i o n r a t e , t h a n i t i s n e c e s s a r y t o m o d i f y E q u a t i o n 21 s o t h a t o v a r i o l e d e a c t i v a t i o n a p p l i e s o n l y t o a c t i v a t e d , f u n c t i o n i n g o v a r i o l e s . One m o d i f i c a t i o n , w h i c h d o e s n o t d e s c r i b e t h e p r o c e s s i n v o l v e d b u t o n l y a p p r o x i m a t e s i t s r e s u l t s , i s t o s e t t h e e x p r e s s i o n d e t e r m i n i n g A ~~ Tiricix d e a c t i v a t i o n (c ) e q u a l t o one u n t i l m o s t o f t h e o v a r i o l e s h a v e b e e n a c t i v a t e d . O v a r i o l e a c t i v a t i o n i s c o m p l e t e A — T s when t h e e x p r e s s i o n c o n t r o l l i n g a c t i v a t i o n (1 - b ) i s e q u a l t o o n e . T h u s , t h e e x p r e s s i o n c o n t r o l l i n g d e a c t i v a t i o n e q u a l s one u n t i l t h e e x p r e s s i o n c o n t r o l l i n g a c t i v a t i o n i s e q u a l t o o n e , w h i c h p r o b a b l y o c c u r s a t t h e age a t maximum e g g p r o d u c t i o n r a t e (Tmax). T h u s , i n l i g h t o f t h e s e two e f f e c t s , E q u a t i o n 21 b e c o m e s : dE/dT = 0.0 dE/dT = a - ( 1 - b A " T s ) [A <_ T s ] [Ts <A < Tmax] (22) JT-./jm tt , A - Ts. A - Tmax dE/dT = a * ( 1 - b )«c Ts < A ^Tmax < A* Where: dE/dT A = r a t e o f e g g p r o d u c t i o n = a ge o f f e m a l e f r o m e c l o s i o n Ts = age of female at f i r s t egg laying Tmax = age of female when maximum egg production rate occurs. Equation 22 was subsequently found to be s i m i l a r to two other functions developed to describe egg production. The three functions d i f f e r only i n the ph y s i o l o g i c a l basis used to develop each (see Appendix II for the development of these two equations) . One was developed by F i t z - E a r l e et. a l . (1969) and McMillan et a l . (1970 a,b) to describe egg produc-t i o n i n D. melanogaster: dE/dT =0.0 [A < Ts] dE/dT = a . ( l - e - ( r - s ) . ( A - T s ) ) B e - s . A [ T f l < A ] ( 2 3 ) where A and Ts are as defined for Equation 22. The other function was developed from consideration of factors e f f e c t i n g 2 2 the de r i v a t i v e of egg production rate (d E/dT ) rather than egg production rate i t s e l f : dE/dT =0.0 [A ^  Tmin] dE/dT = a-(e" X*(A-Tmin)_ e~Y.(A-Tmin) [T min < A ] ( 2 4 ) where A i s as defined for Equation 22 and: Tmin = minimum age at which females can f i r s t lay eggs Three sets of hypotheses generated three egg production functions: Equation 22, 23, and 24. F i t t i n g each equation to t h e d a t a m i g h t g i v e some i n s i g h t a s t o w h i c h h y p o t h e s i s comes c l o s e s t t o d e s c r i b i n g t h e a c t u a l u n d e r l y i n g m e c h a n i s m . H o w e v e r , t h e r e i s no n e e d t o f i t t h e d a t a t o a l l t h r e e e q u a t i o n s b e c a u s e t h e y a r e f u n c t i o n a l l y i d e n t i c a l a s s e e n b y t h e r e l a t i o n s h i p among t h e i r p a r a m e t e r s ( T a b l e 1 3 ) . I f a n y one o f t h e f u n c t i o n s g i v e s a n a d e q u a t e d e s -c r i p t i o n o f t h e d a t a t h i s w i l l n o t d i s p r o v e t h e v a l i d i t y o f a n y o f t h e o t h e r s e t s o f h y p o t h e s e s . The v a l i d i t y o f t h e h y p o t h e s e s c a n o n l y b e shown b y e x p e r i m e n t s t h a t i n v o l v e i n t e r n a l e x a m i n a t i o n o f t h e r e p r o d u c t i v e o r g a n s . N e v e r t h e -l e s s , t h e e q u a t i o n s do a d e q u a t e l y d e s c r i b e t h e d a t a , a n d t h e f a c t t h a t t h e y c a n be d e r i v e d f r o m t h r e e s e p a r a t e s e t s o f h y p o t h e s e s s u g g e s t s t h a t f o r p r e d i c t i v e p u r p o s e s t h e y a r e r o b u s t . A s L e v i n s h a s a r g u e d , t r u t h i s a t t h e i n t e r s e c t i o n o f c o n f l i c t i n g l i e s . P a r a m e t e r E s t i m a t i o n . L i n e a r r e g r e s s i o n c o u l d n o t be u s e d t o e s t i m a t e t h e v a l u e s o f p a r a m e t e r s a , b , a n d c b e c a u s e t h e e g g p r o d u c t i o n f u n c t i o n ( E q u a t i o n 22) c o u l d n o t be l i n e a r -i z e d . H o w e v e r , a s s u m i n g t h a t b y age a t maximum e g g p r o d u c t i o n A Ts r a t e (Tmax) t h e e x p r e s s i o n 1 - b a p p r o a c h e s a v a l u e o f o n e . T h e n , d a t a f o r f e m a l e s o l d e r t h a n Tmax c o u l d be d e s c r i b e d b y : dE/dT = a - c A ' T s (25) w h i c h l i n e a r i z e s t o : T a b l e 13. R e l a t i o n s h i p among parameters of the t h r e e egg p r o d u c t i o n f u n c t i o n s showing the sameness of the t h r e e e q u a t i o n s . E q u a t i o n 22 23 24 a = a = a , -r+s -y+x b = e = e l n ( d E / d T ) = l n ( a ) + (A - T s ) - l n ( c ) (26) As v a l u e s o f I n ( d E / d T ) a r e u n d e f i n e d f o r e g g p r o d u c t i o n r a t e s e q u a l t o z e r o , s u c h v a l u e s w e r e s e t e q u a l t o 1.2 . T h i s v a l u e was l e s s t h a n t h e m a j o r i t y o f s m a l l dE/dT b u t n o t s o s m a l l t h a t i t b i a s e d t h e r e g r e s s i o n . E s t i m a t e s o f p a r a m e t e r c f o r v i r g i n a n d m a t e d f e m a l e s d i f f e r e d ( T a b l e 1 4 ) . B e c a u s e v i r g i n s n o r m a l l y l a y f e w e r e g g s t h a n m a t e d f e m a l e s ( W i l s o n , K i n g , a n d L o w r y 1955) d i f f e r e n t p a r a m e t e r e s t i m a t e s w e r e e x p e c t e d ( F i g u r e 11 c o m p a r e d t o F i g u r e s 7 - 1 0 ) . The d i f f e r e n c e b e t w e e n e s t i m a t e s o f p a r a m e t e r c f o r f e m a l e s m a t e d a t 3 d a y s a n d o t h e r m a t e d f e m a l e s ( T a b l e 14) was u n e x p e c t e d b e c a u s e o f s i m i l a r i t i e s among f e m a l e s m a t e d a t 1, 3, a n d 5 d a y s . The f e m a l e s f o r t h e s e t h r e e a g e s a t c o p u l a t i o n (Tc = 1 , 3, a n d 5) d i d n o t d i f f e r s i g n i f i c a n t l y i n t h e i r a g e a t f i r s t e g g l a y i n g ( T s ) , i n t h e i r maximum e g g p r o d u c t i o n r a t e ( m a x • d E / d T ) , o r i n t h e i r a g e a t maximum e g g p r o d u c t i o n r a t e (Tmax) ( T a b l e 1 2 ) . B e c a u s e t h e r e seemed t o be no r e a s o n f o r t h e f e m a l e s c o p u l a t e d a t 3 d a y s t o d i f f e r i n t h e i r e s t i m a t e o f c f r o m f e m a l e s c o p u l a t e d a t 1 a n d 5 d a y s , u n l e s s one was t o p o s t u l a t e some c y c l i c phenomenon, d a t a f o r f e m a l e s c o p u l a t e d a t 3 d a y s w e r e d i s r e g a r d e d i n s u b s e q u e n t a n a l y s i s . I n a d d i t i o n , d a t a f o r a g e s a t c o p u l a t i o n o f 1 a n d 5 d a y s (Tc = 1 a n d 5) w e r e c o m b i n e d i n t o a s i n g l e b l o c k (Ts = 5) b e c a u s e t h e y w e r e i n d i s t i n g u i s h a b l e ( T a b l e 1 2 ) . 68. Table 14. Analysis for regression , Equation 25, using part of the egg production data in Figures 7-11: individual estimates of parameter c for each treatment (Tc). Results in terms of probability (P) that individual estimates of c for each age at copulation (Tc) gives a significantly better fit than a single estimate of c (c). Combinations of Tc c P Virgin,1,3,5,10,15,30a (all) 0.9523 0.715 % 1,3,5,10,15,30 (no Virgin) 0.9505 1.09 % Virgin,1,5,10,15,30 (no 3) 0.9493 4.28 % 1,5,10,15,30 0.9467 9.04 % (no Virgin or 3) aindividual estimates of c = 0.9625 (Virgin), 0.9376 (Tc = 1), 0.9751 (Tc = 3), 0.9625 (5), 0.9411 (10), 0.9350 (15), 0.9293 (30) . Subsequently, c was estimated by f i t t i n g Equation 25 d i r e c t l y to the data using a non-linear regression program. Because t h i s gave an estimate of c (0.9432) that d i f f e r e d l i t t l e from the f i r s t estimate (0.9467) , the above section on the f i r s t estimate was retained for i t s comparison of the values of c for d i f f e r e n t ages at copulation. The data suggested that the value of parameter b of the egg production function (Equation 22) might vary with age at f i r s t egg laying (Ts). Using the above estimate of c (0.9432) , estimates of b were determined for various com-binations of the four ages at f i r s t egg laying. Two estimates of b, one f o r 5 and 10 days and the other for 15 and 30 days, gave a s i g n i f i c a n t f i t , but separate b for a l l four ages did not (Table 15). Note, values of P are biased estimates (Table 10). However, examination of the data as four un-r e l a t e d blocks showed that, f o r age at f i r s t egg laying of 30 days (Ts = 30), the value of b was not s i g n i f i c a n t l y d i f f e r e n t from zero (Table 16). This resulted i n three estimates of b (Table 17) which gave a greater residue sum of squares (14,103.4) than did two estimates of b (14,096.9). The decline i n b with increasing age at f i r s t egg laying (Ts) resulted from age at maximum egg production rate (Tmax) being independent of age at copulation (Tc) (Table 12). Possibly, females' ovarioles s t a r t to become funct i o n a l shortly a f t e r eclosion, presumably p r i o r to the 70. T a b l e 15. A n a l y s i s o f v a r i a n c e f o r r e g r e s s i o n , E q u a t i o n 22, u s i n g e g g p r o d u c t i o n d a t a i n F i g u r e s 7-10: i n d i v i d u a l e s t i m a t e s o f a a n d b f o r a l l t r e a t m e n t s ( T s ) . R e s i d u a l sum o f s q u a r e s c a l c u l a t e d b y summing a l l sum o f s q u a r e s b e l o w r e g r e s s i o n sum o f s q u a r e s b e i n g e x a m i n e d up t o and i n c l u d i n g f i n a l r e s i d u a l sum o f s q u a r e s . R e s i d u a l d e g r e e s o f f r e e d o m c a l c u l a t e d i n a s i m i l a r manner. S o u r c e SS DF P R e g r e s s i o n f o r a mean 34 ,569 1 0. 0001 % A d d i t i o n a l f o r a , b , and c s i n g l e • 5 ,599. 4 2 0. 0000 % A d d i t i o n a l f o r two b 1 s ° 3 ,216. 7 1 0. 0004 % A d d i t i o n a l f o r f o u r b ' s ^ 91. 4 2 41. 0 % A d d i t i o n a l f o r f o u r a 1 s e 32. 8 3 88. 8 % F i n a l r e s i d u a l 13 ,972. 7 271 T o t a l 57 ,482 280 a M e a n egg p r o d u c t i o n r a t e (dE/dT) = 11.1 \ P a r a m e t e r e s t i m a t e s : a = 25.1; b =0.7821; c = 0.9432 . c P a r a m e t e r e s t i m a t e s : a = 25.1; b = 0.8316 (Ts = 5&10), 0.0043 (15&30); c =0.9432 . d P a r a m e t e r e s t i m a t e s : a = 25.1; b = 0.8144 (Ts = 5 ) , 0.8619 (10), 0.0067 (15), 0.0007 (30); c = 0.9432 . e P a r a m e t e r e s t i m a t e s : a = 26.0 (Ts = 5 ) , 23.0 (10), 26.0 (15), 24.8 (30); b = 0.8304 (5), 0.8331 (10), 0.0090 (15), 0.0005 (30); c = 0.9432 . 71. T a b l e 16. A n a l y s i s o f v a r i a n c e f o r r e g r e s s i o n , E q u a t i o n 22, u s i n g e g g p r o d u c t i o n d a t a i n F i g u r e 10. S o u r c e SS DF P R e g r e s s i o n 14.9 1 45.5 % R e s i d u a l 5 486.5 19_ T o t a l b 501.4 20 a P a r a m e t e r e s t i m a t e s : a =24.8; b = 0.0005; c = 0.9432 ^ R e s i d u a l f o r f i t t i n g E q u a t i o n 25; p a r a m e t e r e s t i m a t e s a = 25.1; c = 0.9432 . 72. minimum age of f i r s t egg laying (Tmin). Eggs are developed to the p r e f e r t i l i z a t i o n stage and are retained i n the ovarioles u n t i l copulation stimulates o v i p o s i t i o n (Wilson et al. 1955). For greater ages at copulation more ovarioles have started functioning and are awaiting the stimulus of mating to begin o v i p o s i t i o n . Thus, for greater ages at copu-l a t i o n there i s a shorter time, and a smaller value of b, to maximum egg production. By the minimum age possible for maximum egg production rate (16.4 days), most ovarioles are functional and are f i l l e d with mature eggs. The longer the delay i n copulation, the older the eggs become, and the more l i k e l y they are to lose t h e i r v i a b i l i t y . This implies that the f e r t i l i t y of the f i r s t few eggs l a i d by females would decline with increasing age at f i r s t egg laying (Ts). This i s exactly what happened as i s shown below i n the section on egg f e r t i -l i z a t i o n (Figure 19a). F i t t i n g separate values of a for each age at f i r s t egg laying (Ts) was not s i g n i f i c a n t (Table 15), probably because a i s approximately equal to the maximum egg production rate (a - max. dE/dT) which was not s i g n i f i c a n t l y d i f f e r e n t f or the d i f f e r e n t ages at copulation (Table 12). The predictions generated by the egg production model (Equations 19, 20 and 22) using the parameter estimates i n Table 17 are shown as s o l i d l i n e s i n Figures 7-10. / 73 v Table 17. Comparison of parameter estimates and c e r t a i n independent variables for the egg production model (Equations 19, 20, 22, 27, and 28). Estimates are f o r a l l Drosophila melanogaster females. Ts = female age at f i r s t egg laying; Tmin = minimum age at f i r s t egg laying; Tmax = age of female when maximum egg production rate occurs. Ts Parameters Tmin Tmax a b c F i r s t experiment V i r g i n 9.2 0.0000 0.9702 — 25.6 5 25.1 0.8316 0.9432 4.7 16.4 10 25.1 0.8316 0.9432 4.7 16.4 15 25.1 0.0067 0.9432 4.7 16.4 30 25.1 0.0000 0.9432 4.7 30.4 Second experiment 23.7 0.9455 0.9422 4.7 a 40.8 aUsed estimate from f i r s t experiment. 74. VIRGIN FEMALES Model Development. The female D. melanogaster has been used as a p h y s i o l o g i c a l black box. Dif f e r e n t inputs of an eth o l o g i c a l and e c o l o g i c a l nature were applied to the box: si x d i f f e r e n t ages at copulation, which resulted i n d i f f e r i n g outputs: ages at s t a r t of egg laying, maximum rates of egg production and ages at these maximums, rates of decline i n egg production, etc. Based on these r e s u l t s and in s i g h t gained from the l i t e r a t u r e , c e r t a i n assumptions were made about what might be occurring inside the black box. Not mating a female can be considered as a new input i n t o the black box. The r e s u l t i n g output can be examined to see i f i t follows l o g i c a l l y from the hypothesis b u i l t for the mated females. In a sense, therefore, these independently c o l l e c t e d data can serve to t e s t some of the predictions of the model. The age at f i r s t egg laying for v i r g i n s , Ts = 20.2, occurred shortly a f t e r the e a r l i e s t age at which maximum egg production rate can occur for mated females, Tmax = 16.4 (Table 12). I t was assumed that by age 16.4 days most ovarioles were activated and f i l l e d with eggs awaiting stimulus of copulation for o v i p o s i t i o n (MATED FEMALESi Model Develop-ment) . With t h i s backlog of eggs, i t would be easier to spontaneously overcome the i n h i b i t i o n to ovipo s i t i o n i n g that i s usually overcome by copulation. Thus, i t might be expected that v i r g i n s would begin laying eggs shortly a f t e r 16.4 days. The age at maximum egg production rate (Tmax) for mated females was either 16.4 days or age at f i r s t egg laying (Ts) i f the l a t t e r occurred a f t e r 16.4 days (Equation 20). Age at f i r s t egg layin g f o r v i r g i n s was 20.2 days. I t i s expected that v i r g i n s would s t a r t laying eggs at a maximum rate. This i s what was observed (Table 12 and Figure 11). The maximum egg production rate (max. dE/dT) was much lower f o r v i r g i n s (11.8) than for mated females (24.4). Perhaps for mated,females, eggs are f e r t i l i z e d and ov i p o s i -ted as r a p i d l y as they are produced. For v i r g i n s i t may be necessary to maintain the backlog of eggs (Wilson et a l 1955 p. 234-235) needed to overcome the i n h i b i t i o n to o v i p o s i t i o n . This assumed need to maintain a backlog would explain the observation of maximum egg production rate being less f or v i r g i n s than for mated females. Because v i r g i n s started laying eggs between 15 and 30 days, the general trend of v i r g i n data i s expected and was found to be, s i m i l a r to that for females copulated at 15 and 30 days (Tc = 15 and 30) (Figure 11 compared to Figures 9 and 10). I t appears that the experimental r e s u l t s for v i r g i n females follow l o g i c a l l y from the hypotheses b u i l t for mated females. I t i s appropriate, therefore, to f i t the data on egg production rate for v i r g i n females to the model developed f o r mated females (Equations 19, 20 and 22). Parameter Estimation. Using methods described previously, the v i r g i n data on egg production rate was f i t t e d to the egg production function (Equation 22). The data were adequately described by the two para-meters a and c with age at maximum egg production rate set equal to age at f i r s t egg laying (Tmax = Ts). The addition of the t h i r d parameter, b, was not s i g n i f i c a n t (Table 18). This was expected as 1) the model for mated females predicts a value of b near zero for age at f i r s t egg laying greater than 16.4 days (b - 0.0 for Ts > 16.4) and 2) f i r s t egg laying occurred at 20.2 days f o r v i r g i n s . The value of a i s c l o s e l y associated with maximum egg production rate which i s greater for mated females than for v i r g i n s . That the estimate of a for v i r g i n s was less than the estimate of a for mated females was, therefore, expected (Table 17). The hypotheses underlying the mated female model do not suggest whether estimates of parameter c for v i r g i n females should be greater or less than estimates for mated females. The estimate of parameter c was greater for v i r g i n s , i n d i -cating that ovarioles of v i r g i n s are deactivated at a slower rate (Table 17). A hypothesis f o r t h i s lower rate of ovariole d e a c t i -vation for v i r g i n s i s that females with a lower' basic egg production rate may not put the same s t r a i n on t h e i r ovarioles and, therefore, t h e i r ovarioles are not deactivated as Table 18. Analysis of variance for regression, Equation 22, using egg production data i n Figure 11. Residual sum of squares calculated by summing a l l sums of squares below regression sum of squares being examined up to and including f i n a l r e s i d u a l sum of squares. Residual degrees of freedom calculated i n a s i m i l a r manner. Source SS DF P Regression f o r meana 603. 1 1 0 .0002 % Additional for parameters a and c*5 89. 3 1 1 .42 % Additional for parameter b 30. 2 1 14 .2 % F i n a l r e s i d u a l 720. 4 5_3 Total 1443 56 Mean egg production rate (dE/dT) =3.3 . ^Parameter estimates: a = 9.2; c = 0.9702 . °Parameter estimates: a = 23.0; b = 0.2113; c = 0.9418 r a p i d l y . Assuming parameter a i s a measure of the basic rate of ovariole production and parameter c i s an inverse measure of ovariole deactivation, then, the lower the estimate of parameter a, the higher the estimate of para-meter c. This supposed r e l a t i o n s h i p between basic egg production rate (a) and ovariole deactivation (c) i s extremely tentative as i t i s based on only two sets of estimates of a and c, one for mated females and one for v i r g i n s . A further study of t h i s proposed r e l a t i o n s h i p , using egg production data from the l i t e r a t u r e (Appendix I I I ) , showed that i t probably does not e x i s t . SECOND EGG PRODUCTION EXPERIMENT Introduction. The r e s u l t s of the f i r s t experiment showed that,instead of a negatively accelerated r i s e to a plateau, egg production rate increases with age to a maximum and then begins to decline. There was, however, an i n d i c a t i o n that as older and older females are copulated, the change i n egg production s t a b i l i z e s , but t h i s observation was li m i t e d to the range of ages at copulation of 15 to 30 days, which corresponds to the age at which v i r g i n females spontaneously begin to lay eggs (Ts = 20.2, Table 12). Thus, data were needed on the ef f e c t s of ages at copulation (Tc) well beyond the range of ages at which v i r g i n s begin to lay. It i s assumed that once a female reaches an age of approximately 16.4 days, a l l ovarioles are activated and, either egg production rate i s at i t s maximum, i f previously copulated, or, with no previous copulation, the same maximum egg production rate w i l l be reached once copulation occurs. This assumes the capacity of a female to reach maximal egg production rate i s undiminished with age. If a female s t a r t s to lay eggs as a v i r g i n , i t i s reasonable to assume that t h i s would reduce the capacity to reach the same maximum egg production rate as a female, copulated at the same age, that had not started laying eggs as a v i r g i n . Two females i n the f i r s t experiment l a i d eggs p r i o r to copulation, but t h i s occurred just before copulation 80. s o t h a t t h e d u r a t i o n o f e g g l a y i n g was t o o s h o r t t o a f f e c t p o s t c o p u l a t o r y e g g p r o d u c t i o n . T h u s , d a t a w e r e n e e d e d f o r f e m a l e s m a t e d a t v a r i o u s a g e s a f t e r b e g i n n i n g e g g l a y i n g a s v i r g i n s . M e t h o d s . I n p l a c e o f u s i n g a c o h o r t o f f e m a l e s o f t h e same d a t e o f e c l o s i o n f r o m one s t o c k o f D. m e l a n o g a s t e r a s i n t h e f i r s t e x p e r i m e n t , a s i n g l e m a l e a n d f e m a l e f r o m two d i f f e r e n t i n b r e d s t r a i n s o f D. m e l a n o g a s t e r , t h e v g a n d v g ; e s t r a i n s * , w e r e m a t e d . F e m a l e o f f s p r i n g o f t h i s m a t i n g w e r e u s e d a s t h e e x p e r i m e n t a l a n i m a l s . T h i s p r o c e d u r e was u s e d t o i n s u r e u n i f o r m h e t e r o z y g o s i t y , t h u s , d e c r e a s i n g g e n e t i c v a r i -a b i l i t y a n d i n c r e a s i n g f e c u n d i t y a n d v i a b i l i t y ( D a v i d a n d M e r l e 1968; M c D a n i e l a n d Grimwood 1971). I n i t i a l l y , d i s h e s w e r e c h a n g e d e v e r y d a y , e v e r y f o u r t h d a y f o r v i r g i n s , t o more a c c u r a t e l y m e a s u r e age r e l a t e d p h e n o -menon. L a t e r , t h e v i r g i n m e t h o d o f c h a n g i n g d i s h e s e v e r y f o u r t h d a y was a d o p t e d f o r a l l f e m a l e s , a s c h a n g i n g d i s h e s e v e r y d a y i n h i b i t e d e g g l a y i n g o f f e m a l e s . D a t a f o r a g e s a t c o p u l a t i o n 1-5 a n d 10 d a y s w e r e d i s c a r d e d b e c a u s e t h i s i n h i b i t i o n t o e g g l a y i n g a f f e c t e d t h e s e f i r s t t w o g r o u p s o f m a t e d f e m a l e s , a s c a n b e s e e n b y c o m p a r i n g t h e l i n e o f p r e -d i c t i o n f o r t h e o t h e r t r e a t m e n t s w i t h t h e d a t a f o r f e m a l e s m a t e d a t 1-5 a n d 10 d a y s ( F i g u r e s 12 a n d 13). I n i t i a l l y , t h e *From t h e s t o c k s o f D.T. S u z u k i , D e p t . o f Z o o l o g y , UBC, V a n c o u v e r 8, B.C., C a n a d a . Figure 12. E f f e c t of age of female Drosophila  melanogaster (A) on rate of egg production (dE/dT) for age at copulation of 1 to 5 days (Tc = 1-5): second experiment. Each point represents the mean of 16 or less r e p l i c a t e s + 1 SE. The s o l i d l i n e of predi c t i o n i s generated by the egg production function (Equation 22) using parameter estimates from Table 17. 81. Figure 13. E f f e c t of age of female Drosophila  melanogaster (A) on rate of egg production (dE/dT) for age at copulation of 10 days (Tc = 10): second experiment. Each point represents the mean of 17 or less r e p l i c a t e s + 1 SE. The s o l i d l i n e of p r e d i c t i o n i s generated by the egg pro-duction function (Equation 22) using parameter estimates from Table 17. 82 83. data i s much lower than the norm, but a f t e r the change i n methods, the data f a l l s r i g h t along the l i n e of p r e d i c t i o n . Females were placed with males according to a random scheme weighted by the p r o b a b i l i t i e s of dying and of previous copulation. This was done to insure equal r e p l i c a t i o n at each treatment l e v e l . Ages at copulation were changed by adding groups at ages beyond 30 days as well as sub s t i t u t i n g day 20 for day 15: Tc = V i r g i n , 1-5, 10, 20, 30, 40, 50, 60, 70, and 80 days. F l i e s i n group Tc = 1-5 days were randomly mated 1-5 days a f t e r e c l o s i o n . Results from the f i r s t experi-ment indicated that these females would a l l respond as i f mated at age f i v e days. This had to be confirmed for females i n the second experiment. Instead of examining and reexamining dishes a f t e r the female had been removed, a l l eggs were transferred to a wet piece of f i l t e r paper i n an enclosed p e t r i dish. If t o t a l number of eggs exceeded f o r t y , they were subsampled to include the greater of 20 eggs or 1/3 of the t o t a l . The f i l t e r paper was moistened d a i l y . The dishes were examined a f t e r eight days and a record made of t o t a l egg number, number of eggs which had hatched, and number of eggs which had started to develop but had not hatched. Results. What i s most s t r i k i n g about the data for the second experiment, i n contrast to the f i r s t , i s that v i r g i n females l a i d eggs i n the same number and i n the same pattern as t h e i r 84. mated s i s t e r s (Figure 14a compared to Figures 14b-h). Virgins t y p i c a l l y lay fewer eggs than mated females (Wilson et a l 1955) and t h i s was the pattern i n the f i r s t experiment. In the second experiment, mating appeared to have no e f f e c t on egg production rate, as can be seen by comparing v i r g i n s from both experiments (Figure 15* and Figure 11) to mated females from f i r s t experiment (Figures 7-10). This suggests that a genetic s t r a i n was i n i t i a t e d from the single p a i r of progenitors that had a very d i f f e r e n t egg-laying t a c t i c . Some mechanism i n h i b i t s o v i p o s i t i o n u n t i l mating or u n t i l the age when a v i r g i n spontaneously begins to ov i p o s i t i s reached. Probably due to some mutation, t h i s mechanism was not functioning i n the second experiment. The l a t e ages at copulation (Tc = 40 to 80 days) were to have shown e f f e c t s on egg production rate (dE/dT) of mating females that had started laying eggs as v i r g i n s . The second experiment d i d show t h i s e f f e c t , but v i r g i n s i n t h i s experiment were laying eggs at the same rate as mated females and not at a reduced rate as i n the f i r s t experiment. Model Development. The egg laying t a c t i c adapted by females form the second experiment, copulation independent egg produc-t i o n , d i f f e r e d from the t a c t i c adopted by the f i r s t experiment *Figure 15 shows, i n d i f f e r e n t format, the same data as i n Figure 14a. F i g u r e 14. E f f e c t o f age of female D r o s o p h i l a  melanogaster (A) on egg p r o d u c t i o n r a t e (dE/dT) f o r v a r i o u s ages a t c o p u l a t i o n ( T c ) : second experiment. Each p o i n t r e p r e s e n t s the mean of a number of r e p l i c a t e s + 1 SE: A. V i r g i n s , 70 or l e s s ; r e p l i c a t e s ; B. Tc = 20 days, 8 or l e s s r e p l i c a t e s ; C. Tc = 30 days, 8 or l e s s r e p l i c a t e s ; D. Tc = 40 days, 6 or l e s s r e p l i c a t e s ; E. Tc = 50 days, 5 or l e s s r e p l i c a t e s ; F. Tc = 60 days, 3 or l e s s r e p l i c a t e s ; G. Tc = 70 days 2 or l e s s r e p l i c a t e s ; H. Tc = 80 days, 2 or l e s s r e p l i c a t e s . The common s o l i d l i n e of p r e d i c t i o n i s generated by the egg p r o d u c t i o n f u n c t i o n (Equation 22) u s i n g parameter e s t i m a t e s from T a b l e 17. Figure 15. E f f e c t of age of v i r g i n female Drosophila melanogaster (A) on egg production rate (dE/dT): second experiment. Each point represents the mean of 70 or less r e p l i c a t e s + 1 SE. The s o l i d l i n e of pre d i c t i o n i s generated by the egg production function (Equation 22) using parameter estimates from Table 17. H 1 1 1 1 1 1 h 1-0»0 50-0 100-0 AGE (DAYS) 87. females, copulation dependent egg production. Through simulation models of both t a c t i c s , r e l a t i v e advantages and disadvantages of each could be explored. The general trend of the data was much the same as for the early ages at f i r s t egg laying i n the f i r s t experi-ment (Ts = 5 and 10 days). Assuming the same physiology of ovariole a c t i v a t i o n , production, and deactivation i s at work (MATED FEMALE: Model Development), the egg production model (Equations 19, 20, and 22) was used to describe the data from both experiments. However, Equation 19 had to be modified because age at f i r s t egg laying (Ts) i s equal to age at which a female i s f i r s t capable of laying eggs (Tmin) and i s independent of age at copulation (Tc): Ts = Tmin (27) Equation 20 had to be modified as well, because age at maximum egg production rate (Tmax) i s also a constant*, inde-pendent of age at copulation: Tmax = 40.8 (28) Parameter Estimation. Similar non-linear regression methods described previously were used to estimate a and c from the second experiment's data. Again, these methods gave biased *Tmax = 40.8 + 1.3 (SE), n = 4. 88. estimates of P (Table 10). These estimates of a and c gave a f i t l i t t l e better than when parameter a was set equal to the maximum egg production rate (Tmax) and c was estimated by i t s e l f . The l a t t e r estimates of a and c were used as norms fo r the remainder of the analysis because s e t t i n g a equal to maximum egg production (a = Tmax) give a b i o l o g i c a l meaning to a. (Table 19). Separate estimates of parameter c for each age at copulation (Tc) did not give a s i g n i f i c a n t l y better f i t when compared to a single estimate of c (Table 20). Using sing l e estimates of a and c (Table 19), a single estimate of parameter b was made for a l l data (Table 21). Separate estimates of parameter b fo r each age at copulation d i d not give a s i g n i f i c a n t f i t compared to t h i s single estimate of b (Table 22) . Using sing l e estimates of b and c (Table 21), i n d i v i d u a l estimates of parameter a were made for each age at copulation. Individual estimates of a were only marginally favored over a single estimate of a, and i t was decided to use a single e s t i -mate of parameter a (Table 22). The r e s u l t s of the above analysis were single estimates of parameters a f b, and c (Table 17) which were independent of age at copulation (Tc). The common l i n e of predic t i o n based on these estimates and the egg production model (Equations 22, 27 and 28) are shown as s o l i d l i n e s i n Figure 14. 89. Table 19. Analysis of variance for regression, Equation 25, using part of the egg production data i n Figure 14: single estimate of a and c f o r a l l treatments. A comparison of parameter a estimated by regression and of a set equal to maximum egg production rate (max. dE/dT). Source Parameter Estimates a c SS DF P Regression 24.0 0.9416 2870.3 1 0. 0000 % Residual 988.9 74 T o t a l a 3859.2 75 Regression 23.7 b 0.9422 2869.6 1 0. 0000 % Residual 989.6 74 T o t a l a 3859.2 75 a R e s i d u a l for f i t t i n g mean egg production rate (dE/dT) = 6.64 . Parameter a set equal to observed maximum egg production rat e : max. dE/dT = 23.7 ± 2.7 (SE), n = 4. Table 20. Analysis of variance for regression, Equation 25, using part of the egg production data i n Figure 14: i n d i v i d u a l estimates of a and c for each treatment (Tc). Source SS DF F i t t i n g i n d i v i d u a l c's (for single a) 83.3 7 54.8 % Additional for i n d i v i d u a l a's 173.2 7 7.67 ' F i t t i n g i n d i v i d u a l a's (for single c ) c 196.2 7 3.73 Addition a l f o r i n d i v i d u a l c's 60.3 7 68.7 ' Residual 733.1 58 T o t a l d 989.6 72 aParameter estimates: a = 23.7; c (Tc = V i r g i n , 20, 30...80) = 0.9471, 0.9383, 0.9439, 0.9507, 0.9397, 0.9156, 0.9326, 0.9430 . ^Parameter estimates: a and c (Tc = Virgin...80) = 25, 0.9444; 19, 0.9509; 32, 0.9369; 4, 0.9861; 4000, 0.8012; 11000, 0.8141 . °Parameter estimates: a (Tc = Virgin...80) = 25.2, 19.9, 27.5, 27.0, 23.0, 11.7, 17.4, 25.4; c = 09.422 . Residual for f i t t i n g a = 23.7 and c = 0.9422 from Table 19. Table 21. Analysis of variance for regression, Equation 22, using egg production data i n Figure 14: single estimate of a, b, and c for a l l treatments. 7 = Source SS DF P Regression 5663 2 0.0000 % Res i d u a l 3 1372 119 Total 7035 121 aParameter estimates: a = 23.7; b = 0.9455; c = 0.9422 Residual for f i t t i n g mean dE/dT =8.03 . Table 22. Analysis of variance for regression, Equation 22, using egg production data i n Figure 14: i n d i v i d u a l estimates of a and b for each treatment (Tc). Source SS DF F i t t i n g i n d i v i d u a l b's (for single a and c ) a 134.3 7 10.8 % Additional for i n d i v i d u a l a's 132.4 7 9.55% F i t t i n g i n d i v i d u a l a's (for single b and c) 178.3 7 3.57 % Additional for i n d i v i d u a l b ' s b 88.4 7 31.0 % Residual 1105.3 105 T o t a l d 1372 119 aParameter estimates: a = 23.7; b (Tc = V i r g i n ...80) = 0.9436; 0.9533, 0.9484, 0.6255, 0.*9270, 0.9884, 0.755; c = 0.9422 . ^Parameter estimates: a and b (Tc = Virgin...80) = 33, 0.9687; 25, 0.9580; 44, 0.9819; 30, 0.9458; 23, 0.8829; 9950, 1.0000; 17, 0.7300; 25, 0.7600; c = 0.9422 . Parameter estimates: a (Virgin...80) = 24.9, 22.4, 24.3, 30.0, 24.4, 12.2, 17.7, 25.7; b = 0.9455; c = 0.9422 . Residual for f i t t i n g a = 23.7, b = 0.9455, and c = 0.9422 from Table 21. 93. CONCLUSION OF PART II In summary, Equation 19, 20 and 22 (or Equations 22, 27 and 28) represent part of the model f o r the egg produc-t i o n process. The equations include the components for ovariole a c t i v a t i o n , - ovariole production, and ovariole de-a c t i v a t i o n , but not the component for v i t e l l o g e n e s i s . The functions were used to pre d i c t and describe the e f f e c t s of age of female (A) and age of female at copulation (Tc) on egg production rate (dE/dT). The functions are: Ts = Tmin Ts = Tc [Tc <_ Tmin] [Tmin < Tc] (19) Tmax =16.4 [Tc<16.4] [16.4 < Tc] (20) Tmax = Tc dE/dT =0.0 [ A - T s ] A — dE/dT = a* (1 - b ) [Ts <_ A <_ Tmax] (22) / J r T, ,, , A - Ts. A - Tmax ,Ts < A , dE/dT = a* (1 - b ) «c [_ „ A ' ' Tmax < A Where: a i s the female's maximum r e a l i z e d egg production; b as redefined above, i s a measure of the lag between copulation and maximum rate of egg production and i s a function of age at copulation (Tc); and ' J 9 4 . c i s the p r o b a b i l i t y that a functioning ovariole w i l l remain functioning. and where: dE/dT = rate of egg production A = age of female from eclosion Tc = age at copulation Ts = age of female at f i r s t egg laying Tmin = minimum age at f i r s t egg laying Tmax = age of female when maximum egg production rate occurs o 95. PART III EGG FERTILIZATION INTRODUCTION AND METHODS I t i s possible to develop equations f o r the egg f e r t i -l i z a t i o n process s i m i l a r to those for egg production. F e r t i l e egg production rate would be a function of female age and age at copulation: dFE/dT = f(A,Tc). I t was decided, however, to make egg f e r t i l i z a t i o n a function of egg produc-t i o n r a t e : dFE/dT = f(dE/dT), and to develop equations that would predict the proportion of eggs produced that become f e r t i l e . This approach involves two components f o r the egg f e r t i l i z a t i o n process: sperm storage and sperm release (Figure 16). Once mated, a female has to store some or a l l of the sperm received. Also, sperm must be maintained to prevent loss i n v i a b i l i t y . This i s the component of sperm storage. Many animals practice external f e r t i l i z a t i o n , i n which case the male i s responsible for sperm release. Sperm storage i s , therefore, a subsidiary component of egg f e r t i l i z a t i o n . The component of sperm release involves getting stored sperm together, for f e r t i l i z a t i o n , with the eggs that have been produced. Within l i m i t s imposed by the number of sperm stored, sperm release determines f e r t i l e egg produc-t i o n rate. Figure 16. Flow diagram of the relationships among the components of egg f e r t i l i z a t i o n . Rec-tangles represent v a r i a b l e s , hexagons represent components. 96. U n f e r t i l i z e d E g g s Sperm A v a i l a b l e t o F e m a l e s Sperm R e l e a s e y<-^ Sperm S t o r a g e " ^ Sperm S t o r e d by F e m a l e s EGG F E R T I L I Z A T I O N F e r t i l i z e d E g g s Methods used to c o l l e c t data on f e r t i l e egg production rate are described under methods for the two egg production studies. 98. SPERM STORAGE Female D. melanogaster can only store a fixed number of sperm. Sperm stored are a f r a c t i o n of sperm received during insemination. Females then use most of these stored sperm for f e r t i l i z a t i o n (Lefevre and Johnson 1962). If a female's storage a b i l i t y does not change with age and a l l sperm stored are used, then t o t a l number of f e r t i l e eggs l a i d (F) would be the same as number of sperm stored (n) and would be a constant independent of age at copulation (Tc). Plots of t o t a l f e r t i l e eggs l a i d (F) on age at copulation (Tc) for both experiments (Figures 17 and 18) showed that t o t a l f e r t i l e eggs (F) was i n i t i a l l y constant but began to decline at a steady rate (m) for ages at copulation (Tc) greater than some minimum age (Td). The drop i n t o t a l f e r t i l e eggs f o r older females could be due to a number of causes: decreased v i a b i l i t y of eggs, i n a b i l i t y to maintain sperm v i a b i l i t y , or decrease i n sperm storage a b i l i t y . Whatever the cause, i t appears that t o t a l f e r t i l e eggs l a i d (F) i s a constant for females copulated while young and i s a function of age at copulation for females copulated when older: F = n [Tc <_ Td] (29) „ -m.(Tc-Td) r m , m , F = n«e [Td < Tc] Figure 17. Effect of age at first egg laying (Ts) on total number of fertile eggs laid (F) by mated Drosophila melanogaster; first experiment. Each point represents the mean of 9 or less replicates + 1 SE. The solid line of prediction is generated by the sperm storage function (Equation 29) using parameter estimates from Table 24. 0.0 50*0 100. AGE AT FIRST EGG LAYING Figure 18. E f f e c t of age at copulation (Tc) on t o t a l number of f e r t i l e eggs l a i d (F) by mated female Drosophila melanogaster; second experi-ment. Each point represents the mean of 8 or less r e p l i c a t e s + 1 SE. The s o l i d l i n e of pr e d i c t i o n i s generated by the sperm storage function (Equation 29) using parameter estimates from Table 24. 101. where: F = maximum possible number of f e r t i l e eggs l a i d during female's l i f e t i m e n = sperm storage a b i l i t y * m = measure of decline i n a b i l i t y to produce f e r t i l e eggs Td = age when decline begins Tc = age at copulation While Equation 29 deals s p e c i f i c a l l y with sperm storage, i t also a f f e c t s sperm release by putting an upper l i m i t on number of eggs that can be f e r t i l i z e d . *based on the assumption that a l l stored sperm are used for f e r t i l i z a t i o n (Lefevre and Johnson 1962) . 102. SPERM RELEASE D i f f e r i n g modes o f s p e r m r e l e a s e w o u l d r e s u l t i n d i f f e r i n g t r e n d s i n f e r t i l e e g g p r o d u c t i o n r a t e ( d F E / d T ) . I f r a t e o f s p e r m r e l e a s e i s a c o n s t a n t , t h e n f e r t i l e e g g p r o d u c t i o n r a t e w o u l d a l s o b e c o n s t a n t p r o v i d e d t h a t e g g p r o d u c t i o n r a t e e x c e e d e d t h i s c o n s t a n t . I f s p e r m r e l e a s e i s s u c h t h a t e v e r y e g g p r o d u c e d h a d t h e same p r o b a b i l i t y o f b e i n g f e r t i l i z e d a s l o n g a s a n y s p e r m a r e l e f t , t h e n f e r t i l e e g g p r o d u c t i o n r a t e w o u l d be a c o n s t a n t p r o p o r t i o n (q) o f e g g p r o d u c t i o n r a t e . I f s p e r m r e l e a s e i s p r o p o r t i o n a l t o number o f s p e r m l e f t , t h e n f e r t i l e e g g p r o d u c t i o n r a t e w o u l d b e e i t h e r a n e v e r d e c r e a s i n g number, o r a n e v e r d e c r e a s i n g p r o p o r t i o n o f e g g p r o d u c t i o n r a t e . The p r o p o r t i o n o f e g g s f e r t i l i z e d (q) r e m a i n e d f a i r l y h i g h a n d c o n s t a n t u n t i l j u s t b e f o r e t h e l a s t f e r t i l e e g g s w e r e l a i d . I t t h e n d e c r e a s e d s h a r p l y t o a l o w l e v e l . The d e c r e a s e t o o k p l a c e j u s t b e f o r e c u m u l a t i v e f e r t i l e e g g s l a i d ( t h e i n t e r g a l o f dFE/dT o r /dFE/dT) e q u a l e d maximum p o s s i b l e f e r t i l e e g g s l a i d (F f r o m E q u a t i o n 2 9 ) . ( F i g u r e 1 9 ) . The p r o p o r t i o n o f e g g s f e r t i l i z e d (q) c a n b e a p p r o x i -m a t e d by a c o n s t a n t u n t i l c u m u l a t i v e f e r t i l e e g g s l a i d (/dFE/dT) e q u a l s maximum p o s s i b l e f e r t i l e e g g s l a i d ( F ) . Once t h i s o c c u r s t h e p r o p o r t i o n becomes z e r o . T h i s a s s u m e s t h e d u r a t i o n o f t h e d e c l i n e i n f e r t i l i z a t i o n i s s h o r t e n o u g h t o b e a p p r o x i m a t e d b y a s t e p f u n c t i o n . T h u s : F i g u r e 19. P r o p o r t i o n o f eggs f e r t i l i z e d (q) as a f u n c t i o n o f the percentage of the t o t a l number of f e r t i l e eggs l a i d (100%•/dFE/dT/F). The d o t t e d l i n e s r e p r e s e n t an average v a l u e of q. Data f o r mated female D r o s o p h i l a melanogaster: A. F i r s t experiment, average q = 0.604; B. Second experiment, average q = 0.829. S t a r r e d v a l u e s not used i n c a l c u l a t i n g q. PERCENTAGE OF FERTILE EGGS LAID 104. Where; dFE/dT = dE/dT«g dFE/dT = 0.0 [/dFE/dT <_ F] [F < /dFE/dT] (30) d F E / d T = f e r t i l e e g g p r o d u c t i o n r a t e i n f e r t i l e e g g s p e r d a y dE/dT = e g g p r o d u c t i o n r a t e q = p r o p o r t i o n o f e g g s t h a t a r e f e r t i l i z e d /dFE/dT = i n t e r g a l o f d F E / d T , o r c u m u l a t i v e f e r t i l e e g g s p r o d u c e d F = maximum p o s s i b l e number o f f e r t i l e e g g s l a i d d u r i n g f e m a l e ' s l i f e t i m e ( E q u a t i o n 29) 105. PARAMETER ESTIMATES P a r a m e t e r v a l u e s f o r E q u a t i o n 29 a n d 30 w e r e e s t i m a t e d f o r d a t a f r o m b o t h e x p e r i m e n t s ( T a b l e 24). F o r t h e s p e r m s t o r a g e f u n c t i o n ( E q u a t i o n 29), t h e s e c o n d e x p e r i m e n t s d a t a g a v e a s i g n i f i c a n t f i t w h i l e t h e f i r s t e x p e r i m e n t ' s d a t a d i d n o t ( T a b l e 23). T h i s l a c k o f s i g n i f i c a n c e was p r o b a b l y due t o t h e s m a l l number o f d a t a p o i n t s (n=4) r a t h e r t h a n a v i o l a -t i o n o f t h e a s s u m p t i o n s o f t h e s p e r m s t o r a g e f u n c t i o n . The d a t a d i d f a l l f a i r l y c l o s e l y a l o n g t h e p r e d i c t i o n l i n e ( s o l i d l i n e i n F i g u r e 17) a n d i t h a d a f o r m s i m i l a r t o t h e d a t a f r o m t h e s e c o n d e x p e r i m e n t w h i c h d i d g i v e a s i g n i f i c a n t f i t ( F i g u r e 18). T h e r e f o r e , t h e s p e r m s t o r a g e f u n c t i o n ( E q u a t i o n 29) a n d i t s p a r a m e t e r e s t i m a t e s w e r e r e t a i n e d f o r t h e f i r s t e x p e r i m e n t ' s d a t a . F e m a l e s f r o m t h e s e c o n d e x p e r i m e n t h a d h i g h e r e s t i m a t e s o f s p e r m s t o r a g e a b i l i t y (n) a n d o f p r o p o r t i o n o f e g g s f e r t i l i -z e d ( q ) . T h i s was e x p e c t e d a s " h y b r i d s o f D r o s o p h i l a h a v e - b e e n shown t o e x h i b i t h e t e r o s i s e x p r e s s e d i n t e r m s o f f e c u n d i t y (Gowen 1952; V i t u k h i v a n d B e a r d m o r e 1959" ( M c D a n i e l a n d G r i m w o o d 1971). Table 23. Analysis of variance for regression, Equation 29, using f e r t i l e egg data i n Figures 17 and 18. Source SS DF P F i r s t experiment Regression 5,197.3 2 28.7 % Res i d u a l 5 467.7 1 T o t a l b 5,665 3 Second experiment Regression 123,704 2 0.0410 % Residual 0 2,556 4_ T o t a l d 126,260 6 aParameter estimates: n = 173; m = 0.0501; Td = 16.4 . ^Residual for f i t t i n g mean number of t o t a l f e r t i l e eggs l a i d (F) = 152.1 . CParameter estimates: n = 349; m =0.0681; Td =29.0 . Residual for f i t t i n g mean number of t o t a l f e r t i l e eggs l a i d (F) = 146.9 . 107. Table 24. Comparison of parameter estimates for the egg f e r t i l i z a t i o n model (Equations 29 and 30). Estimates are for a l l mated Drosophila melanogaster females. Parameters n m Td q F i r s t experiment 173 0.0501 16.4 0.604 Second experiment 349 0.0681 29.0 0.829 108 CONCLUSION OF PART III The process of egg f e r t i l i z a t i o n i s summarized by a model that includes a sperm storage function and a sperm release function: F = n F = n . e " m * ( T c - T d ) dFE/dT = dE/dT-q [/dFE/dT <_ F] (30) dFE/dT =0.0 [F < /dFE/dT] where: F = maximum possible number of f e r t i l e eggs l a i during a female's l i f e t i m e n = sperm storage a b i l i t y m = measure of decline i n a b i l i t y to produce f e r t i l e eggs Td = age when decline begins Tc = age at copulation dFE/dT = f e r t i l e egg production rate dE/dT = egg production rate q = proportion of eggs that are f e r t i l i z e d /dFE/dT = i n t e r g a l of dFE/dT, or cumulative f e r t i l e eggs produced [Tc <_ Td] [Td < Tc] (29) 109. The model has two basic assumptions: 1) proportion of eggs f e r t i l i z e d i s a constant (q) u n t i l p o t e n t i a l f e r t i l e egg production (F) i s achieved, and 2) for females copulated late i n l i f e , p o t e n t i a l f e r t i l e egg production (F) declines with increasing age at copulation (Tc). A TEST OF THE P R E D I C T I V E POWERS OF MODELS DEVELOPED I N PARTS I I AND I I I The p a r a m e t e r s o f t h e e g g f e r t i l i t y m o d e l w e r e e s t i m a t e d n o t d i r e c t l y f r o m d a t a o n f e r t i l e e g g p r o d u c t i o n r a t e (dFE/dT) b u t i n d i r e c t l y f r o m c u m u l a t i v e f e r t i l e e g g s ( i n t e r g a l o f dFE/dT o r /dFE/dT) a n d p r o p o r t i o n f e r t i l e ( ( d F E / d T ) / ( d E / d T ) ) . T h e r e f o r e , d a t a o n f e r t i l e e g g p r o -d u c t i o n r a t e (dFE/dT) c a n s e r v e a s a c o m b i n e d t e s t o f t h e p r e d i c t i v e p o w e r s o f b o t h t h e e g g p r o d u c t i o n a n d e g g f e r t i l i z a t i o n m o d e l s . A v i s u a l c h e c k o f t h e p r e d i c t i v e p o w e r s o f t h e t w o m o d e l s i s made b y p l o t t i n g o b s e r v e d v e r s u s p r e d i c t e d f e r t i l e e g g p r o d u c t i o n r a t e s ( F i g u r e s 20 a n d 21). I n s p e c t i o n i n d i c a t e s t h a t t h e m o d e l makes b e t t e r p r e d i c t i o n s f o r t h e s e c o n d e g g p r o d u c t i o n e x p e r i m e n t ( F i g u r e 20) t h a n f o r t h e f i r s t e x p e r i m e n t ( F i g u r e 21). , A s t a t i s t i c a l a n a l y s i s o f g o o d n e s s o f f i t c a n be made u s i n g a l i n e a r r e g r e s s i o n o f o b s e r v a t i o n o n p r e d i c t i o n . A s s u m i n g p e r f e c t a g r e e m e n t b e t w e e n o b s e r v a t i o n (Y) a n d p r e -d i c t i o n ( X ) , t h e n a l i n e a r r e g r e s s i o n (Y = A + BX) s h o u l d g i v e a n i n t e r c e p t o f z e r o (A = 0.0) a n d a s l o p e o f one (B = 1.0). I f t h e o b s e r v e d i n t e r c e p t a n d s l o p e d i f f e r s i g n i f i c a n t l y f r o m t h e s e v a l u e s , t h i s w o u l d i n d i c a t e t h a t t h e m o d e l s a n d / o r d a t a n e e d t o be r e a s s e s s e d . P r e f o r m i n g t h e r e g r e s s i o n a n d d e t e r m i n i n g i f A a n d B a r e s i g n i f i c a n t l y Figure 20. Pl o t of observed on expected f e r t i l e egg production rate (dFE/dT) for mated female Drosophila me1anogasfer; second experiment. 0 - 0 1 0 . 0 EXPECTED DFE/DT B > 0 Figure 21. Pl o t of observed on expected f e r t i l e egg production rate (dFE/dT) for mated female Drosophila melanogaster: f i r s t experiment. EXPECTED DFE/DT d i f f e r e n t , r e s p e c t i v e l y , from 0.0 and 1.0 w i l l henceforth be c a l l e d the regression t e s t . A chi-square t e s t might seem more appropriate i n dealing with observed and expected values. However c h i -square can only be used for counts, while the data here i s for rates. C o r r e l a t i o n c o e f f i c i e n t s were also calculated, but they were always so highly s i g n i f i c a n t as to indicate that they would not be good tests of the models' pr e d i c t -a b i l i t y . The regression t e s t for data from the second egg production experiment (Table 25a) shows that the model does not give good p r e d i c t i o n . It can be asked i f t h i s i s due to the egg production model or the egg f e r t i l i t y model. The regression t e s t f o r data on egg production (Table 25b) shows that the egg production model gives good pre d i c t i o n , suggesting that the problem l i e s with the egg f e r t i l i t y model. While most of the data on proportion of eggs f e r t i l e (Figure 19b) f a l l s near the mean value of propor-t i o n f e r t i l e (q = 0.829), the data f o r females copulated at 40 days i s much lower than the other groups. Removal of these data from the analysis gives good pr e d i c t i o n f o r both the egg f e r t i l i t y data (Table 25c) and the egg produc-t i o n data (Table 25d). The model d i d not give good predi c t i o n e i t h e r for the f e r t i l e egg production data from the f i r s t egg production experiment (Figure 21 and Table 25a) or for the egg T a b l e 25. L i n e a r r e g r e s s i o n p a r a m e t e r s (A and B) f o r c o m p a r i n g e x p e r i m e n t a l o b s e r v a t i o n (Y) f r o m s e c o n d e x p e r i m e n t t o p r e d i c t i o n (X) o f t h e egg p r o d u c t i o n and egg f e r t i l i z a t i o n m o d e l s . D a t a a r e f r o m F i g u r e 20. P i s t h e p r o b a b i l i t y t h a t A = 0.0 o r t h a t B = 1.0; dFE/dT = f e r t i l e egg p r o d u c t i o n r a t e ; dE/dT = egg p r o d u c t i o n r a t e ; T c = age a t c o p u l a t i o n . P a r a m e t e r E s t i m a t e SE DF P A. P r e d i c t i o n o f dFE/dT f o r a l l mated f e m a l e s : A 0.2450 0.2954 90 40.9% B 0.9036 0.0415 90 2.25 % B. P r e d i c t i o n o f dE/dT f o r a l l mated f e m a l e s : A -0.7549 0.5829 90 19.9% B 1.0816 0.0594 90 17.7 % C. P r e d i c t i o n o f dFE/dT f o r a l l mated f e m a l e s e x c e p t T c = 40: A 0.2028 0.3438 73 55.7 % B 0.9439 0.0477 73 24.7 % D. P r e d i c t i o n o f dE/dT f o r a l l mated f e m a l e s e x c e p t T c = 40: A -0.8454 0.5830 73 15.1 % B 1.037 0.0590 73 53.3 % 115. production data (Table 26b). An examination of data for females mated at 5 days (Figure 7) shows that there is a large discrepancy at age 4.5 days. If this one point is eliminated from analysis, the model will give good prediction for the egg production data (Table 26c) but not for the fertile egg production data (Table 26d). This inability of the egg fertilization model to give good prediction for the first experiment may be due to the assumption of proportion fertile being constant. Closer examination of the data (Figure 19a), shows that proportion fertile (q) may initially be a function of age at first egg laying (Tc). The low levels for age at first egg laying of 15 and 30 days are probably significant as mentioned above (MATED FEMALES: Parameter Estimation). This again points out the need for more intensive study of the effects of copulation beyond 15 days. As mentioned before, the second experiment was performed to f i l l this gap but failed to because the egg production rate was the same for all females regardless of when or if copulation occurred. 116. T a b l e 26. L i n e a r r e g r e s s i o n p a r a m e t e r s (A a n d B) f o r c o m p a r i n g e x p e r i m e n t a l o b s e r v a t i o n (Y) f r o m f i r s t e x p e r i m e n t t o p r e d i c t i o n (X) o f t h e e g g p r o d u c t i o n a n d e g g f e r t i l i z a t i o n m o d e l s . D a t a a r e f r o m F i g u r e 21. P i s t h e p r o b a b i l i t y t h a t A =0.0 o r t h a t B = 1.0; dFE/dT = f e r t i l e e g g p r o d u c t i o n r a t e ; dE/dT = e g g p r o d u c t i o n r a t e ; T s = age a t f i r s t e g g l a y i n g . P a r a m e t e r E s t i m a t e SE DF P A. P r e d i c t i o n o f dFE/dT f o r a l l m a t e d f e m a l e s : A 0.9932 0.3360 74 0.418 % B 0.7101 0.0664 74 0.0043 % B. P r e d i c t i o n o f dE/dT f o r a l l m a t e d f e m a l e s : A. 1.640 0.8203 74 4.93 % B 0.8647 0.0794 74 9.27 % C. P r e d i c t i o n o f dE/dT f o r a l l m a t e d f e m a l e s e x c e p t one p o i n t f o r Ts = 5: A 1.274 0.8026 73 11.7 % B 0.8915 0.0771 73 16.4 % D. P r e d i c t i o n o f dFE/dT f o r a l l m a t e d f e m a l e s e x c e p t one p o i n t f o r T s = 5: A 0.7121 0.2733 73 1.11 % B 0.7781 0.0534 73 0.0092 % 117. OTHER PROCESSES Aging. Anderson (1965; c i t e d by Watt 1968 p. 309) pointed out that chronological time and b i o l o g i c a l time need not have a one to one r e l a t i o n s h i p or even a constant r e l a t i o n -ship, but, i t i s assumed that the former r e l a t i o n s h i p holds for these experiments. Thus, i n the simulation model, at the end of each chronological day the f l i e s are aged one b i o l o g i -c a l day. Mortality. For t h i s study mortality i s considered to be one of two types: age related mortality (natural mortality) and other mortality (lumped under the heading "predation"). Natural mortality of D. melanogaster had been found to take the shape of the cumulative normal d i s t r i b u t i o n when cumula-t i v e number dead i s plotted against age (Erk and Samis 1970; VanHerrewege and David 1970). To construct a natural mortality function for females from the experiments on egg production and egg f e r t i l i z a t i o n , only mean age at death (Ds) and the standard deviation (s) needed to be calculated (Table 27). Corresponding data were not c o l l e c t e d on natural male mortality. However, the data of David et a l . (1970) indicated that females l i v e approximately 7.3 days longer than males. Thus, mean age at death for males i s estimated by subtracting 7.3 days from mean age at death for females Table 27. Comparison of parameter estimates for the mortality function (Equation 31). Estimates are for a l l Drosophila melanogaster females from the egg production experiments. Parameters Ds Dr a s F i r s t experiment 57.1 49.8 18.7 Second experiment 65.8 58.5 17.4 See text for determination of Dr. (Dr = Ds - 7.3) ( T a b l e 2 7 ) . The same s t a n d a r d d e v i a t i o n was u s e d f o r b o t h s e x e s . The p r o b a b i l i t y (Pd) o f a f e m a l e d y i n g d u r i n g one d a y i s a f u n c t i o n ( g a u s e ( •)*) o f t h e f e m a l e ' s a ge (A) a n d t h e p a r a m e t e r ' s mean age a t d e a t h (Ds) a n d s t a n d a r d d e v i a -t i o n o f age a t d e a t h ( s ) : P d = g a u s e ( A ^ ) - g a u s e ( ( A " 1 ) " D s ) (31) s s a n d , t h u s , f e m a l e d e n s i t y ( M s ( A ) * * ) b e c o m e s : Ms(A) = M s ( A - l ) . ( l - Pd) (32) M a l e m o r t a l i t y i s h a n d l e d i n t h e same way. O t h e r m o r t a l i t y , " p r e d a t i o n " , i s a c a t c h - a l l f o r a l l o t h e r s o u r c e s o f m o r t a l i t y . D o b z h a n s k y a n d W r i g h t (1943) e s t i m a t e d a 9.2% t o t a l m o r t a l i t y f o r a n a t u r a l p o p u l a t i o n o f D. p s e u d o o b s u r a . F o r s i m u l a t i o n , " p r e d a t i o n " i s v a r i e d f o r d i f f e r e n t r u n s t o s t u d y i t s e f f e c t on t h e m o d e l p o p u l a -t i o n ' s r e p r o d u c t i v e r a t e . S p e c i a l a t t e n t i o n was p a i d t o v a l u e s o f o t h e r m o r t a l i t y , " p r e d a t i o n " , n e a r t h e e s t i m a t e b y D o b z h a n s k y a n d W r i g h t (19*43) o f a 9.2% t o t a l m o r t a l i t y . X—m *The f u n c t i o n g a u s e (—§—) g i v e s t h e c u m u l a t i v e a r e a u n d e r t h e n o r m a l p r o b a b i l i t y c u r v e f r o m n e g a t i v e i n f i n i t y t o X, w h e r e X i s a n o b s e r v a t i o n f r o m a p o p u l a t i o n w i t h mean m a n d s t a n d a r d d e v i a t i o n s. *'* Ms (A) r e p r e s e n t s t h e v e c t o r o f f e m a l e d e n s i t i e s (Ms) f o r g i v e n a g e s ( A ) . 120. PART IV THE MODEL OF SEXUAL REPRODUCTION INTRODUCTION Once a series of functions have been developed through experimental components analysis, the functions can be linked together into a simulation model. This section deals with the mechanics of transforming the various functions into a computer model, the simulations done with the model, and the analysis of the simulation r e s u l t s . 121. CONSTRUCTION AND PROGRAMMING The m o d e l o f s e x u a l r e p r o d u c t i o n c a n be s u m m a r i z e d b y a l i s t i n g o f t h e f r a g m e n t a r y e q u a t i o n s d e v e l o p e d a b o v e . * (Compare t h i s l i s t i n g w i t h F i g u r e 22). A t t h e b e g i n n i n g o f e a c h d a y t h e p o p u l a t i o n s a r e a g e d : Mr (A) = Mr (A - 1) (33) M s ( A , T c ) = M s ( A - l , T c ) (34) Tt-W T h e n number o f m a t i n g s i s d e t e r m i n e d : T> m i v , , i Tm-Co> ,, Tm«Co« Co = Ac-Tt.Mr-W- (1 - T t . M r ) • (1 - m^. r. T ) b a s e d o n m a l e d e n s i t y : Mr = EMr(A) o n r e c e p t i v e f e m a l e d e n s i t y : W = Z M s ( A , T c ) - R R = 0.0 R = g R = g . e - h * ( A - T f ) a n d o n r a t e o f s u c c e s s f u l s e a r c h : A c = p - ( 2 D ) . 1 . 4 1 . ( e + d*(Mr+Ms)) A c = p * ( 2 D ) ' 1 . 4 1 ' f Ms = E M s (A,Tc) [A <_ T r ] [ T r < A: <_ T f ] [ T f < A] « [Mr+Ms <_ 106.6] [106.6 < Mr+Ms] (8) (18) (17) (16) (15) (35) • D e f i n i t i o n s o f p a r a m e t e r s o f t h e s e e q u a t i o n s a r e g i v e n i n A p p e n d i x I V . F i g u r e 22. F l o w d i a g r a m o f t h e m o d e l o f s e x u a l r e p r o d u c t i o n . R e c t a n g l e s r e p r e s e n t v a r i a b l e s . Hexagons r e p r e s e n t p r o c e s s e s ; t h e numbers i n t h e h e x a g o n s c o r r e s p o n d t o e q u a t i o n s s u m m a r i z i n g t h e p r o c e s s . 122. Male Density by Age Female Density by Age and, Except f o r V i r g i n s , Age at Copulation Aging 33,34 Mating 8,18,17,16,15,35 Egg Production 22,19,20 K Egg F e r t i l i z a t i o n 30,29 F e r t i l e Eggs Laid Eggs Laid < M o r t a l i t y \ ; Aj-32,31,36 / ' ^7 123. Then egg production i s determined; dE/dT =0.0 dE/dT = a. (1 - b A " T s ) dE/dT = a*(1 -.Ts = Tmin Ts = Tc and Tmax = 16.4 Tmax = Tc , A - Ts, A b ) • c - Tmax [A <_ Ts] [Ts < A <_ Tmax] [Tmax < A] [Tc <_ Tmin] [Tmin < Tc] [Tc <_ 16.4] [16.4 < Tc] (22) (19) (20) Next f e r t i l e egg production i s calculated: dFE/dT = dE/dT-q dFE/dT =0.0 F = n F = n«e -m«(Tc-Td) [/dFE/dT <_ F] [F < /dFE/dT] [Tc <_ Td] [Td < Tc] (30) (29) F i n a l l y , c a l c u l a t i o n s are made for death of f l i e s due to natural mortality: Ms(A,Tc) = Ms(A-l,Tc)•(1 - Pd) _., ','A-Ds,' ,'(A-T)'-Ds» Pd = gause (——) - gause ( ) and due to "predation" (Pp): Ms(A,Tc) = Ms(A-l,Tc)•(1 - Pp) (32)** (31)** (36.) ** *For females of the second egg production experiment both are constant: Ts =4.7 and Tmax = 40.5 **Similar mortality equations can be written for males by sub s t i t u t i n g Mr(A) for Ms(A,Tc) and Dr for Ds. 124. A v i s u a l representation of the program (Figure 23) shows i t i s broken into several subprograms, some of which correspond to one or more components outlined i n the above l i s t i n g of the fragmentary equations (Figure 22). Other subprograms handle the l o g i c a l flow of the program and the i n i t i a l determination of the male density vector, Mr(A), and the female density matrix, Ms(A,Tc). Also, each sub-program can contain a d d i t i o n a l commands dealing with 1) t r a n s l a t i o n of the fragmentary equations into FORTRAN IV, 2) transformation of equations dealing with i n d i v i d u a l s to ones dealing with populations, and 3) the input and output of v a r i a b l e s . Output of some models depends on t h e i r i n i t i a l conditions. This i s not true of the present model. However, the model does have to go through a number of cycles before the output reaches a steady state. During simulation, once the i n i t i a l populations are set up, each of the subprograms i s c a l l e d successively. After completing the l a s t subprogram, the cycle i s restarted i f a steady state has not been reached, otherwise the simula-t i o n of the program i s terminated (Figure 23). The model can not be considered a true representation of sexual reproduction. The fragmentary equations are i n many cases only tent a t i v e , but they do indicate the nature of the r e l a t i o n s h i p they t r y to represent. The same can not be said of the simplifying assumptions made i n order to t i e the frag-mentary equations together into a whole model. Figure 23. Flow diagram of the computer program for sexual reproduction. 125. SEXFY < CALL START < CALL PARTI > <(CALL PART 4 ^ <(CALL PART 3 ^ <(CALL PART4/»-1S f e r t i l e egg production steady state?. START 1) set male and female age groups to zero PARTI (AGING) 1) age a l l male age groups 2) add constant density of males to f i r s t age group PART4 (MORTALITY) 1) k i l l of appropriate density of males (uses DIE,PRED)  PARTI (AGING,MATING) 1) start new day 2) age a l l male and female age groups 3) add constant density of males and females to f i r s t age groups PART2 (MATING) 1) determine density of newly mated females PART4 (MORTALITY) 1) k i l l of appropriate density of males and females (uses DIE,PRED) PART3 (EGG PRODUCTION,EGG F E R T I L I Z A T I O N ) 1) determine density eggs and density f e r t i l e eggs la i d during this day (uses BFUNT,FEGGS) -YES E X I T 126. The model i s not a closed system. There i s no feed-back between f e r t i l e egg production and addition of new adults to the population. Each day a constant number of adults i s added to the youngest age groups of males and females (Figure 23). The model does not include the l i n k between female's n u t r i t i o n a l state and her egg production. In f a c t the model assumes that no ingestion of food takes place.* The environment the simulated populations occupy can best be described as an i n f i n i t e p l a i n with a uniform cover-ing of f r u i t f l y media and with very l i t t l e f l u c t u a t i o n i n l i g h t , temperature, or humidity. In r e a l i t y , D. melanogaster have discrete feeding s i t e s . The f l i e s probably tend to congregate at these s i t e s making t h e i r e f f e c t i v e density much higher than t h e i r actual density. •Nut r i t i o n i s considered i n Appendix I I I . 127. SIMULATION The m o d e l was a l l o w e d t o r u n t h r o u g h a number o f d a i l y c y c l e s u n t i l a s t e a d y s t a t e was a c h i e v e d . T h i s u s u a l l y o c c u r r e d a f t e r 80 t o 100 d a y s . S t e a d y s t a t e was t a k e n a s t h e t i m e when t o t a l d a i l y f e r t i l e e g g p r o d u c t i o n became c o n s t a n t . T o t a l f e r t i l e e g g p r o d u c t i o n o f t e n became c o n s t a n t b e f o r e t o t a l e g g p r o d u c t i o n d i d . T h i s r e s u l t e d f r o m t h e a s s u m p t i o n o f a f i x e d number o f s p e r m b e i n g made a v a i l a b l e t o t h e f e m a l e a t m a t i n g . Once a c e r t a i n l e v e l o f e g g p r o d u c -t i o n was a c h i e v e d , a l l s p e r m w e r e b e i n g u s e d . T h u s , w h i l e e g g p r o d u c t i o n c o u l d go h i g h e r , f e r t i l e e g g p r o d u c t i o n c o u l d n o t a n d r e m a i n e d a c o n s t a n t . A t o t a l o f 42 s i m u l a t i o n s w e r e r u n . T h i s number r e s u l t e d f r o m t h e c o m b i n a t i o n o f t w o p r e d a t i o n l e v e l s , t h r e e p o p u l a t i o n t y p e s , a n d s e v e n p o p u l a t i o n d e n s i t i e s . A t e a c h o f t h e t w o p r e d a t i o n l e v e l s , l o w a n d medium, m a l e p r e d a t i o n was a s s u m e d t o be h i g h e r . A l t h o u g h m a l e s a n d f e m a l e s move i n much t h e same way, m a l e s h a v e a g r e a t e r a v e r a g e v e l o c i t y ( M a n n i n g 1963). T h e r e f o r e , m a l e s a r e more l i k e l y t o e n c o u n t e r a p r e d a t o r . The p r e d a t i o n v a l u e s , i n p r o p o r t i o n o f p o p u l a t i o n d y i n g , w e r e 0.01 a n d 0.02 a t l o w l e v e l p r e d a t i o n a n d 0.08 a n d 0.10 a t medium l e v e l p r e d a t i o n f o r f e m a l e s a n d m a l e s r e s p e c t i v e l y . The v a l u e s a t t h e medium l e v e l w e r e c e n t e r e d a r o u n d t h e e s t i m a t e o f t o t a l m o r t a l i t y , 0.092, made b y D o b z h a n s k y a n d W r i g h t (1943) (OTHER PROCESSES: 128. M o r t a l i t y ) . Three population types (A, B, and C) were simulated (Table 28). Types A and C represent the two experimental populations from, re s p e c t i v e l y , the second and f i r s t egg production experiments. Type B i s a hybrid using the t a c t i c s of the second experiment (copulation independent egg production) and the parameter estimates of the f i r s t experiment. A fourth population type with the f i r s t experiment's t a c t i c s (copulation dependent egg production) and the second experiment's para-meter estimates was not used because the f i r s t t a c t i c requires parameter estimates not included i n the second set of estimates. If simulations were done only for types A and C i t would not be possible to say i f differences i n r e s u l t s were due to differences i n t a c t i c s or to differences i n parameter estimates. However, a comparison of r e s u l t s from types A and B would show differences i n e f f e c t s of the two sets of para-meter estimates independent of t a c t i c e f f e c t s . Comparing r e s u l t s from types B and C would show differences between t a c t i c s independent of parameter estimates; i t would show differences between copulation independent egg production (Type B) and copulation dependent egg production (Type C). Table 28. C l a s s i f i c a t i o n of population types f o r model simulation. Population Type Egg Production 3 , T a c t i c Parameter*5 Estimates A 2nd experiment 2nd experiment B 2nd experiment 1st experiment C 1st experiment 1st experiment 1st experiment: copulation dependent egg production; 2nd experiment: copulation independent egg production. See Tables 17, 24, and 27 for actual estimates. 130. RESULTS The output of the 42 simulations i s given i n the form of a p l o t of reproductive rate as a function of female density (Figure 24). Reproductive rate i s defined as the population's d a i l y f e r t i l e egg production divided by the number of new adults added to the population each day. The inverse of the reproductive rate gives the proportion of f e r t i l e eggs that must survive to adulthood i n order to prevent the population's e x t i n c t i o n . With increasing density, reproductive rate i n i t i a l l y shows a p o s i t i v e l y accelerated r i s e and then a l e v e l i n g o f f to a plateau f o r a l l combinations of predation l e v e l and population type (Figure 24). This pattern i s not an a r t i f a c t of using a semi-log p l o t . There was a decline i n reproductive rate with increasing mortality, "predation", for a l l combinations of population type and density. At low "predation" there appeared to be l i t t l e d i f ference between t a c t i c s (population type B compared to type C) but there d i d appear to be differences between para-meter estimates (type A compared to type B). Medium "predation" gave a r e v e r s a l of these r e s u l t s ; t a c t i c s d i f f e r e d (type B to type C), but parameter estimates did not (type A to type B). If there i s no pre-adult mortality, the model predicts that a population w i l l become extinct i f reproductive rate f a l l s below 1.0 f e r t i l e eggs per day/new adult per day, assuming Figure 24. Simulation r e s u l t s for the sexual reproduction model. Reproductive rate shown as a function of female density for three popula-t i o n types and two "predation" (mortality) l e v e l s . Populations f a l l i n g below the c r i t i c a l reproductive rate (dotted line) w i l l go e x t i n c t . See text for d e t a i l s . 131. 132. a one to one sex r a t i o (dotted l i n e Figure 24). The density required to maintain t h i s c r i t i c a l reproductive rate was four times as great f o r a population subjected to medium predation as compared to a population subjected to low pre--3 2 dation. Thus, at densxtxes near 2.0 x 10 females per m. , a low predation population w i l l survive but a medium pre-dation population w i l l become extinct. Given any preadult mortality, then higher d e n s i t i e s w i l l be required to maintain the population, but i n a l l cases the low adult predation population w i l l have an advantage. 133. DISCUSSION I f Figure 24 represented the r e s u l t s of an experiment, then, i n order to understand how the r e s u l t s occurred, more det a i l e d experiments would be required. However, these are not experimental r e s u l t s but r e s u l t s from simulation of a model constructed from d e t a i l e d experiments. To understand how the r e s u l t s occurred, i t i s only necessary to examine the l o g i c and equations of the model. This becomes progress-i v e l y more d i f f i c u l t as models become larger and larger. To the extent that the model i s an adequate represen-t a t i o n of r e a l i t y , the "how" of the model r e s u l t s w i l l be the same as the "how" of the actual b i o l o g i c a l process. At t h i s stage, however, the model i s s t i l l t e n tative. Thus, the following discussion of how the r e s u l t s occurred i s best considered as a guide to further study rather than any explanation of r e a l i t y . The i n i t i a l r i s e i n reproductive rate with increasing density was due to two f a c t o r s : 1) the proportion of females mated increases, thus, increasing t o t a l d a i l y f e r t i l e egg production; 2) females are on average mated at an e a r l i e r age which r e s u l t s i n a higher f e r t i l e egg production rate per female. These two density dependent factors gave the i n i t i a l exponential r i s e i n reproductive rate (Figure 24). Reproductive rate reaches a plateau at high densities because these two factors become constants: proportion of 134. f e m a l e s m a t e d c o u l d be no g r e a t e r t h a n o n e ; t h e r e i s a minimum ag e l i m i t t o t h e a b i l i t y t o mate a n d p r o d u c e e g g s . As p r e d a t i o n i n c r e a s e s f e m a l e s do n o t l i v e l o n g e n o u g h t o r e a c h t h e i r maximum e g g p r o d u c t i o n r a t e . T h u s , r e p r o d u c t i v e r a t e d e c l i n e s w i t h i n c r e a s i n g p r e d a t i o n . How-e v e r , i t i s p o s s i b l e t h a t i n c r e a s e d p r e d a t i o n w i l l , up t o a p o i n t , i n c r e a s e a p o p u l a t i o n ' s r e p r o d u c t i v e a b i l i t y ( L a c k 1954 p. 180). D i f f e r e n c e s i n p a r a m e t e r e s t i m a t e s o f t h e n a t u r a l m o r t a l i t y m o d e l a n d n o t d i f f e r e n c e s i n p a r a m e t e r e s t i m a t e s o f t h e e g g p r o d u c t i o n a n d f e r t i l i z a t i o n m o d e l s c a u s e d t h e d i f f e r e n c e i n r e p r o d u c t i v e r a t e b e t w e e n t h e two s e t s o f p a r a m e t e r e s t i m a t e s ( F i g u r e 24: l o w p r e d a t i o n , t y p e s A a n d B ) . T h i s i s shown by t h e l a c k o f d i f f e r e n c e b e t w e e n t h e two s e t s o f p a r a m e t e r e s t i m a t e s a t medium p r e d a t i o n w h e r e n a t u r a l m o r t a l i t y h a s l i t t l e e f f e c t ( F i g u r e 24: medium p r e -d a t i o n , t y p e s A a n d B ) . A t l o w p r e d a t i o n f e m a l e s l i v e d l o n g e n o u g h t o u s e up a l l t h e i r a v a i l a b l e s p e r m , t h u s , t h e r e was no d i f f e r e n c e i n r e p r o d u c t i v e r a t e b e t w e e n t h e t a c t i c s ( F i g u r e 24: l o w p r e -d a t i o n , t y p e s B a n d C ) . H o w e v e r , a t medium p r e d a t i o n , a d i f f e r e n c e d i d o c c u r ( F i g u r e 24: medium p r e d a t i o n , t y p e s B a n d C ) . F e m a l e s l i v e d o n l y l o n g e n o u g h t o u s e a p a r t o f t h e i r a v a i l a b l e s p e r m a n d , t h u s , a f e m a l e ' s e g g p r o d u c t i o n r a t e o v e r h e r l i m i t e d l i f e s p a n d e t e r m i n e d w h i c h t a c t i c u s e d up s p e r m t h e f a s t e s t . 135. C o p u l a t i o n i n d e p e n d e n t e g g p r o d u c t i o n (B) h a d a h i g h e r s h o r t t e r m r a t e b e c a u s e i t l a c k e d t h e l a g e f f e c t o f c o p u l a -t i o n d e p e n d e n t e g g p r o d u c t i o n ( C ) . I n c o p u l a t i o n i n d e p e n d e n t e g g p r o d u c t i o n e g g l a y i n g s t a r t e d a t 4.7 d a y s a n d r o s e t o a maximum a t 16.4 d a y s . I f c o p u l a t i o n o c c u r r e d a f t e r 4.7 d a y s , e g g p r o d u c t i o n w o u l d h a v e a l r e a d y s t a r t e d a n d f e r t i l e e g g p r o d u c t i o n r a t e w o u l d be h i g h e r t h a n i f e g g p r o d u c t i o n s t a r t e d a f t e r c o p u l a t i o n . F u j i t a (1954) c l a s s i f i e d t h e r e s p o n s e s o f i n s e c t e g g p r o d u c t i o n t o c h a n g i n g d e n s i t y a s one o f t h r e e t y p e s : D r o s o p h i l a t y p e ( F i g u r e 2 5 a ) , i n t e r m e d i a t e t y p e ( F i g u r e 3 5 b ) , a n d A l l e e t y p e ( F i g u r e 2 5 c ) . He c o n c l u d e d t h a t t h e f i r s t t wo w e r e l i m i t i n g c a s e s o f t h e more g e n e r a l A l l e e t y p e ( F i g u r e 2 5 d ) . The c u r v e g e n e r a t e d b y my m o d e l h a s some s i m i l a r i t y t o t h e A l l e e t y p e ( F i g u r e 24 c o m p a r e d t o F i g u r e 2 5 c ) , b u t i t h a s some s t r i k i n g d i f f e r e n c e s : 1) a n "S" s h a p e d r i s e ; t o i t s maximum i n s t e a d o f a n e g a t i v e l y a c c e l e r a t e d r i s e ; 2) no d e c l i n e f r o m t h e maximum; 3) t h e r e p r o d u c t i v e r a t e t e n d s t o w a r d z e r o a n d n o t some p o s i t i v e v a l u e a s d e n s i t y g o e s t o z e r o . T h e r e a r e e x p l a n a t i o n s f o r t h e s e d i f f e r e n c e s . I f F i g u r e 24 p l o t t e d r e p r o d u c t i v e r a t e a g a i n s t d e n s i t y i n s t e a d o f l o g d e n s i t y t h e i n i t i a l e x p o n e n t i a l r a t e w o u l d be m a s k e d , an d t h e c u r v e w o u l d a p p e a r t o be n e g a t i v e l y a c c e l e r a t e d . A l s o , b o t h F u j i t a ' s m o d e l (1954) a n d W a t t ' s m o d e l (1968 Figure 25. Three hypotheses for the e f f e c t population density on the egg production of in s e c t s : A. Drosophila type; B. Intermediate type; C. A l l e e type; D. Overlay of Drosophila, intermediate and A l l e e types. A l l three types a f t e r F u j i t a (1954). 137. p. 306-307) p r e d i c t a n "S" s h a p e d r i s e f o r s p e c i e s w i t h a s m a l l " c o p u l a t i o n t e n d e n c e " . The d e c l i n e f r o m t h e maximum was a t t r i b u t e d t o com-p e t i t i o n f o r o v i p o s i t i o n s i t e s b y F u j i t a (1954), a n d t o o v i p o s i t i o n s i t e c o m p e t i t i o n a n d a d i r e c t d e n s i t y e f f e c t o n f e c u n d i t y b y W a t t (1968 p. 288-311), w h i l e Conway (1969 p. 123-130) i n c l u d e d b o t h o f W a t t ' s e f f e c t s a n d a d d e d a t h i r d : m a l e i n t e r f e r e n c e . My m o d e l , i n t e n t i o n a l l y , d i d n o t i n c l u d e a n y o f t h e s e c o m p o n e n t s ; t h e r e f o r e , i t i s n o t u n e x p e c t e d t h a t i t d o e s n o t p r e d i c t a d e c l i n e . O v i p o s i t i o n s i t e c o m p e t i t i o n a n d a n y i n t e r f e r e n c e w e r e d e l i b e r a t e l y e l i m i n a t e d f r o m t h e e g g p r o d u c t i o n e x p e r i m e n t s ( F I R S T EXPERIMENT: R e s u l t s ) . The m o d e l w o u l d show e f f e c t s o f c o m p e t i t i o n f o r o v i p o s i t i o n s i t e s i f t h e s u b m o d e l on o v i -p o s i t i o n s i t e s e l e c t i o n h a d b e e n d e v e l o p e d (MATING: A p p l i c a t i o n o f t h e M o d e l t o R e p r o d u c t i o n i n D r o s o p h i l a  m e l a n o g a s t e r ) a n d i n t e r f e r e n c e c o u l d be i n c l u d e d a s o u t l i n e d a b o v e (MATING: F u r t h e r D e v e l o p m e n t o f t h e M o d e l ) . My m o d e l c o u l d p r e d i c t t h a t r e p r o d u c t i v e r a t e g o e s t o a v a l u e g r e a t e r t h a n z e r o a s d e n s i t y g o e s t o z e r o , w h i c h i s w h a t t h e A l l e e c u r v e p r e d i c t s , b u t i t c o u l d o n l y do s o i f r e p r o d u c t i v e r a t e was e x p r e s s e d i n t e r m s o f e g g p r o d u c t i o n i n s t e a d o f f e r t i l e e g g p r o d u c t i o n . F u j i t a (1954) d o e s n o t make c l e a r i f he i s u s i n g t h e A l l e e c u r v e f o r e g g p r o d u c t i o n o r f o r f e r t i l e e g g p r o d u c t i o n . 138. Of course any parthenogenic species could have a non zero intercept possibly g i v i n g them a greater chance to sur-vive (greater r e s i l i e n c e ) with a change i n population density. Species that are both parthenogenic and sexual have, at d i f f e r e n t times, the best of both worlds: greater r e s i l i e n c e as well as genetic recombination. However, a non zero intercept i s u n r e a l i s t i c i n the present study as the fecundity of a l l sexually reproducing animals approaches zero as density goes to zero. The r e l a t i v e importance of the differences between A l l e e curve and my model can best be considered i n l i g h t of the i n t e r a c t i o n between fecundity (percent increase i n popu-l a t i o n size) and mortality (percent decrease i n population size) over a range of population densities (Figures 26a and b). The mortality curves are a f t e r Holling's (1965 p. 43) type 3, "vertebrate" predator responses to prey density* with the addition of a d i r e c t density e f f e c t causing mortality to tend toward 100% at high population d e n s i t i e s . The fecundity curves represent two populations with s i m i l a r c h a r a c t e r i s t i c s , but with one C'high" fecundity) more fecund than the other ("low" fecundity). Figure 26a shows fecundity curves generated by my model: an "S" shaped r i s e from zero to a plateau with no decline. Figure 26b again *Holling's type 1, " f i l t e r feeder", and type 2, "invertebrate", are not included. Figure 26. Interaction of various fecundity (broken lines) and mortality ( s o l i d lines) curves. See text for explanation. 139. POPULATION DENSITY 140. shows an "S" shaped, r i s e to a peak but with a l t e r n a t i v e declines, one for the "high" and the other for the "low fecundity curve. Each i n t e r s e c t i o n of the mortality and fecundity curves represents one of two kinds of thresholds: EQ = stable equilibrium density; ET = threshold equilibrium density between two stable equilibrium or between ex t i n c t i o n and a stable equilibrium. The ET equilibriums are unstable equilibrium from which populations tend to go e i t h e r to e x t i n c t i o n (EX) or to a stable equilibrium (EQ). The dotted l i n e represents the fecundity l e v e l (the " c r i t i c a l " fecundity) below which the population w i l l go e x t i n c t i r r e s p e c t i v e of any predation e f f e c t s . The major difference between my model's pr e d i c t i o n (26a) and the A l l e e type fecundity curve (26b) i s the l a t t e r s i n c l u s i o n of a decline at high density. The addition of t h i s decline i n fecundity with increasing density (Figure 26b) has a dramatic lowering of the equilibrium d e n s i t i e s . The two types of decline represent the two t a c t i c s used by the female D. melanogaster i n the egg production experiments: copulation dependent egg production and copulation independent egg production. In Figure 26b "high " fecundity i s equated with copulation independent egg production and "low" fecundity with copula-t i o n dependent egg production. 141. The copulation independent t a c t i c i s advantageous at low population d e n s i t i e s . However, i f the decline i n fecundity at high d e n s i t i e s i s caused by l i m i t e d number of egg s i t e s or l i m i t e d food supply, and i f many females are unmated due to interference, then a population using the copulation independent t a c t i c would have both mated and v i r g i n females competing for o v i p o s i t i o n s i t e s and for food. This would put the population at a disadvantage compared to one using the copulation dependent t a c t i c where v i r g i n s would not compete as much for o v i p o s i t i o n s i t e s and food. Under these assumptions, the "high" fecundity, copulation indepen-dent t a c t i c population would s u f f e r a sharper decline i n fecundity having an equilibrium lower than that of the "low" fecundity, copulation dependent t a c t i c population (Figure 26b). The dotted l i n e i n Figures 24 and 26 was c a l l e d the c r i t i c a l reproductive rate. Populations with a reproductive rate below t h i s l i n e would go extinct because t h e i r replace-ment r a t i o would be less than one. As predation increases there i s an increase i n the density needed to maintain the reproductive rate above the c r i t i c a l l e v e l . Predation was considered to be of the type modeled by Hol l i n g (1965, 1966) which predicts a chance i n the propor-t i o n of a population l o s t to predation as that population's density changes. In the simulation, predation was assumed to be a constant independent of density. Thus, the model needs'to be expanded to include the ef f e c t s on fecundity of 142. mortality as i t acts through changes i n the population 1s age structure. Then the interactions between mortality and fecundity curves that determines equilibrium d e n s i t i e s can be studied more r e a l i s t i c a l l y (Figure 26) . The medium predation l e v e l (Figure 23) i s based on the f i e l d estimates of t o t a l mortality (9.2%) for D. pseudoobscura (Dobzhansky and Wright 1943). They also estimated the density of a natural population to be 0.05 to 2 0.10 f l i e s / m . . The reproductive rate of a population with these estimates of t o t a l mortality and density, 13 to 19 f e r t i l e eggs per day/new adult per day, i s s u f f i c i e n t to maintain a population with a preadult mortality rate of 90-95% (Figure 23). This reproductive rate i s not, however, the maximum possible. At a higher density the population would have a higher re-productive rate and could maintain i t s e l f with a much higher preadult mortality. However, i t i s possible that populations do not t r y for the "best" s i t u a t i o n but are w i l l i n g to s e t t l e for a t o l e r a b l e s i t u a t i o n . I t i s also possible that at high d e n s i t i e s reproductive rate begins to decline, as i n Figure 25b. Thus, the population could a c t u a l l y be at some "best" i n t e r -mediate density. 143. GENERAL CONCLUSION Experimental components analysis was used to develop a model of sexual reproduction. The technique of d i s -aggregating sexual reproduction into processes and these processes into components gave a framework to guide the experimental work. This was e s p e c i a l l y true for mating, and the model developed for t h i s process gives a basis f o r future experimental studies of any two i n t e r a c t i n g popula-t i o n s . The experiments analyzing the ef f e c t s of age and age at copulation on the processes of egg production and egg f e r t i l i z a t i o n were f i r s t steps of a study of these processes. The r e s u l t s suggested new experiments which would give greater i n s i g h t i n t o the processes. No experimental basis was given to the mating model, and parameters were estimated by using available data from the l i t e r a t u r e . Because the model of sexual reproduction i s not yet complete, the r e s u l t s of the simulations are speculative. However, speculation can be useful i f i t provides a c l e a r guide for future work. Certain of the r e s u l t s from the model are important enough to warrant further consideration and study: the sigmoid response of reproductive rate to increasing density, and the apparent advantage at high predatory mortality of copulation independent egg produc-t i o n , while at high prey densities copulation dependent egg production appears advantageous. 144. BIBLIOGRAPHY Anderson, F.S. 1957. The e f f e c t of density on the b i r t h and death rate. Reprinted from annual report 1954-1955, Govt, pest i n f e s t a t i o n lab, Springforbi, Denmark, p. 1-27. Anderson, F.S. 1965. Simple population models and t h e i r a p p l i c a t i o n to the formation of complex models. Proc. XII Int. Congr. Entomel., London, p. 620-622. Bastock, M. and A. Manning. 1955. 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The e f f e c t s of a r t i f i c i a l s e l e c t i o n for mating speed i n Drosophila melanogaster. Anim. Behav. 11: 82-92. 148. Manning, A. 1967. The control of sexual r e c e p t i v i t y i n female Drosophila. Anim. Behav. 15: 239-250. Mukerji, M.K. and E.J. LeRoux. 1965. Laboratory rearing of a Quebec s t r a i n of the pentatomid predator, Podisus maculiventris (Say) Hemiptera: Pentratomidae). Phytoprotection 46: 40-60. Mukerji, M.K. and E.J. LeRoux. 1969. A quantitative study of food consumption and growth of Podisus maculiventris (Hemiptera: Pentatomidae). Canad. Ent. 101: 387-403. Odurn, E.P. 1971. Fundamentals of ecology. W.B. Saunders Co., Philadelphia. 574 p. Pear l , R. 1932. The influence of density of population upon egg production i n Drosophila melanogaster. J . Exp. Zool. 63: 57-84. Pearl, R. and S. Parker. 1922. On the influence of density of population upon rate of reproduction i n Drosophila. Proc. Nat. Acad. S c i . 8: 212-219. Paulik, G.J. and J.W. Greenough, J r . 1966. Management analysis for a salmon resource system, 215-252. In K.E.F. Watt (Ed.), Systems analysis i n ecology. Academic Press, New York. 149. S k e l l a n , J .G. 1958. The m a t h e m a t i c a l f o u n d a t i o n s u n d e r -l y i n g t h e u s e o f l i n e t r a n s e c t s i n a n i m a l e c o l o g y . B i o m e t r i c s 14: 385-400. V a n H e r r e w e g e , J . a n d J . D a v i d . 1970. R e d u c t i o n o f l o n g e v i t y i n D r o s o p h i l a by c h r o n i c i n g e s t i o n o f a b a c t e r i a l t o x i n . E x p . G e r o n t . 5: 131-143. V e t u k h i v , M.O. a n d J . A . G e a r d m o r e . 1959. E f f e c t o f e n v i r o n -ment u p o n t h e m a n i f e s t a t i o n o f h e t e r o s i s a n d homeo-s t a s i s i n D r o s o p h i l a p s e u d o o b s c u r a . G e n e t i c s 44: 759-768. V o l t e r r a , V. 1928. V a r i a t i o n a n d f l u c t u a t i o n o f t h e number i n d i v i d u a l s i n a n i m a l s p e c i e s l i v i n g t o g e t h e r . J . C o n s . perm. i n t . E x p l o r . M e r . 3: 3-51. W a t t , K.E.F. 1968. E c o l o g y a n d r e s o u r c e management. McGraw-H i l l Book C o . , New Y o r k , 450 p. W i l s o n , L . P . , R.C. K i n g , a n d J.D. L o w r y . 1955. S t u d i e s on t h e t u s t r a i n o f D r o s o p h i l a m e l a n o g a s t e r . I . P h e n o t y p i c a n d g e n o t y p i c c h a r a c t e r i z a t i o n . G r o w t h 19: 215-244. 150. APPENDIX I ESTIMATING FLY VELOCITY KNOWING THE NUMBER OF CONTACTS OF FLIES WITH A 0.018 m. DIAMETER CONTACT AREA* As each f l y moves at a v e l o c i t y of Vm i t sweeps out a contact zone of width 2«R, where R i s the radius of the contact area. That i s , i f a f l y approaches within R units of the centre of the contact area i t w i l l be recorded as a contact. Consider now a f l y moving during time T. During t h i s time i t moves a distance Vm«T and sweeps out an area: 2-R.Vm-T + 3.14.R2 I f 2 Mr+Ms = number of male and female f l i e s per m and A = area of the j a r then t o t a l area swept out by a l l f l i e s i n the j a r , i . e . by (Mr+Ms)«A f l i e s , i s (2«R.Vm-T + 3'14-R 2)•(Mr+Ms)«A Since there i s only one contact area i n the j a r , the density of contact areas i s 1/A. Therefore, Nc, number of times a f l y enters the contact area, i s given by: Nc = (2-R'Vm-T + 3•14«R 2)•(Mr+Ms)-A-(1/A) *From Holling 1966 p. 79. 151. f r o m w h i c h Nc - 3-14-R 2-(Mr+Ms) V m = 2.R.T.(Mr+Ms) ( 1 0 ) I n t h e p r e s e n t c a s e Nc i s e x p r e s s e d a s number o f c o n t a c t s p e r h o u r s o t h a t T = 1 h r . a n d s i n c e R = 0.0009 m. t h e r e f o r e : _ No - 0.002544-(Mr+Ms) 0.018-(Mr+Ms) 152. APPENDIX II PHYSIOLOGICAL BASIS OF THE TWO ALTERNATIVE EGG PRODUCTION FUNCTIONS: EQUATIONS 23 AND 24 Equation 23. F i t z - E a r l e et a l . (1969) and McMillan et al. (1970 a, b) studied d a i l y egg production rates for females kept continually with males form eclosion. They noted a s i m i l a r r i s e to a maximum egg production rate followed by a decline as seen i n Figures 7 and 8. They argued that egg production i s a two stage process. Number of oocytes (x) i s f i x e d at the time females f i r s t begin laying eggs (Ts). Oocytes become eggs at a constant instantaneous rate (r) and eggs are oviposited at a constant instantaneous rate (s). I t can be shown (Fit z - E a r l e et al.1969) that the equation describing the above i s : d E / d T = ;( r' a-;-' e R' A) • d - e - ( r ~ s ) • < A- T m i n>) . e - s * A If a, defined as the p o t e n t i a l maximum d a i l y egg production rate, i s substituted for the f i r s t term i n the above function, then: dE/dT =0.0 [A<_ Ts] JT, / j m n - (r-s) • (A-TminK -S-A R M , -, dE/dT = a*(1-e ' v )*e [Ts < A] where: dE/dT = egg production rate i n eggs per day A = age of female from eclosion i n days Ts = age of female at f i r s t egg laying (23) 153. E q u a t i o n 24. F a c t o r s c o n t r o l l i n g c h a n g e s i n e g g p r o d u c t i o n r a t e (dE/dT) c a n be c o n s i d e r e d i n t e r m s o f t h e i r e f f e c t s o n 2 2 t h e d e r i v a t i v e o f e g g p r o d u c t i o n r a t e (d E/dT ) . A s more a n d more e g g s (E) a r e l a i d , t h e f e m a l e i s l e s s a n d l e s s a b l e t o l a y e g g s a n d h e r e g g p r o d u c t i o n r a t e d e c r e a s e s . 2 2 T h u s , a s number o f e g g s l a i d (E) i n c r e a s e s , d E/dT becomes n e g a t i v e . F u r t h e r , i t i s a s s u m e d t h a t a l o w e g g p r o d u c t i o n r a t e s t i m u l a t e s a n i n c r e a s e i n e g g p r o d u c t i o n r a t e up t o some m a x i m a l r a t e (max. d E / d T ) . T h u s , as e g g p r o d u c t i o n r a t e (dE/dT) i n c r e a s e s t o w a r d s i t s maximum r a t e (max. d E / d T ) , 2 2 d E/dT becomes l e s s a n d l e s s p o s i t i v e . T h i s s e c o n d e f f e c t i s e v e n t u a l l y m a s k e d b y t h e f i r s t e f f e c t o f e v e r i n c r e a s i n g v a l u e o f t o t a l e g g s l a i d ( E ) . As t o t a l e g g s l a i d (E) i n c r e a s e s , i t e v e n t u a l l y d r i v e s e g g p r o d u c t i o n r a t e (dE/dT) t o z e r o . T h e s e a s s u m p t i o n s c a n be e x p r e s s e d m a t h e m a t i c a l l y b y : d 2 E / d T 2 = - t - E + u- (max. dE/dT - dE/dT). w h i c h i n t e g r a t e s t o ( G i l b e r t p e r s . comm.): dE/dT = v . e - x - ( A - T m i n ) + w . Q - y . ( A - T m i n ) a n d a s s u m i n g a = v = -w w i t h f e m a l e s c o n s i d e r e d t o be i m m a t u r e up t o some minimum a g e ( T m i n ) : dE/dT = 0.0 [A <_ Tmin] (24) ,^ , -,m , - x . (A-Tmin) - y ( A - T m i n ) . r m . v ' dE/dT = a«(e -e 2 ) [T m i n < A] where dE/dT and A are as defined for Equation 23 and Tmin = minimum age at which females can f i r s t eggs 155. APPENDIX III EFFECT OF FOOD INTAKE ON PARAMETERS OF THE EGG PRODUCTION FUNCTION Introduction. I t was suggested above (VIRGIN FEMALES: Parameter Estimation) that f o r the egg production function: dE/dT =0.0 [A <_ Ts] dE/dT = a- (1 - b A " T s ) [Ts < A <_ Tmax] (22) dE/dT = a-(1 - b A " T s ) - c A " T m a x [Tmax < A] there might be a r e l a t i o n s h i p between parameters a and c. Parameter a i s a measure of how f a s t each ovariole i s working, and c i s r e l a t e d to the rate at which ovarioles wear out. The suggested hypothesis was that the f a s t e r ovarioles work the faster they wear out ( i . e . as a increases, c decreases). A t e s t of the hypothesis i s to have several blocks of data each with d i f f e r i n g estimates of parameters a and c and to c a l c u l a t e the c o r r e l a t i o n c o e f f i c i e n t between the parameters If the c o r r e l a t i o n i s s i g n i f i c a n t l y negative the hypothesis can not be rejected, and there may be a r e l a t i o n s h i p between ovariole production and ovariole deactivation. . Two sets of data met the requirements; both were studies of d a i l y fecundity as a function of age and food 156. intake: 1) Mukeji and Holling's study (unpublished data) of Podisus meculiventris as analyzed by Kitching, Gallopin, and Reynolds (unpublished manuscript); and 2) David, Van Herrewege, and F o u i l l e t ' s (1971) study of D. melanogaster. In addition to t e s t i n g for a r e l a t i o n s h i p between parameters a and c, these two sets of data allow the ef f e c t s of food intake on the egg production process to be studied. S p e c i f i c a l l y , they enable the v i t e l l o g e n e s i s component to be included. The egg production function (Equation 22) was f i t t e d to both sets of data as before, with two exceptions: 1) parameter a was not estimated by regression but was set equal to the observed maximum egg production rate (a = max. dE/dT) as previous work had shown t h i s approxi-mation made l i t t l e d i fference to the analysis (Table 19); 2) f o r the P..meculiventris data no t e s t was performed to measure the si g n i f i c a n c e of a single estimate of a, b, and c for a l l blocks of data as previously a single estimate was always s i g n i f i c a n t (Tables 15 and 21). Data for Podisus maculiventris. Mukeji and Holling's methods are s i m i l a r to those of Mukeji and LeRoux (1965). B r i e f l y , females were given a surplus of food for a l l preadult stages and f o r the f i r s t 5 days of adulthood. During these 5 days a male was kept with the female. On the 6th day the female was i s o l a t e d and put on one of f i v e food l e v e l s u n t i l death 157. ( T a b l e 29). T h e r e w e r e f i v e r e p l i c a t e s a t e a c h f o o d l e v e l . E g g s w e r e c o l l e c t e d e v e r y 6 d a y s . P l o t t i n g e g g p r o d u c t i o n r a t e (dE/dT) o n a g e (A) showed much t h e same r e s u l t s a s f o r my e x p e r i m e n t s : a r i s e a maximum f o l l o w e d b y a d e c l i n e ( F i g u r e 26). An a n a l y s i s o f t h r e s h o l d s a n d maximums showed t h a t a g e a t f i r s t e g g l a y i n g ( T s ) * a n d age a t maximum e g g p r o d u c t i o n r a t e (Tmax) w e r e i n d e p e n d e n t o f f o o d l e v e l , b u t maximum e g g p r o d u c t i o n r a t e (max, dE/dT) was n o t t h e same f o r a l l f o o d l e v e l s ( T a b l e 30) I t was a s s u m e d p a r a m e t e r a i s a p p r o x i m a t e l y e q u a l t o maximum e g g p r o d u c t i o n r a t e (a = max. d E / d T ) . Maximum e g g p r o d u c t i o n r a t e v a r i e d w i t h f o o d l e v e l , t h e r e f o r e , p a r a m e t e r a d o e s t o o . T h u s , i n t h e i n i t i a l f i t o f t h e e g g p r o d u c t i o n f u n c t i o n ( E q u a t i o n 22) t o t h e d a t a , i n d i v i d u a l e s t i m a t e s o f a f o r e a c h f o o d l e v e l a n d a s i n g l e e s t i m a t e o f b a n d c f o r a l l f o o d l e v e l s w e r e made a n d w e r e a s s u m e d t o be s i g n i f i c a n t I n d i v i d u a l e s t i m a t e s f o r e a c h f o o d l e v e l o f b o t h p a r a m e t e r s b a n d c w e r e f o u n d t o g i v e a s i g n i f i c a n t l y b e t t e r f i t t h a n a s i n g l e e s t i m a t e o f e i t h e r b o r c . ( T a b l e 3 1 ) . The s o l i d l i n e s i n F i g u r e 27 w e r e g e n e r a t e d b y t h e s e e s t i m a t e s ( F o o t -n o t e b , T a b l e 31) a n d t h e e g g p r o d u c t i o n f u n c t i o n ( E q u a t i o n 22). A s a b o v e , t h e e s t i m a t e s o f P i n T a b l e 31 a r e b i a s e d ( s e e T a b l e 1 0 ) . U s i n g t h e s e e s t i m a t e s o f a a n d c ( T a b l e 3 1 ) , t h e h y p o t h e s i s o f t h e i n v e r s e r e l a t i o n s h i p b e t w e e n a a n d c was t e s t e d b y d e t e r m i n i n g t h e c o r r e l a t i o n b e t w e e n them. * I n t h e m a i n t e x t Ts was t h e f i r s t d a y o f e g g p r o d u c -t i o n ; i n A p p e n d i x I I I i t i s t a k e n t o be t h e l a s t d a y b e f o r e e g g p r o d u c t i o n . 158. Table 29. Relationship between food a v a i l a b i l i t y and food intake (both dry weight) for female Podlsus  maculiventris. Data are means ± 1 SE for females 6-54 days old from Kitching, Gallopin, and Reynolds (unpublished manuscript). Food Treatment Food Food Ingested Level Available (mgm./6 days) (mgm./6 days) 1 1 larva/4 days 3.39 ± .015 3.44 ± .75 2 1 larva/day 13.56 ± .06 9.72 ± .31 3 2 larvae/day 21.12 ± .12 16.2 ± 1 . 3 4 4 larvae/day 54.24 ± .24 31.3 ± 2 . 0 5 2 large larvae/day 323.28 ± .49 41.6 ± 5 . 8 Figure 27. E f f e c t of age of female Podisus  maculiventris (A) on egg production rate (dE/dT) for various food intake l e v e l s : A. Food l e v e l 1: 1 larva/4 days; B. Food l e v e l 2: 1 larva/day; C. Food l e v e l 3:2 larvae/day; D. Food l e v e l 4: 4 larvae/day; E. Food l e v e l 5: 2 large larvae/day. Each point represents the mean of 5 or less r e p l i -cates + 1 SE. Data are from Mukeji and H o l l i n g (unpublished data). The s o l i d l i n e s of p r e d i c t i o n are generated by the egg production function (Equation 22) using parameter estimates from Table 31. T a b l e 30. R e l a t i o n s h i p b e t w e e n f o o d l e v e l a n d : 1) a g e a t f i r s t e g g l a y i n g ( T s ) , 2) maximum e g g p r o d u c t i o n r a t e (max. d E / d T ) , a n d 3) a g e a t maximum e g g p r o d u c t i o n r a t e (Tmax). D a t a a r e means ± 1 SE f o r P o d i s u s m a c u l i v e n t r i s f e m a l e s f r o m M u k e j i a n d H o l l i n g ( u n p u b l i s h e d d a t a ) . F o o d Number o f T s a max. dE/dT Tmax L e v e l F e m a l e s 1 5 5.4 ± 1.5 4.6 ± 1.3 19.8 ± 4.4 2 5 4.2 ± 1.2 10.3 ± 1.7 25.8 ± 5.5 3 5 3.0 ± 0.0 14.9 ± 1 . 5 19.8 ± 2.9 4 5 4.2 ± 1 . 2 22.0 ± 2 . 0 22.2 ± 4 . 4 5 5 4.2 ± 1 . 2 15.7 ± 2 . 0 17.4 ± 2 . 4 F o r a l l f o o d l e v e l s , a v e r a g e Ts =4.2 ± 0.5 . b F o r a l l f o o d l e v e l s , a v e r a g e Tmax = 21.0 ± 1.8 . 161. Table 31. Analysis of variance f o r regression, Equation 22, using egg production data i n Figure 27: i n d i v i d u a l estimates of b and c for each treatment (food l e v e l ) . Source SS DF P F i t t i n g i n d i v i d u a l b's (for i n d i v i d u a l a's and single c) 335.2 4 0.0622 Additional f o r i n d i v i d u a l c's 197.8 4 1.64 % F i t t i n g i n d i v i d u a l c's (for i n d i v i d u a l a's and single b) 340.2 4 0.0541 Additional f o r i n d i v i d u a l b's 192.8 4 1.86 % Residual : 3553.6 223 T o t a l d 4086.7 231 aParameter estimates: a (food l e v e l 1-5) = 4.6, 10.3, 14.9, 22.0, 15.7; b (food l e v e l 1-5) = 0.9714, 0.9514, 0.9089, 0.9005, 0.8652; c = 0.9661 . ^Parameter estimates: a (food l e v e l 1-5) = 4.6, 10.3, 14.9, 22.0, 15.7; b = 0.9629, 0.9534, 0.8995, 0.9111, 0.8605; c = 0.9490, 0.9690, 0.9533, 0.9776, 0.9557 . Parameter estimates: a (food l e v e l 1-5) = 4.6, 10.3, 14.9, 22.0, 15.7; b = 0.9061; c = 0.9623, 0.9256, 0.9537, 0.9773, 0.9575 . Residual for f i t t i n g a (food l e v e l 1-5) = 4.6, 10.3, 14.9, 22.0, 15.7; b = 0.9061; c = 0.9661 . 162. Instead of being negative as predicted by the hypothesis, the c o r r e l a t i o n c o e f f i c i e n t was p o s i t i v e , but not s i g n i f i -cantly so (R = 0.644, P = 24.1%, N = 5). Data for Drosophila melanogaster. Complete methods are to be found i n David et a l . (1971). B r i e f l y , larvae were reared with a surplus of food (food l e v e l 5). Upon emergence, as adults, groups of four females and f i v e males were put on one or f i v e food l e v e l s : Yeast-Cornmeal Food Level Concentration 1 2 3 4 5 There were three r e p l i c a t e groups for each food l e v e l . Food was renewed and eggs counted every day. The t y p i c a l r i s e to a maximum with a decline was shown when egg production rate (dE/dT) was plotted against ate (A) (Figure 28). As only means of the data were a v a i l -able, no s t a t i s t i c a l comparison of age at f i r s t egg laying (Ts), age at maximum egg production rate (Tmax), and maxi-mum egg production rate (max. dE/dT) could be made. I t appeared (Figure 28) that these f i r s t two were constants independent of food l e v e l (Ts = 0.0, Tmax = 12.0), while maximum egg production rate varied with food l e v e l (max. 1% 2% 4% 8% 16% Figure 28. E f f e c t of age of female Drosophila  melanogaster (A) on egg production rate (dE/dT) for various food intake l e v e l s : A Food l e v e l 1: 1% Yeast -Cornmeal media; <3 Food l e v e l 2: 2% Yeast -Cornmeal media; V Food l e v e l 3 : 4% Yeast--Cornmeal media; > Food l e v e l 4: 8% Yeast -Cornmeal media; X Food l e v e l 5: 16% Yeast -Cornmeal media. Each point represents the mean of 3 or less r e p l i -cates. Data are from David, Van Herrewege, and F o u i l l e t (1971). The s o l i d l i n e s of pr e d i c t i o n are generated by the "new" egg production function (Equation 37) using parameter estimates from Table 34. 164. dE/dT = 5.6, 9.6, 43.5,48.3, a n d 96.2 f o r f o o d l e v e l s 1-5 r e s p e c t i v e l y ) . S i n g l e e s t i m a t e s o f a , b , a n d c g a v e a s i g n i f i c a n t f i t , a s d i d i n d i v i d u a l e s t i m a t e s o f a (a = max. dE/dT) f o r e a c h f o o d l e v e l ( T a b l e 32). I n d i v i d u a l e s t i m a t e s o f c f o r e a c h f o o d l e v e l w e r e s i g n i f i c a n t , b u t i n d i v i d u a l e s t i m a t e s o f b w e r e n o t s i g n i f i c a n t ( T a b l e 33). R e e x a m i n i n g t h e d a t a ( F i g u r e 28) shows 1) t h a t a p l a t e a u i n e g g p r o d u c t i o n may e x i s t f o r " s u b o p t i m a l " f o o d l e v e l s ( f o o d l e v e l s 1-4); a n d 2) t h a t , a s n o t e d b y D a v i d e t a l (1971), t h e d a t a m e r g e s i n t o a common l i n e a f t e r 35-45 d a y s . T h i s s u g g e s t s t h a t a s i n g l e c u r v e e x i s t s ( a s i n g l e v a l u e o f a , b , a n d c ) w i t h a maximum c u t o f f (d) o f e g g p r o d u c t i o n r a t e (dE/dT) f o r l e v e l s 1-4. The p a r a m e t e r e s t i m a t e s o f t h i s h y p o t h e s i s ( T a b l e 34 a n d E q u a t i o n 37 b e l o w ) h a d a much s m a l l e r sum o f s q u a r e s (4,675.8, d f = 108) t h a n d i d i n d i v i d u a l e s t i m a t e s o f p a r a m e t e r s a , b , a n d c , f o r e a c h f o o d l e v e l (6,096.5, d f = 100, T a b l e 33). T h e s e e s t i -m a t e s ( T a b l e 34) w e r e u s e d t o g e n e r a t e t h e s o l i d l i n e s o f p r e d i c t i o n i n F i g u r e 28. A s t h e a n a l y s i s r e s u l t e d i n o n l y one e s t i m a t e o f a a n d c , no c o r r e l a t i o n c o e f f i c i e n t c o u l d be d e t e r m i n e d a s a t e s t o f t h e h y p o t h e s i s o f t h e i n v e r s e r e l a t i o n s h i p b e t w e e n a a n d c . Table 32. Analysis of variance for regression, Equation 22, using egg production data i n Figure 28: i n d i v i d u a l estimates of a for each treatment (food l e v e l ) • Residual sum of squares calculated by summing a l l sum of squares below regression sum of squares being examined up to and including f i n a l r e s i d u a l sum of squares. Residual degrees of freedom calculated i n a s i m i l a r manner. Source SS DF P Regression for single a, b, and c a 15,100 2 0.0002 % Additional f o r i n d i v i d u a l a " s b 47,474 4 0.0000 % F i n a l r e s i d u a l 8,476 108 Total° 71,050 114 Parameter estimates: a = 40.6; b = 0.6369; c = 0.9683 . ^Parameter estimates: a = max. dE/dT for food l e v e l s 1-5 = 5.6, 9.6, 43.5, 48.3, 96.2; b = 0.6109; c = 0.9558 Residual for f i t t i n g mean egg production rate (dE/dT) = 25.5 . Table 33. Analysis of variance for regression, Equation 22, using egg production data in Figure 28: individual estimates of b and c for each treatment (food level). Source SS DF Fitting individual b's (for individual a's and single c) 460.3 4 21.0 % Additional for individual c's 1919.3 4 0.0015 % Fitting individual c's (for individual a's and single b) c 1906.4 4 0.0022 % Additional for individual b'sb 473.2 4 11.0 % Residual 6096.5 100 Total d 8476.1 108 Parameter estimates: a (food level 1-5) = 5.6, 9.6, 43.5, 48.3, 96.2; b (food level 1-5) = 0.7350, 0.7799, 0.8479, 0.5875, 0.5868; c = 0.9557 . Parameter estimates: a = 5.6, 9.6, 43.5, 48.3, 96.2; b = 0.7277, 0.8029, 0.8516, 0.5880, 0.5865; c = 0.8831, 0.9857, 0.9584, 0.9726, 0.9491 . Parameter estimates: a = 5.6, 9.6, 43.5, 48.3, 96.2; b = 0.6109; c =.8783, 0.9856, 0.9572, 0.9726, 0.0491 . Final residual from Table 32. 167. Ta b l e 34. Comparison o f parameter e s t i m a t e s and c e r t a i n independent v a r i a b l e s f o r the "new" egg p r o d u c t i o n f u n c t i o n (Equation 37). Es t i m a t e s are f o r D r o s o p h i l a  melanogaster females s t u d i e d by David, Van Herrewege, and F o u i l l e t (1971). Ts = age of female a t f i r s t egg l a y i n g ; Tmax = age o f female a t maximum egg p r o d u c t i o n r a t e . Food Parameters Ts Tmax L e v e l a b c d 1 96.2 .5869 .9464 4.0 0.0 12.0 2 96.2 .5869 .9464 6.4 0.0 12.0 3 96.2 .5869 .9464 24.8 0.0 12.0 4 96.2 .5869 .9464 46.4 0.0 12.0 5 96.2 .5869 .9464 _ 0.0 12.0 168. D i s c u s s i o n . The a n a l y s i s o f t h e D. m e l a n o g a s t e r d a t a shows how t h e v i t e l l o g e n e s i s component (d) c o u l d be i n c o r p o r a t e d i n t o t h e egg p r o d u c t i o n f u n c t i o n ( E q u a t i o n 22) w h i c h now o n l y i n c l u d e s t h e components o f o v a r i o l e a c t i v a t i o n ( b ) , o v a r i o l e d e a c t i v a t i o n ( c ) , and o v a r i o l e p r o d u c t i o n ( a ) : dPE/dT = 0.0 [A _< T s ] dPE/dT = a- (1 - b ) [Ts < A <_ Tmax] dPE/dT = a - ( l - b A " T s ) - c A " T m a x [Tmax < A] (37) dE/dT = dPE/dT [dPE/dT <^  d] dE/dT = d [d < dPE/dT] where: a i s t h e f e m a l e ' s maximum p o t e n t i a l egg p r o d u c t i o n ; b i s a me a s u r e o f t h e l a g between c o p u l a t i o n a n d p o t e n t i a l maximum egg p r o d u c t i o n r a t e and i s a f u n c t i o n o f age a t c o p u l a t i o n ( T c ) ; c i s t h e p r o b a b i l i t y t h a t a f u n c t i o n i n g o v a r i o l e w i l l r e m a i n f u n c t i o n i n g ; and d i s t h e maximum p o s s i b l e e g g p r o d u c t i o n r a t e and i s a f u n c t i o n o f f o o d i n t a k e and e n e r g y p a r t i -t i o n i n g ( F i g u r e s 1 a n d 5 ) . and where: dE/dT = r e a l i z e d e g g p r o d u c t i o n r a t e dPE/dT = p o t e n t i a l e g g p r o d u c t i o n r a t e 170. A = age of female from end of juvenile stage Ts = age of female at f i r s t egg laying Tmax = age of female when po t e n t i a l maximum egg production rate can occur I t i s possible that the P. maculiventris data also f i t t h i s new model. The low food l e v e l s (levels 1-3, Figure 29) do show some signs of a plateau, but the data do not converge for older females. This i s primarily due to the data for food l e v e l 4, but these data have such large standard errors for older females that i t may not be s i g n i f i c a n t l y d i f f e r e n t from the data for the other four food l e v e l s . These l a t t e r data tend to converge for older females. The manner of the addition of v i t e l l o g e n e s i s to the egg production function (Equation 37) implies that the de-ac t i v a t i o n of functioning ovarioles i s a constant and i s independent of egg production rate. Thus, the hypothesis that high ovariole production (a) re s u l t s i n rapid ovariole deactivation (c) i s not supported by the data as i n t e r -preted by the "new" egg production function (Equation 37). F i g u r e 2 9 . E f f e c t o f age o f f e m a l e P o d i s u s  m a c u l i v e n t r i s (A) o n e g g p r o d u c t i o n r a t e (dE/dT) f o r v a r i o u s f o o d i n t a k e l e v e l s : F o o d l e v e l 1: 1 l a r v a / 4 d a y s ; F o o d l e v e l 2: 1 l a r v a / d a y ; F o o d l e v e l 3: 2 l a r v a e / d a y ; F o o d l e v e l 4: 4 l a r v a e / d a y ; F o o d l e v e l 5: 2 l a r g e l a r v a e / d a y . E a c h p o i n t r e p r e s e n t s t h e mean o f 5 o r l e s s r e p l i c a t e s + 1 SE. D a t a a r e f r o m M u k e j i a nd H o l l i n g ( u n p u b l i s h e d d a t a ) . . OO 50-0 100.0 AGE (DAYS) 172. APPENDIX I V SYMBOLS FOR EQUATIONS I N PART I V A l l l o w e r c a s e l e t t e r s d e n o t e c o n s t a n t s ( a , c , e, q , e t c . ) . U p p e r c a s e l e t t e r s u s u a l l y d e n o t e v a r i a b l e s (A, C o , dE / d T , M r , e t c . ) , a l t h o u g h t h e y o c c a s i o n a l l y d e n o t e c o n -s t a n d s (D, D r , D s , a n d F ) . a = f e m a l e ' s maximum r e a l i z e d e g g p r o d u c t i o n A = age f r o m e c l o s i o n A c = r a t e o f s u c c e s s f u l s e a r c h b = m e a s u r e o f l a g b e t w e e n c o p u l a t i o n a n d maximum e g g p r o d u c t i o n r a t e c = p r o b a b i l i t y t h a t a f u n c t i o n i n g o v a r i o l e r e m a i n s f u n c t i o n a l Co = d e n s i t y o f c o p u l a t i o n s d = r a t e o f c h a n g e i n f l y v e l o c i t y D = r e a c t i v e d i s t a n c e o f m a l e s t o f e m a l e s dE/dT = e g g p r o d u c t i o n r a t e dFE/dT = f e r t i l e e g g p r o d u c t i o n r a t e D r = mean a g e a t d e a t h f o r m a l e s Ds = mean a g e a t d e a t h f o r f e m a l e s e = minimum f l y v e l o c i t y f = maximum f l y v e l o c i t y F = maximum p o s s i b l e number o f f e r t i l e e g g s l a i d d u r i n g f e m a l e ' s l i f e t i m e 173. g = i n i t i a l h i g h l e v e l o f r e c e p t i v i t y o f v i r g i n s h = r a t e o f d e c l i n e i n v i r g i n f e m a l e ' s r e c e p t i v i t y m = m e a s u r e o f d e c l i n e i n f e m a l e ' s a b i l i t y t o p r o d u c e f e r t i l e e g g s Mr = t o t a l m a l e d e n s i t y M r ( A ) = m a l e d e n s i t y b y age Ms = t o t a l f e m a l e d e n s i t y M s ( A , T c ) = f e m a l e d e n s i t y by age a n d age a t c o p u l a t i o n n = s p e r m s t o r a g e a b i l i t y p = p r o b a b i l i t y o f a m a l e - r e c e p t i v e f e m a l e c o n t a c t r e s u l t i n g i n c o p u l a t i o n P d = p r o b a b i l i t y o f d y i n g due t o age r e l a t e d m o r t a l i t y Pp = p r o b a b i l i t y o f d y i n g d u e t o o t h e r m o r t a l i t y ( " p r e d a t i o n " ) q = p r o p o r t i o n o f e g g s t h a t a r e f e r t i l i z e d R = p r o b a b i l i t y o f a v i r g i n f e m a l e b e i n g r e c e p t i v e s = s t a n d a r d d e v i a t i o n o f age a t d e a t h Tc = age o f f e m a l e a t c o p u l a t i o n Td = age when d e c l i n e i n f e m a l e ' s a b i l i t y t o p r o d u c e f e r t i l e e g g s b e g i n s T f = age a t s t a r t o f d e c l i n e i n v i r g i n f e m a l e ' s r e c e p t i v i t y Tm = h a n d l i n g t i m e Tmax = f e m a l e ' s a ge when maximum e g g p r o d u c t i o n r a t e o c c u r s T m i n = minimum age a t f i r s t e g g l a y i n g 174, Tr = minimum age for v i r g i n female r e c e p t i v i t y Ts = female's age at f i r s t egg laying Tt = time exposed (time males and females together) W = density of receptive females 

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